maximum likelihood extracting unbiased information from complex networks the choice of free parameters in network models is subjective since it depends on what topological properties are being monitored however we show that the maximum likelihood ml principle indicates a unique statistically rigorous parameter choice associated with a well defined topological feature we then find that if the ml condition is incompatible with the built in parameter choice network models turn out to be intrinsically ill defined or biased to overcome this problem we construct a class of safely unbiased models we also propose an extension of these results that leads to the fascinating possibility to extract only from topological data the hidden variables underlying network organization making them no longer hidden we test our method on world trade web data where we recover the empirical gross domestic product using only topological information

degree dispersion increases the rate of rare events in population networks there is great interest in predicting rare and extreme events in complex systems and in particular understanding the role of network topology in facilitating such events in this letter we show that degree dispersion the fact that the number of local connections in networks varies broadly increases the probability of large rare fluctuations in population networks generically we perform explicit calculations for two canonical and distinct classes of rare events network extinction and switching when the distance to threshold is held constant and hence stochastic effects are fairly compared among networks we show that there is a universal exponential increase in the rate of rare events proportional to the variance of a network s degree distribution over its mean squared

research problems from the aveiro workshop on graph spectra abstract this is a collection of open problems presented at the aveiro workshop on graph spectra held at the university of aveiro portugal from april 10 12 2006

fast computation of von neumann entropy for large scale graphs via quadratic approximations abstract the von neumann graph entropy vnge can be used as a measure of graph complexity which can be the measure of information divergence and distance between graphs however computing vnge is extensively demanding for a large scale graph we propose novel quadratic approximations for fast computing vnge various inequalities for error between the quadratic approximations and the exact vnge are found our methods reduce the cubic complexity of vnge to linear complexity computational simulations on random graph models and various real network datasets demonstrate superior performance

selecting graph cut solutions via global graph similarity graph cut is a common way of clustering nodes on similarity graphs as a clustering method it does not give a unique solution under usually used loss functions we specifically show the problem in similarity graph based clustering setting that the resulting clusters might be even disconnected this is counter intuitive as one wish to have good clustering solutions in the sense that each cluster is well connected and the clusters are balanced the key property of good clustering solutions is that the resulting graphs after clustering share large components with the original ones we wish to detect this case by deriving a graph similarity measure that shows high similarity values to the original graph for good clustering solutions the similarity measure considers global connectivities of graphs by treating graphs as distributions in potentially different euclidean spaces the global graph comparison is then turned into distribution comparison simulation shows that the similarity measure could consistently distinguish different qualities of clustering solution beyond what could be done with the usually used loss functions of clustering algorithms

spectral distances of graphs abstract let 1 g 2 g n g be the adjacency spectrum of a graph g on n vertices the spectral distance g 1 g 2 between n vertex graphs g 1 and g 2 is defined by g 1 g 2 i 1 n i g 1 i g 2 here we provide some initial results regarding this quantity first we give some general results concerning the spectral distances between arbitrary graphs and compute these distances in some particular cases certain relation with the theory of graph energy is identified the spectral distances bounded by a given constant are also considered next we introduce the cospectrality measure and the spectral diameter and obtain specific results indicating their relevance for the theory of cospectral graphs finally we give and discuss some computational results and conclude the paper by a list of conjectures

inverse mermin wagner theorem for classical spin models on graphs in this paper we present the inversion of the mermin wagner theorem on graphs by proving the existence of spontaneous magnetization at finite temperature for classical spin models on transient on the average graphs i e graphs where a random walker returns to its starting point with an average probability overline f l1 this result which is here proven for models with o n symmetry includes as a particular case n 1 providing a very general condition for spontaneous symmetry breaking on inhomogeneous structures even for the ising model

entropies of complex networks with hierarchically constrained topologies the entropy of a hierarchical network topology in an ensemble of sparse random networks with hidden variables associated with its nodes is the log likelihood that a given network topology is present in the chosen ensemble we obtain a general formula for this entropy which has a clear interpretation in some simple limiting cases the results provide keys with which to solve the general problem of fitting a given network with an appropriate ensemble of random networks

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

construction of minimal n 2 n encoders for any n the encoding problem rumelhart and mcclelland 1986 is an important canonical problem it has been widely used as a benchmark here we have analytically derived minimal sized nets necessary and sufficient to solve encoding problems of arbitrary size the proofs are constructive we construct n 2 n encoders and show that two hidden units are also necessary for n 2 moreover the geometric approach employed is general and has much wider applications for example this method has also helped us derive lower bounds on the redundancy necessary for achieving complete fault tolerance phatak and koren 1992a b

distance between the normalized laplacian spectra of two graphs abstract let g v e be a simple graph of order n the normalized laplacian eigenvalues of graph g are denoted by 1 g 2 g n 1 g n g 0 also let g and g be two nonisomorphic graphs on n vertices define the distance between the normalized laplacian spectra of g and g as n g g i 1 n i g i g p p 1 define the cospectrality of g by c s n g min n g g g not isomorphic to g let c s n n max c s n g g a graph on n vertices in this paper we give an upper bound on c s n g in terms of the graph parameters moreover we obtain an exact value of c s n n an upper bound on the distance between the normalized laplacian spectra of two graphs has been presented in terms of randic energy as an application we determine the class of graphs which are lying closer to the complete bipartite graph than to the complete graph regarding the distance of normalized laplacian spectra

a new dissimilarity measure for comparing labeled graphs abstract we use spectral graph theory to compare graphs that share the same node set taking into account global graph structures we propose a general framework using eigendecomposition of graph laplacians we show its special cases and propose a new dissimilarity measure that avoid problems of spectral analysis the new dissimilarity emphasizes the importance of small eigenvalues which are known to carry the main information on graphs general properties of the dissimilarity are discussed the dissimilarity provides an efficient and intuitive tool for graph analysis

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

distance between spectra of graphs richard brualdi proposed in stevanivic 2007 10 the following problem problem awgs 4 let gngn and gn be two nonisomorphic graphs on n vertices with spectra 1 2 nand 1 2 n respectively define the distance between the spectra of gngn and gn as gn gn i 1n i i 2 or use i 1n i i define the cospectrality of gngn by cs gn min gn gn gn not isomorphic to gn let csn max cs gn gn a graph on n vertices problem a investigate cs gn cs gn for special classes of graphs problem b find a good upper bound on csncsn in this paper we completely answer problem b by proving that csn 2csn 2 for all n 2n 2 whenever csncsn is computed with respect to any lplp norm with 1 p 1 p and csn 1csn 1 with respect to the l l norm the cospectrality cs km n cs km n of the complete bipartite graph km nkm n for all positive integers m and n with m n m 1 2m 1 is computed as a consequence of the latter result it is proved that in general there is no constant upper bound on e e e e where e and e e denote the number of edges of the graphs g and g g respectively such that cs g g g cs g g g in contrast we show that for such graphs g and g g we have e e 1 in particular e e 3e e e 3e

solution of the two star model of a network the p star model or exponential random graph is among the oldest and best known of network models here we give an analytic solution for the particular case of the two star model which is one of the most fundamental of exponential random graphs we derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition

nonlinear autoassociation is not equivalent to pca a common misperception within the neural network community is that even with nonlinearities in their hidden layer autoassociators trained with backpropagation are equivalent to linear methods such as principal component analysis pca our purpose is to demonstrate that nonlinear autoassociators actually behave differently from linear methods and that they can outperform these methods when used for latent extraction projection and classification while linear autoassociators emulate pca and thus exhibit a flat or unimodal reconstruction error surface autoassociators with nonlinearities in their hidden layer learn domains by building error reconstruction surfaces that depending on the task contain multiple local valleys this interpolation bias allows nonlinear autoassociators to represent appropriate classifications of nonlinear multimodal domains in contrast to linear autoassociators which are inappropriate for such tasks in fact autoassociators with hidden unit nonlinearities can be shown to perform nonlinear classification and nonlinear recognition

general formulation of long range degree correlations in complex networks we provide a general framework for analyzing degree correlations between nodes separated by more than one step i e beyond nearest neighbors in complex networks one joint and four conditional probability distributions are introduced to fully describe long range degree correlations with respect to degrees k and k of two nodes and shortest path length l between them we present general relations among these probability distributions and clarify the relevance to nearest neighbor degree correlations unlike nearest neighbor correlations some of these probability distributions are meaningful only in finite size networks furthermore as a baseline to determine the existence of intrinsic long range degree correlations in a network other than inevitable correlations caused by the finite size effect the functional forms of these probability distributions for random networks are analytically evaluated within a mean field approximation the utility of our argument is demonstrated by applying it to real world networks

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

epidemic extinction paths in complex networks we study the extinction of long lived epidemics on finite complex networks induced by intrinsic noise applying analytical techniques to the stochastic susceptible infected susceptible model we predict the distribution of large fluctuations the most probable or optimal path through a network that leads to a disease free state from an endemic state and the average extinction time in general configurations our predictions agree with monte carlo simulations on several networks including synthetic weighted and degree distributed networks with degree correlations and an empirical high school contact network in addition our approach quantifies characteristic scaling patterns for the optimal path and distribution of large fluctuations both near and away from the epidemic threshold in networks with heterogeneous eigenvector centrality and degree distributions

large deviations of random walks on random graphs we study the rare fluctuations or large deviations of time integrated functionals or observables of an unbiased random walk evolving on erdos renyi random graphs and construct a modified biased random walk that explains how these fluctuations arise in the long time limit two observables are considered the sum of the degrees visited by the random walk and the sum of their logarithm related to the trajectory entropy the modified random walk is used for both quantities to explain how sudden changes in degree fluctuations similar to dynamical phase transitions are related to localization transitions for the second quantity we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk

cospectrality of graphs richard brualdi proposed in stevanivic 2007 6 the following problem problem awgs 4 let gngn and gn be two nonisomorphic graphs on n vertices with spectra 1 2 nand 1 2 n respectively define the distance between the spectra of gngn and gn as gn gn i 1n i i 2 or use i 1n i i define the cospectrality of gngn by cs gn min gn gn gn not isomorphic to gn let csn max cs gn gn a graph on n vertices problem a investigate cs gn cs gn for special classes of graphs problem b find a good upper bound on csncsn in this paper we study problem a and determine the cospectrality of certain graphs by the euclidian distance let knkn denote the complete graph on n vertices nk1nk1 denote the null graph on n vertices and k2 n 2 k1k2 n 2 k1 denote the disjoint union of the k2k2 with n 2n 2 isolated vertices where n 2n 2 in this paper we find cs kn cs kn cs nk1 cs nk1 cs k2 n 2 k1 cs k2 n 2 k1 n 2n 2 and cs kn n cs kn n

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

enhancement of large fluctuations to extinction in adaptive networks during an epidemic individual nodes in a network may adapt their connections to reduce the chance of infection a common form of adaption is avoidance rewiring where a noninfected node breaks a connection to an infected neighbor and forms a new connection to another noninfected node here we explore the effects of such adaptivity on stochastic fluctuations in the susceptible infected susceptible model focusing on the largest fluctuations that result in extinction of infection using techniques from large deviation theory combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state we are able to predict and analyze large fluctuations and extinction in adaptive networks we find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero adaption furthermore we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate even when holding the average number of infected nodes constant

a new metric for probability distributions we introduce a metric for probability distributions which is bounded information theoretically motivated and has a natural bayesian interpretation the square root of the well known spl chi sup 2 distance is an asymptotic approximation to it moreover it is a close relative of the capacitory discrimination and jensen shannon divergence

spectral embedding of graphs in this paper we explore how to embed symbolic relational graphs with unweighted edges in a pattern space we adopt a graph spectral approach we use the leading eigenvectors of the graph adjacency matrix to define eigenmodes of the adjacency matrix for each eigenmode we compute vectors of spectral properties these include the eigenmode perimeter eigenmode volume cheeger number inter mode adjacency matrices and intermode edge distance we embed these vectors in a pattern space using two contrasting approaches the first of these involves performing principal or independent components analysis on the covariance matrix for the spectral pattern vectors the second approach involves performing multidimensional scaling on the l2 norm for pairs of pattern vectors we illustrate the utility of the embedding methods on neighbourhood graphs representing the arrangement of corner features in 2d images of 3d polyhedral objects two problems are investigated the first of these is the clustering of graphs representing distinct objects viewed from different directions the second is the identification of characteristic views of single objects these two studies reveal that both embedding methods result in well structured view spaces for graph data extracted from 2d views of 3d objects

comparing large scale graphs based on quantum probability theory abstract in this paper a new measurement to compare two large scale graphs based on the theory of quantum probability is proposed an explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions it is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues but also about the corresponding eigenvectors moreover we prove that the proposed distance is graph invariant and sub structure invariant examples with various graphs are given and distances between graphs with few vertices are checked computational results for real large scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap

entropy rate of diffusion processes on complex networks we introduce the concept of entropy rate to characterize a diffusion process on a complex network the entropy rate represents the minimal amount of information necessary to describe the diffusion on the network and is a quantity extremely sensitive to the network topology and dynamics by opportunely tuning the kind of diffusion the entropy rate allows one to extract different properties of the network structure moreover entropy maximization indicates how to design optimal diffusion processes providing a new theoretical tool with applications to social technological and communication systems

statistical theory of learning curves under entropic loss criterion the present paper elucidates a universal property of learning curves which shows how the generalization error training error and the complexity of the underlying stochastic machine are related and how the behavior of a stochastic machine is improved as the number of training examples increases the error is measured by the entropic loss it is proved that the generalization error converges to h0 the entropy of the conditional distribution of the true machine as h0 m 2t while the training error converges as h0 m 2t where t is the number of examples and m shows the complexity of the network when the model is faithful implying that the true machine is in the model m is reduced to m the number of modifiable parameters this is a universal law because it holds for any regular machine irrespective of its structure under the maximum likelihood estimator similar relations are obtained for the bayes and gibbs learning algorithms these learning curves show the relation among the accuracy of learning the complexity of a model and the number of training examples