Loops of any size and Hamilton cycles in random scale-free networks
Loops are subgraphs responsible for the multiplicity of paths going from one to another generic node in a given network. In this paper we present an analytic approach for the evaluation of the average number of loops in random scale-free networks valid at fixed number of nodes $N$ and for any length $L$ of the loops. We bring evidence that the most frequent loop size in a scale-free network of $N$ nodes is of the order of $N$ like in random regular graphs while small loops are more frequent when the second moment of the degree distribution diverges. In particular, we find that finite loops of sizes larger than a critical one almost surely pass from any node, thus casting some doubts on the validity of the random tree approximation for the solution of lattice models on these graphs. Moreover we show that Hamiltonian cycles are rare in random scale-free networks and may fail to appear if the power-law exponent of the degree distribution is close to $2$ even for minimal connectivity $k_{min}\geq 3$.
iop_large_loops.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    INSTITUTE OF PHYSICS PUBLISHING                                   %
%                                                                      %
%   `Preparing an article for publication in an Institute of Physics   %
%    Publishing journal using LaTeX'                                   %
%                                                                      %
%    LaTeX source code `ioplau2e.tex' used to generate `author         %
%    guidelines', the documentation explaining and demonstrating use   %
%    of the Institute of Physics Publishing LaTeX preprint files       %
%    `iopart.cls, iopart12.clo and iopart10.clo'.                      %
%                                                                      %
%    `ioplau2e.tex' itself uses LaTeX with `iopart.cls'                %
%                                                                      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% First we have a character check
%
% ! exclamation mark    " double quote  
% # hash                ` opening quote (grave)
% & ampersand           ' closing quote (acute)
% $ dollar              % percent       
% ( open parenthesis    ) close paren.  
% - hyphen              = equals sign
% | vertical bar        ~ tilde         
% @ at sign             _ underscore
% { open curly brace    } close curly   
% [ open square         ] close square bracket
% + plus sign           ; semi-colon    
% * asterisk            : colon
% < open angle bracket  > close angle   
% , comma               . full stop
% ? question mark       / forward slash 
% \ backslash           ^ circumflex
%
% ABCDEFGHIJKLMNOPQRSTUVWXYZ 
% abcdefghijklmnopqrstuvwxyz 
% 1234567890
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\documentclass[12pt]{iopart}
% Uncomment next line if AMS fonts required
%\usepackage{epsf}
\usepackage[dvipdf]{graphicx}
\begin{document}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bc{\begin{center}} 
\def\ec{\end{center}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\title[Loops of any size and Hamilton cycles in random scale-free networks]{Loops of any size and Hamilton cycles in random scale-free networks}

\author{Ginestra Bianconi\dag\ \ddag\ 
\footnote[3]{e.mail: gbiancon@ictp.trieste.it}
and Matteo Marsili\dag\  
\footnote[4]{e.mail: marsili@ictp.trieste.it}
}

\address{\dag\ The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy}

\address{\ddag\ INFM, UdR Trieste, via Beirut 2-4, 34014, Trieste,Italy }


\begin{abstract} 
Loops are subgraphs responsible for the multiplicity of paths going
  from one to another generic node in a given network.  In this paper
  we present an analytic approach for the evaluation of the average
  number of loops in random scale-free networks valid at fixed number
  of nodes $N$ and for any length $L$ of the loops.  We bring evidence
  that the most frequent loop size in a scale-free network of $N$
  nodes is of the order of $N$ like in random regular graphs while
  small loops are more frequent when the second moment of the degree distribution diverges. In particular, we find that finite
  loops of sizes larger than a critical one almost surely pass from
  any node, thus casting some doubts on the validity of the random
  tree approximation for the solution of lattice models on these
  graphs. Moreover we show that Hamiltonian
  cycles are rare in random scale-free networks and may fail to appear
  if the power-law exponent of the degree distribution is close to $2$
  even for minimal connectivity $k_{min}\geq 3$. 

\end{abstract}
\pacs{: 89.75.Hc, 89.75.Da, 89.75.Fb} 

%\submitto{\JSTAT}

% Comment out if separate title page not required
\maketitle

The scale-free network structure has been found in a number of social,
technological and biological networks as the skeleton of their
interaction \cite{RMP,Doro_book,Internet}.  The main property of
scale-free networks is to have a power-law degree distribution $P(k)
\sim k^{-\gamma}$ and second diverging moment, i.e. $\gamma\in(2,3]$.
To distinguish between different scale-free networks, recently, much
attention has been devoted to network motifs
\cite{UAlon,Milo,Laszlo_m}, i.e. subgraphs that recur with higher
frequency than in maximally random graphs with the same degree
distribution.  Among those, the most simple types of subgraphs are
loops \cite{Loops,Guido_cycles,Avraham}, i.e.  closed paths
of various length that visit each node only once.  Loops (or cycles)
are interesting because they account for the multiplicity of paths
between any two nodes. Therefore, they encode the redundant
information in the network structure.  Another discriminant aspect of
real scale-free networks is the presence of degree correlations
between linked nodes. Characteristic motifs in a graph and degree
correlations are in many real graphs not independent phenomena but
they depend on each other as it has been shown for small (up to
maximal connectivity) size subgraphs in
\cite{Loops_last,Dorogovtsev,Vazquez_m,VazquezBa}. Last but not least,
it has been observed that the distribution of loop sizes is intimately
connected with the thermal properties of lattice models defined on
that graph \cite{Parisi_book_stat_phys,Marinari}. On the other hand,
the analytic approach to these models relies on the assumption that
locally a random graph can be considered to have a tree like structure
\cite{Dorog_solution_Potts,Leone_etal_Ising,Ehrhardt_Marsili_JSTAT},
i.e. that loops of finite size are rare.

In this paper we present an analytic derivation of the average number
of loops of any size in a random scale-free network. Our motivation is
that the results on random networks provide a reference picture which
often captures key intuition which extends to correlated networks (see
e.g.  \cite{epidemics,epidemics_corr}). In addition, it provides
valuable information for the statistical mechanics of lattice model on
random graphs~ \cite{Parisi_book_stat_phys,Marinari}.

Let us first recall the classic results for regular random graphs,
i.e. random graphs with fixed connectivity of the nodes $k_i=c$ for
each node $i$.  A regular random graph contains a finite number of small
loops of size $L\ll \log(N)$, with average expected number 

\be 
{\cal
  N}_L=\frac{1}{2L}(c-1)^L \label{reg.small.eq} 
\ee 
and Poisson
fluctuations around the mean.  On the contrary for large loop sizes
$L\sim O(N)$ the number of loops goes as 

\be 
{\cal N}_L=\exp(N
\sigma(\ell))
\label{scaling.eq}
\ee
where $\ell=L/N$ and $\sigma(\ell)$ is a function having the maximum
at $\ell_{max}=c/(c+1)$ whose expression can be found in the
literature \cite{Janson,Marinari}. 
Regarding  Hamilton cycles, i.e. loops that span the entire
network $L=N$  their expected number for a large regular random graph
is diverging with the system size as long as $c\geq 3$. For $c=2$ the
average number of Hamilton cycles  goes to zero as the system size
diverges \cite{Janson}. 
Coming to the scale-free network literature, Ref.\cite{Avraham}
analyzes the number of loops of any size on a pseudo-fractal
scale-free graph and report the scaling behavior 

\be
\log {\cal N}_L=Lf(L/L^*)
\label{Nl}
\ee
with $L^*=N^{1/(\gamma-1)}$. No result were presented on Hamilton
cycles, to our knowledge, so far. 

In the following we characterize the statistics of loops in random
scale-free networks. We find a larger number of small loops with
respect to regular random graphs. In particular, we compute the
expected number of loops of a given size passing through a 
node and find that when $\gamma\in (2,3)$ this number diverges with
the network size, beyond a finite loop size. This raises some doubts
on the solution of lattice models on these graphs based on the local
tree approximation
\cite{Dorog_solution_Potts,Leone_etal_Ising,Ehrhardt_Marsili_JSTAT}.
We also find that loops have a 
 characteristic size $L^*\sim N$.  In other words, our results are consistent with the
scaling $(\ref{Nl})$ with $L^*\sim N$ and with Eq. (\ref{scaling.eq})
for regular graphs. This suggests that the result of Ref.
\cite{Avraham} crucially depends on the peculiar correlations of the
ensemble they consider. Special attention will be given to Hamilton
cycles that in networks with a small $\gamma$ exponent can fail to
exist unless the lower cutoff of the distribution is large enough.

There are different ensembles of random networks one can consider. The classical one follows the prescription of Molloy and Reed\cite{MR}: First, to each node $i$ of the network is assigned a connectivity $k_i$ drawn from the chosen probability distribution, secondly edges are randomly  matched.
This ensemble indeed generates networks of given degree distribution but it may yields networks with multiply occupied links. More precisely, the distribution of 
the links between two nodes of connectivity $k_i$, $k_j$ is a Poisson variable with mean $k_i k_j/(c N)$, where henceforth $c=\langle k \rangle$ will denote the average connectivity. Hence the probability of no multiply occupied links is
\be
\Pi_{i> j}\left(1+\frac{k_i k_j}{cN}\right)e^{-\frac{k_i k_j}{cN}}\simeq 
e^{-\frac{1}{4}\left(\frac{\gamma-2}{3-\gamma} m^{\gamma-2}K^{(3-\gamma)}\right)^2 }
\ee
where the right hand side refers to a scale free random graph with degree distribution $P(k)= a k^{-\gamma}$ with 
$k\in [m,K]$. Taking a structural cutoff $K\sim N^{1/2}$ \cite{poli1,FS_vesp}, we conclude that double links appear with probability one for $\gamma\in (2,3]$ as $N \rightarrow \infty$ in the  Molloy-Reed ensemble.
When counting loops of a network this effect becomes relevant and undesirable. Thus we consider also another ensemble where double links cannot appear: the static fitness network\cite{HV,HV2}.
In the fitness ensemble nodes are assigned a random variable (fitness)
$q$ drawn from a $\rho(q)$
distribution function and every couple of nodes is linked  with a probability depending on the fitness of the considered nodes $p(q,q')$.
When $\rho(q)$ is power-law distributed and $p(q,q')=\frac{q q'}{\langle q\rangle N}$ the resulting graph is  a random scale-free graph characterized by the same exponent of the fitness distribution. In these graph the connectivity of every node is a Poisson variable with expected value $\langle k(q)\rangle=q$.
This ensemble doesn't allow for networks with double links but instead
may give rise to networks with isolated nodes ($k_i=0$) or to nodes
connected with a single link ($k_i=1$) to the others. The presence
of such nodes rules out the possibility to find Hamilton cycles,
hence we shall take this effect into account when discussing Hamilton
cycles.
 
Consequently, the Molloy-Reed ensemble and the fitness ensemble are
not equivalent and have intrinsic properties that could perturb
sensitively the counting of the number of loops.  In order to
understand the dependence on the details of how graphs are generated
in the following we are going to study the expected number of loops in
the two ensembles.

\section{Loops in the fitness ensemble}
The prescription of Ref.\cite{HV} to generate a class of random scale-free networks with exponent $\gamma$ is the following: {\em i)} assign to each node $i$ of the graph a hidden continuous variable $q_i$ distributed according to a scale-free distribution
$\rho(q)=\rho_0 q^{-\gamma}$ for $q\in [m,Q]$ with $\rho_0=(\gamma-1)/(m^{1-\gamma}-Q^{1-\gamma})$ the normalizing constant. 
Then {\em ii)} each pair of nodes with hidden variables $q,q'$ are linked with probability $qq'/(cN)$, where
$c=\langle q\rangle$ is the expected value of $q$. 
For $\gamma\leq 3$ the cutoff $Q\sim N^{1/2}$ is needed to keep the linking probability smaller than one, i.e. $Q^2/(cN)<1$ while for $\gamma>3$ the cutoff is the natural one $Q\sim N^{\frac{1}{\gamma-1}}$.
By construction the expected value of the connectivity of a node with hidden variable $q$ is $\langle k|q\rangle=q$ and there are no multiple connections between nodes. 
 Notice that the average connectivity of the graph is given by

\begin{equation}\label{avgk.eq}
  \langle k\rangle =\langle q\rangle = c\to \frac{\gamma-1}{\gamma-2} m
\end{equation}
in the limit $N\to\infty$.

A loop of size $L$ is an ordered set of distinct nodes $\{1_i,\ldots,i_L\}$. For each choice of the nodes, the
probability that they are connected in a loop is
\[
\frac{q_{i_1}q_{i_2}}{Nc}\frac{q_{i_2}q_{i_3}}{Nc}\cdots
\frac{q_{i_L}q_{i_1}}{Nc}=\prod_\ell \frac{q_{i_\ell}^2}{Nc}.
\]
The total number of possible loops joining these $L$ nodes, in any possible way is $L!/(2L)$ where the factor $2L$ comes from the fact that the initial node of the loop can be chosen in $L$ ways and that there are two orientations. In order to count loops, let us lump together nodes with hidden variable $q_i\in[q,q+\Delta q)$, where $\Delta q $ is a small interval of $q$. In each interval of $q$ there are $N_q\simeq NP(q)\Delta q$ nodes of the network. For each choice of the $L$ nodes, let $n_q$ be the number of nodes with $q_{i_\ell}\in [q,q+\Delta q)$. Then the average number of loops of size $L $ in the graph is given by the number of ways we can choose $\{n_q\}$ nodes multiplied by the probability that these nodes are connected in all distinguishable orderings. Consequently we have 
\be
{\cal N}_L=\frac{L!}{2 L} \sum_{\{n_q\}}  \prod_q \frac{N_q!}{n_q! (N_q-n_q)!}\left(\frac{q^2}{Nc}\right)^{n_q}
\label{uno.eq}
\ee
where the sum is extended over all $\{n_q\}$ such that $\sum_q n_q=L$. Introducing this constraint 
with a delta function and using its integral representation, we find
\be
{\cal N}_L=\frac{L!}{2 L}
\int_{-\infty}^\infty \frac{dx}{2\pi} e^{iLx+N\left\langle \log\left[1+q^2e^{-ix}/(Nc)\right]\right\rangle}.
\label{exact.eq}
\ee 
Notice that in Eq. (\ref{exact.eq}) one can safely take the limit $\Delta q\rightarrow 0$ and that the average over the $P(q)$ distribution is taken assuming that we focus on the limit $N\to\infty$. In what follows, we will evaluate Eq. (\ref{exact.eq}) in different ranges of $L\le N$ in the limit $N\to\infty$.
 
\subsection{Small loops}
For $L$ finite but large, the integral in Eq. (\ref{exact.eq}) is dominated by values $x\simeq -iz^*$ where
\begin{equation}\label{apprx.small.eq}
  e^{-z^*}\simeq\frac{\langle q^2\rangle}{Lc}\left(1-\frac{\langle q^4\rangle}{\langle q^2\rangle^2}\frac{L}{N}+\ldots\right)
\end{equation}
where we have neglected all terms beyond the first leading correction when $N\to\infty$. The argument of the exponential in Eq. (\ref{exact.eq}) can be expanded around $x\sim -iz^*$ yielding 
$N\left\langle \log\left[1+q^2e^{-ix}/cN\right]\right\rangle+Lix\simeq 
L\left[1-z^*-\frac{1}{2}(x-iz^*)^2+O(x-iz^*)^3\right]$.
Hence the integral can be estimated by saddle point for $L$ large. Using the asymptotic expression $L!\simeq\sqrt{2\pi L}L^L e^{-L}$, we find to leading order

\begin{equation}\label{Nsmall.eq}
  {\cal N}_L\simeq \frac{1}{2L}\left(\frac{\langle q^2\rangle}{c}\right)^L
\end{equation}
This approximation is valid as long as the leading correction in Eq. (\ref{apprx.small.eq}) is small.
Using that $\langle q^n\rangle=\rho_0(Q^{n-\gamma+1}-m^{n-\gamma+1})/(n-\gamma+1)$ for $\gamma\ne n$ and that 
$Q\sim \min(N^{1/2},N^{\frac{1}{\gamma-1}})$ we find that the expression above for ${\cal N}_L$ holds when 
\begin{equation}\label{small.diseq}
  L\ll N\frac{\langle q^2\rangle^2}{\langle q^4\rangle}\sim\left\{\begin{array}{cc}
    N & \gamma>5\\
    N^{\frac{\gamma-3}{\gamma-1}} & \gamma>3 \\
    N^{(3-\gamma)/2} & 2<\gamma<3 \
  \end{array}\right.
\end{equation}
with logarithmic corrections for $\gamma=3$ and $5$.
Note that strictly speaking the expansion $(\ref{apprx.small.eq})$ is converging only for  $N\langle q^2\rangle/L \gg N$, i.e. $L\ll N^{(3-\gamma)/2}  $ for $2<\gamma<3$ and $L\ll N^{(\gamma-3)/(\gamma-1)}$  for $\gamma>3$.
Nevertheless Eq.$(\ref{Nsmall.eq})$ remains valid in the limits $(\ref{small.diseq})$ as an asymptotic expansion. 
 For $\gamma>3$ we obtain a result very similar to Eq. (\ref{reg.small.eq}) for regular graphs. On the contrary, for 
$2<\gamma<3$ we have $\langle q^2\rangle\simeq a N^{(3-\gamma)/2}$, with $a$ a constant, 
hence the number of finite loops 
\begin{equation}\label{Nsmall.sf.eq}
  {\cal N}_L\simeq \frac{1}{2L}\left(\frac{a}{c}\right)^L N^{\frac{3-\gamma}{2}L}
\end{equation}
diverges as $N\to\infty$.

\subsection{Intermediate loop sizes and the most frequent loops} 

It is convenient, at this point, to write Eq. (\ref{exact.eq}) as

\begin{equation}\label{f.eq}
  {\cal N}_L\simeq\int_{-\infty}^\infty \frac{dx}{2\sqrt{\pi L}}e^{Nf(Nce^{ix},L/N)}
\end{equation}
where we have used Stirling's approximation and

\begin{equation}\label{fyl.eq}
  f(y,\ell)=\left\langle \log\left[1+q^2/y\right]\right\rangle+\ell\log (\ell y/c)-\ell.
\end{equation}

The integral can be computed by saddle point method, deforming the contour of integration so as to pass from the point
where $f$ is stationary. The condition $\partial_yf(y,\ell)=0$ yields 
\be
\left\langle\frac{q^2}{q^2+y}\right\rangle=\ell.
\ee
Let $y^*(\ell)$ be the value of $y$ which solves this equation. We can expand $f(Nce^{ix},\ell)$ around the
corresponding (complex) value $x^*$ of $x$
\bea
f(Nce^{ix},\ell)&=&f(y^*,\ell)-\frac{{y^*}^2}{2}\frac{\partial^2 f}{\partial y^2}(x-x^*)^2+\ldots\\
&=&f(y^*,\ell)-\frac{y^*}{2}\left\langle \frac{q^2}{\left[q^2+y^*\right]^2}\right\rangle (x-x^*)^2+\ldots
\nonumber
\eea
As long as 
\[
Ny^*\left\langle \frac{q^2}{\left[q^2+y^*\right]^2}\right\rangle\gg 1
\]
we can neglect higher order terms. This yields the leading behavior

\begin{equation}\label{gen.eq}
  {\cal N}_L\simeq \frac{1}{2}
  \left[L Ny^*\left\langle \frac{q^2}{\left[q^2+y^*\right]^2}\right\rangle\right]^{-1/2}e^{Nf(y^*,\ell)}
\end{equation}

The number of loops ${\cal N}_L$ takes its maximum for loops of length $L=N\ell_{\max}$ where

\bea\label{lmax.eq}
  \left\langle\frac{q^2}{c+q^2\ell_{\max}}\right\rangle =1.
\eea
The solution $\ell_{\max}$ is plotted in Fig. \ref{lmax.fig} against $\gamma$ for scale-free graphs and different values of $m$, for $N=10^6$. Notice that as $\gamma\to 2^+$, the size of most probable loops vanishes as $\ell_{\max}\sim \gamma-2$.  Around the maximum, ${\cal N}_L$ takes a form similar to that for regular graphs (see Eq. \ref{scaling.eq}) which is consistent with the scaling form Eq. (\ref{Nl}) with $L^*\sim N$.

\begin{figure}
\begin{center}
%\epsfbox{width=75mm, height=55mm}{lmax.eps}
%\end{center}
\includegraphics[width=75mm, height=55mm]{lmax}
\end{center}
\caption{\label{lmax.fig}The behavior of $\ell_{\max}$ as a function of $\gamma$ for  $m=1,2,3$ and $N=10^6$ for  fitness ensemble (solution of the Eq. $(\ref{lmax.eq})$ in solid lines) and for the Molloy-Reed ensemble (solution of the  Eq. $(\ref{lmax_MR.eq} )$ in dashed lines). }
\end{figure}

For scale-free random graphs with $2<\gamma<3$ there is an
intermediate range of loop sizes $L\sim N^{(3-\gamma)/2}$ which is
related to the solutions with $y^*=\mu N$ with $\mu> 1$. More
precisely, we find that for loops of size
$L=\chi(\mu)N^{(3-\gamma)/2}$ we have
\[
{\cal N}_L\simeq \frac{G(\mu)}{L} e^{L[\log(\mu L/c)-1+H(\mu)]}
\]
with $G(\mu)$ a function of $\mu$ and
\bea
\chi(\mu)=(\gamma-1)m^{\gamma-1}\sum_{n=0}^{\infty}(-1)^n\frac{\mu^{-n-1}}{3-\gamma+2n}\nonumber \\
H(\mu)=\frac{1}{\chi(\mu)}(\gamma-1)m^{\gamma-1}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\mu^{-n}}{n(1-\gamma+2n)}.
\eea
Notice that ${\cal N}_L$ does not satisfy a simple scaling form such as Eq. (\ref{Nl}) in this intermediate region.


\subsection{Hamilton cycles}

From Eq. $(\ref{uno.eq})$ we can  easily calculate  the number of  Hamilton cycles. Indeed for $L=N$ we have the asymptotic behavior
\be
{\cal N}_N=\sqrt{\frac{\pi}{2N}}  e^{N[2\langle\log q\rangle-1-\log c]}.
\label{h.eq}
\ee
This is the expected number of Hamiltonian cycles over all the
networks of the fitness ensemble including networks with nodes of low
degree $k_i=0,1$, which by definition cannot have an Hamilton cycle. 
It seems a sensible thing to compute the number of Hamilton cycles in
networks with a  minimal degree connectivity  grater than $3$, i.e. $k_i\ge=3$. 
In fact it  is well known that for a regular random graph of
connectivity $c=2$ the expected number of Hamilton cycles goes to zero
in the $N\rightarrow \infty$ limit whereas regular graphs are
Hamiltonian when $c\ge 3$. 
Taking this as a reference result, we 
normalize ${\cal N}_N$ by the probability $\pi$ that all the nodes
have at least 
$3$ connections. Since the connectivity of each node in the fitness network is aPoisson variable with expected value $q$ the probability that all the
nodes have connectivity $k\geq 3$ is simply given by
$\pi=e^{N\lambda(m,\gamma)}$, where 

\be
\lambda(m,\gamma)=\langle\log(1-(1+q+q^2/2)e^{-q})\rangle 
\ee
In the limit $N\to\infty$ we find 
\be
\frac{1}{N}\log\frac{{\cal N}_N}{\pi}\to
\log\left(\frac{\gamma-2}{\gamma-1}m\right)+\frac{3-\gamma}{\gamma-1}-\lambda(m,\gamma).
\label{hsf.eq}
\ee
This implies that if random scale-free graphs have Hamilton cycles
only for $m>m_c(\gamma)$ where $m_c(\gamma)$ is the value of $m$ for
which Eq. (\ref{hsf.eq}) vanishes. Conversely, for $m<m_c(\gamma)$ a
random scale-free graphs has almost surely no Hamilton cycle. 

In Fig. $\ref{Hc.fig}$ we report the critical value $m_c(\gamma)$ as a
function of $\gamma$.  Notice that $m_c\sim 1/(\gamma -2)\to\infty$ as
$\gamma\to 2^-$. Consequently, if we consider only the networks of the nesemble with $k_{min}\ge 3$, we find that random
graphs with $\gamma<\gamma_{\rm Fit}^*=2.16\dots$ (where
$m_c(\gamma_{\rm Fit}^*)=3$) do not have Hamiltonian cycles.
Considering that regular random graphs with $k=c\ge3$ are Hamiltonian,
this may seem a surprising result, at first sight. The basic intuition
to explain this apparent paradox is that most paths pass through well
connected nodes. Hence even if $k_i\ge 3$ it is very unlikely to
have a path spanning the entire network which is not passing through
the most connected nodes more than once.

\subsection{Loops passing through a node}

In order to  count of the number of loops of size $L$ passing through a given
node, with fitness value $q_i$, we can repeat the previous
outlined above, without taking the average over $q_i$. 
For $\gamma<3$  and short loops sizes $L\ll N^{\frac{3-\gamma}{2}}$
this gives the expected number 

\be
{\cal N}_L({q_i})\simeq \frac{{q_i}^2}{cN}\frac{1}{2L}\left(\frac{a}{c}\right)^{(L-1)}N^{\frac{3-\gamma}{2}(L-1)}.
\ee
Focusing on nodes with $q_i\simeq N^{\alpha}$, we find that the
number of loops of size 
\be
L\geq L_0\equiv 1+\frac{2}{3-\gamma}(1-2\alpha)
\ee
diverges with the network's size $N$. For example, nodes with a finite
$q_i$ have an infinite number of $L=5$ loops passing through it in
networks
with $\gamma<2.5$ but at most a finite number of loops of size $L=3$
if $\gamma>2$. The most connected nodes ($\alpha=1/2$) instead have an
infinite number of loops of any size $L\ge 3$ passing through it.
Notice that $L_0\to\infty$ as $\gamma\to 3$ in order to
match the behavior $L_0\sim \log N$ of regular graphs. Conversely, 
in a finite graph of $N$ nodes, only the large fitness nodes with 
\be
q_i\gg N^{\frac{1}{2}-\frac{3-\gamma}{2}(L-1)}
\ee 
belong to a significant number of loops of size $L$.

\section{Loops in the Molloy-Reed ensemble}

The counting of the number of loops in the Molloy-Reed\cite{MR}
follows a procedure much similar to the one considered for the fitness
ensemble nevertheless giving different results. 
To construct a Molloy-Reed network one proceed as follows:
{\it i)} a degree is assigned to each node of the network following
the desired degree distribution with cutoff $K\sim\min(N^{1/2},N^{\frac{1}{\gamma-1}})$. Degree distributions which do not
satisfy the parity of $cN=\sum_i k_i$ are disregarded; 
{\it ii)} the edges  coming out of the nodes are randomly matched
until all edges are connected. When this procedure ends with nodes
having links to themselves (tadpoles), the whole network is rejected
and the procedure is started anew.

To calculate ${\cal N}_L$ in this ensemble first one has to count in
how many ways it is possible to have a loop of size $L$ in the network
and weight the results with the fraction of possible networks in the
ensemble which  contains the loop. 
Let us first state that the total number of graphs in the Molloy-Reed
ensemble is given by $(cN-1)!!$. Indeed when constructing the network
by linking $cN$ unconnected edges one start by taking one edge at
random and connecting it to one of the $(cN-1)$ possible
connections. Then one proceed taking another edge and linking it to
one of the remaining  $(cN-3)$ possible connections thus giving rise
of one of the $(cN-1)!!$ possible networks. 
By similar arguments one shows that the total number of networks
containing a given loop of size $L$ are $(cN-2L-1)!!$. 
On the other side the total number of loops of size $L$ in the
Molloy-Reed ensemble are given by the number of ways one can choose an
ordered set of $L$ nodes  $\{1_i,\ldots,i_L\}$  of connectivity
$\{k_1,k_2,\dots,k_L\}$ and connect them on a loop. As for the fitness
network the total number of possible loops joining these $L$ nodes, in
any possible way is $L!/(2L)$. 
The number of ways one can choose the edges coming out of the nodes to form the loop is given by
\[
\Pi_{i=1}^L k_i(k_i-1).
\]
Consequently the average number of loops in the Molloy-Reed ensemble will be given by
\be
{\cal N}_L=\frac{L!}{2 L} \sum_{\{n_k\}}  \prod_{k=m}^K \frac{N_k!}{n_k! (N_k-n_k)!}\left({k(k-1)}\right)^{n_k}W_{N,L}
\label{unoMR.eq}
\ee
where $N_k=NP(k)$ ($n_k$) is the number of nodes with connectivity $k$
present in the network (loop), $K$ is the  cutoff of the degree distribution and  the sum over $\{n_k\}$ is restricted to  $\{n_k\}$
such that $\sum_k n_k=L$. Moreover we use the definition $W_{N,L}={(cN-2L-1)!!}/{(cN-1)!!}$.
If we use the Stirling approximation for  $W_{N,L}$ 
we get the expression
\be
 W_{N,L}\sim (cN)^{-L}  e^{Ng(\ell)}
\ee
with $\ell=L/N$ and
\be
g(\ell)=\frac{1}{2}(c-2\ell)\log\left(\frac{c-2\ell}{c}\right)+\ell.
\label{g.eq}
\ee
Thus we get
\be
{\cal N}_L=\frac{L!}{2 L} \sum_{\{n_k\}}  \prod_{k=m}^K \frac{N_k!}{n_k! (N_k-n_k)!}\left(\frac{{k(k-1)}}{cN}\right)^{n_k}e^{N g(\ell)}
\ee
which except for the substitution $q^2\rightarrow k(k-1)$ and the factor $\exp(Ng(\ell))$ is equivalent to the expression $(\ref{exact.eq})$  of the average number of loops of size $L$ in the fitness ensemble.
Following the same steps as in the fitness ensemble, we get 
\be
{\cal N}_L=\frac{L!}{2 L}
\int_{-\infty}^\infty \frac{dx}{2\pi} e^{iLx+N\left\langle \log\left[1+k(k-1)e^{-ix}/(Nc)\right]\right\rangle+N g(\ell)}
\label{exactMR.eq}
\ee 
with  $g(\ell)$ given by Eq. (\ref{g.eq}).

\subsection{Small loop size} The number of small loops in the Molloy-Reed ensemble is given by
\begin{equation}\label{NsmallMR.eq}
  {\cal N}_L\simeq \frac{1}{2L}\left(\frac{\langle k(k-1)\rangle}{c}\right)^L
\end{equation}
where this approximation is valid asymptotically for loops sizes satisfying Eq.$(\ref{small.diseq})$.
Note that as in the fitness ensemble for $\gamma\in (2,3)$ short loops diverge as ${\cal N}_L\sim N^{\frac{3-\gamma}{2}}$.
\subsection{Intermediate loops sizes} For intermediate loops  in the Molloy-Reed ensemble a similar expression to Eq. $(\ref{fyl.eq})$ holds 
with 
\be
f'(y,\ell)=\langle \log[1+k(k-1)/y]\rangle+\ell\log(\ell y/c)-\ell+g(\ell).
\ee 
The calculations of the  average number of loops is very similar for the Molloy-Reed and fitness ensemble, with a difference for the equation of the loops of maximal size which in the Molloy-Reed ensemble satisfy
\be\label{lmax_MR.eq}
   \left\langle\frac{k(k-1)}{c-2\ell_{max}+k(k-1)\ell_{\max}}\right\rangle =1.
\ee
In Fig. $\ref{lmax.fig}$ we report the value of $\ell_{max}$ in the Molloy-Reed
networks as a function of $\gamma$ for different value of the minimal connectivity $m$ for $N=10^6$.

\subsection{Hamiltonian cycles} Starting with expression
 $(\ref{exactMR.eq})$ one can easily evaluate the expected number of
 Hamiltonian cycles in Molloy-Reed networks. 
 Indeed for $L=N$  and  $c>2$ one can use the Stirling approximation
 to find the asymptotic behavior ($N\rightarrow \infty$) 
\be
\frac{1}{N}\log({\cal N}_N)={\langle\log( k(k-1)/c)\rangle+\frac{1}{2}(c-2)\log(1-2/c)}.
\label{hcmr.eq}
\ee
If we approximate $k(k-1)$ with $k^2$ which is possible close to
 $\gamma\rightarrow 2$ in the limit $c\rightarrow \infty $ we recover
 the same behavior as in the fitness ensemble: if the minimal
 connectivities  $m$ is smaller than the value $m_c(\gamma)$ for which
 Eq. (\ref{hcmr.eq}) vanishes, then a scale-free network is typically not Hamiltonian.
 In Fig. $\ref{Hc.fig}$ we report $m_c(\gamma)$ for $2<\gamma<3$ and
 we confirm the behavior $m_c\sim 1/(\gamma-2)$ for $\gamma\to 2$. 
 For example, we find that Molloy-Reed random scale-free graphs with minimal connectivity
 $m=3$ are typically not Hamiltonian if $\gamma<\gamma^*_{\rm MR}=2.27\dots$.
As for the fitness ensemble, the intuition is that it is not possible
to extract from a random-scale-free graphs a
 subgraph which is a regular random graph with fixed connectivity
 $c\geq 3$ if $m<m_c(\gamma)$.

\begin{figure}
\begin{center}
%\epsfbox{Hc.eps}
%\end{center}
\includegraphics[width=75mm, height=55mm]{Hc}
\end{center}
\caption{\label{Hc.fig} 
Dependence of  $m_c(\gamma)$ on $\gamma$ in the fitness (solid line) and Molloy-Reed ensemble (dashed line) in the limit $N\rightarrow \infty$. Observe that for $\gamma<\gamma^*$  we have $m_c(\gamma)>3$ in both ensembles.}
\end{figure}

\subsection{Loops passing through a node} To count the loops of size $L$
passing through a given node of connectivity $k$ we must fix it and
choose other $L-1$ nodes to form the loop. 
For short loops sizes and exponent $\gamma<3$ this gives the expected number
\be
{\cal N}_L(k)\simeq \frac{k(k-1)}{cN}\frac{L-1}{L} {\cal N}_{L-1}
\ee
with ${\cal N}_L\sim N^{\frac{3-\gamma}{2}L}$. The same results
derived for the fitness ensemble hold: There is a critical finite loop size
$L_0(k)$ such that there are infinitely many loops of size $L>L_0$
passing through a given node of connectivity $k$. 
On the contrary, in a finite but large network of $N$ nodes, 
loops of size $L$ becomes significant for  nodes of connectivity
\be
k\gg N^{\frac{1}{2}-\frac{3-\gamma}{4}(L-1)}.
\ee

\section{Numerical results}

We compare the analytic results derived so far with the direct count
of the number of loops in a sample of computer generated random graphs
in both the fitness and the Molloy-Reed ensemble. This is important
because ${\cal N}_L$ is a fluctuating quantity which takes
exponentially large values. In other words, the analytic calculation
of the expected number of loops may be dominated by (exponentially in
$N$) rare realization of graphs with an exponentially large number of
loops. In this case the number of loops of a typical realization of a
graph would differ from our estimate.  We have chosen the fastest
known algorithm for calculating the total number of loops exactly
\cite{Johnson} as in Ref. \cite{Marinari}.  This algorithm has a upper
time bound of $O(N{\cal L})$ where ${\cal L}$ is the total number of
loops in the network.  The simulations performed in this way enable
one only to consider small networks sizes $N<50$ and small $m\le 3$ as
the total number of loops in such graphs increases exponentially with
the system size.  Note that for such small sizes the degree
distribution contains nodes of very similar degree since the upper
cutoff is $K\sim6 $ for $\gamma=2.1$.  Moreover in order to compare
the direct counting with the analytical calculation, we have chosen a
fixed degree (fitness) distribution $N_k=NP(k)$ to reduce fluctuations
that become relevant for such small sizes.  In Fig.  $\ref{Fit.fig}$ we report the
analytic prediction of the average number of loops of a given size in
a fitness network of $N=30$ nodes.  This results are compared with
direct counting of the loops in computer realizations of these
networks were data are averaged over $50$ realizations. We found
strong sample to sample fluctuations which we believe are responsible
for the deviation from the analytical results.

On the contrary, for the Molloy-Reed networks of same system size 
the direct count of loops is very close to the analytic
prediction. Fig. $\ref{MR.fig}$ reports the direct count of loops for
Molloy-Reed networks\cite{Nota} with $N=30$ and several degree distributions 
and it compares it with the corresponding analytic prediction.

\begin{figure}
%\begin{center}
%\epsfbox{fit.eps}
%\end{center}
\begin{center}
\includegraphics[width=75mm, height=55mm]{fit}
\end{center}
\caption{\label{Fit.fig}Number of loops for fitness networks of
  $N=30$ nodes and given distribution  $\rho(q)$ with $m=3$. The
  average is taken over $50$ realizations.} 
\end{figure}



\begin{figure}
\begin{center}

\includegraphics[width=75mm, height=55mm]{MR}
\end{center}
\caption{\label{MR.fig}Number of loops for  MR networks without double
  links of $N=30$ nodes and different $\gamma$. The direct count
  averaged on $10$ realizations is compared to the analytic prediction
  for the same degree distribution (full lines).} 
\end{figure}


\section{Conclusions}

In conclusion we have computed analytically the expected number of
loops of any size in a scale-free network.  We found that scale-free
graphs have a very large number of small loops compared to regular
random graphs.  On the contrary we have shown that, also with a
minimal connectivity $k_{min}\geq 3$ the expected number of Hamilton
cycles can be zero in the $N \rightarrow \infty $ limit provided that
$\gamma$ is sufficiently close to $2$. The reason for this is that
paths connecting many nodes need to pass frequently on nodes with high
connectivity. Put differently, it is not possible to embed a regular
graph of connectivity $c\ge 3$, which would have an Hamilton cycle, in
scale free networks if $\gamma$ is too small, even if all nodes have
$k_i\ge c$.  In the intermediate region of relatively large loops we
found that the expected number of loops attains its maximum for loops
of size $L\sim N$.  These results are derived both in the fitness and
in the Molloy-Reed ensembles. While the generic picture is the same,
the results in the two ensembles differ quantitatively highlighting
that the loop size distribution is somewhat sensitive to the precise
prescription for drawing random graphs. Moreover we have checked the
results with direct counting of computer generated scale-free networks
belonging to the two ensembles.  It would be desirable to derive
similar results for ensembles of correlated scale-free networks. 
\ack
G. B.  was partially supported by EVERGROW and by EU grant
HPRN-CT-2002-00319, STIPCO. We wish to thank F. Tria for useful
discussions.
\section*{References}
\begin{thebibliography}{60}
\bibitem{RMP}
  R. Albert and A.-L. Barab\'asi, Statistical mechanics of complex networks, 2002 {\it Rev. Mod. Phys.} {\bf 74} 47 
\bibitem{Doro_book}
S. N. Dorogovtsev and J. F. F. Mendes, {\it Evolution of Networks}
 (Oxford University Press,Oxford,2003).
\bibitem{Internet}
 R. Pastor-Satorras and A. Vespignani,  {\it Evolution and Structure of the Internet }(Cambridge University Press,Cambridge,2004).
\bibitem{UAlon}
S. Itzkovitz, R. Milo, N. Kashtan, G. Ziv and U. Alon, Subgraphs in random networks, 2003 {\it Phys. Rev. E} \textbf{ 68}  026127
\bibitem{Milo}
R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and  U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, 2002
 {\it Science} {\bf 298} 824 
\bibitem{Laszlo_m}
 R. Dobrin, Q. K. Beg, A.-L. Barab\'asi and Z. N. Oltvai, Aggregation of topological motifs in the Escherichia coli transcriptional network, 2004
{\it BMC Bioinformatics} {\bf 5} 10 
\bibitem{Loops}
 G. Bianconi and A. Capocci, Number of Loops of Size $h$ in Growing Scale-Free Networks, 2003 {\it Phys. Rev. Lett.} {\bf 90 } 078701 
\bibitem{Guido_cycles}
G. Caldarelli, R. Pastor-Satorras and A. Vespignani,  Structure of Cycles and local ordering in complex networks, 2002 {\it Eur. Phys. Jour. B } {\bf 38} 183
\bibitem{Avraham}
H. D. Rozenfeld, J. E. Kirk, E. M. Bollt and D. ben-Avraham,Statistics of Cycles: How Loopy is your Network?, 2004 [cond-mat/0403536] 
\bibitem{Dorogovtsev}
S. Dorogovstev, Clustering of correlated networks, 2004 {\it Phys. Rev. E} {\bf 69} 027104 
\bibitem{Loops_last}
G. Bianconi, G. Caldarelli and A. Capocci, Loops structure of the Internet at the Autonomous System Level, 2004  [cond-mat/0408349] 
\bibitem{Vazquez_m}
 A. Vazquez, R. Dobrin, D. Sergi, J.-P. Eckmann, Z. N. Oltvai, A.-L. Barab\'asi,The topological relationship between the large-scale attributes and local interactions patterns of complex networks, 2004 {\it PNAS} {\bf 101} 17940 
\bibitem{VazquezBa}
A. Vazquez, J. G. Oliveira and A.-L. Barab\'asi, The inhomogeneous evolution of subgraphs and cycles in complex networks, 2005 [ cond-mat/0501399].
\bibitem{epidemics}
R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, 2001 {\it Phys. Rev. Lett.} {\bf 86} 3200
\bibitem{epidemics_corr}
M. Bogu\~na, R. Pastor-Satorras and A. Vespignani, Epidemic Spreading in Scale-Free Networks, 2003 {\it Phys. Rev. Lett.} {\bf 90} 028701 
\bibitem{Janson}
S. Janson, T. Luczak, A. Rucinski, {\it  Random graphs} (John Wiley \& Sons,2000).
\bibitem{Marinari}
E. Marinari and R. Monasson, Circuits in random Graphs: form local trees to global loops, 2004 {\it J. Stat. Mech.} P09004
\bibitem{Parisi_book_stat_phys}
G. Parisi {\it Statistical Field Theory} (Addison Wesley,RedwoodCity, USA,1988)
\bibitem{Dorog_solution_Potts}
S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Potts model on complex networks, 2004 {\it Eur. Phys. J. B }{\bf 38},177 
\bibitem{Leone_etal_Ising}
M. Leone, A. Vazquez, A. Vespignani and R. Zecchina, Ferromagnetic ordering in graphs with arbitrary degree distribution, 2002 {\it  Eur. Phys. J. B} {\bf 28} 191 
\bibitem{Ehrhardt_Marsili_JSTAT}
G. Ehrhardt and M. Marsili, Potts model on random trees, 2005 {\it J. Stat. Mech.} P02006 
\bibitem{HV}
G Caldarelli, A. Capocci, P. De Los Rios and M. A. Mu\~noz, Scale-free networks form varying vertex intrisic fitness,  2002 {\it Phys. Rev. Lett.} {\bf 89}, 258702 
\bibitem{HV2}
M. Bogu\~na and R. Pastor-Satorras, Class of correlated random networks with hidden variables, 2003 {\it Phys. Rev. E} {\bf 68} ,036112 
\bibitem{MR}
M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, 1995 {\it Random Structures and Algorithms}  {\bf 6}, 161
\bibitem{poli1}
Z. Burda, A. Krzywicki, Uncorrelated random networks, 2003 {\it  Phys. Rev. E} {\bf 67} 046118
\bibitem{FS_vesp}
M. Bogu\~na, R. Pastor-Satorras and A. Vespignani, Cut-offs and finite size effects in scale-free networks, 2004 {\it Eur. Phys. Jour.} {\bf 38}, 205(2004).
\bibitem{Johnson}
D. B. Johnson, Finding elementary circuits of a directed graph,1975 {\it SIAM J. Comput.} {\bf 4},77  
\bibitem{Nota}Note that in the considered Molloy-Reed networks double links are not allowed. This correction is supposelly quite small for the considered system sizes.
\end{thebibliography}
\end{document}
\section{Introduction}
Many authors use \LaTeX\ to 
produce their typescripts
and we can use the source code to produce the printed version;  this gives 
more rapid publication with a smaller chance of typographical 
error.

We explain here the procedures and specific requirements for 
the preparation and presentation of text and illustrations for 
articles in \LaTeXe\ using the IOPP class file 
\verb"iopart.cls". We also illustrate \verb"iopart.cls" through 
its use in the creation of this document. The
class file and accompanying documentation are available to all authors 
and copies can be 
obtained (free of charge)
\begin{itemize}
\item from the World Wide Web (\verb"http://www.iop.org/Journals/texstyle")
by downloading zipped files for PCs or tar compressed format files for
Unix
\item from the World Wide Web by using our 
Author Enquiry Service\\ (\verb"http://www.iop.org/Journals/AES")
to request that the files be sent on disk
\item by contacting the Electronic Services Specialist,
Institute of Physics Publishing, Dirac House, Temple Back, 
Bristol BS1~6BE, UK (\verb"esub@ioppublishing.co.uk") and requesting 
the files be sent by e-mail or on 3$1\over2$ inch PC or Macintosh disk
\end{itemize}

There are 10 and 12 point versions of the style, but the 12pt
version should be used for preprints and is selected by opening the file
with the initial line:
\begin{verbatim}
\documentclass[12pt]{iopart}
\end{verbatim}
If the \verb"[12pt]" is omitted the article will be set in 10pt type (i.e.
in the same size type as in the printed journal).
The 12pt version 
of the style gives a `preprint' form with a page width and type size 
1.2 times larger than that for normal single-column journals
with extra spacing between lines. The page depth is less than 1.2 times
the normal page depth so that articles will fit on the page on both 
A4 and Letter paper.
This form is the one required for 
the initial submission of a typescript for 
refereeing and copy editing. Authors need not aim to optimize the line and
page breaks as they will inevitably change when converted to the final
format for printing.

The 10 pt version has the same page dimensions and type sizes as a
single-column journal and shows approximately 
how the text would appear in print. It can also be used to produce 
camera-ready copy for journal special issues.

Other compatible
\LaTeXe\ packages can be used if they are available in the normal
distribution of \LaTeXe; if it is essential to use
a non-standard package then the extra files needed to process the 
article must also be sent in. Authors should be aware that the final 
version will be printed on a different page size and using different fonts 
to the preprint version so that any special effects used should not
contain material that is not easily scalable.

We recommend that authors use the IOPP \LaTeXe\ class file; 
however articles prepared using almost any  version of \TeX\ or \LaTeX\ 
can be 
handled (${\cal AMS}$\TeX, L${\cal AMS}$TeX, PHYZZX, etc) 
and authors not using the IOPP 
style files
can submit their source code in the way described 
below. 
Alterations to the source code will be made in-house by the 
Production Department in order to conform to IOPP house style and 
journal format.

We aim to maintain our normal standards 
for articles published from \TeX\ and \LaTeX\ files so we reserve 
the right to make small alterations  to clarify and improve the 
English where necessary and to put the article into IOPP house style.

\subsection{Double-column journals}
Authors writing for double-column journals should also use the IOPP preprint
macros. Conversion from the single-column format to the double-column 
output required for printing will be done by the Production Department 
at Institute of Physics
Publishing.


\section{Submitting new articles}

\subsection{Sending in files for submission to internally refereed journals}
\label{int}
Articles for consideration for publication in the journals listed in 
table~\ref{e-mail} may be submitted electronically without the need to 
supply hard copies of the manuscript. 
Those journals are refereed internally. 
Articles for consideration by all other IOPP journals must be submitted in hard copy form. Electronic files can then be sent in on acceptance or when requested by the editor.

Authors are asked to prepare their articles using the 12pt 
style and submit by World Wide Web upload, e-mail, FTP or on PC or Mac 
floppy disk.
Please send all files necessary to reproduce your article, 
including the article itself and all figure files. 
Combine all your files into a single archive using a compression or archiving 
program. Utilities like WinZIP or StuffIt will both archive and compress your 
files. Alternatively you could archive your files using a program like TAR 
and compress the resulting file using gzip.


\subsection{By World Wide Web upload}
Users of Netscape (version 2 or higher) or Internet Explorer 
(version 4 or higher) can submit using our World Wide Web upload form.
This can be found at 
\verb"http://www.iop.org/Journals/authorsubs"
Complete the on-line form with information about your submission and then upload 
your article in one of two ways.

\begin{enumerate}
\item Having created a single compressed file containing an archive of your text 
and graphics files, use the `Browse...' button on the form to select that file.

\item If your article has already been posted to the LANL e-print archive, select 
the archive name and enter the e-print number. ({\it Note}: We cannot download 
your paper from the LANL e-print server until it has been publicly released, 
usually by 10 pm on the day you submitted it there.) Use the `Transfer File' 
button at the bottom of the form to submit your file.
\end{enumerate}


\subsection{By E-mail} 
\label{byemail}
Files should be sent to the relevant journal mailbox (see table~\ref{e-mail}).
Please send articles as attachments to e-mail messages. 
Any additional files such as figure files should be combined into a single archive with the article file as described in section~\ref{int}. 
All information about the submission should be included in the main body of the message.
This must include:
\begin{itemize}
\item Your full name
\item The name of the journal to which you are submitting
\item The full title of the article
\item The full list of authors
\item The article type (e.g. Paper, Letter to the Editor etc.)
\item The status of the article (i.e. new revised or accepted)
\item Your postal address
\item Your e-mail address
\item Your telephone number
\item Your fax number (if applicable)
\item The article file format (i.e. \LaTeXe)
\item The number of separate figure files
\end{itemize}

\begin{table}
\caption{\label{e-mail}E-mail addresses for all the journals to which authors can 
submit electronically.}
\lineup
\begin{tabular}{@{}ll}
\br
Journal&{\rm E-mail address}\\
\mr
Journal of Physics A: Mathematical and General&jphysa@ioppublishing.co.uk\\
Journal of Physics B: Atomic, Molecular and Optical Physics
&jphysb@ioppublishing.co.uk\\
Journal of Physics: Condensed Matter&jpcm@ioppublishing.co.uk\\
Journal of Physics D: Applied Physics&jphysd@ioppublishing.co.uk\\
Journal of Physics G: Nuclear and Particle Physics&jphysg@ioppublishing.co.uk\\ 
Classical and Quantum Gravity&cqg@ioppublishing.co.uk\\ 
Combustion Theory and Modelling&ctm@ioppublishing.co.uk\\ 
European Journal of Physics&ejp@ioppublishing.co.uk\\ 
Inverse Problems&ip@ioppublishing.co.uk\\ 
High Performance Polymers&hpp@ioppublishing.co.uk\\
Journal of Micromechanics and Microengineering&jmm@ioppublishing.co.uk\\ 
Journal of Optics A: Pure and Applied Optics&jopa@ioppublishing.co.uk\\ 
Journal of Optics B: Quantum and Semiclassical Optics&jopb@ioppublishing.co.uk\\
Measurement Science and Technology&mst@ioppublishing.co.uk\\ 
Nanotechnology&nano@ioppublishing.co.uk\\ 
Nonlinearity&non@ioppublishing.co.uk\\ 
New Journal of Physics&njp@ioppublishing.co.uk\\
Physics Education&ped@ioppublishing.co.uk\\ 
Physics in Medicine and Biology&pmb@ioppublishing.co.uk\\
Physiological Measurement&pmea@ioppublishing.co.uk\\ 
Plasma Physics and Controlled Fusion&ppcf@ioppublishing.co.uk\\ 
Reports on Progress in Physics&rop@ioppublishing.co.uk\\ 
Semiconductor Science and Technology&sst@ioppublishing.co.uk\\ 
Superconductor Science and Technology&sust@ioppublishing.co.uk\\ 
Waves in Random Media&wrm@ioppublishing.co.uk\\  
\br
\end{tabular}
\end{table}

\subsection{By FTP}
Create a file named \verb"readme.txt" containing all your submission information 
(e.g. your name, the name of the journal you are submitting to etc. See section ~\ref{byemail} 
for full list).
Create a single compressed file containing an archive of your text and graphics 
files along with the file \verb"readme.txt".
Files should be sent by anonymous FTP to \verb"ftp.iop.org". 
Please logon with the username {\it anonymous} and use your e-mail address as 
the password.
Change to the directory \verb"/incoming" and then change to the directory of the
journal to which you wish to submit and
upload your files. 
Please do not create any new directories on our server.
Your files will remain in the \verb"/incoming" directory for no more than 90 minutes 
before they are moved to a secure location.

\subsection{On Floppy Disk}
Files may be sent in on 3.5 inch PC or Apple Macintosh formatted disk. 
Please include a file named \verb"readme.txt" containing all your submission 
information as described above.


\section{Preparing your article}
Using \LaTeX\ with the \verb"iopart" class file 
provides a simple way of 
producing an article in a form suitable for publication in one 
of the IOPP journals. Authors may add their own macros 
at the start of an article 
provided they do not overwrite existing definitions and 
that they send copies of their new macros with their text file.
\verb"iopart" can be used with other package files such
as those loading the AMS extension fonts 
\verb"msam" and \verb"msbm" (these fonts provide the 
blackboard bold alphabet 
and various extra maths symbols as well as symbols useful in figure 
captions); an extra style file \verb"iopams.sty" is provided to load these
packages and provide extra definitions for bold Greek letters.

In preparing your article you are 
requested to follow these guidelines as closely as possible; this will 
minimize the amount of file editing required and will hasten the 
production process. This is particularly important with regard to the 
reference list.

The file name can be up to eight characters long with the suffix 
\verb".tex". 
Please use files names that are likely to be unique, and include 
commented material to identify the journal, author and reference number if 
known. The first non-commented line should be 
\verb"\documentclass[12pt]{iopart}"  to load the preprint class 
file. Other standard predeclared option files can be included in 
square brackets;
copies of any non-standard options must be sent in with the source code.
Omitting \verb"[12pt]" produces an article with the normal journal
page and type sizes. Macros for the individual paper not included 
in a style file
should be inserted in the preamble to the paper with comments to
describe any complex or non-obvious ones.
The start of the article text is signalled by 
\verb"\begin{document}".
Authors of very long articles may find it convenient to separate 
their article 
into a series of files each containing a section, each of which is called 
in turn by the primary file. 

\section{The title and abstract page} 
The code for setting the title page information is slightly different from
the normal default in \LaTeX. 

\subsection{Titles and article types}
The title is set in bold unjustified type using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter 
of the title should be capitalized with the rest in lower case. 
Mathematical expressions within the title may be left in light-face type 
rather than bold because the Computer Modern bold maths and symbol
fonts may not be available at the size required for the title.  
The final 
printed version will have bold mathematical expressions 
in the title. If the title is unsuitable for use as a running head a short
form can be provided as an optional argument (in square brackets)
before the full title, i.e. \verb"\title[Short title]{Full title}".
A short title is required 
when the title itself is too long to 
be used 
as the short title or when the title contains a footnote. 

For article types other than papers the IOPP class file, \verb"iopart.cls", 
allows 
a generic heading \verb"\article[Short title]{TYPE}{Full title}" 
and the specific 
definitions given in table~\ref{arttype}. In each case (apart from Letters
to the Editor) an 
optional argument can be used immediately after the control sequence name
to specify the short title; where no short title is given the full title
will be used as the running head at the top of each page apart from the
first. 
For Letters no short title is required as 
the running head is automatically defined to be {\it Letter to the Editor}.
The generic heading could be used for 
articles such as those presented at a conference or workshop, e.g.
\begin{verbatim}
\article[Short title]{WORKSHOP ON HIGH-ENERGY PHYSICS}{Title}
\end{verbatim}


\begin{table}
\caption{\label{arttype}Types of article defined in the {\tt iopart.cls} 
class file.}
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Command&Type&Heading on first page\\
\mr
\verb"\title{#1}"&Paper&---\\
\verb"\review{#1}"&Review&REVIEW\\
\verb"\topical{#1}"&Topical review&TOPICAL REVIEW\\
\verb"\comment{#1}"&Comment&COMMENT\\
\verb"\note{#1}"&Note&NOTE\\
\verb"\paper{#1}"&Paper&---\\
\verb"\prelim{#1}"&Preliminary communication&PRELIMINARY COMMUNICATION\\
\verb"\rapid{#1}"&Rapid communication&RAPID COMMUNICATION\\
\verb"\letter{#1}"&Letter&LETTER TO THE EDITOR\\
\verb"\article{#1}{#2}"&Other articles&Whatever is entered as {\tt 
\#1}\\
\br
\end{tabular*}
\end{table}

\subsection{Authors' names and addresses}
The next information required is the list of authors' names and 
their affiliations. For the authors' names type \verb"\author{#1}", 
where \verb"#1" is the 
list of all authors' names. The style for the names is initials then
surname, with a comma after all but the last 
two names, which are separated by `and'. Initials should {\it not} have 
full stops. First names may be used if 
desired. If the authors are at different addresses one of the symbols 
\dag, \ddag, \S, $\|$, \P, $^+$, *, $\sharp$ should be used after each 
surname to reference an author to his/her address. 
The symbols should be used in the order given.  If 
an author has additional information to appear as a footnote, such as 
a permanent address, and the
footnote symbols are not being used to identify an address,
the footnote should be entered after the surname
as a normal \LaTeX\ footnote, without
specifying a sign. Where footnote symbols are being used to indicate
which address the author is at, the 
symbol used for a footnote should be the next one from the list given 
above and has to be selected individually using the command
\verb"\footnote[<num>]{Text of footnote}", where \verb"<num>" is a
number representing the position of the desired symbol in the list above,
i.e.\ 1 for \dag, 2 for \ddag, etc.

The addresses of the authors' affiliations follow the list of authors. 
Each address is set by using
\verb"\address{#1}" with the address as the single parameter in braces. 
If there is more 
than one address then the appropriate symbol should come at the start of
the address.
 
Please also add the corresponding author's e-mail address if applicable. This is done by inserting the 
command \verb"\ead{#1}" before the command \verb"\maketitle" where \verb"#1" is the e-mail address.  
See section~\ref{startsample} for sample coding. For more than one e-mail address, please use the command 
\verb"\eads{\mailto{#1}, \mailto{#2}}" with \verb"\mailto" surrounding each e-mail address. These 
macros allow us to convert e-mail addresses to `mailto' links when articles are published on our 
website.

\subsection{The abstract}
The abstract follows the addresses and
should give readers concise information about the content 
of the article and indicate the main results obtained and conclusions 
drawn. It should be complete in itself with no table numbers, figure 
numbers or references included and should not normally exceed 200 
words.
To indicate the start 
of the abstract type \verb"\begin{abstract}" followed by the text of the 
abstract (not in braces).  The abstract should normally be restricted 
to a single paragraph and is terminated by the command
\verb"\end{abstract}"

\subsection{Subject classification numbers}
Following the abstract come any
Physics and Astronomy Classification System (PACS) codes
or American Mathematical Society 
(AMS) classification scheme numbers.  
The command 
\verb"\pacs{#1}", with the subject classification numbers from the  
Physics and Astronomy Classification Scheme as the parameter, defines 
the subject area of the paper (or for a single number \verb"\pacno{#1}").  
If PACS numbers are not readily 
available, {\it Physics Abstracts\/} classification scheme numbers can be 
given instead. If this command  is omitted the 
classification numbers for indexing
will be allocated by IOPP staff. It is 
unnecessary to supply PACS numbers for {\it Inverse 
Problems} and {\it Physics in Medicine and Biology}.
AMS classification numbers may be given as well as, or instead of, PACS 
numbers for mathematical articles, they are specified using the 
command \verb"\ams{#1}".
After any classification numbers the command
\verb"\submitto{#1}" can be inserted, where \verb"#1" is the journal name written in full or the appropriate control sequence as
given in 
~\ref{jlab1}. This will print a line indicating that the article has been submitted to one of the Institute of Physics journals. The command is not essential to the running of the file.  


\subsection{Making a separate title page}
The command \verb"\maketitle" forces a page break after the point where it
is inserted and so to keep the header material on a separate page from the
body of the text insert \verb"\maketitle" or \verb"\newpage" after the
classification codes 
or the end of the abstract. 
If \verb"\maketitle" is not included the text of the
article will start immediately after the abstract.  


\subsection{Sample coding for the start of an article}
\label{startsample}
The code for the start of a title page of a typical paper might read:
\begin{verbatim}
\documentclass[12pt]{iopart}
\begin{document}
\title[The anomalous magnetic moment of the 
neutrino]{The anomalous magnetic moment of the 
neutrino and its relation to the solar neutrino problem}

\author{P J Smith\dag, T M Collins\ddag, 
R J Jones\ddag\footnote[3]{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road, 
Bristol BS8 1TS, UK.} and Janet Williams\P}

\address{\dag\ Mathematics Faculty, Open University, 
Milton Keynes MK7~6AA, UK}
\address{\ddag\ Department of Mathematics, 
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{\P\ Department of Computer Science, 
University College London, Gower Street, London WC1E~6BT, UK}

\ead{williams@ucl.ac.uk}

\begin{abstract}
...
\end{abstract}

\submitto{\JPG}
\pacs{1315, 9440T}

\maketitle
\end{verbatim}

\subsection{AMS macros}
Please do not use the style file \verb"amsmath.sty" (part of the AMSTeX package) in conjunction with \verb"iopart.cls". This will result in several errors. To make use of the macros defined in \verb"amsmath.sty", we have provided the file \verb"setstack.sty" which reproduces the following useful macros from \verb"amsmath.sty":
\begin{verbatim}
\overset
\underset
\sideset
\substack
\boxed
\leftroot
\uproot
\dddot
\ddddot
\varrow
\harrow
\end{verbatim}
\section{The text}
\subsection{Sections, subsections and subsubsections}
The text of papers and reviews, but not comments or letters, should be 
divided into sections, subsections and, where necessary, 
subsubsections. To start a new section, end the previous paragraph and 
then include \verb"\section" followed by the section heading within braces. 
Numbering of sections is done {\it automatically} in the headings: 
sections will be numbered 1, 2, 3, etc, subsections will be numbered 
2.1, 2.2,  3.1, etc, and subsubsections will be numbered 2.3.1, 2.3.2, 
etc.  Cross references to other sections in the text should, where
possible, be made using 
labels (see section~\ref{xrefs}) but can also
be made manually. See section~\ref{eqnum} for information on the numbering of displayed equations. Subsections and subsubsections are 
similar to sections but 
the commands are \verb"\subsection" and \verb"\subsubsection" respectively. 
Sections have a bold heading, subsections an italic heading and 
subsubsections an italic heading with the text following on directly.
\begin{verbatim}
\section{This is the section title}
\subsection{This is the subsection title}
\end{verbatim}


The first section is normally an introduction,  which should state clearly 
the object of the work, its scope and the main advances reported, with 
brief references to relevant results by other workers. In long papers it is 
helpful to indicate the way in which the paper is arranged and the results 
presented.

For articles not divided into sections, precede the 
start of the text (without leaving a blank line)
with the command \verb"\nosections", which provides the 
appropriate space and causes the paragraph indentation to be cancelled 
for the first paragraph.

Footnotes should be avoided whenever possible. If required they should be 
used only for brief notes that do not fit conveniently into the text. The 
standard \LaTeX\ macro \verb"\footnote" should be used and will normally
give an appropriate symbol; if a footnote sign needs to be specified
directly \verb"\footnote[<num>]{Text}" can be used instead where
\verb"<num>" is the number of the appropriate symbol as discussed in 
relation to authors (1~=~\verb"\dag", 2~=~\verb"\ddag", 3~=~\verb"\S", 
etc).  

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from 
colleagues, special work by technical staff or financial support from 
organizations should do so in an unnumbered Acknowledgments section 
immediately following the last numbered section of the paper. The 
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section. For Letters 
\verb"\ack" does not set a heading but leaves a line space and does not 
indent the next paragraph.

\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts 
the flow of the article, may be consigned to an appendix. 
Any appendices should be included at the end of the main text of the paper, after the acknowledgments section (if any) but before the reference list.
If there are 
two or more appendices they will be called Appendix A, Appendix B, etc. 
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"\appendix" is used to signify the start of the
appendixes. Thereafter \verb"\section", \verb"\subsection", etc, will 
give headings appropriate for an appendix. To obtain a simple heading of 
`Appendix' use the code \verb"\section*{Appendix}". If it contains
numbered equations, figures or tables the command \verb"\appendix" should
precede it and \verb"\setcounter{section}{1}" must follow it. 
 
\subsection{Some matters of style}
It will help your readers if your article is written in a clear,
consistent and concise manner. The Production Department at Institute of Physics
Publishing will try to make sure that your work is presented to its
readers in the best possible way without sacrificing the individuality of
your writing.

The main elements of consistency and style we look for 
are presented in the booklet 
{\it Notes for Authors} (available upon request from Institute of Physics 
Publishing, 
Dirac House, Temple Back, Bristol, BS1 6BE, UK or from our World Wide
Web site (\verb"http://www.iop.org/Journals/nfa/"). Some recommended 
points to note, however, are the following.
\begin{enumerate}
\item Authors are often inconsistent in the use of `ize' and `ise' endings.
We recommend using `-ize' spellings (diagonalize, 
renormalization, minimization, etc) but there are some common 
exceptions to this, for example: devise, 
promise and advise.

\item English spellings are preferred (colour, flavour, behaviour, 
tunnelling, artefact, focused, focusing, fibre, etc). We write of a 
computer program on disk; otherwise, we use `programme' and `disc'.

\item The words table, figure, equation and reference should be written 
in full and {\bf not} contracted to Tab., fig., eq. and ref.
\end{enumerate}

Please check your article carefully for accuracy, consistency and clarity before
submission. Remember that your article will probably be read by many
people whose native language is not English and who may not therefore 
be aware of many of the subtle meanings of words or idiomatic phases
present in the English language. It therefore helps if you try and keep
sentences as short and simple as possible.

\section{Mathematics}
\subsection{Two-line constructions}
The great advantage of \LaTeX\ 
over other text processing systems is its 
ability to handle mathematics to almost any degree of complexity. However, 
in order to produce an article suitable for publication within a journal, 
authors should exercise some restraint on the constructions used.
For simple fractions in the text the solidus \verb"/", as in 
$\lambda/2\pi$, should be used instead of \verb"\frac" or \verb"\over", 
care being taken to use parentheses where necessary to avoid ambiguity, 
for example to distinguish between $1/(n-1)$ and $1/n-1$. Exceptions to 
this are the proper fractions $\frac12$, $\frac13$, $\frac34$, 
etc, which are better left in this form. In displayed equations 
horizontal lines are preferable to solidi provided the equation is 
kept within a height of two lines. A two-line solidus should be 
avoided where possible; the construction $(\ldots)^{-1}$ should be 
used instead; for example use:
\begin{equation*}
\frac{1}{M_{\rm a}}\left(\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}\right)^{-1}
\end{equation*}
instead of
\begin{equation*}
\frac{1}{M_{\rm a}}\biggl/\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}.
\end{equation*}


\subsection{Roman and italic in mathematics}
In mathematics mode \LaTeX\ automatically sets variables in an italic 
font. In most cases authors should accept this italicization. However, 
there are some cases where it is better to use a Roman font; for 
instance, IOPP journals use a Roman d for a differential d, a Roman e 
for an exponential e and a Roman i for the square root of $-1$. To 
accommodate this and to simplify the  typing of equations we have 
provided some extra definitions. \verb"\rmd", \verb"\rme" and \verb"\rmi" 
now gives Roman d, e and i respectively for use in equations, 
e.g.\ $\rmi x\rme^{2x}\rmd x/\rmd y$ 
is obtained by typing \verb"$\rmi x\rme^{2x}\rmd x/\rmd y$". 
 

Certain other common mathematical functions, such as cos, sin, det and 
ker, should appear in Roman type. \LaTeX\ provides macros for 
most of these functions 
(in the cases above, \verb"\cos", \verb"\sin", \verb"\det" and \verb"\ker" 
respectively); we have also provided 
additional definitions for $\Tr$, $\tr$ and 
$\Or$ (\verb"\Tr", \verb"\tr" and \verb"\Or", respectively). 

Subscripts and superscripts should be in Roman type if they are labels 
rather than variables or characters that take values. For example in the 
equation
\[
\epsilon_m=-g\mu_{\rm n}Bm
\]
$m$, the $z$ component of the nuclear spin, is italic because it can have 
different values whereas n is Roman because it 
is a label meaning nuclear ($\mu_{\rm n}$ 
is the nuclear magneton).

\subsection{Alignment of mathematics}
\subsubsection{Alignment on the secondary margin.}
IOPP style for displayed mathematics in single-column journals
is not to centre equations, 
as \LaTeX\ normally does, but to have each equation indented to a 
secondary margin a fixed 
distance from the left-hand margin of the text, except for long 
equations that will just fit on one line, or need to be continued on 
subsequent lines, which start full left.  
Any continuation lines are indented the fixed amount. 
The macros in the IOPP preprint style automatically 
line equations up on the 
secondary margin unless they are set within double dollar signs.
Thus the use of double dollar signs should be avoided and the alternative
\verb"\[ ... \]" should be used instead for unnumbered equations.
The equation environment should normally be used for numbered single-line
equations and the eqnarray array environment for multiline equations.
It is then only necessary to indicate which lines 
should start full left and this is done by including \verb"\fl" (full
left) at the start of the lines. 
Thus the equations:
\begin{eqnarray}
\phi_{k}(\vec{r})=(2\pi)^{-3/2} \exp(\rmi\vec{k}\cdot\vec{r}) \\
N^+=\exp(\case12\pi\nu)\Gamma(1-\rmi\nu).
\end{eqnarray} 
are set with the code
\begin{verbatim}
\begin{eqnarray}
\phi_{k}(\vec{r})=(2\pi)^{-3/2} \exp(\rmi\vec{k}\bdot\vec{r}) \\
N^+=\exp(\case12\pi\nu)\Gamma(1-\rmi\nu).
\end{eqnarray} 
\end{verbatim}
Where an equation will not fit on a line if indented but would if it 
were not, then the equation is started full left and this is achieved 
simply by adding \verb"\fl" to the start of the line. For example
the equation
\begin{equation}
\fl R_{\rho l m,\rho'l'm'}(E)=\frac{1}{r_0}\sum_{i,j}\;\langle\rho l m r_0
\mid\Phi_i\rangle\; [(H_{\Omega}+B)-ES_{\Omega}]^{-1}_{i,j}\;\langle\Phi_j
\mid\rho'l'm'r_0\rangle. 
\end{equation}
does not fit on the line if indented to the secondary margin but fits in 
comfortably when full left.


For equations which do not fit on one line, even if started full left,
the first line should be set full left with the turnover lines at the 
secondary margin. This is achieved by using the eqnarray
environment
and adding \verb"\fl" at the start 
of the first line and \verb"\\" at the end of each line (apart from
the final line of the equation). Equations should be split at 
mathematically sound points, often at =, + or $-$ signs or between 
terms multiplied together. The connecting signs are not repeated and 
appear only at the beginning of the turned-over line. A multiplication 
sign should be added to the start of turned-over lines where the break 
is between two multiplied terms. Where an equation is broken at an 
equals sign (or similar, i.e.\ $\equiv$, $\le$, $\sim$, etc) the sign 
is made more prominent by aligning it to the left of the secondary 
margin; where it is a +, $-$ or $\times$ the sign goes to the right. 
Alignment to the left of the secondary margin is achieved by adding 
\verb"\lo" in front of the sign (and enclosing the sign within braces if it 
consists of more than one character or control sequence, e.g.\ 
\verb"\lo{:=}"). An example demonstrating these features is: 
\begin{eqnarray} 
\fl\langle\cos(q\Omega_s)\rangle=\frac12\int^\infty_0 
\frac{k_s(b)}{k^{\rm tot}_s}\{\cos[q\Omega^{\rm o}_S(b_f,R^s_{x})]+
\cos[q\Omega^{\rm i}_S(b_f,R^s_{x})]\}
2\pi b\, \rmd b\nonumber\\ 
\lo=\sum_c{(\mu^s_c)^2/\vert\Delta V'_s(R^s_{\rm c})\vert \over \sum_n 
(R^s_n\mu^s_n)^2(1-V^s_n/E)^{1/2}/\vert\Delta'_s(R^s_n)\vert}\nonumber\\ 
\times \frac12\int^{b_{\rm max}}_0 
\{\cos[q\Omega^{\rm o}_S(b_f,R^s_{\rm c})]
+\cos[q\Omega^{\rm i}_S(b_f,R^s_{\rm c})]\} 
b\, \rmd b/v_s(b,R^s_{\rm c}). 
\end{eqnarray} 
where a simplified version of the 
code used is: 
\begin{verbatim} 
\begin{eqnarray} 
\fl    <first line>  \nonumber\\ 
\lo=   <second line> \nonumber\\
\times <third line>  
\end{eqnarray} 
\end{verbatim} 

Note that alignment at the secondary margin normally takes 
precedence over aligning equals signs so there is usually no need 
to include any ampersands within the 
\verb"eqnarray" environment.


\subsubsection{Secondary alignment.}
While the primary alignment either on the secondary 
margin or full left is adequate in most cases 
there are examples where additional 
alignment is desirable. Firstly, for repeated series of short 
equations, secondly for
equations with attached conditions and thirdly for connected 
series of equations with a short left-hand side which together
occupy more than a full line but where each individual 
part is short. In these cases the \verb"eqnarray" environment 
should be used; there will still be alignment at the 
secondary margin but ampersands should be positioned to 
provide the secondary alignment. For equations with conditions the
space separating the longest part from its condition is provided by 
\verb"\qquad". Examples of equations requiring secondary alignment are:
\begin{eqnarray}
A^{(3/2)}=A^{(+)}-A^{(-)}&(I=\case32)\\
A^{(1/2)}=A^{(+)}+2A^{(-)}\qquad&(I=\case12)\\            
A^{(0)}&({\rm otherwise}).
\end{eqnarray}
which is obtained with the code
\begin{verbatim}
\begin{eqnarray}
A^{(3/2)}=A^{(+)}-A^{(-)}&(I=\case32)\\
A^{(1/2)}=A^{(+)}+2A^{(-)}\qquad&(I=\case12)\\            
A^{(0)}&({\rm otherwise}).
\end{eqnarray}
\end{verbatim}
and
\begin{eqnarray}
C(12)&=[\vec\pi(x)\cdot\vec\phi(x+r)]\nonumber\\
&\simeq 1-{\rm const}{r^2\over L^2}\int^L_r{x\, \rmd x\over 
x^2}+\cdots\\
&\simeq 1-{\rm const}{r^2\over L^2}\ln\left({L\over r}\right)+\cdots.
\end{eqnarray}
for which the code is
\begin{verbatim}
\begin{eqnarray}
C(12)&=[\vec\pi(x)\cdot\vec\phi(x+r)]\nonumber\\
&\simeq 1-{\rm const}{r^2\over L^2}\int^L_r{x\, \rmd x\over 
x^2}+\cdots\\
&\simeq 1-{\rm const}{r^2\over L^2}\ln\left({L\over r}\right)+\cdots.
\end{eqnarray}
\end{verbatim}

\subsection{Displayed equations in double-column journals}
The way equations are displayed in the Institute of Physics Publishing
double-column journals differs from that in single-column journals. 
However
authors submitting to double-column journals can produce their equations
as described above for single-column journals
and conversion to the proper double-column format will
take place here as proofs are being prepared. 

\subsection{Special characters for mathematics}
Bold italic characters are used in our journals to signify vectors (rather
than using an upright bold or an over arrow). To obtain this effect
use \verb"\bi{#1}" within maths mode, e.g. $\bi{ABCdef}$. If upright 
bold characters are required in maths use \verb"\mathbf{#1}" within maths
mode, e.g. $\mathbf{XYZabc}$. The calligraphic (script) uppercase alphabet
is obtained with \verb"\mathcal{AB}" or \verb"\cal{CD}" 
($\mathcal{AB}\cal{CD}$).

The American Mathematical Society provides a series of extra symbol fonts
to use with \LaTeX\ and packages containing the character definitions to
use these fonts. Authors wishing to use Fraktur 
\ifiopams$\mathfrak{ABC}$ \fi
or Blackboard Bold \ifiopams$\mathbb{XYZ}$ \fi can include the appropriate
AMS package (e.g. amsgen, amsfonts, amsbsy, amssymb) with a 
\verb"\usepackage" command or add the command \verb"\usepackage{iopams}"
which loads the four AMS packages mentioned above and also provides
definitions for extra bold characters (all Greek letters and some
additional other symbols). 

The package iopams uses the definition \verb"\boldsymbol" in amsbsy
which allows individual non-alphabetical symbols and Greek letters to be 
made bold within equations.
The bold Greek lowercase letters \ifiopams$\balpha \ldots\bomega$,\fi 
are obtained with the commands 
\verb"\balpha" \dots\ \verb"\bomega" (but note that
bold eta\ifiopams, $\bfeta$,\fi\ is \verb"\bfeta" rather than \verb"\beta")
and the capitals\ifiopams, $\bGamma\ldots\bOmega$,\fi\ with commands 
\verb"\bGamma" \dots\
\verb"\bOmega". Bold versions of the following symbols are
predefined in iopams: 
bold partial\ifiopams, $\bpartial$,\fi\ \verb"\bpartial",
bold `ell'\ifiopams, $\bell$,\fi\  \verb"\bell", 
bold imath\ifiopams, $\bimath$,\fi\  \verb"\bimath", 
bold jmath\ifiopams, $\bjmath$,\fi\  \verb"\bjmath", 
bold infinity\ifiopams, $\binfty$,\fi\ \verb"\binfty", 
bold nabla\ifiopams, $\bnabla$,\fi\ \verb"\bnabla", 
bold centred dot\ifiopams, $\bdot$,\fi\  \verb"\bdot", other 
characters are made bold using 
\verb"\boldsymbol{\symbolname}".

Table~\ref{math-tab2} lists some other macros for use in 
mathematics with a brief description of their purpose.
Both \verb"\ms" (medium space) and \verb"\bs" (big space) can be used to
provide extra spacing between lines of a displayed equation or table.
This space may be necessary when several separate equations are within the
same equation environment. 

\begin{table}
\caption{\label{math-tab2}Other macros defined in IOPP macros for use in maths.}
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus
12pt}}l}}
\br
Macro&Result&Description\\
\mr
Spaces\\
\verb"\fl"&&Start line of equation full left\\
\verb"\ms"&&Spread out lines in displayed equations slightly ($\sim$3pt)\\
\verb"\bs"&&Bigger space ($\sim$6pt) to separate lines in displays\\
\verb"\ns"&&Small negative space between lines in displays\\
\bs
\multispan{3}{For symbols to left of 5 pica indent\hfill}&\\
\verb"\lo{#1}"&$\#1$&Any symbol overhanging to left\\
\verb"\eql"&$=$&Left overhanging equals sign\\
\verb"\lsim"&$\sim$&Left overhanging tilde\\ 
\verb"\lsime"q&$\simeq$&Left overhanging approximately equals\\
\verb"\lequiv"&$\equiv$&Left overhanging equivalent sign\\
\bs
Miscellaneous\\
\verb"\case{#1}{#2}"&$\case{\#1}{\#2}$&Text style fraction in display\\
\verb"\Tr"&$\Tr$&Roman Tr (Trace)\\
\verb"\tr"&$\tr$&Roman tr (trace)\\
\verb"\Or"&$\Or$&Roman O (of order of)\\
\verb"\tdot{#1}"&$\tdot{x}$&Triple dot over character\\
\verb"\lshad"&$\lshad$&Text size left shadow bracket\\
\verb"\rshad"&$\rshad$&Text size right shadow bracket\\
\br
\end{tabular*}
\end{table}

\subsection{Miscellaneous points}
Exponential expressions, especially those containing subscripts or 
superscripts, are clearer if the notation $\exp(\ldots)$ is used except for 
simple examples. For instance $\exp[\rmi(kx-\omega t)]$ and $\exp(z^2)$ are 
preferred to $\e^{\rmi(kx-\omega t)}$ and $\e^{z^2}$, but 
$\e^2$ 
is acceptable. Similarly the square root sign $\sqrt{\phantom{b}}$ should 
only be used with relatively
simple expressions, e.g.\ $\sqrt2$ and $\sqrt{a^2+b^2}$;
in other cases the 
power $1/2$ should be used.

It is important to distinguish between $\ln = \log_\e$ and $\lg 
=\log_{10}$. Braces, brackets and parentheses should be used in the 
following order: $\{[(\;)]\}$. The same ordering of brackets should be 
used within each size. However, this ordering can be ignored if the
brackets have a 
special meaning (e.g.\ if they denote an average or a function).  Decimal 
fractions should 
always be preceded by a zero: for example 0.123 {\bf not} .123. For long 
numbers commas are not inserted but instead a thin space is added after 
every third character away from the position of the decimal point unless 
this leaves a single separated character: e.g.\ $60\,000$, $0.123\,456\,78$ 
but 4321 and 0.7325.

Equations that are referred to in the text should be numbered with 
the number on the right-hand side. 


\subsection{Equation numbering}
\label{eqnum}
\LaTeX\ provides facilities for automatically numbering equations 
and these should be used where possible. Sequential numbering (1), (2), 
etc, is the default numbering system although, if the command
\verb"\eqnobysec" is included in the preamble, equation numbering
by section is obtained, e.g.\ 
(2.1), (2.2), etc. In articles with several appendixes equation numbering
by section is useful in the appendixes even when sequential numbering has
been used throughout the main body of the text and is switched on by the 
\verb"\appendix" command. Equation numbering by section {\it must} 
be used for {\it Reports on Progress in 
Physics}. When referring to an equation in the text, either put 
the equation number, in brackets, e.g.\ `as in (2)', or spell out the 
word equation in full, e.g.\ `if equation (2) is factorized'; do not 
use abbreviations such as eqn or eq.
When cross-referencing is used, \verb"\ref{<label>}"
 will produce `(\verb"<eqnum>")',
\verb"\eref{<label>}" produces `equation (\verb"<eqnum>")' and
\verb"\Eref{<label>}" produces `Equation (\verb"<eqnum>")', 
where \verb"<label>" 
is the
label to produce equation number \verb"<eqnum>".


If an equation number is centred between lines then the 
command \verb"\eqalign{...}" 
can be used within the `equation' environment.
After \verb"\begin{equation}" enclose the lines over 
which the number is
to be centred 
within \verb"\eqalign{...}"  with 
\verb"\\" or \verb"\cr"
at the end of each line. Ampersands are unnecessary within the
\verb"\eqalign" but can be used for secondary alignment if necessary.
The code 
\begin{verbatim}
\begin{equation}
\eqalign{T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}}
\end{equation}
\end{verbatim}
gives four equations with a centred 
number:
\begin{equation}
\eqalign{T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}}
\end{equation}
Note that the secondary alignment at the equals signs would not normally
be necessary but is included here for demonstration purposes.


Sometimes it is useful to number equations as parts of the same
basic equation. This can be accomplished by inserting the 
commands \verb"\numparts" before the equations concerned and 
\verb"\endnumparts" when reverting to the normal sequential numbering.
The equations below show the previous equations numbered as separate parts
using \verb"\numparts ... \endnumparts" and the \verb"eqnarray"
environment
\numparts
\begin{eqnarray}
T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}
\end{eqnarray}
\endnumparts


\subsection{Miscellaneous extra commands for displayed equations}
The \verb"\cases" command of Plain \TeX\ is available
for use with \LaTeX\ but has been amended slightly to 
increase the space between the equation and the condition. 
\Eref{cases} 
demonstrates simply the output from the \verb"\cases" command
\begin{equation}
\label{cases}
X=\cases{1&for $x \ge 0$\\
-1&for $x<0$\\}
\end{equation}
The code used was:
\begin{verbatim}
\begin{equation}
\label{cases}
X=\cases{1&for $x \ge 0$\\
-1&for $x<0$\\}
\end{equation}
\end{verbatim}

To obtain text style fractions within displayed maths the command 
\verb"\case{#1}{#2}" can be used (see equations (2) and (5)) instead
of the usual \verb"\frac{#1}{#2}" command or \verb"{#1 \over #2}".

When two or more short equations are on the same line they should be 
separated by a `qquad space' (\verb"\qquad"), rather than
\verb"\quad" or any combination of \verb"\,", \verb"\>", \verb"\;" 
and \verb"\ ".
 


\section{Referencing}
Two different styles of referencing are in common use: the Harvard 
alphabetical system and the Vancouver
numerical system. All the IOPP journals allow 
the use of the Harvard system but the numerical system should {\bf not} be 
used in {\it Physics in Medicine and Biology}.
Brief descriptions of the use of the two 
referencing systems are given below.

\subsection{Harvard system}
In the Harvard system the name of the author appears in the text together 
with the year of publication. As appropriate, either the date or the name 
and date are included within parentheses. Where there are only two authors 
both names should be given in the text; if there are more than two 
authors only the first name should appear followed by `{\it et al}' 
(which can be obtained by 
typing \verb"\etal"). When two or 
more references to work by one author or group of authors occur for the 
same year they should be identified by including a, b, etc after the date 
(e.g.\ 1986a). If several references to different pages of the same article 
occur the appropriate page number may be given in the text, e.g.\ Kitchen 
(1982, p 39).

The reference list at the end of an article consists of an 
unnumbered section containing an
alphabetical listing by authors' names and in date order for each 
author or group of identical authors. The reference list in the 
preprint style is started by including the command
\verb"\section*{References}" and then
\verb"\begin{harvard}".
There will be two basic types of 
entries within the reference list: (i) those to journal articles and 
(ii) those to books, conference proceedings and reports. For both of 
these types of references \verb"\item[]"
is required before the start of an individual reference.
The reference list is completed with \verb"\end{harvard}".
There is also a shortened form of the coding; \verb"\section*{References}"
and \verb"\begin{harvard}" can be replaced by the single command
\verb"\References" and \verb"\end{harvard}" can be shortened to
\verb"\endrefs".


\subsubsection{References to journal articles.}
A normal reference to a journal article contains three changes of 
font:
the authors and date appear in Roman type, the journal title in 
italic, the volume number in bold and the page numbers in Roman again. 
A typical journal entry would be:

\smallskip
\begin{harvard}
\item[] Cisneros A 1971 {\it Astrophys.\ Space Sci.} {\bf 10} 87
\end{harvard}
\smallskip

\noindent which would be obtained by typing, within the references
environment 
\begin{verbatim}
\item[] Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\end{verbatim}

Features to note are the following.

(i) The authors should be in the form surname (with only the first 
letter capitalized) {\bf followed} by the initials with {\bf no} 
periods after the initials. Authors should be separated by a comma 
except for the last two which should be separated by `and' with no 
comma preceding it. For journals that accept titles of articles in the 
reference list,  the title should be in Roman (upright)
lower case letters, except for an initial 
capital, and should follow the date.

(ii) The journal is in italic and is abbreviated. \ref{jlabs} 
gives a list of 
macros that will give the correct abbreviation for 
many of the common journals. If a journal has several parts denoted by 
different letters the part letter
should be inserted after the journal in Roman type, e.g.\ 
{\it Phys.\ Rev.\ \rm A}. 

(iii) The volume number is bold; the page number is Roman.
 Both the initial and final page 
numbers should be given where possible. The final page number should be in 
the shortest possible form and separated from the initial page number by an 
en rule (\verb"--"), e.g.\ 1203--14.

(iv) Where there are two or more references with identical authors, 
the authors' names should not be repeated but should be replaced by 
\verb"\dash" on the second and following occasions. Thus
\begin{verbatim}
\item[]Davis R, Mann A K and Wolfenstein L 1989  {\it Ann. Rev. Nucl.
Part. Sci.} {\bf 39} 467
\item[]\dash 1990 Private communication 
\end{verbatim}

\subsection{Electronic journal references}
These do not always follow the conventional {\it year-journal-volume-page numbers} pattern.
The journals may be electronic {\it and} print (e.g. {\it Phys. Rev.} D) or electronic only (e.g. {\it Living Reviews in Relativity}).
Electronic journals with unique referencing systems are given below.

\begin{itemize}
\item {\it Phys. Rev. (parts A, C and D).} Each article has a six-digit unique reference number which is cited after the volume number. So, for example, 
\smallskip
\numrefs{1}
\item Carlip S and Vera R 1998 {\it Phys. Rev.} D {\bf 58} 011345 
\endnumrefs
\smallskip
Note that page numbers are no longer used.

\item {\it Journal of High Energy Physics.}
This journal is primarily electronic, with a print version for archive purposes.
Papers are cited as per the following example.
\smallskip
\numrefs{1}
\item Davies K and Brown G 1997 {\it J. High Energy Phys.}
JHEP12(1997)002
\endnumrefs
\smallskip
with no spaces, where JHEP is obvious, 12 is the month (December in this case), 1997 is the year and 002 is the paper number.

\item {\it Living Reviews in Relativity.}
This journal is electronic only. The short form for this journal is {\it Liv. Rev. Rel.} and papers are cited simply by a number consisting of year and article number. So, for example,
\smallskip
\numrefs{1}
\item Gundlach C 1999 {\it Liv. Rev. Rel.} 1994-4
\endnumrefs
\smallskip
\end{itemize}

\subsection{Preprint references}
Since the introduction of the LANL preprint server this class of references has grown in importance. 
To reflect this and to maximize  the number of hyperlinks available through IOPP's Hypercite service in our electronic journals, each preprint reference should be given in as much detail as possible. Such references should include authors, year of work and preprint number (preceded by the word {\it Preprint}). 
An example would be
\smallskip
\numrefs{1}
\item Harrison M 1999 Dipheomorphism-invariant manifolds {\it Preprint} hep-th/9909196
\endnumrefs
\smallskip
If the preprint has been published then the full published reference is given first and the preprint details can be added in brackets after, on a separate line. So, for example,
\smallskip
\numrefs{1}
\item Neilson D and Choptuik M 2000 {\it Class. Quantum Grav.} {\bf 17} 761
\item[](Neilson D and Choptuik M 1998 {\it Preprint} gr-qc/9812053)
\endnumrefs
\smallskip

\subsubsection{References to books, conference proceedings and reports.}

References to books, proceedings and reports are similar, but have only two
changes of font. The authors and date of publication are in Roman, the 
title of the book is in italic, and the editors, publisher, 
town of publication 
and page number are in Roman. A typical reference to a book and a
conference paper might be

\smallskip
\begin{harvard}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc.\
Int.\ Conf.\ on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\end{harvard}
\smallskip

\noindent which would be obtained by typing
\begin{verbatim}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. on Nuclear 
Physics (Munich)} vol~1 (Amsterdam: North-Holland/American 
Elsevier) p~517
\end{verbatim}
\noindent respectively.


Features to note are the following.

(i) Book titles are in italic and should be spelt out in full with 
initial capital letters for all except minor words. Words such as 
Proceedings, Symposium, International, Conference, Second, etc should 
be abbreviated to Proc., Symp., Int., Conf., 2nd, 
respectively, but the rest of the title should be given in full, 
followed by the date of the conference and the 
town or city where the conference was held. For 
Laboratory Reports the Laboratory should be spelt out wherever 
possible, e.g.\ {\it Argonne National Laboratory Report}.

(ii) The volume number as, for example, vol~2, should be followed by 
the editors, if any, in a form such as ed~A~J~Smith and P~R~Jones. Use 
\etal if there are more than two editors. Next comes the town of 
publication and publisher, within brackets and separated by a colon, 
and finally the page numbers preceded by p if only one number is given 
or pp if both the initial and final numbers are given.

Cross referencing between the text and the 
reference list is not necessary for alphabetic referencing 
in the Harvard system as adding or deleting a reference 
does not normally change any of the other references.

\subsection{Numerical system}
In the numerical system references are numbered sequentially 
throughout the text. The numbers occur within square brackets and one 
number can be used to designate several references. A numerical 
reference list in the preprint style is started by including the 
command \verb"\section*{References}" and then
\verb"\begin{thebibliography}{<num>}", where \verb"<num>" is the largest
number in the reference list (or any other number with the same number
of digits).  The 
reference list gives the references in 
numerical, not alphabetical, order and is completed by
\verb"\end{thebibliography}". Short forms of the commands are again
available: \verb"\Bibliography{<num>}" can be used at the start of the
references section and \verb"\endbib" at the end.
(Note that footnotes should not be 
part of a numerical reference system, but should be included in the 
text using the symbols \dag, \ddag, etc.)

References to journals and books are similar to those in the Harvard 
system, except that two or more references with identical authors are 
spelt out in full, i.e.\ they are {\bf not} replaced with \verb"\dash". 
When one number covers two or more separate references \verb"\nonum"
or \verb"\par\item[]" should be included at
the start of each reference in a group after the first.
A typical numerical reference list might begin

\smallskip

\numrefs{1}
\item Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} (Moscow: 
Moscow State University Press) p~103
\item Caplar R and Kulisic P 1973 {\it Proc.\ Int.\ Conf.\ 
on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\item Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\endnumrefs
\smallskip

\noindent which would be obtained by typing

\begin{verbatim}
\item Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. 
on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\item Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\end{verbatim}

The point to note is that this is identical to the entries in the 
Harvard system except that square brackets following
\verb"\item" are no longer required.
If you are using \LaTeX's cross-referencing system then \verb"\item" would be replaced by \verb"\bibitem{label}" corresponding to \verb"\cite{label}" in the text (see section \ref{refs}).
 
\subsection{Reference lists}
A complete reference should provide the reader with enough information to 
locate the article concerned and should consist of: name(s) and initials, 
date published, title of journal or book, volume number, editors, if any, 
and town of publication and publisher in parentheses for books, 
and finally the 
page numbers. Titles of journal articles may also be included.
Up to twenty authors may be given in a particular reference; where 
there are more than twenty only the first should be given followed by 
`{\it et al}'. Abbreviations of the names of periodicals used by Institute
of Physics Publishing are usually the same as those 
given in British Standard 
BS 4148: 1985. If an author is unsure of an abbreviation and the 
journal is not given in~\ref{jlabs}, 
it is best to leave the title in 
full. The terms {\it loc.\ cit.\ }and {\it ibid.\ }should not be used. 
Unpublished conferences and reports should generally not be included 
in the reference list and articles in the course of publication should 
be entered only if the journal of publication is known. 
A thesis submitted for a higher degree may be included 
in the reference list if it has not been superseded by a published 
paper and is available through a library; sufficient information 
should be given for it to be traced readily.

\section{Cross referencing\label{xrefs}}
The facility to cross reference items in the text is very useful when 
composing articles the precise form of which is uncertain at the start 
and where revisions and amendments may subsequently be made. When using 
cross referencing, labels are given to elements in the text, for 
instance sections, figures, tables or equations and the elements may 
be referred to elsewhere
in the text by using the label. When the article is 
first processed the labels are read in and assigned, e.g.\ 2.1 for a 
subsection or (4) for an equation number. When the article is 
processed a second or subsequent time the label assignments are read 
in at the start of the file and the correct values given in the text. 
\LaTeX\ provides excellent facilities for doing cross-referencing
and these can be very useful in preparing articles.

\subsection{References}
\label{refs}
Cross referencing is useful for numeric reference lists because, if it 
is used, adding 
another reference to the list does not then involve renumbering all 
subsequent references. It is not necessary for referencing 
in the Harvard system where the final reference list is alphabetical 
and normally no other changes are necessary when a reference is added or
deleted.
Two passes are necessary initially to get the cross references right 
but once they are correct a single run is usually sufficient provided an 
\verb".aux" file is available and the file 
is run to the end each time.
\verb"\cite" and \verb"\bibitem" are used to link citations in the text
with their entry in the reference list;
if the 
reference list contains an entry \verb"\bibitem{label}", 
then \verb"\bibitem{label}" 
will produce the correct number in the reference list and 
\verb"\cite{label}" will produce the number within square brackets in the 
text. \verb"label" may contain alphabetic letters, 
or punctuation characters but must not contain spaces or commas. It is also
recommended that the underscore character \_{} is not used in cross
referencing. 
Thus labels for the form 
\verb"eq:partial", \verb"fig:run1", \verb"eq:dy'", 
etc, may be used. When several 
references occur together in the text \verb"\cite" may be used with 
multiple labels with commas but no spaces separating them; 
the output will be the 
numbers within a single pair of square brackets with a comma and a 
thin space separating the numbers. Thus \verb"\cite{label1,label2,label4}"
would give [1,\,2,\,4]. Note that no attempt is made to sort the 
labels and no shortening of groups of consecutive numbers is done.
Authors should therefore try to use multiple labels in the correct 
order.

The numbers for the cross referencing are generated in the order the 
references appear in the reference list, so that if the entries in the 
list are not in the order in which the references appear in the text 
then the 
numbering within the text will not be sequential. To correct this 
change the ordering of the entries in the reference list and then 
rerun {\it twice}.

\subsection{Equation numbers, sections, subsections, figures and 
tables}
Cross references can be made to equation numbers, sections, 
subsections, figures and tables or
any numbered environment
and this is a very useful feature when 
writing a document as its final structure is often not fully defined 
at the start. Thus a later section can be referred to by a label 
before its precise number is known and when it is defined there is no 
need to search back through the document to insert the correct value 
manually. For this reason the use of cross referencing 
can save considerable time. 

Labels for equation numbers, sections, subsections, figures and tables 
are all defined with the \verb"\label{label}" command and cross references 
to them are made with the \verb"\ref{label}" command. The \verb"\label" 
macro 
identifies the type of environment it is used in and converts \verb"label" 
into the correct form for that type of environment, thus \verb"\ref{label}" 
might give (2.3) for an equation number but 3.1 for a subsection and 2 
for a figure or table number. 

Any section, subsection, subsubsection, appendix or subappendix 
command defines a section type label, e.g. 1, 2.2, A2, A1.2 depending 
on context. A typical article might have in the code of its introduction 
`The results are discussed in section\verb"~\ref{disc}".' and
the heading for the discussion section would be:
\begin{verbatim}
\section{Results}
\label{disc}
\end{verbatim}
Labels to sections, etc, may occur anywhere within that section except
within another numbered environment. 
Within a maths environment labels can be used to tag equations which are 
referred to within the text. 
An example of an equation with a label and a reference to it 
is:
\begin{verbatim}
\begin{equation}
X=a\cos\theta+ b\sin\phi. \label{cossin}
\end{equation}
Equation (\ref{cossin}) ...
\end{verbatim}
which produces
\begin{equation}
X=a\cos\theta+ b\sin\phi. \label{cossin}
\end{equation}
Equation (\ref{cossin}) ...

In addition to the standard \verb"\ref{<label>}" the abbreviated
forms given in the \tref{abrefs}
are available for reference to standard parts of the text

\Table{\label{abrefs}Alternatives to the normal references $\backslash${\tt ref} 
and the text generated by
them. Note it is not normally necessary to include the word equation
before an equation number except where the number starts a sentence. The
versions producing an initial capital should only be used at the start of
sentences.} 
\br
Reference&Text produced\\
\mr
\verb"\eref{<label>}"&equation (\verb"<num>")\\
\verb"\Eref{<label>}"&Equation (\verb"<num>")\\
\verb"\fref{<label>}"&figure \verb"<num>"\\
\verb"\Fref{<label>}"&Figure \verb"<num>"\\
\verb"\sref{<label>}"&section \verb"<num>"\\
\verb"\Sref{<label>}"&Section \verb"<num>"\\
\verb"\tref{<label>}"&table \verb"<num>"\\
\verb"\Tref{<label>}"&Table \verb"<num>"\\
\br
\endTable

\section{Tables and table captions}
Tables are numbered serially and referred to in the text 
by number (table 1, etc, {\bf not} tab. 1). Each table should have an 
explanatory caption which should be as concise as possible. If a table 
is divided into parts these should be labelled \pt(a), \pt(b), 
\pt(c), etc but there should be only one caption for the whole 
table, not separate ones for each part.

In the preprint style the tables may be included in the text 
or listed separately after the reference list 
starting on a new page. 

\subsection{The basic table format}
The standard form for a table is:
\begin{verbatim}
\begin{table}
\caption{\label{label}Table caption.}
\begin{indented}
\item[]\begin{tabular}{@{}llll}
\br
Head 1&Head 2&Head 3&Head 4\\
\mr
1.1&1.2&1.3&1.4\\
2.1&2.2&2.3&2.4\\
\br
\end{tabular}
\end{indented}
\end{table}
\end{verbatim}

Points to note are:
\begin{enumerate}
\item The caption comes before the table. It should have a full stop at
the end.

\item Tables are normally set in a smaller type than the text.
The normal style is for tables to be indented in the same way as
equations. This is accomplished
by using \verb"\begin{indented}" \dots\ \verb"\end{indented}"
and putting \verb"\item[]" before the start of the tabular environment.
Omit these
commands for any tables which will not fit on the page when indented.

\item The default alignment of columns should be aligned left and 
adding \verb"@{}" omits the extra space before the first column.

\item Tables have only horizontal rules and no vertical ones. The rules at
the top and bottom are thicker than internal rules and are set with
\verb"\br" (bold rule). 
The rule separating the headings from the entries is set with
\verb"\mr" (medium rule).

\item Numbers in columns should be aligned on the decimal point;
to help do this a control sequence \verb"\lineup" has been defined 
which sets \verb"\0" equal to a space the size of a digit, \verb"\m"
to be a space the width of a minus sign, and \verb"\-" to be a left
overlapping minus sign. \verb"\-" is for use in text mode while the other
two commands may be used in maths or text.
(\verb"\lineup" should only be used within a table
environment after the caption so that \verb"\-" has its normal meaning
elsewhere.) See table~\ref{tabone} for an example of a table where
\verb"\lineup" has been used.
\end{enumerate}

\begin{table}
\caption{\label{tabone}A simple example produced using the standard table commands 
and $\backslash${\tt lineup} to assist in aligning columns on the 
decimal point. The width of the 
table and rules is set automatically by the 
preamble.} 

\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{7}{l}}
\br                              
$\0\0A$&$B$&$C$&\m$D$&\m$E$&$F$&$\0G$\cr 
\mr
\0\023.5&60  &0.53&$-20.2$&$-0.22$ &\01.7&\014.5\cr
\0\039.7&\-60&0.74&$-51.9$&$-0.208$&47.2 &146\cr 
\0123.7 &\00 &0.75&$-57.2$&\m---   &---  &---\cr 
3241.56 &60  &0.60&$-48.1$&$-0.29$ &41   &\015\cr 
\br
\end{tabular}
\end{indented}
\end{table}

\subsection{Simplified coding and extra features for tables}
The basic coding format can be simplified using extra commands provided in
the \verb"iopart" class file. The commands up to and including 
the start of the tabular environment
can be replaced by
\begin{verbatim}
\Table{\label{label}Table caption}
\end{verbatim}
this also activates the definitions within \verb"\lineup".
The final three lines can also be reduced to \verb"\endTable" or
\verb"\endtab". Similarly for a table which does not fit in when indented
\verb"\fulltable{\label{label}caption}" \dots\ \verb"\endfulltable" or \verb"\endtab"
can be used. \LaTeX\ optional positional parameters can, if desired, be added after 
\verb"\Table{\label{label}caption}" and \verb"\fulltable{\label{label}caption}".


\verb"\centre{#1}{#2}" can be used to centre a heading 
\verb"#2" over \verb"#1" 
columns and \verb"\crule{#1}" puts a rule across 
\verb"#1" columns. A negative 
space \verb"\ns" is usually useful to reduce the space between a centred 
heading and a centred rule. \verb"\ns" should occur immediately after the 
\verb"\\" of the row containing the centred heading (see code for
\tref{tabl3}). A small space can be 
inserted between rows of the table 
with \verb"\ms" and a half line space with \verb"\bs" 
(both must follow a \verb"\\" but should not have a 
\verb"\\" following them).
   
\Table{\label{tabl3}A table with headings spanning two columns and containing notes. 
To improve the 
visual effect a negative skip ($\backslash${\tt ns})
has been put in between the lines of the 
headings. Commands set-up by $\backslash${\tt lineup} are used to aid 
alignment in columns. $\backslash${\tt lineup} is defined within
the $\backslash${\tt Table} definition.}
\br
&&&\centre{2}{Separation energies}\\
\ns
&Thickness&&\crule{2}\\
Nucleus&(mg\,cm$^{-2}$)&Composition&$\gamma$, n (MeV)&$\gamma$, 2n (MeV)\\
\mr
$^{181}$Ta&$19.3\0\pm 0.1^{\rm a}$&Natural&7.6&14.2\\
$^{208}$Pb&$\03.8\0\pm 0.8^{\rm b}$&99\%\ enriched&7.4&14.1\\
$^{209}$Bi&$\02.86\pm 0.01^{\rm b}$&Natural&7.5&14.4\\
\br
\end{tabular}
\item[] $^{\rm a}$ Self-supporting.
\item[] $^{\rm b}$ Deposited over Al backing.
\end{indented}
\end{table}

Units should not normally be given within the body of a table but 
given in brackets following the column heading; however, they can be 
included in the caption for long column headings or complicated units. 
Where possible tables should not be broken over pages. 
If a table has related notes these should appear directly below the table
rather than at the bottom of the page. Notes can be designated with
footnote symbols (preferable when there are only a few notes) or
superscripted small roman letters. The notes are set to the same width as
the table and in normal tables follow after \verb"\end{tabular}", each
note preceded by \verb"\item[]". For a full width table \verb"\noindent"
should precede the note rather than \verb"\item[]". To simplify the coding 
\verb"\tabnotes" can, if desired, replace \verb"\end{tabular}" and 
\verb"\endtabnotes" replaces
\verb"\end{indented}\end{table}".

If all the tables are grouped at the end of a document
the command \verb"\Tables" is used to start a new page and 
set a heading `Tables and table captions'.
  
\section{Figures and figure captions}
Figures may be included in an article as encapsulated PostScript files or
using the \LaTeX\ picture environment. Alternatively authors may send in 
high quality printed versions of their figures (fair copies) and 
attach copies of the fair copies to each 
typescript. The fair copies should be in black 
Indian ink or printing on tracing paper, plastic or white card or 
paper, or glossy photographs.

Each figure should have a brief caption describing it and, if 
necessary, interpreting the various lines and symbols on the figure. 
As much lettering as possible should be removed from the figure itself and 
included in the caption. If a figure has parts, these should be 
labelled ($a$), ($b$), ($c$), etc. 
\Tref{blobs} gives the definitions for describing symbols and lines often
used within figure captions (more symbols are available
when using the optional packages loading the AMS extension fonts).

\begin{table}
\caption{\label{blobs}Control sequences to describe lines and symbols in figure 
captions.}
\begin{indented}
\item[]\begin{tabular}{@{}lllll}
\br
Control sequence&Output&&Control sequence&Output\\
\mr
\verb"\dotted"&\dotted        &&\verb"\opencircle"&\opencircle\\
\verb"\dashed"&\dashed        &&\verb"\opentriangle"&\opentriangle\\
\verb"\broken"&\broken&&\verb"\opentriangledown"&\opentriangledown\\
\verb"\longbroken"&\longbroken&&\verb"\fullsquare"&\fullsquare\\
\verb"\chain"&\chain          &&\verb"\opensquare"&\opensquare\\
\verb"\dashddot"&\dashddot    &&\verb"\fullcircle"&\fullcircle\\
\verb"\full"&\full            &&\verb"\opendiamond"&\opendiamond\\
\br
\end{tabular}
\end{indented}
\end{table}

Unless the figures files are incorporated into the text 
the captions should be listed at the end of the article.

\subsection{Figure captions where figure files are not available}
The command \verb"\Figures" starts a new page and an unnumbered section
with the heading `Figure captions'. 
The captions should then be set with the commands:
\begin{verbatim}
\begin{figure}
\caption{\label{label}Figure caption.}
\end{figure}
\end{verbatim}
or more simply
\begin{verbatim}
\Figure{\label{label}Figure caption.}                                         
\end{verbatim}
The caption should finish with a full stop and the printed version will be 
indented as in Institute of Physics Publishing single-column journals.

\subsection{Inclusion of graphics files}
If graphics files are available as encapsulated PostScript (EPS) files 
(or are created within the \LaTeX\ picture environment) they may
be included within the body of the text at an appropriate point using a
standard graphics inclusion package. Authors should ensure EPS files meet
the following criteria.
\begin{itemize}
\item The Bounding Box should indicate the area of the figure 
with a minimum of white space around it and not the dimensions of the
page.
\item Any fonts used should be from the standard PostScript set (Times,
Helvetica, Courier and Symbol).
\item Scanned images should be of 600 dpi resolution for line art (black
and white) and 150 dpi resolution for grayscale or colour.
\item Captions and labels (e.g.\ Figure 1) should not be included in the
EPS file although part letters (e.g.\ ($a$)) are acceptable provided they
are placed close or within the boundary of the figure. 
\end{itemize}

The precise coding required will depend on the graphics package being used
and the printer driver. We use a printer driver compatible with DVIPS but
authors should avoid using special effects generated by including verbatim
PostScript code within the \LaTeX\ file with specials other than the
standard figure inclusion ones.

Using the epsf package figures can be included using code such as:
\begin{verbatim}
\begin{figure}
\begin{center}
\epsfbox{file.eps}
\end{center}
\caption{\label{label}Figure caption}
\end{figure}
\end{verbatim}


\clearpage

\appendix
\section{List of macros for formatting text, figures and tables}


\begin{table}[hb]
\caption{Macros available for use in text. Parameters in square brackets 
are optional.}
\footnotesize\rm
\begin{tabular}{@{}*{7}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\title[#1]{#2}"&Title of article and short title (optional)\\
\verb"\paper[#1]{#2}"&Title of paper and short title (optional)\\
\verb"\letter{#1}"&Title of Letter to the Editor\\
\verb"\comment[#1]{#2}"&Title of Comment and short title (optional)\\
\verb"\topical[#1]{#2}"&Title of Topical Review and short title 
(optional)\\
\verb"\review[#1]{#2}"&Title of review article and short title (optional)\\
\verb"\note[#1]{#2}"&Title of Note and short title (optional)\\
\verb"\prelim[#1]{#2}"&Title of Preliminary Communication \& short title\\
\verb"\author{#1}"&List of all authors\\
\verb"\article[#1]{#2}{#3}"&Type and title of other articles and 
short title (optional)\\
\verb"\address{#1}"&Address of author\\
\verb"\pacs{#1}"&PACS classification codes\\
\verb"\pacno{#1}"&Single PACS classification code\\
\verb"\ams{#1}"&American Mathematical Society classification code\\
\verb"\submitto{#1}"&`Submitted to' message\\
\verb"\maketitle"&Creates title page\\
\verb"\begin{abstract}"&Start of abstract\\
\verb"\end{abstract}"&End of abstract\\
\verb"\nosections"&Inserts space before text when no sections\\
\verb"\section{#1}"&Section heading\\
\verb"\subsection{#1}"&Subsection heading\\
\verb"\subsubsection{#1}"&Subsubsection heading\\
\verb"\appendix"&Start of appendixes\\
\verb"\ack"&Acknowledgments heading\\
\verb"\References"&Heading for reference list\\
\verb"\begin{harvard}"&Start of alphabetic reference list\\
\verb"\end{harvard}"&End of alphabetic reference list\\
\verb"\begin{thebibliography}{#1}"&Start of numeric reference list\\
\verb"\end{thebibliography}"&End of numeric reference list\\
\verb"\etal"&\etal for text and reference lists\\
\verb"\dash"&Rule for repeated authors in alphabetical reference list\\
\verb"\nonum"&Unnumbered entry in numerical reference list\\
\br
\end{tabular}
\end{table}


\begin{table}
\caption{Macros defined within {\tt iopart.cls}
for use with figures and tables.}
\begin{indented}
\item[]\begin{tabular}{@{}l*{15}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\Figures"&Heading for list of figure captions\\
\verb"\Figure{#1}"&Figure caption\\
\verb"\Tables"&Heading for tables and table captions\\
\verb"\Table{#1}"&Table caption\\
\verb"\fulltable{#1}"&Table caption for full width table\\
\verb"\endTable"&End of table created with \verb"\Table"\\
\verb"\endfulltab"&End of table created with \verb"\fulltable"\\
\verb"\endtab"&End of table\\
\verb"\br"&Bold rule for tables\\
\verb"\mr"&Medium rule for tables\\
\verb"\ns"&Small negative space for use in table\\
\verb"\centre{#1}{#2}"&Centre heading over columns\\
\verb"\crule{#1}"&Centre rule over columns\\
\verb"\lineup"&Set macros for alignment in columns\\
\verb"\m"&Space equal to width of minus sign\\
\verb"\-"&Left overhanging minus sign\\
\verb"\0"&Space equal to width of a digit\\
\br
\end{tabular}
\end{indented}
\end{table}


\clearpage
\section{Control sequences for journal
abbreviations}
\label{jlabs}
\begin{table}[hb]
\caption{\label{jlab1}Abbreviations for the IOPP journals.}
\begin{indented}
\item[]
\begin{tabular}{@{}lll}
\br
Macro name&{\rm Short form of journal title}&Years relevant\\
\mr
{\it Current journals}\\
\mr
\verb"\CQG"&Class. Quantum Grav.\\
\verb"\CTM"&Combust. Theory Modelling\\
\verb"\EJP"&Eur. J. Phys.\\
\verb"\HPP"&High Performance Polymers\\
\verb"\IP"&Inverse Problems\\
\verb"\JHEP"&J. High Energy Phys.&1999 and onwards\\
\verb"\JMM"&J. of Michromech. and Microeng.\\
\verb"\JOA"&J. Opt. A: Pure and Applied Opt.&1998 and onwards\\
\verb"\JOB"&J. Opt. B: Quantum and Semiclass. Opt.&1999 and onwards\\
\verb"\JPA"&J. Phys. A: Math. Gen.\\
\verb"\jpb"&J. Phys. B: At. Mol. Opt. Phys.&1988 and onwards\\
\verb"\JPCM"&J. Phys: Condens. Matter&1989 and onwards\\
\verb"\JPD"&J. Phys. D: Appl. Phys.\\
\verb"\JPF"&J. Phys. F: Met. Phys.\\
\verb"\jpg"&J. Phys. G: Nucl. Part. Phys.&1989 and onwards\\
\verb"\MSMSE"&Modelling Simul. Mater. Sci. Eng.\\
\verb"\MST"&Meas. Sci. Technol.&1990 and onwards\\
\verb"\NET"&Network:Comput. Neural Syst.\\
\verb"\NJP"&New J. Phys.&1999 and onwards\\
\verb"\NL"&Nonlinearity\\
\verb"\NT"&Nanotechnology\\
\verb"\PM"&Physiol. Meas.\\
\verb"\PMB"&Phys. Med. Biol.\\
\verb"\PPCF"&Plasma Physics and Controlled Fusion\\
\verb"\PSST"&Plasma Sources Sci. Technol.\\
\verb"\PUS"&Public Understand. Sci.\\
\verb"\RPP"&Rep. Prog. Phys.\\
\verb"\SST"&Semicond. Sci. Technol.\\
\verb"\SUST"&Supercond. Sci. Technol.\\
\verb"\WRM"&Waves in Random Media\\
\mr
{\it No Longer Published}\\
\mr
\verb"\BIO"&Bioimaging&1998\\
\verb"\DSE"&Distributed Systems Eng.&1993--1999\\
\verb"\JO"&J. Opt.&1970--1998\\
\verb"\JPB"&J. Phys. B: At. Mol. Phys.&1968--1987\\
\verb"\JPC"&J. Phys. C: Solid State Phys.&1968--1988\\
\verb"\JPE"&J. Phys. E: Sci. Instrum.&1968--1989\\
\verb"\JPG"&J. Phys. G: Nucl. Phys.&1975--1988\\
\verb"\PAO"&Pure and Applied Opt.&1992--1998\\
\verb"\QO"&Quantum Opt.&1989--1994\\
\verb"\QSO"&Quantum and Semiclass. Opt.&1995--1998\\
\br
\end{tabular}
\end{indented}
\end{table}



\begin{table}

\caption{\label{jlab2}Abbreviations for some more 
common non-IOPP journals.}
\begin{indented}
\item[]\begin{tabular}{@{}ll}
\br
Macro name&{\rm Short form of journal}\\
\mr
\verb"\AC"&Acta Crystallogr.\\
\verb"\AM"&Acta Metall.\\
\verb"\AP"&Ann. Phys., Lpz\\
\verb"\APNY"&Ann. Phys., NY\\
\verb"\APP"&Ann. Phys., Paris\\
\verb"\CJP"&Can. J. Phys.\\
\verb"\GRG"&Gen. Rel. Grav.\\
\verb"\JAP"&J. Appl. Phys.\\
\verb"\JCP"&J. Chem. Phys.\\
\verb"\JJAP"&Japan. J. Appl. Phys.\\
\verb"\JMMM"&J. Magn. Magn. Mater.\\
\verb"\JMP"&J. Math. Phys.\\
\verb"\JOSA"&J. Opt. Soc. Am.\\
\verb"\JP"&J. Physique\\
\verb"\JPhCh"&J. Phys. Chem.\\
\verb"\JPSJ"&J. Phys. Soc. Japan\\
\verb"\JQSRT"&J. Quant. Spectrosc. Radiat. Transfer\\
\verb"\NC"&Nuovo Cimento\\
\verb"\NIM"&Nucl. Instrum. Methods\\
\verb"\NP"&Nucl. Phys.\\
\verb"\PF"&Phys. Fluids\\
\verb"\PL"&Phys. Lett.\\
\verb"\PR"&Phys. Rev.\\
\verb"\PRL"&Phys. Rev. Lett.\\
\verb"\PRS"&Proc. R. Soc.\\
\verb"\PS"&Phys. Scr.\\
\verb"\PSS"&Phys. Status Solidi\\
\verb"\PTRS"&Phil. Trans. R. Soc.\\
\verb"\RMP"&Rev. Mod. Phys.\\
\verb"\RSI"&Rev. Sci. Instrum.\\
\verb"\SSC"&Solid State Commun.\\
\verb"\SPJ"&Sov. Phys.--JETP\\
\verb"\ZP"&Z. Phys.\\
\br
\end{tabular}
\end{indented}
\end{table}

\end{document}


cond-mat0502552.extracted_bib
\begin{thebibliography}{60}
\bibitem{RMP}
  R. Albert and A.-L. Barab\'asi, Statistical mechanics of complex networks, 2002 {\it Rev. Mod. Phys.} {\bf 74} 47 
\bibitem{Doro_book}
S. N. Dorogovtsev and J. F. F. Mendes, {\it Evolution of Networks}
 (Oxford University Press,Oxford,2003).
\bibitem{Internet}
 R. Pastor-Satorras and A. Vespignani,  {\it Evolution and Structure of the Internet }(Cambridge University Press,Cambridge,2004).
\bibitem{UAlon}
S. Itzkovitz, R. Milo, N. Kashtan, G. Ziv and U. Alon, Subgraphs in random networks, 2003 {\it Phys. Rev. E} \textbf{ 68}  026127
\bibitem{Milo}
R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and  U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, 2002
 {\it Science} {\bf 298} 824 
\bibitem{Laszlo_m}
 R. Dobrin, Q. K. Beg, A.-L. Barab\'asi and Z. N. Oltvai, Aggregation of topological motifs in the Escherichia coli transcriptional network, 2004
{\it BMC Bioinformatics} {\bf 5} 10 
\bibitem{Loops}
 G. Bianconi and A. Capocci, Number of Loops of Size $h$ in Growing Scale-Free Networks, 2003 {\it Phys. Rev. Lett.} {\bf 90 } 078701 
\bibitem{Guido_cycles}
G. Caldarelli, R. Pastor-Satorras and A. Vespignani,  Structure of Cycles and local ordering in complex networks, 2002 {\it Eur. Phys. Jour. B } {\bf 38} 183
\bibitem{Avraham}
H. D. Rozenfeld, J. E. Kirk, E. M. Bollt and D. ben-Avraham,Statistics of Cycles: How Loopy is your Network?, 2004 [cond-mat/0403536] 
\bibitem{Dorogovtsev}
S. Dorogovstev, Clustering of correlated networks, 2004 {\it Phys. Rev. E} {\bf 69} 027104 
\bibitem{Loops_last}
G. Bianconi, G. Caldarelli and A. Capocci, Loops structure of the Internet at the Autonomous System Level, 2004  [cond-mat/0408349] 
\bibitem{Vazquez_m}
 A. Vazquez, R. Dobrin, D. Sergi, J.-P. Eckmann, Z. N. Oltvai, A.-L. Barab\'asi,The topological relationship between the large-scale attributes and local interactions patterns of complex networks, 2004 {\it PNAS} {\bf 101} 17940 
\bibitem{VazquezBa}
A. Vazquez, J. G. Oliveira and A.-L. Barab\'asi, The inhomogeneous evolution of subgraphs and cycles in complex networks, 2005 [ cond-mat/0501399].
\bibitem{epidemics}
R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, 2001 {\it Phys. Rev. Lett.} {\bf 86} 3200
\bibitem{epidemics_corr}
M. Bogu\~na, R. Pastor-Satorras and A. Vespignani, Epidemic Spreading in Scale-Free Networks, 2003 {\it Phys. Rev. Lett.} {\bf 90} 028701 
\bibitem{Janson}
S. Janson, T. Luczak, A. Rucinski, {\it  Random graphs} (John Wiley \& Sons,2000).
\bibitem{Marinari}
E. Marinari and R. Monasson, Circuits in random Graphs: form local trees to global loops, 2004 {\it J. Stat. Mech.} P09004
\bibitem{Parisi_book_stat_phys}
G. Parisi {\it Statistical Field Theory} (Addison Wesley,RedwoodCity, USA,1988)
\bibitem{Dorog_solution_Potts}
S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Potts model on complex networks, 2004 {\it Eur. Phys. J. B }{\bf 38},177 
\bibitem{Leone_etal_Ising}
M. Leone, A. Vazquez, A. Vespignani and R. Zecchina, Ferromagnetic ordering in graphs with arbitrary degree distribution, 2002 {\it  Eur. Phys. J. B} {\bf 28} 191 
\bibitem{Ehrhardt_Marsili_JSTAT}
G. Ehrhardt and M. Marsili, Potts model on random trees, 2005 {\it J. Stat. Mech.} P02006 
\bibitem{HV}
G Caldarelli, A. Capocci, P. De Los Rios and M. A. Mu\~noz, Scale-free networks form varying vertex intrisic fitness,  2002 {\it Phys. Rev. Lett.} {\bf 89}, 258702 
\bibitem{HV2}
M. Bogu\~na and R. Pastor-Satorras, Class of correlated random networks with hidden variables, 2003 {\it Phys. Rev. E} {\bf 68} ,036112 
\bibitem{MR}
M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, 1995 {\it Random Structures and Algorithms}  {\bf 6}, 161
\bibitem{poli1}
Z. Burda, A. Krzywicki, Uncorrelated random networks, 2003 {\it  Phys. Rev. E} {\bf 67} 046118
\bibitem{FS_vesp}
M. Bogu\~na, R. Pastor-Satorras and A. Vespignani, Cut-offs and finite size effects in scale-free networks, 2004 {\it Eur. Phys. Jour.} {\bf 38}, 205(2004).
\bibitem{Johnson}
D. B. Johnson, Finding elementary circuits of a directed graph,1975 {\it SIAM J. Comput.} {\bf 4},77  
\bibitem{Nota}Note that in the considered Molloy-Reed networks double links are not allowed. This correction is supposelly quite small for the considered system sizes.
\end{thebibliography}