document: the study of complex networks has notably increased in the last years with applications to a variety of fields ranging from computer science@xcite and biology to social science@xcite and finance@xcite . 
   a central problem in network science @xcite is the study of the random walks ( rw ) on a graph , and in particular of the relation between the topological properties of the network and the properties of diffusion on it . 
   this subject is not only interesting from a purely theoretical perspective , but it has important implications to various scientific issues ranging from epidemics @xcite to the classification of web pages through pagerank algorithm @xcite . 
   finally , rw theory is also used in algorithms for community detection  @xcite .    in this paper 
   we set up a new framework for the study of topologically biased random walks on graphs . 
   this allows to address problems of community detection and synchronization  @xcite in the field of complex networks @xcite . 
   in particular by using topological properties of the network to bias the rws we explore the network structure more efficiently . a similar approach but with different focus can be found in @xcite . in this research 
   we are motivated by the idea that biased random walks can be efficiently used for community finding . 
   to this aim we introduce a set of mathematical tools which allow us an efficient investigation of the `` bias parameters '' space . 
   we apply this tools to uncover some details in the spectra of graph transition matrix , and use the relation between spectra and communities in order to introduce a novel methodology for an efficient community finding . 
   the paper is organized as follows : in the first section we define the topologically biased random walks ( tbrw ) . 
   we then develop the mathematical formalism used in this paper , specifically the perturbation methods and the parametric equations of motion , to track the behaviour of different biases . in the second section we focus on the behavior of spectral gap in biased random walks . 
   we define the conditions for which such a spectral gap is maximal and we present numerical evidence that this maximum is global . in the third section we present an invariant quantity for the biased random walk ; 
   such constant quantity depends only upon topology for a broad class of biased random walks . 
   finally , in the fourth section we present a general methodology for the application of different tbrw in the community finding problems . 
   we then conclude by providing a short discussion of the material presented and by providing an outlook on different possible applications of tbrw . 
   rws on graphs are a sub - class of markov chains  @xcite . the traditional approach deals with the connection of the _ unbiased _ rw properties to the spectral features of _ transition operators _ associated to the network @xcite . 
   a generic graph can be represented by means of the adjacency matrix @xmath0 whose entries @xmath1 are @xmath2 if an edge connects vertices @xmath3 and @xmath4 and @xmath5 otherwise . 
   here we consider undirected graphs so that @xmath0 is symmetric . the _ normal matrix _ 
   @xmath6 is related to @xmath0 through @xmath7 , where @xmath8 is a diagonal matrix with @xmath9 , i.e. the degree , or number of edges , of vertex @xmath3 . in the following we use uppercase letters for non - diagonal matrices and lowercase letters for the diagonal ones . 
   note that by definition @xmath10 . 
   consequently @xmath11 with @xmath12 _ iif _ @xmath13 , i.e. if @xmath3 and @xmath4 are nearest neighbors vertices . 
   the matrix @xmath14 defines the transition probabilities for an _ unbiased _ random walker to pass from @xmath4 to @xmath3 . 
   in such a case @xmath15 has the same positive value for any of the neighbors @xmath3 of @xmath4 and vanishes for all the other vertices@xcite . in analogy to the operator defining the single step transition probabilities in general markov chains , 
   @xmath6 is also called the transition _ matrix _ of the unbiased rw . 
   a _ biased _ rw on a graph can be defined by a more general transition matrix @xmath16 where the element @xmath15 gives again the probability that a walker on the vertex @xmath4 of the graph will move to the vertex @xmath3 in a single step , but depending on appropriate weights for each pair of vertex @xmath17 . a genuine way to write these probabilities 
   is to assign weights @xmath18 which represent the rates of jumps from vertex @xmath4 to vertex @xmath3 and normalize them : [ probabpasage ] t_ij=.    in this paper we consider biases which are self - consistently related to graph topological properties . 
   for instance @xmath18 can be a function of the vertex properties ( the network degree , clustering , etc . ) or some functions of the edge ones ( multiplicity or shortest path betweenness ) or any combination of the two . 
   there are other choices of biases found in the literature such as for instance maximal entropy related biases  @xcite . 
   some of the results mentioned in this paper hold also for biases which are not connected to graph properties as will be mentioned in any such case . 
   our focus on graph properties for biases is directly connected with application of biased random walks in examination of community structure in complex networks . 
   let us start by considering a vertex property @xmath19 of the vertex @xmath3 ( it can be either local as for example the degree , or related to the first neighbors of @xmath3 as the clustering coefficient , or global as the vertex betweenness ) . 
   we choose the following form for the weights : [ probbias ] w_ij = a_ije^x_i , where the parameter @xmath20 tunes the strength of the bias . 
   for @xmath21 the unbiased case is recovered . by varying @xmath22 
   the probability of a walker to move from vertex @xmath4 to vertex @xmath3 will be enhanced or reduced with respect to the unbiased case according to the property @xmath19 of the vertex @xmath3 . 
   for instance when @xmath23 , i.e. the degree of the vertex @xmath3 , for positive values of the parameter @xmath22 the walker will spend more time on vertices with high degree , i.e. it will be attracted by hubs . 
   for @xmath24 it will instead try to `` avoid '' traffic congestion by spending its time on the vertices with small degree . 
   the entries of the transition matrix can now be written as : [ transitionentries ] t_ij(,)=. for this choice of bias we find the following results : ( i ) we have a unique representation of any given network via operator @xmath25 , i.e. knowing the operator , we can reconstruct the graph ; ( ii ) for small @xmath26 we can use perturbation methods around the unbasied case ; ( iii ) this choice of bias permits in general also to visit vertices with vanishing feature @xmath27 , which instead is forbidden for instance for a power law @xmath28 ; ( iv ) this choice of biases is very common in the studies of energy landscapes , when biases represent energies @xmath29 ( see for example  @xcite and references therein ) .    in a similar way one can consider a _ 
   edge property @xmath30 ( for instance edge multiplicity or shortest path betweenness ) as bias . in this case 
   we can write the transition probability as : [ transitionentriesedge ] t_ij(,)=. the general case of some complicated multiparameter bias strategy can be finally written as [ transitionentriesmulti ] t_ij(,,)= .    while we mostly consider biased rw based on vertex properties , as shown below , most of the results can be extended to the other cases . 
   the transition matrix in the former case can also be written as : @xmath31 where the diagonal matrices @xmath32 and @xmath33 are such that @xmath34 and @xmath35 . 
   the frobenius - perron theorem implies that the largest eigenvalue of @xmath36 is always @xmath37  @xcite . 
   furthermore , the eigenvector @xmath38 associated to @xmath39 is strictly positive in a connected aperiodic graph . 
   its normalized version , denoted as @xmath40 , gives the asymptotic stationary distribution of the biased rw on the graph . 
   assuming for it the form @xmath41 , where @xmath42 and @xmath43 is a normalization constant , and plugging this in the equation @xmath44 we get : @xmath45 hence the equation holds _ iif _ @xmath46 . 
   therefore the stable asymptotic distribution of vertex centerd biased rws is [ explidistro ] p_i()=()^-1 e^x_iz_i ( ) . 
   for @xmath21 we have the usual form of the stationary distribution in an unbiased rw where @xmath47 and @xmath48 . for general @xmath22 it can be easily demonstrated that the asymptotic solution of edge biased rw is @xmath49 , while for multiparametric rw the solution is @xmath50 .    using eqs . 
   ( [ explidistro ] ) and ( [ transitionentries ] ) we can prove that the detailed balance condition @xmath51 holds . 
   at this point it is convenient to introduce a different approach to the problem @xcite . 
   we start by symmetrizing the matrix @xmath25 in the following way : @xmath52^{-1/2}\bsy{\hat{t}}(\bsy{x } , \beta ) [ \bsy{\hat{p}}(\beta)]^{1/2}\,,\ ] ] where @xmath53 is the diagonal matrix with the stationary distribution @xmath54 on the diagonal . 
   the entries of the symmetric matrix for vertex centered case are given by @xmath55 the symmetric matrix @xmath56 shares the same eigenvalues with the matrix @xmath25 ; anyhow the set of eigenvectors is different and forms a complete orthogonal basis , allowing to define a meaningful  distance  between vertices 
   . such distance can provide important additional information in the problem of community partition of complex networks . 
   if @xmath57 is the @xmath58 eigenvector of the asymmetric matrix @xmath25 associated to the eigenvalue @xmath59 ( therefore @xmath60 ) , the corresponding eigenvector @xmath61 of the symmetric matrix @xmath56 , can always be written as @xmath62 . 
   in particular for @xmath63 we have @xmath64 . 
   the same transformation ( [ symmat ] ) can be applied to the most general multiparametric rw . in that case 
   the symmetric operator is @xmath65    this form also enables usage of perturbation theory for hermitian linear operators . 
   for instance , knowing the eigenvalue @xmath66 associated to eigenvector @xmath67 , we can write the following expansions at sufficiently small @xmath68 : @xmath69 and @xmath70 . 
   it follows that for a vertex centered bias [ firstorderlambda ] _ 
   ^(1)()=^s(1 ) ( , ) , where , @xmath71\ ] ] with @xmath72 being the anticommutator operator . 
   operator @xmath73 and @xmath74 are diagonal matrices with @xmath75 and @xmath76 which is the expected value of @xmath27 that an random walker , will find moving from vertex @xmath3 to its neighbors . in the case of edge bias 
   the change of symmetric matrix with parameter @xmath22 can be written as @xmath77 , and @xmath78 represents the schur - hadamard product i.e. element wise multiplication of matrix elements . 
   the eigenvector components in @xmath79 at the first order of expansion in the basis of the eigenvectors at @xmath22 are given by ( for @xmath80 ) : [ firstordvecs ] = . 
   for @xmath81 the product @xmath82 vanishes and eqs . 
   ( [ derivation ] ) and ( [ firstordvecs ] ) hold only for non - degenerate cases . in general , 
   usual quantum mechanical perturbation theory can be used to go to higher order perturbations or to take into account degeneracy of eigenvalues . 
   we can also exploit further the formal analogy with quantum mechanics using parametric equations of motion ( pem )  @xcite to study the @xmath22 dependence of the spectrum of @xmath83 . 
   if we know such spectrum for one value of @xmath22 , we can calculate it for any other value of @xmath22 by solving a set of differential equations corresponding to pem in quantum mechanics . 
   they are nothing else the expressions of eqs . 
   ( [ firstorderlambda ] ) and ( [ firstordvecs ] ) in an arbitrary complete orthonormal base @xmath84 . 
   first the eigenvector is expanded in such a base : @xmath85 . 
   we can then write @xmath86 where @xmath87 ( @xmath88 ) is a column ( row ) vector with entries @xmath89 and @xmath90 is the matrix with entries @xmath91 . 
   let us now define the matrix @xmath92 whose rows are the copies of vector @xmath93 . 
   the differential equation for the eigenvectors in the basis @xmath94 is then  @xcite @xmath95    a practical way to integrate eqs . 
   ( [ pemlambda ] ) and ( [ pemvector ] ) can be found in  @xcite . in order to calculate parameter dependence of eigenvectors and eigenvalues , 
   the best way to proceed is to perform an lu decomposition of the matrix @xmath96 as the product of a lower triangular matrix @xmath97 and an upper triangular matrix @xmath98 , and integrate differential equations of higher order which can be constructed in the same way as equations ( [ pemlambda ] ) and ( [ pemvector ] )  @xcite . 
   a suitable choice for the basis is just the ordinary unit vectors spanned by vertices , i.e. @xmath99 . 
   we found that for practical purposes , depending on the studied network , it is appropriate to use pem until the error increases to much and then diagonalize matrix again to get better precision . 
   pem efficiently enables study of the large set of parameters for large networks due to its compatitive advantage over ordinary diagonalization . 
   vs @xmath22 for networks of 10 communities with 10 vertices each ( the probability for an edge to be in a community is @xmath100 while outside of the community it is @xmath101 ) . 
   solid points represent the solutions computed via diagonalization , while lines report the value obtained through integration of pem . 
   different bias choice have been tested . 
   circles ( blue ) are related to degree - based strategy , square ( red ) are related to clustering - based strategies , diamonds ( green ) multiplicity - based strategies . the physical quantities to get the variable @xmath27 in eq . 
   ( [ probbias ] ) in these strategies have been normalized with respect to their maximum value.,width=302,height=245 ] 
   a key variable in the spectral theory of graphs is the _ spectral gap _ 
   @xmath102 , i.e. the difference between first unitary and the second eigenvalues . the spectral gap measures 
   how fast the information on the rw initial distribution is destroyed and the stationary distribution is approached . 
   the characteristic time for that is @xmath103\simeq 1/\mu$ ]  @xcite . 
   we show in fig .  1 the dependence of spectral gap of simulated graphs with communities for different strategies ( degree , clustering and multiplicity based ) at a given value of parameter @xmath22 . in all investigated cases 
   the spectral gap has its well defined maximum , i.e. the value of parameter @xmath22 for which the random walker converges to stationary distribution with the largest rate . 
   the condition of maximal spectral gap implies that it is a stationary point for the function @xmath104 , i.e. that its first order perturbation coefficient vanishes at this point : @xmath105\ket{v_2(\beta_m)},\end{aligned}\ ] ] where @xmath73 and @xmath106 are defined above . 
   the squares of entries @xmath107 of the vector @xmath108 in the chosen basis @xmath109 , define a particular measure on the graph . 
   equation ( [ maximalspectralgap ] ) can be written as    @xmath110 . 
   thus we conclude that the local spectral maximum is achieved if the average difference between property @xmath19 and its expectation @xmath111 , with respect to this measure , in the neighborhood of vertex @xmath3 vanishes . 
   we have studied behavior of spectral gap for different sets of real and simulated networks ( barabsi - albert model with different range of parameters , erds - rnyi model and random netwroks with given community structure ) and three different strategies ( degree - based , clustering - based and multiplicity - based ) . 
   although in general it is not clear that the local maximum of spectral gap is unique , we have found only one maximum in all the studied networks . 
   this observation is interesting because for all cases the shapes of spectral gap _ vs. _ @xmath22 looks typically gaussian - like . in both limits 
   @xmath112 the spectral gap of heterogeneous network is indeed typically zero , as the rw stays in the vicinity of the vertices with maximal or minimal value of studied property @xmath19 . 
   a fundamental question in the theory of complex networks is how topology affects dynamics on networks . 
   our choice of @xmath22-parametrized biases provides a useful tool to investigate this relationship . 
   a central issue is , for instance , given by the search of properties of the transition matrix @xmath113 which are independent of @xmath22 and the chosen bias , but depend only on the topology of the network . 
   an important example comes from the analysis the determinant of @xmath113 as a function of the bias parameters : @xmath114 for vertex centered bias using eq . 
   ( [ derivation ] ) we have @xmath115 and using the diagonality of the @xmath73 and @xmath116 @xmath117 in other words the quantity @xmath118 is a topological constant which does not depend on the choice of parameters . 
   for @xmath21 we get @xmath119 and it follows that this quantity does not depend on the choice of vertex biases @xmath19 either . 
   it can be shown that such quantity coincides with the determinant of adjacency matrix which must be conserved for all processes . 
   .,width=302,height=245 ]    there are many competing algorithms and methods for community detection  @xcite . despite a significant scientific effort to find such reliable algorithms , 
   there is not yet agreement on a single general solving algorithm for the various cases . in this section instead of adding another precise recipe , we want to suggest a general methodology based on tbrw which could be used for community detection algorithms . to add trouble , 
   the very definition of communities is not a solid one . in most of the cases we define communities as connected subgraphs 
   whose density of edges is larger within the proposed community than outside it ( a concept quantified by modularity @xcite ) . 
   scientific community is therefore thriving to find a benchmark in order to assess the success of various methods . 
   one approach is to create synthetic graphs with assigned community structure ( benchmark algorithms ) and test through them the community detection recipes @xcite . 
   the girvan - newman ( gn ) @xcite and lancichinetti - fortunato - radicchi ( lfr ) @xcite are the most common benchmark algorithms . in both these models several topological properties ( not only edge density ) 
   are unevenly distributed within the same community and between different ones . 
   we use this property to propose a novel methodology creating suitable tbrw for community detection . 
   the difference between internal and external part of a community is related to the `` physical '' meaning of the graph . in many real processes 
   the establishment of a community is facilitated by the subgraph structure . 
   for instance in social networks agents have a higher probability of communication when they share a lot of friends . 
   we test our approach on gn benchmark since in this case we can easily compute the expected differences between the frequency of biased variables within and outside the community .    in this section 
   we will describe how to use tbrw for community detection . for @xmath21 
   our method is rather similar to the one introduced by donetti and muoz @xcite . 
   the most notable difference is that we consider the spectral properties of transition matrix instead of the laplacian one . 
   we decide if a vertex belongs to a community according to the following ideas : _ 
   ( i ) _ we expect that the vertices belonging to the same community to have similar values of eigenvectors components ; _ ( ii ) _ we expect relevant eigenvectors to have the largest eigenvalues . 
   indeed , spectral gap is associated with temporal convergence of random walker fluctuations to the ergodic stationary state . 
   if the network has well defined communities , we expect the random walker to spend some time in the community rather than escaping immediately out of it . 
   therefore the speed of convergence to the ergodic state should be related to the community structure . 
   therefore eigenvectors associated with largest eigenvalues ( except for the maximal eigenvalue 1 ) should be correlated with community structure . 
   coming back to the above mentioned donetti and muoz approach here we use the fact that some vertex properties will be more common inside a community and less frequent between different communities . 
   we then vary the bias parameters trying both to shrink the spectral gap in transition matrix and to maximize the separation between relevant eigenvalues and the rest of the spectra .    , 
   @xmath120 , @xmath121 , @xmath122 . 
   there is a clear gap between `` community '' band and the rest of the eigenvalues . , width=302,height=245 ]     as a function of parameter @xmath123 which biases rw according to degrees of the vertices and parameter @xmath124 which bias rw according to multiplicities of the edges . 
   both degrees and multiplicity values are normalized with respect to the maximal degree and multiplicity ( therefore the largest value is one).,width=302,height=245 ]    for example in the case of gn benchmark the network consist of @xmath125 communities each with @xmath120 vertices i.e. @xmath126 vertices all together . 
   the probability that the two vertices which belong to the same community are connected is @xmath127 . 
   the probability that the two vertices which belong to different communities are connected is @xmath128 . 
   the fundamental parameter @xcite which characterizes the difficulty of detecting the structure is [ mu ] = , where @xmath129 is the mean degree related to inter - community connections and @xmath130 is the mean degree related to edges inside - community . as a rule of thumb we can expect to find well defined communities when @xmath131 , and observe some signature of communities even when @xmath132  @xcite . 
   the probabilities @xmath127 and @xmath128 are related via the control parameter @xmath133 as @xmath134 . 
   we now examine the edge multiplicity . 
   the latter is defined as the number of common neighbors shared by neighbouring vertices . 
   the expected multiplicity of an edge connecting vertices inter - community and inside - communities are respectively @xmath135 on fig . 
   [ params ] we plot the ratio of the quantitites above defined , @xmath136 , _ vs. _ the parameter @xmath133 . 
   we see that even for @xmath137 the ratio remains smaller than @xmath2 implying that the multiplicity is more common in the edges in the same community . 
   based on this analysis for this particular example we expect that if we want to find well - defined communities via tbrw we have to increase bias with respect to the multiplicity . through numerical simulations 
   we find that the number of communities is related to number of eigenvalues in the `` community band '' . 
   namely one in general observes a gap between eigenvalues @xmath138 and the next eigenvalue evident in a network with a strong community structure ( @xmath139 ) . 
   the explanation that we give for that phenomenon can be expressed by considering a network of @xmath140 separated graphs . 
   for such a network there are @xmath140 degenerate eignevalues @xmath141 . 
   if we now start to connect these graphs with very few edges such a degeneracy is broken with the largest eigenvalue remaining @xmath2 while the next @xmath142 eigenvalues staying close to it . 
   the distance between any two of this set of @xmath142 eigenvalues will be smaller than the gap between this community band and the rest of the eigenvalues in the spectrum . 
   therefore , the number of eigenvalues different of @xmath2 which are forming this `` community band '' is always equal to the number of communities minus one , at least for different gn - type networks with different number of communities and different sizes , as long as @xmath139 . 
   for example in the case of 1000 gn networks described with parameters @xmath126 , @xmath120 , @xmath121 , @xmath122 , i.e. @xmath143 , the histograms of eigenvalues are depicted on figure [ band ] .    for our purposes we used two parameters biased rw , 
   in which topological properties are @xmath144 i.e. the normalized degree ( with respect to maximal degree in the network ) and @xmath145 i.e. the normalized multiplicity ( with respect to maximal multiplicity in the network ) . 
   we choosed gn network whose parameters are @xmath126 , @xmath120 , @xmath146 and @xmath147 , for which @xmath148 . being @xmath149 the number of communities , as a criterion for good choice of parameters we decided to use the difference between @xmath150 and @xmath151 , i.e. , we decided to maximize the gap between `` community band '' and the rest of eigenvalues ; checking at the same time that the spectral gap shrinks . in fig . 
   [ lambda3lambda4 ] , we plot such a quantity with respect to different biases . 
   it is important to mention that for every single network instance there are different optimal parameters . 
   this can be seen on figure [ lambdahistograms ] , where we show the difference between unbiased and biased eigenvalues for 1000 gn nets created with same parameters . as shown in the figure 
   the difference between fourth and fifth eigenvalue is now not necessarily the optimal for this choice of parameters . 
   every realization of the network should be independently analyzed and its own parameters should be carefully chosen .    , @xmath152 , @xmath150 and @xmath151 for 1000 gn networks described with parameters @xmath126 , @xmath120 , @xmath146 and @xmath147 . 
   with black colour we indicate the eigenvalues of nonbiased rw , while with red we indicate the eigenvalues of rw biased with parameters @xmath153 and @xmath154 . 
   note how this choice of parameters does not maximize `` community gap '' for all the different realizations of monitored gn network.,width=302,height=245 ]    in the figs . 
   [ unbiased ] and [ biased ] we present instead the difference between unbiased and biased projection on three eigenvectors with largest nontrivial eigenvalues . 
   using 3d view it is easy to check that communities are better separated in the biased case then in the non - biased case .     and @xmath147 . 
   for this choice of parameters @xmath148 . 
   there is a strong dispersion between different vertices which belong to the same community.,width=302,height=245 ]     and @xmath154 . 
   different markers represent four different predefined communities . 
   this is an example of the same gn graph realization with @xmath146 and @xmath147 as the one on the previous figure . for this choice of parameters @xmath148 . 
   one can notice tetrahedral distribution of vertices in which vertices from the same community belong to the same branch of tetrahedron.,width=302,height=245 ] 
   in this paper we presented a detailed theoretical framework to analyze the evolution of tbrw on a graph . by using as bias 
   some topological property of the graph itself allows to use the rw as a tool to explore the environment . 
   this method maps vertices of the graph to different points in the @xmath155-dimensional euclidean space naturally associated with the given graph . in this way we can measure distances between vertices depending on the chosen bias strategy and bias parameters . 
   in particular we developed a perturbative approach to the spectrum of eigenvalues and eigenvectors associated with the transition matrix of the system . 
   more generally we generalized the quantum pem approach to the present case . 
   this led naturally to study the behavior of the gap between the largest and the second eigenvalue of the spectrum characterizing the relaxation to the stationary markov state . in numerical applications of such a theoretical framework 
   we have observed a unimodal shape of the spectral gap _ vs. _ the bias parameter which is not an obvious feature of the studied processes . 
   we have finally outlined a very promising application of topologically biased random walks to the fundamental problem of community finding . 
   we described the basic ideas and proposed some criteria for the choice of parameters , by considering the particular case of gn graphs . 
   we are working further in direction of this application , but the number of possible strategies ( different topological properties we can use for biasing ) and types of networks is just too large to be presented in one paper . 
   furthermore , since in many dynamical systems as the www or biological networks , a feedback between function and form ( topology ) is evident , our framework may be a useful way to describe mathematically such an observed mechanism . in the case of biology , for instance 
   , the shape of the metabolic networks can be triggered not only by the chemical properties of the compounds , but also by the possibility of the metabolites to interact . 
   biased rw can be therefore the mechanism through which a network attains a particular form for a given function . by introducing such approach 
   we can now address the problem of community detection in the graph . 
   this the reason why here we have not introduced another precise method for community detection , but rather a possible framework to create different community finding methods with different _ ad hoc _ strategies . indeed in real situations we expect different types of network to be efficiently explored by use of different topological properties . 
   this explains why we believe that tbrw could play a role in community detection problems , and we hope to stimulate further developments , in the network scientific community , of this promising methodology . 
   * acknowledgments * vinko zlati wants to thanks mses of the republic of croatia through project no . 098 - 0352828 - 2836 for partial support . 
   authors acknowledge support from ec fet open project `` foc '' nr 
   . 255987 . 
   l. lovasz , _ combinatorics _ * 2 * , 1 - 48 , ( 1993 ) d. aldous , j. fill , _ reversible markov chains and random walks on graphs _ , in press , r. pastor - satorras , a. vespignani _ phys . 
   _ * 86 * , 3200 , ( 2001 ) .
summary: we present a new approach of topology biased random walks for undirected networks . 
   we focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks . 
   this analogy is extended through the use of parametric equations of motion ( pem ) to study the features of random walks _ vs. _ parameter values . 
   furthermore , we show an analysis of the spectral gap maximum associated to the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state . applications of these studies allow _ ad hoc _ algorithms for the exploration of complex networks and their communities .