\section{Introduction}

Competition between growth and fragmentation is a common phenomenon for a structured population. It arises for instance in a context of cell division (see, among many others, \cite{adimy, basse, Bekkal1, Bekkal2, Doumic, farkas, GW2, MetzDiekmann, PTou}), polymerization (see \cite{biben, destaing}), telecommunication (see \cite{Baccelli}) or neurosciences (see \cite{PPS}). It is also a mechanism which rules the proliferation of prion's proteins (see \cite{CL1, Greer, LW}). These proteins are responsible of spongiform encephalopaties and appear in the form of aggregates in infected cells. Such polymers grow attaching non infectious monomers and converting them into infectious ones. On the other hand they increase their number by splitting.

To describe such phenomena, we write the following integro-differential equation,
\begin{equation}\label{eq:temporel}\left \{ \begin{array}{l}
\displaystyle\dfrac{\partial}{\partial t} u(x,t) + \dfrac{\partial}{\partial x} \big(\tau(x) u(x,t)\big) + \beta(x) u(x,t) = 2 \int_{x}^{\infty} \beta(y) \kappa (x,y) \, u(y,t) \, dy, \qquad x \geqslant0,
\\
\\
u(x,0)=u_0(x),
\\
\\
u(0,t)=0.
\end{array}\right.\end{equation}
The function $u(x,t)$ represents the quantity of individuals (cells or polymers) of structured variable (size, protein content...) $x$ at time $t.$ These individuals grow (\emph{i.e.}, polymers aggregate monomers, or cells increase by nutrient uptake for instance) with the rate $\tau(x).$ Equation \eqref{eq:temporel} also takes into account the fragmentation of a polymer (or the division of a cell) of size $y$ into two smaller polymers of size $x$ and $y-x.$ This fragmentation occurs with a rate $\beta(y)$ and produce an aggregate of size $x$ with the rate $\kappa(x,y).$ Equation \eqref{eq:temporel} is a particular case of the more general one
\begin{equation}\label{eq:general}\dfrac{\p}{\p t} u(x,t) + \dfrac{\p}{\p x} \big(\tau(x) u(x,t)\big) + [\beta(x)+\mu(x)] u(x,t) = n \int_{x}^{\infty} \beta(y) \kappa (x,y) \, u(y,t) \, dy, \qquad x \geqslant x_0,\end{equation}
with the bound condition $u(x_0,t)=0$ (see \cite{Banasiak, CL1, LW}). Here, polymers are broken in an average of $n>1$ smaller ones by the fragmentation process, there is a death term $\mu(x)\geq0$ representing degradation, and a minimal size of polymers $x_0$ which can be positive. This more general model is biologically and mathematically relevant in the case of prion proliferation and is used in \cite{CL2, CL1, Greer, LW} with a coupling to an ODE. Our results remain true for this generalization.

A fundamental tool to study the asymptotic behaviour of the population when $t\to\infty$ is the existence of eigenelements ($\lb,\U,\phi$) solution of the equation

\begin{equation}
\label{eq:eigenproblem}
\left \{ \begin{array}{l}
\displaystyle \f{\p}{\p x} (\tau(x) \U(x)) + ( \beta (x) + \lb) \U(x) = 2 \int_x^\infty \beta(y)\kappa(x,y) \U(y) dy, \qquad x \geqslant0,
\\
\\
\tau\U(x=0)=0 ,\qquad \U(x)\geq0, \qquad \int_0^\infty \U(x)dx =1,
\\
\\
\displaystyle -\tau(x) \f{\p}{\p x} (\phi(x)) + ( \beta (x) + \lb) \phi(x) = 2 \beta(x) \int_0^x \kappa(y,x) \phi(y) dy, \qquad x \geqslant0,
\\
\\
\phi(x)\geq0, \qquad \int_0^\infty \phi(x)\U(x)dx =1.
\end{array} \right.
\end{equation}

For the first equation (equation on $\U$) we are looking for ${\mathcal D}'$ solutions defined as follows : $\U\in L^1(\mathbb{R}^+)$ is a ${\mathcal D}'$ solution if $\forall\varphi\in{\mathcal C}^\infty_c(\mathbb{R}^+),$
\begin{equation}\label{eq:D'eigenproblem}
\displaystyle -\int_0^\infty\tau(x)\U(x)\p_x\varphi(x)\,dx + \lb\int_0^\infty\U(x)\varphi(x)\,dx =  \int_0^\infty\beta(x)\U(x)\Bigl(2\int_0^\infty\varphi(y)\kappa(y,x)\,dy-\varphi(x)\Bigr)\,dx.
\end{equation}
Concerning the dual equation, we are looking for a solution $\phi\in W^{1,\infty}_{loc}(0,\infty)$ such that the equality holds in $L^1_{loc}(0,\infty),\ i.e.$ almost everywhere.

When such elements exist, the asymptotic growth rate for a solution to \eqref{eq:temporel} is given by the first eigenvalue $\lb$ and the asymptotic shape is given by the corresponding eigenfunction $\U.$ More precisely, it is proved for a constant fragmentation rate $\beta$ that $u(x,t)e^{-\lb t}$ converges exponentially fast to $\rho\U(x)$ where $\rho=\int u_0(y)dy$ (see \cite{LP,PR}). For more general fragmentation rates, one can use the dual eigenfunction $\phi$ and the so-called ''General Relative Entropy`` method introduced in \cite{MMP1,BP}. It provides similar results but without the exponential convergence, namely that $$\int_0^\infty \bigl|u(y,t)e^{-\lambda t}-\langle u_0,\phi\rangle{\mathcal U}(y)\bigr|\phi(y)\,dy\underset{t\to\infty}{\longrightarrow}0$$ where $\langle u_0,\phi\rangle=\int u_0(y)\phi(y)dy$ (see \cite{MMP1,MMP2}).

The eigenvalue problem can also be used in nonlinear cases, such as prion proliferation equations, where there is a quadratic coupling of Equation \eqref{eq:temporel} or \eqref{eq:general} with a differential equation.  In \cite{CL2, CL1, Engler, Pruss} for instance, the stability of steady states is investigated. The use of entropy methods in the case of nonlinear problems remains however a challenging and widely open field (see \cite{PTum} for a recent review).

\

Existence and uniqueness of eigenelements has already been proved for general fragmentation kernels $\kappa(x,y)$ and fragmentation rates $\beta(x),$ but with very particular polymerization rates $\tau(x),$ namely constant ($\tau\equiv1$ in \cite{BP}), homogeneous ($\tau(x)=x^\mu$ in \cite{M1}) or with a compact support ($Supp\,\tau=[0,x_M]$ in \cite{Doumic}).

The aim of this article is to consider more general $\tau$ as \cite{CL1, Silveira} suggest. Indeed, there is no biological justification to consider specific shapes of $\tau$ in the case when $x$ represents a size (mass or volume) or some structuring variable and not the age of a cell (even in this last case it is not so clear that $\f{dx}{dt}=1,$ since biological clocks may exhibit time distorsions). For instance, for the prion proteins, the fact that the small aggregates are little infectious (see \cite{Lenuzza,Silveira}) leads us to include the case of rates vanishing at $x=0.$

Considering fully general growth rates is thus indispensable to take into account biological or physical phenomena in their full diversity.
The proof of \cite{BP} can be adapted for non constant rates but still positive and bounded ($0<m<\tau(x)<M$). The paper \cite{M1} gives results for $\tau(0)=0,$ but for a very restricted class of shape for $\tau.$ The paper \cite{Doumic} gives results for $\tau$ with general shape in the case where there is also an age variable (integration in age then allows to recover Problem \eqref{eq:temporel}), but requires a compact support and regular parameters. Here we consider polymerization rates that can vanish at $x=0,$ with general shape and few regularity for the all parameters ($\tau,\ \beta$ and $\kappa$).

From a mathematical viewpoint, relaxing as far as possible the assumptions on the rates $\tau,\kappa,\beta,$ as we have done in this article, also leads to a better understanding of the intrinsic mechanisms driving the competition between growth and fragmentation.

\begin{theorem}[Existence and Uniqueness]\label{th:eigenelements}
Under assumptions \eqref{as:kappa1}-\eqref{as:betatauinf},
there exists a unique solution $(\lb,\U,\phi)$ (in the sense we have defined before) to the eigenproblem \eqref{eq:eigenproblem} and we have
$$\lb>0,$$
$$x^\al\tau\U\in L^p(\mathbb{R}^+),\quad\forall \al\geq-\gamma,\quad\forall p\in[1,\infty],$$
$$x^\al\tau\U\in W^{1,1}(\mathbb{R}^+),\quad\forall \al\geq0$$
$$\exists k>0\ s.t.\ \f{\phi}{1+x^k}\in L^\infty(\mathbb{R}^+),$$
$$\tau\f{\p}{\p x}\phi\in L_{loc}^\infty(\mathbb{R}^+).$$
\end{theorem}

The end of this paper is devoted to define precisely the assumptions and prove this theorem. It is organized as follows : in Section \ref{se:coefficients} we describe the assumptions and give some examples of interesting parameters. In Section \ref{se:proof} we prove Theorem \ref{th:eigenelements} using $a\ priori$ bounds on weighted norms and then we give some consequences and perspectives in Section \ref{se:csq}. The proof of technical lemmas and theorem can be found in the Appendix.

\

\section{Coefficients}\label{se:coefficients}

\subsection{Assumptions}

For all $y\geq0,\ \kappa(.,y)$ is a nonnegative measure with a support included in $[0,y].$ We define $\kappa$ on $(\mathbb{R}_+)^2$ as follows : $\kappa(x,y)=0\ \text{for}\ x>y.$ We assume that
 for all continuous function $\psi,$ the application $f_\psi:y\mapsto\int\psi(x)\kappa(x,y)\,dx$ is Lebesgue measurable.\\
The natural assumptions on $\kappa$ (see \cite{Greer} for the motivations) are that polymers can split only in two pieces which is taken into account by
\begin{equation}\label{as:kappa1}\int\kappa(x,y) dx = 1.\end{equation}
So $\kappa(y,.)$ is a probability measure and $f_\psi\in L^\infty_{loc}(\mathbb{R}^+).$ The conservation of mass imposes
\begin{equation}\label{as:kappa2}\int x\kappa(x,y) dx = \frac y 2,\end{equation}
a property that is automatically satisfied for a symetric fragmentation ($i.e.\ \kappa(x,y)=\kappa(y-x,y)$) thanks to \eqref{as:kappa1}. For the more general model \eqref{eq:general}, assumption \eqref{as:kappa2} becomes $\int x\kappa(x,y) dx = \frac y n$ to preserv the mass conservation.\\
We also assume that the second moment of $\kappa$ is less than the first one
\begin{equation}\label{as:kappa3}\int\f{x^2}{y^2} \, \kappa(x,y) dx \leq c < 1/2\end{equation}
(it becomes $c<1/n$ for model \eqref{eq:general}). We refer to the Examples for an explanation of the physical meaning.

\

For the polymerization and fragmentation rates $\tau$ and $\beta,$ we introduce the set
$${\mathcal P}:=\bigl\{f\geq0\,:\,\exists\mu,\nu\geq0,\ \limsup_{x\to\infty}x^{-\mu}f(x)<\infty\ \text{and}\ \liminf_{x\to\infty}x^\nu f(x)>0\bigr\}$$
and the space
$$L^1_0:=\bigr\{f,\ \exists a>0,\ f\in L^1(0,a)\bigl\}.$$
We consider
\begin{equation}\label{as:betatauspace}\beta\in L^1_{loc}(\mathbb{R}^{+*})\cap{\mathcal P},\qquad
\exists\al_0\geq0\ s.t.\ \tau\in L^\infty_{loc}(\mathbb{R}^{+},x^{\al_0}dx)\cap{\mathcal P}\end{equation}
satisfying
\begin{equation}\label{as:taupositivity}\forall K\ \text{compact of}\ (0,\infty),\ \exists m_K>0\quad s.t.\quad\tau(x)\geq m_K\ \text{for}\ a.e.\ x\in K\end{equation}
(if $\tau$ is continuous, this assumption \eqref{as:taupositivity} is nothing but saying that for all $x>0,\ \tau(x)>0$) and
\begin{equation}\label{as:betasupport}\exists b\geq0,\quad Supp\beta=[b,\infty).\end{equation}
Assumption \eqref{as:betasupport} is necessary to prove uniqueness and existence for the adjoint problem.

\

To avoid shattering (zero-size polymers formation, see \cite{Banasiak,LW}), we assume
\begin{equation}\label{as:kappatau}\exists\, C>0,\gamma\geq0\quad s.t.\qquad\int_0^x\kappa(z,y)\,dz\leq \min\Bigl(1,C\Bigl(\f x y\Bigr)^\gamma\Bigr)\qquad\text{and}\qquad\f{x^\gamma}{\tau(x)}\in L^1_0\end{equation}
which links implicitely $\tau$ to $\kappa,$ and also
\begin{equation}\label{as:betatau0}\f{\beta}{\tau}\in L^1_0.\end{equation}
On the other hand, to avoid forming infinitely long polymers (gelation phenomenon, see \cite{EscoMischler1,EscoMischler2}), we assume
\begin{equation}\label{as:betatauinf}\lim_{x\rightarrow +\infty}\f{x\beta(x)}{\tau(x)}=+\infty.\end{equation}

\begin{remark}\label{rk:kappa}
In case when \eqref{as:kappatau} is satisfied for $\gamma>0,$ then \eqref{as:kappa3} is automatically fulfilled (see Lemma \ref{lm:kappa} in the Appendix).
\end{remark}

\

\subsection{Examples}

First we give some examples of coefficients which satisfy or not our previous assumptions.\\
For the fragmentation kernel, we first check the assumptions \eqref{as:kappa1} and \eqref{as:kappa2}. They are satisfied for autosimilar measures, namely $\kappa(x,y)=\f1y\kappa_0(\f xy),$ with $\kappa_0$ a probability measure on $[0,1],$ symmetric in $1/2.$ Now we exhibit some $\kappa_0.$

\

\noindent{\bf General mitosis :} a cell of size $x$ divides in a cell of size $rx$ and one of size $(1-r)x$ (see \cite{M2})
\begin{equation}\label{ex:kappar}\kappa_0^r=\f12(\delta_{r}+\delta_{1-r})\qquad\text{for}\qquad r\in[0,1/2].\end{equation}
Assumption \eqref{as:kappatau} is satisfied for any $\gamma>0$ in the cases when $r\in(0,1/2].$ So \eqref{as:kappa3} is also fulfilled thanks to Remark \ref{rk:kappa}. The particular value $r=1/2$ leads to equal mitosis ($\kappa(x,y)=\delta_{x=\f y2}$).\\
The case $r=0$ corresponds to the renewal equation ($\kappa(x,y)=\f12(\delta_{x=0}+\delta_{x=y})$). In this case, we cannot strictly speak of mitosis because the size of the daughters are $0$ and $x.$ It appears when $x$ is the age of a cell and not the size. This particular case is precisely the one that we want to avoid with assumption \eqref{as:kappa3} ; it can also be studied seperately with different tools (see \cite{PTum} for instance). For such a fragmentation kernel, assumption \eqref{as:kappatau} is satified only for $\gamma=0,$ and the moments $\int z^k\kappa_0(z)dz$ are equal to $1/2$ for all $k>0,$ so \eqref{as:kappa3} does not hold true. However, if we consider a convex combination of $\kappa_0^0$ with another kernel such as $\kappa_0^r$ with $r\in(0,1/2],$ then \eqref{as:kappatau} remains false for any $\gamma>0$ but \eqref{as:kappa3} is fulfilled. Indeed we have for $\rho\in(0,1)$
$$\int z^2(\rho\kappa_0^0(z)+(1-\rho)\kappa_0^r(z))\,dz=\f\rho2+\f{1-\rho}2(r^2+(1-r)^2)=\f12(1-2r(1-r)(1-\rho))<\f12.$$

\noindent{\bf Homogeneous fragmentation :}
\begin{equation}\label{ex:kappaal}\kappa_0^\al(z)=\f{\al+1}{2}(z^\al+(1-z)^\al)\qquad\text{for}\qquad\al>-1.\end{equation}
It gives another class of fragmentation kernels, namely in $L^1$ (unlike the mitosis case). The parameter $\gamma=1+\al>0$ suits for \eqref{as:kappatau} and so \eqref{as:kappa3} is fulfilled. It shows that our assumptions allow fragmentation at the ends of the polymers (called depolymerization, see \cite{Lenuzza}, when $\al$ is close to $-1$) once it is not the extreme case of renewal equation.\\
Uniform repartition ($\kappa(x,y)=\f1y{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{0\leq x\leq y}$) corresponds to $\al=0$ and is also included.

\

This last case of uniform repartition is useful because it provides us with explicit formulas for the eigenelements. For instance, we can consider the two following examples.

\

\noindent{\bf First example :} $\,\tau(x)=\tau_0,\ \beta(x)=\beta_0x.$\\
In this case, widely used by \cite{Greer}, the eigenelements exist and we have
$$\lb=\sqrt{\beta_0\tau_0},$$
$$\U(x)=2\sqrt{\f{\beta_0}{\tau_0}}\Bigl(X+\f{X^2}2\Bigr)e^{-X-\f{X^2}2},\quad\text{with}\ X=\sqrt{\f{\beta_0}{\tau_0}}x,$$
$$\phi(x)=\f12(1+X).$$
\noindent{\bf Second example :} $\ \tau(x)=\tau_0x.$\\
For such $\beta$ for which there exists eigenelements, we have
$$\lb=\tau_0\qquad \text{and}\qquad \phi(x)=\f x{\int y\U(y)}.$$
For instance when $\beta(x)=\beta_0x^n$ with $n\in\mathbb{N}^*,$ then the eigenelements exist and we can compute $\U$ and $\phi$ and we have the formulas in Table \ref{tab:examples}. In this table we can notice that $\U(0)>0$ but the boundary condition $\tau\U(0)=0$ is fulfilled.

\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$n=1$&$\lb=\tau_0$&$\U(x)=\f{\beta_0}{\tau_0}e^{-\f{\beta_0}{\tau_0}x}$&$\phi(x)=\f{\beta_0}{\tau_0}x$\\
\hline
$n=2$&$\lb=\tau_0$&$\U(x)=\sqrt{\f{2\beta_0}{\pi\tau_0}}e^{-\f12\f{\beta_0}{\tau_0}x^2}$&$\phi(x)=\sqrt{\f{\pi\beta_0}{2\tau_0}}x$\\
\hline
$n$ & $\lb=\tau_0$&$\U(x)=\Bigl(\f{\beta_0}{n\tau_0}\Bigr)^{\f1n}\f n{\Gamma(\f1n)}e^{-\f1n\f{\beta_0}{\tau_0}x^n}$&$\phi(x)=\Bigl(\f{\beta_0}{n\tau_0}\Bigr)^{\f1n}\f {\Gamma(\f1n)}{\Gamma(\f2n)}x$\\
\hline
\end{tabular}
\caption{\label{tab:examples}The example $\tau(x)=\tau_0x,\ \beta(x)=\beta_0x^n$ and uniform repartition $\kappa(x,y)=\f1y{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{0\leq x\leq y}.$ The table gives the eigenelements solution to \eqref{eq:eigenproblem}.}
\end{center}
\end{table}

Now we turn to non-existence cases. Let us consider constant fragmentation $\beta(x)=\beta_0$ with an affine polymerization $\tau(x)=\tau_0+\tau_1x,$ and any fragmentation kernel $\kappa$ which satisfies to assumptions \eqref{as:kappa1}-\eqref{as:kappa2}. We notice that \eqref{as:betatauinf} is not satisfied and look at two instructive cases.

\

\noindent{\bf First case :} $\ \tau_0=0.$\\
In this case assumption \eqref{as:betatau0} does not hold true. Assume that there exists $\U\in L^1(\mathbb{R}^+)$ solution of \eqref{eq:eigenproblem} with the estimates of Theorem \ref{th:eigenelements}. Integrating the equation on $\U$ we obtain that $\lb=\beta_0,$ but multiplying the equation by $x$ before integration we have that $\lb=\tau_1.$ We conclude that eigenelements cannot exist if $\tau_1\neq\beta_0.$\\
Moreover, if we take $\kappa(x,y)=\f1y{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{0\leq x\leq y},$ then a formal computation shows that any solution to the first equation of \eqref{eq:eigenproblem} belongs to the plan $Vect\{x^{-1},x^{-\f{2\beta_0}{\tau_1}}\}.$ So, even if $\beta_0=\tau_1$, there does not exist an eigenvector in $L^1.$

\

\noindent{\bf Second case :} $\ \tau_0>0.$\\
In this case \eqref{as:betatau0} holds true but the same integrations than before lead to
$$\int x\U(x)\,dx=\f{\tau_0}{\beta_0-\tau_1}.$$
So there cannot exist any eigenvector $\U\in L^1(x\,dx)$ for $\tau_1\geq\beta_0.$

\

\section{Proof of the main theorem}\label{se:proof}

The proof of Theorem \ref{th:eigenelements} is divided as follows. We begin with a result concerning the positivity of the $a\ priori$ existing eigenvectors (Lemma \ref{lm:positivity}). We then define, in Section \ref{subsec:truncated}, a regularized and truncated problem for which we know that eigenelements exist (see the Appendix \ref{se:KR} for a proof using the Krein-Rutman theorem), and we choose it such that the related eigenvalue is positive (Lemma \ref{lm:lambdapositivity}). In Section~\ref{subsec:estim}, we give a series of estimates that allow us to pass to the limit in the truncated problem and so prove the existence for the original eigenproblem \eqref{eq:eigenproblem}. The positivity of the eigenvalue $\lb$ and the uniqueness of the eigenelements are proved in the two last subsections.

\subsection{A preliminary lemma}

Before proving Theorem \ref{th:eigenelements}, we give a preliminary lemma, useful to prove uniqueness of the eigenfunctions.

\begin{lemma}[Positivity]\label{lm:positivity}
Consider $\U$ and $\phi$ solutions to the eigenproblem \eqref{eq:eigenproblem}.\\
We define $\displaystyle m:=\inf_{x,y}\bigl\{x\,:\,(x,y)\in Supp\,\beta(y)\kappa(x,y)\bigr\}.$ Then we have, under assumptions \eqref{as:kappa1}, \eqref{as:kappa2}, \eqref{as:taupositivity} and \eqref{as:betasupport}
$$Supp\,\U=[m,\infty)\qquad\text{and}\qquad\tau\U(x)>0\quad\forall x>m,$$
$$\phi(x)>0\quad\forall x>0.$$
If additionaly $\f1\tau\in L^1_0,$ then $\phi(0)>0.$
\end{lemma}

\begin{remark}
In case $Supp\,\kappa=\{(x,y)/x\leq y\},$ then $m=0$ and Lemma \ref{lm:positivity} and Theorem \ref{th:eigenelements} can be proved without the connexity condition \eqref{as:betasupport} on the support of $\beta.$
\end{remark}

\begin{proof}
Let $x_0>0,$ we define $F:x\mapsto \tau(x)\U(x)e^{\int_{x_0}^x\f{\lb+\beta(s)}{\tau(s)}ds}.$ We have that
\begin{equation}\label{Upositivity}F'(x)=2e^{\int_{x_0}^x\f{\lb+\beta(s)}{\tau(s)}ds}\int\beta(y)\kappa(x,y)\U(y)\,dy\geq0.\end{equation}
So, as soon as $\tau\U(x)$ once becomes positive, it remains positive for larger $x.$

\

We define $a:=\inf\{x\,:\,\tau(x)\U(x)>0\}.$ We first prove that $a\leq \f b2.$ For this we integrate the equation on $[0,a]$ to obtain
$$\int_0^a\int_a^\infty\beta(y)\kappa(x,y)\U(y)\,dydx=0,$$
$$\int_a^\infty\beta(y)\U(y)\int_0^a\kappa(x,y)\,dxdy=0.$$
Thus for almost every $y\geq\max(a,b),\ \int_0^a\kappa(x,y)\,dx=0.$ As a consequence we have
$$1=\int\kappa(x,y)\,dx=\int_a^y\kappa(x,y)\,dx\leq\f1a\int x\kappa(x,y)\,dx=\f y{2a}$$
thanks to \eqref{as:kappa1} and \eqref{as:kappa2}, and this is possible only if $b\geq2a.$

\

Assume by contradiction that $m<a,$ integrating \eqref{eq:eigenproblem} multiplied by $\varphi,$ we have for all $\varphi\in{\mathcal C}_c^\infty$ such that $Supp\,\varphi\subset[0,a]$
\begin{equation}\label{eq:positivity}\int\int\varphi(x)\beta(y)\kappa(x,y)\U(y)\,dydx=0.\end{equation}
By definition of $m$ and using the fact that $m<a,$ there exists $(p,q)\in(m,a)\times(b,\infty)$ such that $(p,q)\in Supp\,\beta(y)\kappa(x,y).$ But we can choose $\varphi$ positive such that $\varphi(p)\U(q)>0$ and this is a contradiction with \eqref{eq:positivity}.
So we have $m\geq a.$\\
To conclude we notice that on $[0,m],\ \U$ satisfies
$$\p_x(\tau(x)\U(x))+\lb\U(x)=0.$$
So, thanks to the condition $\tau(0)\U(0)=0$ and the assumption \eqref{as:taupositivity}, we have $\U\equiv0$ on $[0,m],$ so $m=a$ and the first statement is proved.

\

For $\phi,$ we define $G(x):=\phi(x)e^{-\int_{x_0}^x\f{\lb+\beta(s)}{\tau(s)}ds}.$ We have that
\begin{equation}\label{eq:phipositivity}G'(x)=-2e^{-\int_{x_0}^x\f{\lb+\beta(s)}{\tau(s)}ds}\beta(x)\int_0^x\kappa(y,x)\phi(y)\,dy\leq0,\end{equation}
so, as soon as $\phi$ vanishes, it remains null. Therefore $\phi$ is positive on an interval $(0,x_1)$ with $x_1\in\mathbb{R}_+^*\cup\{+\infty\}.$ Assuming that $x_1<+\infty$ and using that $x_1>a=m$ because $\int\phi(x)\U(x)dx=1,$ we can find $X\geq x_1$ such that
$$\int_{x_1}^XG'(x)\,dx=-2\int_{x_1}^X\int_0^{x_1}e^{\int_{x_0}^x\f{\lb+\beta(s)}{\tau(s)}ds}\phi(y)\beta(x)\kappa(y,x)\,dy\,dx<0.$$
This contradicts that $\phi(x)=0$ for $x\geq x_1,$ and we have proved that $\phi(x)>0$ for $x>0.$\\
If $\f1\tau\in L^1_0,$ we can take $x_0=0$ in the definition of $G$ and so $\phi(0)>0$ or $\phi\equiv0.$ The fact that $\phi$ is positive ends the proof of the lemma.

\qed
\end{proof}

\

\subsection{Truncated problem}
\label{subsec:truncated}
The proof of the theorem is based on uniform estimates on the solution to a truncated equation. Let $\eta,\ \delta,\ R$ positive numbers and define
$$\tau_\eta(x)=\left\{\begin{array}{ll}\eta&0\leq x\leq\eta\\
                                       \tau(x)&x\geq\eta.
                      \end{array}\right.$$
Then $\tau_\eta$ is lower bounded on $[0,R]$ thanks to \eqref{as:taupositivity} and we denote by $\mu=\mu(\eta,R):=\inf_{[0,R]}\tau_\eta.$
The existence of eigenelements $(\lb_\eta^\delta,\U_\eta^\delta,\phi_\eta^\delta)$ for the following truncated problem when $\delta R<\mu$ is standard (see Theorem \ref{th:KreinRutman} in the Appendix).

\begin{equation}\label{eq:truncated}
\left \{ \begin{array}{l}
\displaystyle \f{\p}{\p x} (\tau_\eta(x) \U_\eta^\delta(x)) + ( \beta(x) + \lb_{\eta}^\delta) \U_\eta^\delta(x) = 2 \int_x^R\beta(y)\kappa(x,y) \U_\eta^\delta(y)\,dy,\qquad 0<x<R,
\\
\\
\tau_\eta\U_\eta^\delta(x=0)=\delta,\qquad \U_\eta^\delta(x)>0 , \qquad \int \U_\eta^\delta(x)dx =1,
\\
\\
\displaystyle -\tau_\eta(x) \f{\p}{\p x} \phi_\eta^\delta(x) + ( \beta(x) + \lb_\eta^\delta) \phi_\eta^\delta(x) - 2 \beta(x) \int_0^x\kappa(y,x) \phi_\eta^\delta(y)\,dy = \delta\phi_\eta^\delta(0),\qquad 0<x<R,
\\
\\
\phi_\eta^\delta(R)=0, \qquad \phi_\eta^\delta(x)>0 , \qquad \int \phi_\eta^\delta(x)\U_\eta^\delta(x)dx =1.
\end{array} \right.
\end{equation}

The proof of the theorem \ref{th:eigenelements} requires $\lb_\eta^\delta>0.$ To enforce it, we take $\delta R=\f\mu2$ and we consider $R$ large enough to satisfy the following lemma.

\begin{lemma}\label{lm:lambdapositivity}
Under assumptions \eqref{as:kappa1}, \eqref{as:betatauspace} and \eqref{as:betatauinf}, there exists a $R_0>0$ such that for all $R>R_0,$ if we choose $\delta=\f\mu{2R},$ then we have $\lb_\eta^\delta>0.$
\end{lemma}

\begin{proof}
Assume by contradiction that $R>0$ and $\lb_\eta^\delta\leq0$ with $\delta=\f\mu{2R}.$ Then, integrating between $0$ and $x>0,$ we obtain
\begin{eqnarray*}
0&\geq&\lb\int_0^x\U(y)\,dy\\
&=&\delta-\tau(x)\U(x)-\int_0^x\beta(y)\U(y)\,dy+2\int_0^x\int_z^R\beta(y)\kappa(z,y)\U(y)\,dy\,dz\\
&=&\delta-\tau(x)\U(x)+\int_0^x\beta(y)\U(y)\,dy+2\int_x^R\Bigl(\int_0^x\kappa(z,y)\,dz\Bigr)\beta(y)\U(y)\,dy\\
&\geq&\delta-\tau(x)\U(x)+\int_0^x\beta(y)\U(y)\,dy.
\end{eqnarray*}
Consequently $$\tau(x)\U(x)\geq\delta+\int_0^x\f{\beta(y)}{\tau(y)}\tau(y)\U(y)\,dy$$
and, thanks to Gr\"onwall's lemma,
$$\tau(x)\U(x)\geq\delta e^{\int_0^x\f{\beta(y)}{\tau(y)}dy}.$$
But assumption \eqref{as:betatauinf} ensures that for all $n\geq0,$ there is a $A>0$ such that $$\f{\beta(x)}{\tau(x)}\geq\f nx,\qquad\forall x\geq A$$
and thus we have
$$\tau(x)\U(x)\geq\delta x^n,\quad\forall x\geq A.$$
Thanks to \eqref{as:betatauspace} we can choose $n$ and $A$ such that $x^{-n}\tau(x)\leq \f\mu4$ for $x\geq A$ and then we have
$$1=\int_0^R\U(x)\,dx\geq\int_A^R\U(x)\,dx\geq\delta\int_A^R\f{x^n}{\tau(x)}\,dx\geq\f2R(R-A)$$
what is a contradiction as soon as $R>2A$; so Lemma \ref{lm:lambdapositivity} holds for $R_0=2A.$

\qed
\end{proof}

\

\subsection{Limit as $\delta\to0$ for $\U_\eta^\delta$ and $\lb_\eta^\delta$}
\label{subsec:estim}
Fix $\eta$ and let $\delta\rightarrow0$ (then $R\to\infty$ since $\delta R=\f\mu2$).

\paragraph{\it First estimate: $\lb_\eta^\delta$ upper bound.}
Integrating equation \eqref{eq:truncated} between $0$ and $R,$ we find
$$\lb_\eta^\delta\leq\delta+\int\beta(x)\U_\eta^\delta(x)\,dx,$$
then the idea is to prove a uniform estimate on $\int\beta\U_\eta^\delta.$ For this we begin with bounding the higher moments $\int x^\al\beta\U_\eta^\delta$ for $\al\geq\max{(2,\al_0+1)}:=m.$\\
Let $\al\geq\text m,$ according to \eqref{as:kappa3} we have
$$\int\f{x^\al}{y^\al}\kappa(x,y)\,dx\leq\int\f{x^2}{y^2}\kappa(x,y)\,dx\leq c<\f12.$$
Multiplying the equation on $\U_\eta^\delta$ by $x^\al$ and then integrating on $[0,R],$ we obtain for all $A\geq\eta$
\begin{eqnarray*}
\int x^\al\bigl((1-2c)\beta(x)\bigr)\U_\eta^\delta(x)\,dx&\leq&\al\int x^{\al-1}\tau_\eta(x)\U_\eta^\delta(x)\,dx\\
&=&\al\int_{x\leq A}x^{\al-1}\tau_\eta(x)\U_\eta^\delta(x)\,dx+\al\int_{x\geq A}x^{\al-1}\tau(x)\U_\eta^\delta(x)\,dx\\
&\leq&\al A^{\al-1-\al_0}\sup_{x\in(0,A)}{\{x^{\al_0}\tau(x)\}}+\omega_{A,\al}\int x^\al\beta(x)\U_\eta^\delta(x)\,dx,
\end{eqnarray*}
where $\omega_{A,\al}$ is a positive number chosen to have $\al\tau(x)\leq\omega_{A,\al} x\beta(x),\ \forall x\geq A.$ Thanks to \eqref{as:kappa3} and \eqref{as:betatauinf}, we can choose $A_\al$ large enough to have $\omega_{A_\al,\al}<1-2c.$
Thus we find
\begin{equation}\label{eq:L1bound1}\forall\al\geq m,\,\exists A_\al:\ \forall\eta,\delta>0,\quad\int x^\al\beta(x)\U_\eta^\delta(x)\,dx\leq\f{\al {A_\al}^{\al-1-\al_0}\sup_{(0,A)}{\{x^{\al_0}\tau(x)\}}}{1-2c-\omega_{A_\al,\al}}:=B_\al.\end{equation}
The next step is to prove the same estimates for $0\leq\al<m$ and for this we first give a bound on $\tau_\eta\U_\eta^\delta.$ We fix $\rho\in(0,1/2)$ and define $x_\eta>0$ as the unique point such that $\int_0^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}dy=\rho.$ It exists because $\beta$ is nonnegative and locally integrable, and $\tau_\eta$ is positive. Thanks to assumption \eqref{as:betatau0}, we know that $x_\eta\underset{\eta\to0}{\longrightarrow}x_0$ where $x_0>0$ satisfies $\int_0^{x_0}\f{\beta(y)}{\tau(y)}dy=\rho,$ so $x_\eta$ is bounded by $0<\underline x\leq x_\eta\leq\overline x.$ Then, integrating \eqref{eq:truncated} between $0$ and $x\leq x_\eta,$ we find
\begin{eqnarray*}
\tau_\eta(x)\U_\eta^\delta(x)&\leq&\delta+2\int_0^{x}\int\beta(y)\U_\eta^\delta(y)\kappa(z,y)\,dy\,dz\\
&\leq&\delta+2\int\beta(y)\U_\eta^\delta(y)\,dy\\
&=&\delta+2\int_0^{x_\eta}\beta(y)\U_\eta^\delta(y)\,dy+2\int_{x_\eta}^\infty\beta(y)\U_\eta^\delta(y)\,dy\\
&\leq&\delta+2\sup_{(0,x_\eta)}\{\tau_\eta\U_\eta^\delta\}\int_0^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\,dy+\f2{x_\eta^m}\int_0^\infty y^m\beta(y)\U_\eta^\delta(y)\,dy\\
&\leq&\delta+2\rho\sup_{(0,x_\eta)}\{\tau_\eta\U_\eta^\delta\}+\f2{x_\eta^m}B_m.
\end{eqnarray*}
Consequently, if we consider $\delta\leq1$ for instance, we obtain
\begin{equation}\label{eq:Linfbound1}\sup_{x\in(0,\underline x)}\tau_\eta(x)\U_\eta^\delta(x)\leq\f{1+2B_m/\underline x^m}{1-2\rho}:=C\end{equation}
so $\tau_\eta\U_\eta^\delta$ is uniformly bounded in a neighborhood of zero.\\
Now we can prove a bound $B_\al$ for $x^\al\beta\U_\eta^\delta$ in the case $0\leq\al<m.$ Thanks to the estimates \eqref{eq:L1bound1} and \eqref{eq:Linfbound1} we have
\begin{eqnarray}\label{eq:L1bound2}
\int x^\al\beta(x)\U_\eta^\delta(x)\,dx&=&\int_0^{\overline x}x^\al\beta(x)\U_\eta^\delta(x)\,dx+\int_{\overline x}^Rx^\al\beta(x)\U_\eta^\delta(x)\,dx\nonumber\\
&\leq&\overline x^\al\sup_{(0,\overline x)}\{\tau_\eta\U_\eta^\delta\}\int_0^{\overline x}\f{\beta(y)}{\tau_\eta(y)}\,dy+\overline x^{\al-m}\int_{\overline x}^Rx^m\beta(x)\U_\eta^\delta(x)\,dx\nonumber\\
&\leq&C\rho\overline x^\al+B_m\overline x^{\al-m}:=B_\al.
\end{eqnarray}
Combining \eqref{eq:L1bound1} and \eqref{eq:L1bound2} we obtain
\begin{equation}\label{eq:L1bound3}\forall\al\geq0,\,\exists B_\al:\ \forall\eta,\delta>0,\quad\int x^\al\beta(x)\U_\eta^\delta(x)\,dx\leq B_\al,\end{equation}
and finally we bound $\lb_\eta^\delta$
\begin{equation}\label{eq:lambdaupperbound}\lb_\eta^\delta=\delta +\int\beta\U_\eta^\delta\leq\delta +B_0.\end{equation}
So the family $\{\lb_\eta^\delta\}_\delta$ belong to a compact interval and we can extract a converging subsequence \mbox{$\lb_\eta^\delta\underset{\delta\to0}{\longrightarrow}\lb_\eta.$}

\

\paragraph{\it Second estimate : $W^{1,1}$bound for $x^\al\tau_\eta\U_\eta^\delta,\ \al\geq0.$}
We use the estimate \eqref{eq:L1bound3}. First we give a $L^\infty$bound for $\tau_\eta\U_\eta^\delta$ by integrating \eqref{eq:truncated} between $0$ and $x$
\begin{equation}\label{eq:Linfbound2}\tau_\eta(x)\U_\eta^\delta(x)\leq\delta+2\int_0^R\beta(y)\U_\eta^\delta(y)\,dy\leq\delta+2B_0:=D_0.\end{equation}
Then we bound $x^\al\tau_\eta\U_\eta^\delta$ in $L^1$ for $\al>-1.$ Assumption \eqref{as:betatauinf} ensures that there exists $X>0$ such that\\
$\tau(x)\leq x\beta(x),\ \forall x\geq X,$ so we have for $R>X$
\begin{eqnarray*}
\int x^\al\tau_\eta(x)\U_\eta^\delta(x)\,dx&\leq&\sup_{(0,X)}\{\tau_\eta\U_\eta^\delta\}\int_0^X x^\al\,dx+\int_X^Rx^{\al+1}\beta(x)\U_\eta^\delta(x)\,dx\\
&\leq&\sup_{(0,X)}\{\tau_\eta\U_\eta^\delta\}\f{X^{\al+1}}{\al+1}+B_{\al+1}:=C_\al.
\end{eqnarray*}
Finally
\begin{equation}\label{eq:L1bound4}
\forall\al>-1,\,\exists C_\al:\ \forall\eta,\delta>0,\quad\int x^\al\tau_\eta(x)\U_\eta^\delta(x)\,dx\leq C_\al
\end{equation}
and we also have that $x^\al\U_\eta^\delta$ is bounded in $L^1$ because $\tau\in{\mathcal P}$ (see assumption \eqref{as:betatauspace}).\\
A consequence of \eqref{eq:L1bound3} and \eqref{eq:L1bound4} is that $x^\al\tau_\eta\U_\eta^\delta$ is bound in $L^\infty$ for all $\al\geq0.$ We already have \eqref{eq:Linfbound2} and for $\al>0$, we multiply \eqref{eq:truncated} by $x^\al,$ integrate on $[0,x]$ and obtain
$$
x^\al\tau_\eta(x)\U_\eta^\delta(x)\leq\al\int_0^R y^{\al-1}\tau_\eta(y)\U_\eta^\delta(y)\,dy+2\int_0^R y^\al\beta(y)\U_\eta^\delta(y)\,dy\leq\al C_\al+2B_\al:=D_\al,
$$
that give immediately
\begin{equation}\label{eq:Linfbound3}
\forall\al\geq0,\,\exists D_\al:\ \forall\eta,\delta>0,\quad\sup_{x>0}x^\al\tau_\eta(x)\U_\eta^\delta(x)\leq D_\al.
\end{equation}
To conclude we use the fact that neither the parameters nor $\U_\eta^\delta$ are negative and we find by the chain rule, for $\al\geq0$
$$\int\bigl|\f\p{\p x}(x^\al\tau_\eta(x)\U_\eta^\delta(x))\bigr|dx\leq\al\int x^{\al-1}\tau_\eta(x)\U_\eta^\delta(x)\,dx+\int x^\al\bigl|\p_x(\tau_\eta(x)\U_\eta^\delta(x))\bigr|\,dx\nonumber$$
\begin{equation}\label{eq:W11bound}\hspace{4cm}\leq\al\int x^{\al-1}\tau_\eta(x)\U_\eta^\delta(x)\,dx+\lb_\eta^\delta\int x^\al\U_\eta^\delta(x)\,dx+3\int x^\al\beta(x)\U_\eta^\delta(x)\,dx
\end{equation}
and all the terms in the right hand side are uniformly bounded thanks to the previous estimates.\\

\

Since we have proved that the family $\{x^\al\tau_\eta\U_\eta^\delta\}_\delta$ is bounded in $W^{1,1}(\mathbb{R}^+)$ for all $\al\geq0,$ then, because $\tau_\eta$ is positive and belongs to ${\mathcal P},$ we can extract from $\{\U_\eta^\delta\}_\delta$ a subsequence which converges in $L^1(\mathbb{R}^+)$ when $\delta\to0.$ Passing to the limit in equation \eqref{eq:truncated} we find that
\begin{equation}\label{eq:truncated2}\left\{\begin{array}{l}\displaystyle\f{\p}{\p x} (\tau_\eta(x) \U_\eta(x))+(\beta(x)+\lb_{\eta})\U_\eta(x) = 2 \int_x^\infty\beta(y)\kappa(x,y) \U_\eta(y)\,dy,\\
\\
\U_\eta(0)=0,\quad\U_\eta(x)\geq0,\quad\int\U_\eta=1,\end{array}\right.\end{equation}
with $\lb_\eta\geq0.$

\

\subsection{Limit as $\eta\to0$ for $\U_\eta$ and $\lb_\eta$}

All the estimates \eqref{eq:L1bound1}-\eqref{eq:W11bound} remain true for $\delta=0.$ So we still know that the family $\{x^\al\tau_\eta\U_\eta\}_\eta$ belongs to a compact set of $L^1,$ but not necessarily $\{\U_\eta\}_\eta$ because in the limit $\tau$ can vanish at zero. We need one more estimate to study the limit $\eta\to0.$

\paragraph{\it Third estimate: $L^\infty$bound for $x^\al\tau_\eta\U_\eta,\ \al\geq-\gamma$.}
We already know that $x^\al\tau_\eta\U_\eta$ is bounded for $\al\geq0.$ So, to prove the bound, it only remains to prove that $x^{-\gamma}\tau_\eta\U_\eta$ is bounded in a neighborhood of zero. Let define $f_\eta:x\mapsto\sup_{(0,x)}\tau_\eta\U_\eta.$ If we integrate \eqref{eq:truncated2} between $0$ and $x'<x,$ we find
$$\tau_\eta(x')\U_\eta(x')\leq2\int_0^{x'}\int\beta(y)\U_\eta(y)\kappa(z,y)\,dy\,dz\leq2\int_0^x\int\beta(y)\U_\eta(y)\kappa(z,y)\,dy\,dz$$
and so for all $x$
$$f_\eta(x)\leq2\int_0^x\int\beta(y)\U_\eta(y)\kappa(z,y)\,dy\,dz.$$
We consider $x_\eta$ and $\underline x$ defined in the first estimate and, using \eqref{as:kappatau} and \eqref{as:betatau0}, we have for all $x<x_\eta$
\begin{eqnarray*}
f_\eta(x)&\leq&2\int_0^x\int\beta(y)\U_\eta(y)\kappa(z,y)\,dy\,dz\\
&=&2\int\beta(y)\U_\eta(y)\int_0^x\kappa(z,y)\,dz\,dy\\
&\leq&2\int_0^\infty\beta(y)\U_\eta(y)\min\Bigl(1,C\Bigl(\f x y\Bigr)^\gamma\Bigr)\,dy\\
&=&2\int_0^x\beta(y)\U_\eta(y)\,dy+2C\int_x^{x_\eta}\beta(y)\U_\eta(y)\Bigl(\f x y\Bigr)^\gamma \,dy+2C\int_{x_\eta}^\infty\beta(y)\U_\eta(y)\Bigl(\f x y\Bigr)^\gamma \,dy\\
&=&2\int_0^x\f{\beta(y)}{\tau_\eta(y)}\tau_\eta(y)\U_\eta(y)\,dy+2Cx^\gamma\int_x^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\f{\tau_\eta(y)\U_\eta(y)}{y^\gamma}\,dy+2C\int_{x_\eta}^\infty\beta(y)\U_\eta(y)\Bigl(\f x y\Bigr)^\gamma \,dy\\
&\leq&2f_\eta(x)\int_0^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\,dy+2Cx^\gamma\int_x^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\f{f_\eta(y)}{y^\gamma}\,dy+2C\|\beta\U_\eta\|_{L^1}\f{x^\gamma}{x_\eta^\gamma}.
\end{eqnarray*}
We set ${\mathcal V}_\eta(x)=x^{-\gamma}f_\eta(x)$ and we obtain
$$(1-2\rho){\mathcal V}_\eta(x)\leq K+2C\int_x^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}{\mathcal V}_\al(y)\,dy.$$
Hence, using Gr\"onwall's lemma, we find that $\displaystyle{\mathcal V}_\eta(x)\leq\f {Ke^{\f{2C\rho}{1-2\rho}}}{1-2\rho}$ and consequently \begin{equation}\label{eq:0Linfbound}x^{-\gamma}\tau_\eta(x)\U_\eta(x)\leq\f {Ke^{\f{2C\rho}{1-2\rho}}}{1-2\rho}:=\wt C,\quad\forall x\in[0,\underline x].\end{equation}

\

This last estimate allows us to bound $\U_\eta$ by $\f{x^\gamma}{\tau}$ which is in $L^1_0$ by the assumption \eqref{as:kappatau}. Thanks to the second estimate, we also have that $\int x^\al\U_\eta$ is bounded in $L^1$ and so, thanks to the Dunford-Pettis theorem (see \cite{Brezis} for instance), $\{\U_\eta\}_\eta$ belong to a $L^1$-weak compact set. Thus we can extract a subsequence which converges $L^1-$weak toward $\U.$ But for all $\e>0,\ \{x^\al\U_\eta\}_\eta$ is bounded in $W^{1,1}([\e,\infty))$ for all $\al\geq1$ thanks to \eqref{eq:W11bound} and so the convergence is strong on $[\e,\infty).$ Then we write
\begin{eqnarray*}
\int|\U_\eta-\U|&=&\int_0^\e|\U_\eta-\U|+\int_\e^\infty|\U_\eta-\U|\\
&\leq&2\wt C\int_0^\e\f{x^\gamma}{\tau(x)}+\int_\e^\infty|\U_\eta-\U|.
\end{eqnarray*}
The first term on the right hand side is small for $\e$ small because $\f{x^\gamma}{\tau}\in L^1_0$ and then the second term is small for $\eta$ small because of the strong convergence. Finally $\U_\eta\underset{\eta\to0}{\longrightarrow}\U$ strongly in $L^1(\mathbb{R}^+)$ and $\U$ solution of the eigenproblem \eqref{eq:eigenproblem}.

\

\subsection{Limit as $\delta,\eta\to0$ for $\phi_\eta^\delta$}

We prove uniform estimates on $\phi_\eta^\delta$ which are enough to pass to the limit and prove the result.

\paragraph{\it Fourth estimate : uniform $\phi_\eta^\delta$-bound on $[0,A]$.}
Let $A>0,$ our first goal is to prove the existence of a constant $C_0(A)$ such that
$$\forall\eta,\delta,\qquad\sup_{(0,A)}{\phi_\eta^\delta}\leq C_0(A).$$
We divide the equation on $\phi_\eta^\delta$ by $\tau_\eta$ and we integrate between $x$ and $x_\eta$ with $0<x<x_\eta,$ where $x_\eta,$ bounded by $\underline x$ and $\overline x,$ is defined in the first estimate. Considering $\delta<\f{\mu(1-2\rho)}{\overline x}$ (fulfilled for $R>\f{\overline x}{2(1-2\rho)}$ since $\delta=\f{\mu}{2R}$), we find
\begin{eqnarray*}
\phi_\eta^\delta(x)&\leq&\phi_\eta^\delta(x_\eta)+2\int_x^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\int_0^y\kappa(z,y)\phi_\eta^\delta(z)\,dz+x_\eta\f\delta \mu\phi_\eta^\delta(0)\\
&\leq&\phi_\eta^\delta(x_\eta)+\sup_{(0,x_\eta)}\{\phi_\eta^\delta\}\Bigl(2\int_0^{x_\eta}\f{\beta(y)}{\tau_\eta(y)}\int_0^y\kappa(z,y)\,dz+x_\eta\f\delta\mu\Bigr)
\end{eqnarray*}
and we obtain
$$\sup_{x\in(0,\underline x)}{\phi_\eta^\delta(x)}\leq\f 1 {1-2\rho-\delta\overline x/\mu}\phi_\eta^\delta(x_\eta).$$
Using the decay of $\phi_\eta^\delta(x)e^{-\int_{\underline x}^x\f{\beta+\lb_\eta^\delta}{\tau_\eta}},$ there exists $C(A)$ such that
$$\sup_{x\in(0,A)}{\phi_\eta^\delta(x)}\leq C(A)\phi_\eta^\delta(x_\eta).$$
Noticing that $\int\phi_\eta^\delta(x)\U_\eta^\delta(x)dx =1,$ we conclude
$$1\geq\int_0^{x_\eta}\phi_\eta^\delta(x)\U_\eta^\delta(x)dx\geq \phi_\eta^\delta(x_\eta)\int_0^{x_\eta}e^{-\int_x^{x_\eta}\f{\beta+\lb_\eta^\delta}{\tau_\eta}}\U_\eta^\delta(x)\,dx,$$
so, as $x_\eta\to x_0$ and $\int_0^{x_0}\U(x)dx>0$ (thanks to Lemma \ref{lm:positivity} and because $x_0>b\geq a$), we have
\begin{equation}\label{eq:phi0bound}\sup_{(0,A)}{\phi_\eta^\delta}\leq C_0(A).\end{equation}

\

\paragraph{\it Fifth estimate : uniform $\phi_\eta^\delta$-bound on $[A,\infty)$.}
Following an idea introduced in \cite{PR} we notice that the equation in \eqref{eq:truncated} satisfied by $\phi_\eta^\delta$ is a transport equation and therefore satisfies the maximum principle (see Lemma \ref{lm:supersolution} in the Appendix). Therefore it remains to build a supersolution $\overline\phi$ that is positive at $x=R,$ to conclude $\phi_\eta^\delta(x)\leq\overline\phi(x)$ on $[0,R].$

This we cannot do on $[0,R],$ but on a subinterval $[A_0,R]$ only. So we begin with an auxiliary function $\overline\vp(x)=x^k+\theta$ with $k$ and $\theta$ positive numbers to be determined. We have to check that on $[A_0,R]$
$$-\tau(x)\f{\p}{\p x}\overline\vp(x)+(\lb_\eta^\delta+\beta(x))\overline\vp(x)\geq 2\beta(x)\int\kappa(y,x)\overline\vp(y)\,dy+\delta\phi_\eta^\delta(0),$$
{\it i.e.}
$$-k\tau(x)x^{k-1}+(\lb_\eta^\delta+\beta(x))\overline\vp(x)\geq\Bigl(2\theta+2\int\kappa(y,x)y^k\,dy\Bigr)\beta(x)+\delta\phi_\eta^\delta(0).$$
For $k\geq2,$ we know that $\int\kappa(y,x)\f{y^k}{x^k}\,dy\leq c<1/2$ so it is sufficient to prove that there exists $A_0>0$ such that we have
\begin{equation}\label{eq:sursolution1}-k\tau(x)x^{k-1}+(\lb_\eta^\delta+\beta(x))(x^k+\theta)\geq(2\theta+2cx^k)\beta(x)+\delta C_0(1)\end{equation}
for all $x>A_0,$ where $C_0$ is defined in \eqref{eq:phi0bound}.
For this, dividing \eqref{eq:sursolution1} by $x^{k-1}\tau(x),$ we say that if we have
\begin{equation}\label{eq:sursolution2}(1-2c)\f{x\beta(x)}{\tau(x)}\geq k+\f{2\theta\beta(x)+\delta C_0(1)}{x^{k-1}\tau(x)},\end{equation}
then \eqref{eq:sursolution1} holds true. Thanks to assumptions \eqref{as:betatauspace} and \eqref{as:betatauinf} we know that there exists $k>0$ such that for any $\theta>0,$ there exists $A_0>0$ for which \eqref{eq:sursolution2} is true on $[A_0,+\infty).$

Then we conclude by choosing the supersolution $\overline\phi(x)=\f{C_0(A_0)}\theta\overline\vp(x)$ so that
$$\overline\phi(x)\geq\phi_\eta^\delta(x)\quad \text{on} \ [0,A_0],$$
and on $[A_0,R],$ we have
\begin{equation}\label{eq:supersol}\left\{\begin{array}{l}
-\tau(x)\f{\p}{\p x}\overline\phi(x)+(\lb_\eta^\delta+\beta(x))\overline\phi(x) \geq 2\beta(x)\int_0^x\kappa(y,x)\overline\phi(y)\,dy+\delta\phi_\eta^\delta(0),
\\
\\
\overline\phi(R)>0,
\end{array}\right.\end{equation}
which is a supersolution to the equation satisfied by $\phi_\eta^\delta.$ Therefore $\phi_\eta^\delta\leq\overline\phi$ uniformly in $\eta$ and $\delta$ and we get
\begin{equation}\label{eq:phibound}\exists k,\theta,C\ s.t.\ \forall\eta,\delta,\quad\phi_\eta^\delta(x)\leq(Cx^k+\theta).\end{equation}

\

Equation \eqref{eq:truncated} and the fact that $\phi_\eta^\delta$ is uniformly bounded in $L^\infty_{loc}(\mathbb{R}^+)$ give immediately that $\p_x\phi_\eta^\delta$ is uniformly bounded in $L^\infty_{loc}(\mathbb{R}^+,\tau(x)dx),$ so in $L^\infty_{loc}(0,\infty)$ thanks to \eqref{as:taupositivity}.

\

Then we can extract a subsequence of $\{\phi_\eta^\delta\}$ which converges ${\mathcal C}^0(0,\infty)$ toward $\phi.$ Now we check that $\phi$ satisfied the adjoint equation of \eqref{eq:eigenproblem}. We consider the terms of \eqref{eq:truncated} one after another.\\
First $(\lb_\eta^\delta+\beta(x))\phi_\eta^\delta(x)$ converges to $(\lb+\beta(x))\phi(x)$ in $L^\infty_{loc}.$\\
For $\p_x\phi_\eta^\delta,$ we have an $L^\infty$ bound on each compact of $(0,\infty).$ So it converges $L^\infty-*weak$ toward $\p_x\phi.$\\
It remains the last term which we write, for all $x>0,$ $$\int_0^x\kappa(y,x)(\phi_\eta^\delta(y)-\phi(y))\,dy\leq\|\phi_\eta^\delta-\phi\|_{L^\infty(0,x)}\underset{\eta,\delta\to0}{\longrightarrow}0.$$
The fact that $\int\phi\U=1$ comes from the convergence $L^\infty-L^1$ when written as
$$1=\int\phi_\eta^\delta(x)\U_\eta^\delta(x)\,dx=\int\f{\phi_\eta^\delta(x)}{1+x^k}(1+x^k)\U_\eta^\delta(x)\,dx\longrightarrow\int\f{\phi(x)}{1+x^k}(1+x^k)\U(x)\,dx=\int\phi\U.$$

\bigskip\bigskip

At this stage we have found $(\lb,\U,\phi)\in\mathbb{R}^+\times L^1(\mathbb{R}^+)\times{\mathcal C}(\mathbb{R}^+)$ solution of \eqref{eq:eigenproblem}. The estimates announced in Theorem \ref{th:eigenelements} also follow from those uniform estimates. It remains to prove that $\lb>0$ and the uniqueness.

\

\subsection{Proof of $\lb>0$}

We prove a little bit more, namely that
\begin{equation}\label{eq:lowerbound}\lb\geq\f12\sup_{x\geq0}\{\tau(x)\U(x)\}. \end{equation}

We integrate the first equation of \eqref{eq:eigenproblem} between $0$ and $x$ and find
\begin{eqnarray*}
0\leq\lb\int_0^x\U(y)\,dy&=&-\tau(x)\U(x)-\int_0^x\beta(y)\U(y)\,dy+2\int_0^x\int_z^\infty\beta(y)\kappa(z,y)\U(y)\,dy\,dz\\
 &\leq&-\tau(x)\U(x)+2\int_0^\infty\int_z^\infty\beta(y)\kappa(z,y)\U(y)\,dy\,dz\\
 &=&-\tau(x)\U(x)+2\int_0^\infty\beta(y)\U(y)\,dy\\
 &=&-\tau(x)\U(x)+2\lb,
\end{eqnarray*}
Hence $2\lb\geq \tau(x)\U(x)$ and \eqref{eq:lowerbound} is proved.

\

\subsection{Uniqueness}

We follow the idea of \cite{M1}. Let $(\lb_1,\U_1,\phi_1)$ and $(\lb_2,\U_2,\phi_2)$ two solutions to the eigenproblem \eqref{eq:eigenproblem}.
First we have
\begin{eqnarray*}
\lb_1\int\U_1(x)\phi_2(x)\,dx&=&\int\Bigl(-\p_x(\tau(x)\U_1(x))-\beta(x)\U_1(x)+2\int_x^\infty\beta(y)\kappa(x,y)\U_1(y)\,dy\Bigr)\phi_2(x)\,dx\\
&=&\int\Bigl(\tau(x)\p_x\phi_2(x)-\beta(x)\phi_2(x)+2\beta(x)\int_0^x\kappa(y,x)\phi_2(y)\,dy\Bigr)\U_1(x)\,dx\\
&=&\lb_2\int\U_1(x)\phi_2(x)\,dx
\end{eqnarray*}
and then $\lb_1=\lb_2=\lb$ because $\int\U_1\phi_2>0$ thanks to Lemma \ref{lm:positivity}. \\
For the eigenvectors we use the General Relative Entropy method introduced in \cite{MMP1,MMP2}. For $C>0,$ we test the equation on $\U_1$ against $\sgn\bigl(\f{\U_1}{\U_2}-C\bigr)\phi_1,$
$$0=\int\Bigl[\p_x(\tau(x)\U_1(x))+(\lb+\beta(x))\U_1(x)-2\int_x^\infty\beta(y)\kappa(x,y)\U_1(y)\,dy\,\Bigr]\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx.$$
Deriving $\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\tau(x)\U_2(x)\phi_1(x)$ we find
$$\begin{array}{l}\displaystyle
\int\p_x(\tau(x)\U_1(x))\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx=\int\p_x\Bigl(\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\tau(x)\U_2(x)\phi_1(x)\Bigr)\,dx\\
\\
\displaystyle\hspace{1cm}+\int\p_x(\tau(x)\U_2(x))\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx-\int\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\p_x(\tau(x)\U_2(x)\phi_1(x))\,dx
\end{array}$$

and then

$$\begin{array}{l}\displaystyle
\int\p_x(\tau(x)\U_1(x))\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx=\\
\\
\displaystyle\hspace{2cm}2\int\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\Bigl[\int_0^x\beta(x)\kappa(y,x)\U_2(x)\phi_1(y)\,dy-\int_x^\infty\beta(y)\kappa(x,y)\U_2(y)\phi_1(x)\,dy\Bigr]\,dx\\
\\
\displaystyle\hspace{4cm}+2\int\int_x^\infty\beta(y)\kappa(x,y)\U_2(y)\,dy\,\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx\\
\\
\displaystyle\hspace{6cm}-\int(\lb+\beta(x))\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\U_2(x)\phi_1(x)\,dx,
\end{array}$$

\

$$\begin{array}{l}\displaystyle
\int\p_x(\tau(x)\U_1(x))\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx=\\
\\
\displaystyle\hspace{2cm}2\int\int\beta(y)\kappa(x,y)\Bigl[\Bigl|\f{\U_1}{\U_2}(y)-C\Bigr|-\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\Bigr]\U_2(y)\phi_1(x)\,dxdy\\
\\
\displaystyle\hspace{4cm}+2\int\int_x^\infty\beta(y)\kappa(x,y)\U_2(y)\,dy\,\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx\\
\\
\displaystyle\hspace{6cm}-\int(\lb+\beta(x))\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\U_2(x)\phi_1(x)\,dx.
\end{array}$$

\

So
$$\begin{array}{l}\displaystyle
0=2\int\int\beta(y)\kappa(x,y)\Bigl[\Bigl|\f{\U_1}{\U_2}(y)-C\Bigr|-\Bigl|\f{\U_1}{\U_2}(x)-C\Bigr|\Bigr]\U_2(y)\phi_1(x)\,dxdy\\
\\
\displaystyle\hspace{4cm}+2\int\int_x^\infty\beta(y)\kappa(x,y)\U_2(y)\,dy\,\f{\U_1}{\U_2}(x)\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx\\
\\
\displaystyle\hspace{6cm}-2\int\int_x^\infty\beta(y)\kappa(x,y)\U_1(y)\,dy\,\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\phi_1(x)\,dx
\end{array}$$

\

$$0=\int\int\beta(y)\kappa(x,y)\U_2(y)\Bigl|\f{\U_1}{\U_2}(y)-C\Bigr|\Bigl[1-\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\sgn\Bigl(\f{\U_1}{\U_2}(y)-C\Bigr)\Bigr]\phi_1(x)\,dxdy.$$

\

Hence $\Bigl[1-\sgn\Bigl(\f{\U_1}{\U_2}(x)-C\Bigr)\sgn\Bigl(\f{\U_1}{\U_2}(y)-C\Bigr)\Bigr]=0$ on the support of $\kappa(x,y)$ for all $C$ thus $\f{\U_1}{\U_2}(x)=\f{\U_1}{\U_2}(y)$ on the support of $\kappa(x,y)$ and
\begin{equation}\label{eq:uniqueness}\p_x\f{\U_1}{\U_2}(x)=\int\beta(y)\kappa(x,y)\Bigl(\f{\U_1}{\U_2}(y)-\f{\U_1}{\U_2}(x)\Bigr)\f{\U_2(y)}{\U_2(x)}\,dy=0\end{equation}
so $\displaystyle\f{\U_1}{\U_2}\equiv cst=1.$

\

We can prove in the same way that $\phi_1=\phi_2$ even if we can have $\U\equiv0$ on $[0,m]$ with $m>0.$ Indeed in this case we know that $\beta\equiv0$ on $[0,m]$ and so
$$\phi_i(x)=\phi_i(0)e^{\int_0^x\f{\lb}{\tau(s)}ds}\quad\forall x\in[0,m],\ i\in\{1,2\}.$$

\

\section{Conclusion, Perspectives}\label{se:csq}

We have proved the existence and uniqueness of eigenelements for the aggregation-fragmentation equation \eqref{eq:temporel} with assumptions on the parameters as large as possible, in order to render out the widest variety of biological or physical models. It gives access to the asymptotic behaviour of the solution by the use of the General Relative Entropy principle.

\

A following work is to study the dependency of the eigenvalue $\lb$ on parameters $\tau$ and $\beta$ (see \cite{M2}). For instance, our assumptions allow $\tau$ to vanish at zero, what is a necessary condition to ensure that $\lb$ tends to zero when the fragmentation tends to infinity. Such results give precious information on the qualitative behaviour of the  solution.

\

Another possible extension of the present work is to prove existence of eigenelements in the case of time-periodic parameters, using the Floquet's theory, and then compare the new $\lb_F$ with the time-independent one $\lb$ (see \cite{Lepoutre}). Such studies can help to choose a right strategy in order to optimize, for instance, the total mass $\int x u(t,x) dx$  in the case of prion proliferation (see \cite{CL1}) or on the contrary minimize the total population $\int u(t,x) dx$ in the case of cancer therapy (see \cite{Lepoutre, Clairambault}).

\

Finally, this eigenvalue problem could be used to recover some of the equation parameters like $\tau$ and $\beta$ from the knowledge of the asymptotic profile of the solution, as introduced in \cite{DPZ, PZ} in the case of symmetric division ($\tau=1$ and $\kappa=\delta_{x=\frac{y}{2}}$), by the use of inverse problems techniques. The method of \cite{PZ} has to be adapted to our general case, in order to model prion proliferation for instance, or yet to recover the aggregation rate $\tau$ ; this is another direction for future research.

\vspace{1cm}

{\bf Aknowledgment}

The authors thank a lot Beno\^it Perthame for his precious help and his corrections.

\newpage{\noindent\LARGE\bf Appendix}


\section{Introduction}
\label{intro}
\textbf{Legislation Networks} were first introduced in 2015 \cite{MKoniaris2017} and further discussed in \cite{NSakhaee2016} \cite{NSakhaee2017}. These networks are essential to explore the relationship between legislation and societies' evolution \cite{NSakhaee2016}. There are many obvious benefits from studying Legislation Networks \cite {NSakhaee2016} \cite{NSakhaee2017} \cite{PZhang2007} \cite{Fowler2007}, but building these networks is not always a straightforward task, as only a few legislation systems provide machine-readable documents or structured databases \cite {EURlex} \cite{Metalex}. The majority of legislation systems supply legal documents in human-readable format. For example the New Zealand Parliamentary Council Office provides machine-readable XML files \cite{PCO} only for the current active Acts, which constitute around ten percent of the entire set of Acts. All other historic documents are scanned and supplied by a third party institute in Portable Document Format (PDF) \cite{NZLII}.

To extract information from legal documents, the first step is the conversion of images into text if the text is not available. This concept is well studied as \textbf{optical character recognition (OCR)} \cite{OCR}, and there are several techniques and tools developed to convert typewritten or handwritten images to text. OCR is the first step of the proposed framework, and for the case study we selected ABBYY FineReader \cite{ABBYY}.

 \textbf{Information Extraction (IE)}  involves locating and extracting specific information from text \cite{IE0}. Information Extraction assumes that in each individual text file there are one or more entities that are similar to those in other text documents but differing in the details \cite{IE2}.

IE approaches in the legal domain are considerably different from other knowledge areas because of the two main characteristics of legal texts. Legal documents exhibit a wide range of internal structure, and they often have a significant amount of manual editorial value added.
One of the earliest information retrieval approaches for legal materials based on searching inside the document was proposed in 1978 \cite{IE1}. Later works mainly used natural supervised learning techniques to retrieve the required data from legal texts, but with a substantial error \cite{Cheng2009} \cite{Textnailing2017}. In the proposed framework of this study several IE tasks are used, and more are described later in this section.

\textbf{Named entity recognition (NER)}
is one of the main sub-tasks of IE. The goal of this task is to find each occurrence of a named entity in the text \cite{NER1}. Entities usually include people, locations, quantities, and organizations but also more specific entities such as the names of genes and proteins \cite{NER2}, the names of college courses \cite{NER3}, and drugs \cite{NER4}. In the New Zealand legislation corpus, entities could be the name of legislative documents such as Acts, Regulations, Bills, Orders, or Case-Laws \cite{NSakhaee2016}. In the case study which is discussed \autoref{Application}, the main required entities inside the text documents are is the names of the New Zealand Acts.

The main traditional NER algorithm that identifies and classifies the named entities is statistical sequence modeling \cite{NER1}. But there are other modern approaches based on combinations of lists, rules, and supervised machine learning \cite{NER5}. To extract the require information for Legislation Network, there are clear rules to identify the named entities, and the classification of the entities is not needed. Therefore the second NER approach is more appropriate and discussed further for the proposed framework.

The next IE task which is used in our study is to detect the relationships that exist among the recognized entities. This task is called \textbf{relation extraction (RE)} \cite{NER1}.
The earliest algorithm for relation extraction is the use of lexico-syntactic patterns \cite{RE1}. This algorithm is still valid and widely used, but there are other algorithms introduced later such as supervised learning \cite{NER1} and bootstrapping \cite{boot}\cite{boot1}.
Considering that legislation texts are well structured, it is assumed that there is a large collection of previously annotated material that can define the rules for classifiers.

\textbf{Approximate string matching} techniques find items in a database when there may be a spelling mistake or other error in the keyword \cite{ASM1}. This is becoming a more relevant issue for fast-growing knowledge areas such as information retrieval \cite{ASM4}. Various techniques are studied to address the identity uncertainty of the objects, and briefly are reviewed in this study. These techniques could be distance based, token based, or a hybrid model of the distance and token based models.\\~
Damerau-Levenshtein metrics are the main approximate string matching techniques to address the distance functions between the two strings \cite{ASM2} \cite{ASM3}. The most famous function in this category is \textit{edit-distance}, and it is defined as the minimum number of changes required to convert one string into the other \cite{ASM3}. Several alternative distance functions to the edit-distance have been proposed such as $q$-gram and maximal matches \cite{ASM5}.\\~
The next set of techniques are token based or probabilistic object identification methods adapted for string matching tasks \cite{FSM1}\cite{FSM2} \cite{ASM1}. \textit{Jaccard similarity} and \textit{cosine similarity} are common token based measures widely used in the information retrieval community\cite{FSM1}. Hybrid techniques combine distance-based and token-based string matching measures such as Jaro-Winkler \cite{FSM4}. All of the string matching algorithms have been developed by filtering and bit-parallelism approaches.\\~
The fastest algorithms use a  combination of filters to discard most of the text by focusing on the potential matches. Hybrid models significantly improve precision and recall reducing the error in a range between $0.1$ to $0.2$ \cite{ASM4}.

Network inferences require high accuracy of data \cite{Error1}.  For this study various string matching techniques are examined for the Legislation Network and comparing the results, a hybrid model of \textit{Jaccard similarity} and \textit{edit-distance} is used as described in the next section.

\textbf{The main contribution of this study} is the proposed Information Extraction framework which engages several processes and enables the researcher to have access to the network information from historic documents. This framework makes it possible to study the Legislation Network as a dynamic graph. In this paper the case study covers all Acts in New Zealand legislation corpus including historic, expired, repealed and consolidated Acts as at end of September 2018. This comes to a set of 23870 PDF files of which about $87\%$ are in scanned image format.

\autoref{sample} shows a sample image of an average quality scanned PDF document. The proposed framework suggests a high-performance procedure to derive network information from such poor quality documents. In the following sections, examples and the experimental results are used to illustrate the framework, its performance, and its potential applications.

In this section a summary of the required Information Extraction processes and methodologies is discussed. In the next section the proposed framework is presented and various examples are explained. Then the case study analysis and the application of the proposed framework are examined. Next, a number of experiments are designed and studied to evaluate the accuracy of the extracted information and to study the robustness of Legislation Network. The study finishes with a quick review on the novelty and the importance of discovering the time-varying behaviour of the Legislation Network.
\begin{figure}
    \centering
    \includegraphics[scale=.7]{docexample.pdf}
    \caption{Married Women Property Protection Act 1860}
    \label{sample}
\end{figure}
\section{The proposed Information Extraction framework}
In this section, the Information Extraction framework to build the Legislation Network is discussed. \autoref{IEF} depicts the overview of the proposed framework.\\~
The process starts with the conversion of non machine readable files to text by using \textit{OCR} available tools. This step is relatively straightforward, but could be time-consuming considering the number of documents in the study. As mentioned earlier, in the case study the tool named ABBYY FineReader \cite{ABBYY} is used. The average accuracy of this step is just above 80 percent and implies the need for a typos analysis step that is discussed in section \ref{Approximate String Matching}.
\subsection{Text Canonicalization}
The next step in the proposed framework is \textit{text canonicalization} \cite{canon}. There are several required tasks to convert all of the text files into a unique format, so the rules can be defined more easily while running the Information Extraction tasks. The text canonicalization step could be implemented via different approaches depending to the text style and language. In this paper, some of the common tasks are suggested, and two potentially required tasks are described.\\~
In the case study the designed system transfers all letters to \textit{Lowercase}. This transition applies a level of consistency across the text documents and the Information Extraction rules.
In the experiments, the system also replaces \textit{Special Characters} with generic tags in the text. The only character which is not replaced is the parenthesis, as it is often used in the title of legislation.
The other suggested generic text canonicalization task is to replace multiple spaces with one space.

\begin{wrapfigure}{l}{5.5cm}
\includegraphics[scale=.7]{IEProcess-3.jpg}
\caption{Legal Text IE Framework}
\label{IEF}
\end{wrapfigure}
Apart from the general text canonicalization steps, there are other potential corrections that shape the text to make it a better input for the Information Extraction process. The first is to remove text margins that OCR mistakenly merges to the main body of text. \autoref{sample} includes examples of these margins that might impact the Information Extraction rules and result in error. As an example in the case study, the phrase \textit{short title} is often used in the margin, and OCR merges it to the nearest part of the text. This might impact the named entity recognition task, so the system removes this phrase and many of its possible misspelled forms from the text.
The next recommended text canonicalization step is to resolve the misspelling issues for the keywords used in the Information Extraction process. As an example in this study, Acts are the main entities, and the rule to recognize them uses the keyword \textit{Act}. So the system corrects some of the possible misspelling forms of the word \textit{-act-}.
\begin{table}[h]
    \centering{}\protect\caption{OCR and Text canonicalization result comparison}
    \label{tab:canonicalization}
    \adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
    \begin{tabular}{l l l}
    \hline
   OCR & I$\sim$ The Short Title' of this' Act shall be, the ((Married Short TItle,'Vomen's 'Property'Protectio\pounds  Aot, 1860." \\
   \hline
   Text canonicalization  & I the short title of this act shall be the ((married vomens propertyprotectio act 1860\\
   \hline
    \end{tabular}
}
\end{table}
To provide a better explanation of this step, \autoref{tab:canonicalization} provides an example referring to third paragraph of the \autoref{sample} image.
As can be seen the text canonicalization converts the text to a simpler unique structure prepares it for the next steps of \textit{named entity recognition} and \textit{relation extraction}. Typo resolution is not expected at this step, being covered under the last step via Approximate String Matching.
\subsection{Named Entity Recognition and Relation Extraction}
As explained, the text canonicalization step normalizes the text files to a unique format and prepares them for in-depth information extraction steps. To extract the network node information, a combined Named Entity Recognition approach is suggested which engages rules and supervised learning. To identify the rules, a sample set of the documents should be reviewed. The sample size is not necessarily large, but a stratified sampling approach is suggested to eliminate the impact of time-period style and author's writing style.

\autoref{Entities} shows examples of the entities in the case study based on the Acts recognition rules. Acts are the main part of the New Zealand legislation system and as explained before the case study only considers the Acts.
\begin{table}[h]
\centering{}\protect\caption{Entities, types and examples}
\label{Entities}
\adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
\begin{tabular}{ l  l  l  l }
\hline
Type & Tag & Sample & Canonicalized text \\
 \hline
 Year & YR & 1860 & the short title of this act shall be the ((married vomens propertyprotectio act 1860 \\
 Act & ACT & married vomens propertyprotectio act 1860 &  the short title of this act shall be the ((married vomens propertyprotectio act 1860\\
 \hline
\end{tabular}
}
\end{table}
In the case study stratified sampling method is used and the strata are five different time periods in a range of more than 200 years. A total number of 55 text files are reviewed, and several clear rules engaging a set of keywords and lists are built to identify the named entities. \autoref{NERrules} provides examples of the rules in each stratum and $y$ represents the year in which the Act is commenced.

\begin{table}[h]
\centering{}\protect\caption{Examples of the Named Entity Recognition rules}
\label{NERrules}
\adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
\begin{tabular}{ l  l  l  l }
\hline
Stratum & Keyword example & Rule example & Sample document  \\
 \hline
 $y<1850$ & \textit{ordinance} & an [\textit{keyword}] to \textit{any phrase} of [\textit{act name}] [date] & Police Magistrates Act 1841 \\
 $1850<y<1900$ & \textit{short title}, \textit{shall be} & the [\textit{keyword}] of this act [\textit{keyword}] the [\textit{act name}] [\textit{year}] & Customs Tariff Act 1873  \\
 $1900<y<1950$ & \textit{amend}, \textit{consolidate} & an act to [\textit{keyword}] \textit{any phrase} of the [\textit{act name}] [date] & Mining Act 1926 \\
 $1950<y<2000$ & \textit{meaning}, \textit{section} & same [\textit{keyword}] as in [\textit{keyword}] [any number] of the [\textit{act name}] [\textit{year}] & Copyright Act 1962 \\
 $2000<y$ & act & this [\textit{keyword}] is the [\textit{act name}] [\textit{year}] & Social Security Act 2018\\
 \hline
\end{tabular}
}
\end{table}

Alongside recognizing the entities, to extract the network edge information, a rule based Relation Extraction approach is suggested considering that legislation texts are contextually structured. To identify the rules, this study suggests to use the same sample set which is used for the Named Entity Recognition. From the case study it is observed that the style of writing legislation has changed considerably over time, so the sampling approach is very important to minimize the impact of various text styles. By reviewing the sample files, there is a large collection of previously annotated material that can define the rules for relation classifiers.

For the case study, as explained total number of 55 text files are reviewed, and several classifier rules engaging a set of keywords are built to identify the relations between the named entities. \autoref{Relations} summarizes the entity relation list for the case study and provides examples.
\begin{table}[h]
\centering{}\protect\caption{Relations example}
\label{Relations}
\adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
\begin{tabular}{ l  l  l  l }
\hline
Relation & Type & Canonicalized text & Sample document \\
 \hline
 Title & TIT & the short title of this act shall be the ((married vomen propertyprotectio act 1860 & Married Women Property Protection Act 1860 \\
 Citation & CIT & within the meaning of section 5 of the companies act 1993 & Trade Marks Act 2002 \\
 Amendment & AMD & section 25.1b amended, by section 5.2 of the trade marks amendment act 2005 & Trade Marks Act 2002\\
 Partial Repeal & PRP & section 5(1) repealed, by section 4(8) of the trade marks amendment act 2011 & Trade Marks Act 2002\\
 Repeal & FRP & acts repealed.
1860, No. 9.the married vyomens pfoperty protection act, 1860. & Married Women Property Protection Act 1880\\
  \hline
\end{tabular}
}
\end{table}
This suggested process can be generalized for any other case study in Legislation Network building process considering that legislation texts are coherently structured. So there is always a large collection of previously annotated material that can define the rules for entity recognition and relation classifiers.
\subsection{Approximate String Matching}\label{Approximate String Matching}
Named entity recognition identifies the Acts and relation extraction recognizes the relationship between them. So these two steps result in an initial version of the node list and the edge list of the intended Legislation Network. However testing this network shows that the extracted data is shoddy with an average error rate of 12 \footnote{To  estimate  this error rate, a  cluster  sampling  method is  used  to  randomly  choose  ten  sets  of  30  entities.  By  manual  check  of  the samples, the rate of incorrectly matched entities is observed.} percent, so another step is required to resolve typo issues and imperfect entities. This poor-quality data implies the need for the approximate string matching step. To run this step two main components are required, the technique and the correct pattern. \autoref{ASMexample} provides an example which shows the first match as the output of the proposed approximate string matching technique.
\begin{table}[h]
\centering{}\protect\caption{Approximate string matching example}
\label{ASMexample}
\adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
\begin{tabular}{ l  l  l }
\hline
Technique & Extracted entity & First match \\
 \hline
Hybrid Model & married vomens propertyprotectio act 1860 & married women property protection act 1860 \\
 \hline
\end{tabular}
}
\end{table}

As mentioned earlier after the implementation of different approximate string matching techniques, a hybrid model of \textit{Jaccard} and \textit{Edit-Distance} is designed and proposed.
\begin{figure}[h]
\centering
\includegraphics[scale=.5]{ASM.pdf}
\caption{Precision and Recall comparison of the approximate string matching techniques}
\label{ASMPic}
\end{figure}
Algorithm \ref{ASM} shows the proposed hybrid model, and \autoref{ASMPic} compares the results of the hybrid model with Edit-Distance and Jaccard techniques in terms of precision and recall of the approximate string matching step. To run this comparison, a stratified sampling technique is used with different time periods being the groups\footnote{Time periods: before 1800, 1800-1850, 1850-1900, 1900-1950, 1950-2000, 2000-2018}.

\begin{algorithm}
\caption{Approximate String Matching of Legislation}\label{ASM}
\begin{algorithmic}[1]
\Procedure{Legislation Name Matching}{}
\State $string1 \gets \textit{Extracted legislation name}$
\State $\textit{masterlist} \gets \text{Open }\textit{Legislation Title Master List}$
\State $j \gets \textit{1}$
\State $\textit{tline} \gets \text{The first line of }\textit{masterlist}$.
\State $GetOut \gets \textit{0}$
\While {$<GetOut \neq 0> \AND <tline \neq 0>$} \\~
\State $string2 \gets \textit{tline}$.
\State $m(j) \gets \textit{Jaccard(string1, string2)}$
\State $n(j) \gets \textit{EditDistance(string1, string2)}$
\If {$<m(j) = 0.5> \OR <n(j) = 0> $}
\State $\textit{GetOut}$
\EndIf
\State $j \gets j+1$.
\State $\textit{tline} \gets \text{The next line of }\textit{masterlist}$
\State \textbf{close};
\EndWhile
\State $[x1 , I1] \gets \textit{max(m)}$
\State $[y1 , I2] \gets \textit{min(n)}$
\If {$y1 \textit{ is smaller than or equal to 5}$}
\State $ match \gets I2$
\ElsIf {$x1 \textit{ is bigger than 0}$}
\State  $ match \gets I1$
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}

The graph shows the error rates in each time sample of documents based on the chosen approximate string matching model. For example for the documents commenced prior to 1850, the first marker point at each graph line shows the false negative error and the precision of the chosen approximate string matching method.
As can be seen samples from this oldest groups of acts show a higher error rate regardless of the approximate string matching method, and Edit-Distance performs slightly better for the old documents comparing to the Jaccard index. In summary the proposed hybrid model performs significantly better than the other two methods for all documents regardless of their age with less than two percent of false-negative error and average precision of more than 98 percent.

In the case study the pattern  which is used for the approximate string matching step is the list of all NZ Acts provided by NZLII \cite{NZLII}. In case of not having access to such a master list, the typo resolution could be more time consuming. Approximate string matching considerably improves the quality of the extracted information, result in reliable edge list and node list. Later in this study the evaluation of the final extracted data set and the robustness of the network is discussed. The robustness study proves the value of a high performing approximate string matching technique which improves the data quality significantly.
\section{Application}\label{Application}
The proposed Information Extraction framework resolves the historic data limitation in previous studies \cite{NSakhaee2016}\cite{NSakhaee2017} and results in a large and reliable dynamic network data set which is called LegiNet and is available at \cite{dataverse}. This dynamic \cite{DynamicGraph} and complex network has a very intersecting range of characteristics and behaviours. To maintain the subject consistency of this paper, more in-depth analysis of network behaviours are delayed to the future studies. In this section, generic network science characteristics of the case studied network are discussed, and an overall view of the structural and node importance evolution is presented.

\autoref{NewvsOld} compares the produced network based on the Information Extraction process with the earlier versions of the network that was built with parsing of limited available XML resources. As illustrated the network size and structure is significantly changed comparing to its earlier versions.
\begin{table}[h]
\centering{}\protect\caption{NZ Legislation Network, this study versus previous studies}
\label{NewvsOld}
\resizebox{\textwidth}{!}{\begin{tabular}{ l  l  l  l  l  l  l}
\hline
Network & Nodes & Edges & Average degree & Average CC\footnote{The average clustering coefficient (CC) is calculated based on the assumption that the network is directed using the approach that discussed in \cite{NSakhaee2016}} & Average path-length & Network type\\
 \hline
This study & 16385 & 137751 & 8.407 & 0.216 & 4.873 & dynamic\\
Previous studies & 3856 & 33884 & 8.878 & 0.39 & 3.569 & one snapshot\\
 \hline
\end{tabular}}
\end{table}
\autoref{graph} and \autoref{measures} capture the overall evolution of the Legislation Network in New Zealand from 1267 to the second quarter of 2018. To visualize the data, a network force-directed approach is used.
\begin{figure}
\begin{center}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1267-1839.jpg}
\subcaption{1267-1839}\label{a1}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1840s.jpg}
\subcaption{1840s}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1850s.jpg}
\subcaption{1850s}\label{c1}
\end{minipage}%
\vspace{-\parskip}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1860s.jpg}
\subcaption{1860s}\label{d1}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1870s.jpg}
\subcaption{1870s}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.82\linewidth]{1900s.jpg}
\subcaption{1880-1909}
\end{minipage}%
\vspace{-\parskip}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=\linewidth]{1910.jpg}
\subcaption{1910s}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=\linewidth]{1920.jpg}
\subcaption{1920s}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=0.9\linewidth]{1950.jpg}
\subcaption{1930-1959}
\end{minipage}%
\vspace{-\parskip}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=\linewidth]{1970.jpg}
\subcaption{1960-1979}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=.97\linewidth]{2000.jpg}
\subcaption{1980-2009}
\end{minipage}%
\begin{minipage}{0.3\linewidth}
\includegraphics[width=\linewidth]{current.jpg}
\subcaption{2010s}
\end{minipage}%
\end{center}
\caption{Overview of the network structure evolution}\label{graph}
\end{figure}
In the layouts in \autoref{graph} each node is placed depending on their connection to the other nodes. As can be seen references between the Acts first appear in the 1840s, but the data-set visually looks like a graph since the 1850s and it gets denser from the 1870s.
\begin{table}[h]
\centering{}\protect\caption{Overview of the network measures evolution}
\label{measures}
\resizebox{\textwidth}{!}{\begin{tabular}{ l  l  l  l  l  l  l  l  l  l  l  l  l}
\hline
Time & 1267-1839 & 1840s & 1850s & 1860s & 1870s & 1880-1909 & 1910s & 1920s & 1930-1959 & 1960-1979 & 1980-2009 & 2010s\\
 \hline
Number of nodes & 28 & 148 & 315 & 939 & 1945 & 4712 & 5473 & 6292 & 8622	& 11940 & 15524 & 16199\\
Number of edges & 0& 12 & 64 & 1252 & 4756 & 14851 & 18767 & 24538 & 44859 & 70683 & 121019 & 130969\\
Average degree & 0 & 0.081 & 0.203 & 1.333 & 2.445 & 3.152 & 3.429 & 3.900 & 5.203 & 5.920 & 7.796 & 8.085\\
Average path length & 0 & 1 & 1.046 & 2.605 & 3.592 & 8.061 & 7.514 & 8.301 & 6.164 & 5.554 & 5.051 & 4.927\\
Directed CC & 0 & 0 & 0.001 & 0.007 & 0.12 & 0.13 & 0.133 & 0.143 & 0.161 & 0.193 & 0.213 & 0.212\\
Small-world\footnote{The small-world sigma $\sigma$ is calculated by comparing clustering coefficient and average path length of each network to 50 equivalent random network with same average degree as suggested by \cite{Sigma}} $\sigma$  & NA & 0 & 0.447 & 1.165 & 15.587 & 75.173 & 82.519 & 80.122 & 24.239 & 16.131 & 195.837 & 208.084\\
 \hline
\end{tabular}}
\end{table}
As can be seen in \autoref{measures} the graphs show some small-world properties from 1860s with $\sigma>1$ and small-world property of the graphs is significant from 1970s comparing to 50 random graphs. As illustrated overlay the network gets denser and the average degree is growing. More significant clusters are observed during the most recent decades which can be seen in \autoref{graph}. These clusters could be the outcome of housekeeping activities such as edge and node removal, or it could be the result of mature referencing approach in legal drafting process. Both of the above hypotheses should be examined in future studies.
\begin{figure}
\begin{center}
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth]{1267-1869chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{1267-1869words.png}
\subcaption{1860s}\label{a2}
\end{minipage}%
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth]{1870-1909chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{1870-1909words.png}
\subcaption{1870-1909}\label{b2}
\end{minipage}%
\vspace{-\parskip}
\begin{minipage}{0.45\linewidth}
\includegraphics[width=.95\linewidth]{1910-1959chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{1910-1959words.png}
\subcaption{1910-1959}\label{c2}
\end{minipage}%
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth]{1960-1979chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{1960-1979words.png}
\subcaption{1960-1979}\label{d2}
\end{minipage}%
\vspace{-\parskip}
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth]{1980-2009chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{1980-2009words.png}
\subcaption{1980-2009}\label{e2}
\end{minipage}%
\begin{minipage}{0.45\linewidth}
\includegraphics[width=\linewidth]{2010-2018chart.pdf}
\centering\includegraphics[width=0.6\linewidth]{2010-2018words.png}
\subcaption{2010-2018}\label{f2}
\end{minipage}%
\end{center}
\caption{Time evolution of the top ten legislation and the top subjects in the top twenty legislation}\label{chart}
\end{figure}

Based on the network structure information provided in \autoref{measures} and \autoref{graph}, six different time periods are chosen for the centrality evolution analysis.
\autoref{chart} captures the time evolution of the top 10 nodes and the most frequent words \footnote{To find the frequent words, Textalyzer Python module is used. The frequent prepositions, conjunctions and articles are excluded from the analysis.} in the top 20 nodes based on Katz prestige centrality measure.

As mentioned earlier, prior to the 1860s the graphs don't show significant small-world properties. The visual presentation in \autoref{a1} to \autoref{c1} also reflects that the network can be considered as a random graph during this period. So the Katz centrality degree distribution is nearly a uniform distribution in these time periods and is excluded from \autoref{chart}.

\autoref{a2} shows the most important nodes with the impression that \textbf{Land} was the most important law subject back at that time.  In the next selected time period the network shows small-world properties, and as can be seen in \autoref{b2} the centrality measure shows a higher kurtosis with the word \textbf{Council} being the most frequent topic in legal domain.

Similarly the other graphs reflect the change in the network structure and highlights the relationship between the laws and the socio-economic requirements of the country. In the current decade, with the new sets of legislation being introduced and referenced to the older documents, the centrality measure is increased comparing to the previous decade, and the hot legal topics show a change which could be a good reflection of the society's needs.
\section{Evaluation and Robustness}
In this section the performance of the proposed framework is discussed. As explained in the previous section, the main goal of the study is to extract the information to build Legislation Network. The framework includes Named Entity Recognition, Relation Extraction, and Approximate String Matching jointly to extract the network's node and edge information. In this section the proposed framework is evaluated and the related errors are calculated. The familiar metrics of recall and precision measures are used to evaluate the system. High \textbf{precision} means that the framework returns substantially more relevant results than irrelevant ones, while high \textbf{recall} means that the process returns most of the relevant results. At the end of this section the impact of the identified errors on the network structure is explained, and the robustness is assessed.
\subsection{Error Estimation, Precision, and Recall}
In the proposed Information Extraction process, Named Entity Recognition is combined with Approximate String Matching to recognize, validate and optimize the entities (the nodes and the edges).
\autoref{Error} illustrates the occurrence of the false-positive and false-negative errors in this process and helps in studying the robustness of the network.

\begin{figure}[h]
\centering
\includegraphics[scale=1]{Error_diagram.jpg}
\caption{Error Diagram}
\label{Error}
\end{figure}
If the Entity Recognition process finds the entities, then there is a possibility that the Approximate String Matching process fails to find the correct match. A type \textit{I} error $\alpha_1$ occurs when the approximate string matching process fails to find the correct match. From one side this issue contributes to the false-positive error because it adds invalid entities to the output. These invalid entities impact the accuracy of the node list and the edge list of Legislation Network. To estimate the $\bar\alpha_1$ in the case study, a cluster sampling method is used to randomly choose ten sets of 30 entities. By manual check of the samples, the rate of incorrectly matched entities is observed. A Kolmogorov-Smirnov test suggests that the estimated error $\bar\alpha_1$ has a normal distribution with the parameters in \autoref{errorrates}.

$\beta_1$ also occurs when approximate string matching system picks a wrong match for the entities. This issue can contribute to the false-negative error because those entities that are wrongly matched to other entities are missing from the data set. The estimation methodology and the estimated value of $\bar\beta_1$ are equal to that of $\bar\alpha_1$ as indicated in \autoref{errorrates}.

$\beta_2$ is measuring the Information Extraction rules' performance. If it fails to recognize entities, then those entities are missed, and it results in another type of false-negative error. The estimation process for $\bar\beta_2$ is different from the previous two errors, and it is harder to address. For the case study a sample set of 30 text files are randomly chosen using cluster sampling method. Then all of the extracted entities for each document is compared to the actual entities in a human involving process. The list of missing entities is categorized into two parts: caused by a typo, or caused by insufficient rules to recognize the entity. The rate of missing entities caused by weak or missing rules is calculated for each document and denoted by $\bar\beta_2$. The Kolmogorov-Smirnov results through all of the 30 documents show that $\bar\beta_2$ has a normal distribution with the parameters in \autoref{errorrates}.\\~

\begin{table}[h]
\centering{}\protect\caption{Errors, sensitivity and specificity}
\label{errorrates}
\renewcommand{\arraystretch}{2}
\begin{tabular}{ l  l  l  l  l | l  l  }
\hline
Measure & $\bar\alpha_1$ & $\bar\beta_1$ & $\bar\beta_2$ & $\bar\beta_3$ & $\bar\alpha$ & $\bar\beta$\\
\hline
 $\mu$ & 0.0160 & 0.0160 & 0.0012 & 0.0007 & 0.0160 & 0.0179\\
 $\sigma$ & 0.0012  & 0.0012 & 0.0001 & 0.0001 &0.0012 & 0.0012\\
 \hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\end{table}
$\beta_3$ addresses the error when typos cause problem for recognizing the entities. The estimation process is very similar to that of $\bar\beta_2$. A sample of 30 text files are collected. Then the rate of missing entities caused by OCR typos is calculated for each document and addressed as $\bar\beta_3$. The Kolmogorov-Smirnov results through the selected documents show that $\bar\beta_3$ has a normal distribution with the parameters in \autoref{errorrates}. In the sample it is observed that the typos that cause the entity recognition failure are only numeric typos. For example OCR might produce an error and convert 1987 to l987 by misspelling number 1 to letter l. Then the Information Extraction rules are impacted to recognize l987 as a year, so the entity is missed.\\~

As \autoref{Error} shows $\alpha_1$ is the only false positive error which contributes to the \textbf{overall false positive error} of the system. \autoref{errorrates} captures $\bar\alpha$, assuming that $\bar\alpha$ estimates the overall type one error.
To estimate the \textbf{overall false negative error} of the system, $\beta_1$, $\beta_2$, and $\beta_3$ are considered as mutually exclusive events. From \autoref{Error} it is also clear that the intersections of each two of these errors are empty, so they are independent. \autoref{errorrates} shows $\bar\beta$, the estimated value for the overall false negative, or the type \textit{II} error.\\~

To calculate the Precision and Recall, \autoref{precision} and \autoref{recall} are used.
\small
\begin{flalign}\label{precision}
    Precision &= \frac{True Positive}{True Positive+False Positive}= \frac{1-\bar\alpha-\bar\beta}{1-\bar\beta}
\end{flalign}
\begin{flalign}\label{recall}
    Recall &= \frac{True Positive}{True Positive+False Negative}= \frac{1-\bar\alpha-\bar\beta}{1-\bar\alpha}
\end{flalign}
\normalsize

Referring to the above equations and  \autoref{errorrates}, the Precision of 98.37\% and Recall of 98.18\% are calculated. These outcomes suggest the high performance of the Proposed Information Extraction framework that results in a high data reliability of the output Legislation Network.

As explained in section \ref{Approximate String Matching}, the proposed hybrid Approximate String Matching technique substantially reduces the errors. It is important to mention that at the earlier stages of the study by using the classic string matching techniques, the error rates were considerably higher, and the accuracy of the network was questionable. A time consuming examination process engaging manual checks was applied to propose the hybrid model which resulted in impressive performance and high precision and recall. The improvement obviously involved a lot of efforts and time, but resulted in accuracy and confidence in Legislation Network studies.

\subsection{Robustness}
With a coherent understanding of the errors, it is very important to study the robustness of the network to the error. The robustness study proves the importance of the data accuracy which supports the value of the proposed hybrid model for the approximate string matching. In this section to study the network robustness, diameter and three major centrality measures are used.

To understand the diameter robustness of the network, attack and failure analysis is required. As discussed earlier Legislation Network in general show scale-free characteristics. So it is expected to observe a reasonable error tolerance of the network as the result of random failures, but vulnerability as the result of attacks\cite{robust2}. To study the network robustness to node failures the method is to randomly remove a fraction of nodes $f$ and recalculate the diameter of the network $d$. To study the network robustness to attack by removing a fraction $f$ of the largest nodes \footnote{Based on their connectivity (total degree)} and observe the change to the the diameter $d$. The results of both failure and attack to the nodes are captured in \autoref{Robust1}.
The observed tolerance to failures and the vulnerability to attacks shows that the connectivity is provided by a few highly connected nodes, and majority of nodes have only few edges.
\begin{figure}[h]
\centering
\includegraphics[scale=.4]{Robust.png}
\caption{Changes of the network diameter $d$ as the function of fraction $f$}\label{Robust1}
\end{figure}
As can be seen the vulnerability to the attacks starts immediately after removing a small fraction $f = 0.3\%$ of the highly connected nodes. This scenario of attack is highly unlikely in Legislation Network considering the high Precision and Recall of the proposed data extraction process.

As discussed in the previous studies \cite{NSakhaee2016} \cite{NSakhaee2017}, the most relevant centrality measure for Legislation Network is the Katz second prestige measure. In recent studies,  the reliability of different centrality measurements against network manipulation has been addressed \cite{CRobust2} \cite{CRobust}, but Katz prestige centrality is not much discussed. In this paper the Katz centrality, betweenness centrality, and degree centrality robustness of Legislation Network against edge deletion error is studied.
To address the robustness, four major measures of accuracy that proposed in \cite{CRobust2} and \cite{CRobust} are used. These measures are Top 1, Top 3, Top 10 percent, and the Pearson correlation to compare the centrality measures between the true network and the manipulated network.

\begin{figure}[h]
\begin{minipage}{0.35\linewidth}
\includegraphics[width=\linewidth]{KatzCentrailtyRobust.png}
\subcaption{Katz Prestige Centrality}
\end{minipage}%
\begin{minipage}{0.35\linewidth}
\includegraphics[width=\linewidth]{BetweennessRobust.png}
\subcaption{Betweenness Centrality}
\end{minipage}%
\begin{minipage}{0.35\linewidth}
\includegraphics[width=\linewidth]{DegreeRobust.png}
\subcaption{In-degree Centrality}
\end{minipage}%
    \caption{Robustness of the top nodes as the function of fraction of manipulated edges \textit{f}}
    \label{centralityrobust}
\end{figure}
The error level is considered as a specific percentage value from the set of {1\%, 5\%, 10\%, 20\%}, that is relative to the number of manipulated edges from the original true network. \autoref{centralityrobust} shows the results of the different Centrality measure as the function of the fraction of manipulated edges $f$. For each fraction level, the test is repeated for 100 times, and the graphs show the average of the all sampled sets. \autoref{pearson} shows the Pearson correlation between the nodes centrality in manipulated network and the original network when 10\% of the edges are randomly deleted.

\begin{table}[h]
\centering{}\protect\caption{Node centrality Pearson correlation between the manipulated network and the original network}
\label{pearson}
\adjustbox{max height=\dimexpr\textheight-6.5cm\relax,
           max width=\textwidth}{
\begin{tabular}{ l  l  l  l }
\hline
Measure & Katz prestige centrality & Betweenness centrality & Degree centrality\\
 \hline
 Significance (p-value) & $2.2e^-16$ & $0.001$ & $0.003$ \\
 Correlation & $0.939$  & $0.948$ & $0.67$ \\
 \hline
\end{tabular}
}
\end{table}

The pattern and level of robustness of the three selected centrality measures considered in this paper are are not as similar as suggested in \cite{CRobust2}. In-degree centrality shows more fragility comparing to betweenness and Katz measures. This difference could be related to the network topology as suggested by \cite{CRobust}. The results also confirm the findings of \cite{CRobust} \cite{CRobust2} that accuracy declines monotonically with increasing error.

As can be seen in the graph, the Katz centrality is fairly robust to the edge deletion when less than 20 percent of the network structure is touched. The graphs indicate a moderate fragility when the network structure is hugely manipulated. For example the removal of 20 percent of the edges somehow impacts the in-degree centrality. However in more than 90 percent of these extreme samples, the top 1 node in the manipulated network is a member of top ten percent of nodes in the original graph. The results imply that centrality measures on Legislation Network are quite robust under small amounts of error (such as 5 percent or under) and to some extent fragile under bigger data errors. So the reliability of the network information is very important for in-depth network studies. As explained earlier the the precision and the recall of the proposed Information Extraction process is above 98 percent, so it is reasonable to compute the centrality measures when studying the Legislation Network.

\section{Conclusion}
This study focused on the \textbf{time} as a very important attribute in understanding and analyzing legislation. Legislation Network has been discussed in recent years, but the importance of having access to the historic legislation was never discussed much. This paper underlined the value of studying legislation as dynamic networks, and proposed a new Information Extraction process to achieve a highly accurate Legislation Network. The performance of the data extraction framework is examined, is compared to the previous studies and proved to be considerably high. This work contributed to the literature of network Information Extraction from old documents, and insisted on the value and applications of the dynamic Legislation Network. The proposed process can be used not only in the legal domain but also in various research areas involving documented knowledge, facts, and cases.

Analyzing a dynamic Legislation Network is a novel approach to understand the underlying process behind the generation of the laws, and to study the behaviour, culture and growth of societies. This subject is very interesting, but mathematically complicated. So it will be discussed in a separate study.