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

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

link prediction based on path entropy abstract information theory has been taken as a prospective tool for quantifying the complexity of complex networks in this paper first we study the information entropy or uncertainty of a path using the information theory after that we apply the path entropy to the link prediction problem in real world networks specifically we propose a new similarity index namely path entropy pe index which considers the information entropies of shortest paths between node pairs with penalization to long paths empirical experiments demonstrate that pe index outperforms the mainstream of link predictors

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

hamiltonian dynamics of preferential attachment prediction and control of network dynamics are grand challenge problems in network science the lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades here we study the dynamics of networks evolving according to preferential attachment pa known to approximate well the large scale growth dynamics of a variety of real networks we show that this dynamics is hamiltonian thus casting the study of complex networks dynamics to the powerful canonical formalism in which the time evolution of a dynamical system is described by hamilton s equations we derive the explicit form of the hamiltonian that governs network growth in pa this hamiltonian turns out to be nearly identical to graph energy in the configuration model which shows that the ensemble of random graphs generated by pa is nearly identical to the ensemble of random graphs with scale free degree distributions in other words pa generates nothing but random graphs with power law degree distribution the extension of the developed canonical formalism for network analysis to richer geometric network models with non degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales

exponential random simplicial complexes exponential random graph models have attracted significant research attention over the past decades these models are maximum entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values here we extend these maximum entropy ensembles to random simplicial complexes which are more adequate and versatile constructions to model complex systems in many applications we show that many random simplicial complex models considered in the literature can be casted as maximum entropy ensembles under certain constraints we introduce and analyze the most general random simplicial complex ensemble with statistically independent simplices our analysis is simplified by the observation that any distribution on any collection of objects including graphs and simplicial complexes is maximum entropy subject to the constraint that the expected value of is equal to the entropy of the distribution with the help of this observation we prove that ensemble is maximum entropy subject to the two types of constraints which fix the expected numbers of simplices and their boundaries

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

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

lp contractive projections and the heat semigroup for differential forms a projection operator that is contractive on lp for two distinct values of p is shown to be contractive for all values of p and the range must be of a special form this result is used to show that the heat semigroup for k forms on many manifolds with nontrivial cohomology in dimension k cannot be contractive on any lp for p 2

noncommutative operations on wiener functionals and feynman s operational calculus the authors recently introduced a family att 0 of banach algebras of functionals on wiener space and showed that for every f in at the functional integrals k t f exist and are given by a perturbation expansion which serves to disentangle in the sense of feynman s operational calculus the operator k t f when is purely imaginary k t f is the analytic operator valued feynman integral of f in this paper noncommutative operations and on wiener functionals are defined which enable the authors to provide a precise and rigorous interpretation of certain aspects of feynman s operational calculus for noncommuting operators in particular the authors show that if f at1and g at2then f g at1 t2 and k t1 t2 f g k t1 f k t2 g hence the product of operators which can be disentangled using their recent results can itself be disentangled the noncommutative operations the disentangling algebras at and the functional integrals possess a rich interlocking structure

configuration model for correlation matrices preserving the node strength correlation matrices are a major type of multivariate data to examine properties of a given correlation matrix a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices such as those derived from random matrix theory that partially preserve properties of the original matrix we propose a model to generate such reference correlation and covariance matrices for the given matrix correlation matrices are often analyzed as networks which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node given this background the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes akin to configuration models for conventional networks our algorithm is derived from the maximum entropy principle we will apply the proposed algorithm to measurement of clustering coefficients and community detection both of which require a null model to assess the statistical significance of the obtained results

fourier asymptotics of fractal measures abstract a measure on r n will be called locally uniformly dimensional if br x cr for all r 1 and all x where br x denotes the ball of radius r about x for l 2 d the measure d is in i so d is well defined we show it is locally in l2 and sup r 1 r n b r y f d 2 d c f 2 under additional hypotheses we show that lim r r n b r y f d 2 2 d is comparable in size to 2 2 a number of other related results are established the special case when is an integer and is the surface measure on a c1 manifold was treated by s agmon and l hormander j analyse math 30 1976 1 38

clustering of random scale free networks we derive the finite size dependence of the clustering coefficient of scale free random graphs generated by the configuration model with degree distribution exponent 2l ensuremath gamma l3 degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets we also find that for values of ensuremath gamma ensuremath approx 2 clustering is virtually size independent and at the same time becomes a de facto non self averaging topological property this implies that a single instance network is not representative of the ensemble even for very large network sizes

statistical mechanics of multiedge networks statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones there are however subtle yet important considerations to be made regarding the nature of the weights used in this generalization weights can be either continuous or discrete magnitudes and in the latter case they can additionally have undistinguishable or distinguishable nature this fact has not been addressed in the literature insofar and has deep implications on the network statistics in this work we face this problem introducing multiedge networks as graphs where multiple distinguishable connections between nodes are considered we develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multiedges per link and their binary projection the latter case is particularly interesting as we show that binary projections can be understood from multiedge processes the implications of these results are important as many real agent based problems mapped onto graphs require this treatment for a proper characterization of their collective behavior

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

explicit solutions of maxwell s equations on a space of constant curvature abstract an explicit solution is given to the cauchy problem for the source free maxwell s equations in a vacuum on a space time of the form r 1 x m3 where m3 is a 3 manifold of constant curvature this solution satisfies huyghens principle that all electromagnetic radiation propagates at exactly the speed of light the solution is obtained by harmonic analysis on m3 and in the process a generating class of plane wave solutions is found these solutions approximate the flat space plane wave solutions in a neighbourhood of a point but their global properties are somewhat different the solutions obtained are easily transplanted to the robertson walker models of general relativity by re scaling the time variable

characterizing complex networks through statistics of mobius transformations abstract it is well known now that dynamics of large populations of globally all to all coupled oscillators can be reduced to low dimensional submanifolds ws transformation and oa ansatz marvel et al 2009 described an intriguing algebraic structure standing behind this reduction oscillators evolve by the action of the group of mobius transformations of course dynamics in complex networks of coupled oscillators is highly complex and not reducible still closer look unveils that even in complex networks some possibly overlapping groups of oscillators evolve by mobius transformations in this paper we study properties of the network by identifying mobius transformations in the dynamics of oscillators this enables us to introduce some new statistical concepts that characterize the network in particular the notion of coherence of the network or subnetwork is proposed this conceptual approach is meaningful for the broad class of networks including those with time delayed noisy or mixed interactions in this paper several simple random graphs are studied illustrating the meaning of the concepts introduced in the paper

entropy of spatial network ensembles we analyze complexity in spatial network ensembles through the lens of graph entropy mathematically we model a spatial network as a soft random geometric graph i e a graph with two sources of randomness namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function the pair connection function of their mutual distance we consider the general case where randomness arises in node positions as well as pairwise connections i e for a given pair distance the corresponding edge state is a random variable classical random geometric graph and exponential graph models can be recovered in certain limits we derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft probabilistic pair connection functions under this formalism we derive the connection function that yields maximum entropy under general constraints finally we apply our analytical framework to study two practical examples ad hoc wireless networks and the us flight network through the study of these examples we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles

path integral solution of the schrodinger equation in curvilinear coordinates a straightforward procedure a new axiomatic formulation of path integrals is used to construct a path integral solution of the schrodinger equation in curvilinear coordinates an important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral consequently a transformation from euclidean to curvilinear coordinates is simple to handle there is no need to introduce quantum corrections into the action functional furthermore the paths are differentiable hence issues related to stochastic paths do not arise the procedure for constructing the path integral solution of the schrodinger equation is straightforward the case of the schrodinger equation in spherical coordinates for a free particle is presented in detail

quantum statistics in complex networks in this work we discuss the symmetric construction of bosonic and fermionic networks and we present a case of a network showing a mixed quantum statistics this model takes into account the different nature of nodes described by a random parameter that we call energy and includes rewiring of the links the system described by the mixed statistics is an inhomogeneous system formed by two class of nodes in fact there is a threshold energy epsilon s such that nodes with lower energy epsilon epsilon s decrease their connectivity in time

symmetric laplacians quantum density matrices and their von neumann entropy abstract we show that the normalized symmetric laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite hilbert space one part corresponding to the vertices the other corresponding to the edges this suggests an interpretation of the symmetric laplacian s von neumann entropy as a measure of bipartite entanglement present between the two parts of the state we then study extreme values for a connected graph s generalized renyi p entropy specifically we show that 1 the complete graph achieves maximum entropy 2 the 2 regular graph a achieves minimum renyi 2 entropy among all k regular graphs b is within log 4 3 of the minimum renyi 2 entropy and log 4 2 3 of the minimum von neumann entropy among all connected graphs c achieves a von neumann entropy less than the star graph point 2 contrasts sharply with similar work applied to normalized combinatorial laplacians where it has been shown that the star graph almost always achieves minimum von neumann entropy in this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound

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

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

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

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

tailored graph ensembles as proxies or null models for real networks i tools for quantifying structure we study the tailoring of structured random graph ensembles to real networks with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically beyond the level of degree statistics our family of ensembles can produce graphs with any prescribed degree distribution and any degree degree correlation function its control parameters can be calculated fully analytically and as a result we can calculate asymptotically formulae for entropies and complexities and for information theoretic distances between networks expressed directly and explicitly in terms of their measured degree distribution and degree correlations

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

mapping and discrimination of networks in the complexity entropy plane complex networks are usually characterized in terms of their topological spatial or information theoretic properties and combinations of the associated metrics are used to discriminate networks into different classes or categories however even with the present variety of characteristics at hand it still remains a subject of current research to appropriately quantify a network s complexity and correspondingly discriminate between different types of complex networks like infrastructure or social networks on such a basis here we explore the possibility to classify complex networks by means of a statistical complexity measure that has formerly been successfully applied to distinguish different types of chaotic and stochastic time series it is composed of a network s averaged per node entropic measure characterizing the network s information content and the associated jenson shannon divergence as a measure of disequilibrium we study 29 real world networks and show that networks of the same category tend to cluster in distinct areas of the resulting complexity entropy plane we demonstrate that within our framework connectome networks exhibit among the highest complexity while e g transportation and infrastructure networks display significantly lower values furthermore we demonstrate the utility of our framework by applying it to families of random scale free and watts strogatz model networks we then show in a second application that the proposed framework is useful to objectively construct threshold based networks such as functional climate networks or recurrence networks by choosing the threshold such that the statistical network complexity is maximized

image of the schwartz space under spectral projection let x g k be a riemannian symmetric space of non compact type and rank one the spectral projection p f of a function f on x can be written p f f where is the elementary spherical function corresponding to the complex parameter we characterize the image of the schwartz space mathcal s p x under the spectral projection for 0 p 2

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

a null model for dunbar s circles abstract an individual s social group may be represented by their ego network formed by the links between the individual and their acquaintances ego networks present an internal structure of increasingly large nested layers or circles of decreasing relationship intensity whose size exhibits a precise scaling ratio starting from the notion of limited social bandwidth and assuming fixed costs for the links in each layer we propose a null model built on a grand canonical ensemble that generates the observed hierarchical social structure the observed internal structure of ego networks becomes a natural outcome to expect when we assume the existence of layers demanding different amounts of resources in the thermodynamic limit reached when the number of ego network copies is large the specific layer degrees follow a poisson distribution we also find that under certain conditions equispaced layer costs are necessary to obtain a constant group size scaling our model presents interesting analogies to a bose einstein gas that we briefly discuss finally we fit and compare the model with an empirical social network

tolerating the community detection resolution limit with edge weighting communities of vertices within a giant network such as the world wide web are likely to be vastly smaller than the network itself however fortunato and barthelemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than l 2 edges where l is the number of edges in the entire network this resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks fortunato and barthelemy s argument can be extended to networks with weighted edges as well and we derive this corollary argument we conclude that weighted modularity algorithms may fail to resolve communities with less than we 2 total edge weight where w is the total edge weight in the network and e is the maximum weight of an intercommunity edge if e is small then small communities can be resolved given a weighted or unweighted network we describe how to derive new edge weights in order to achieve a low e we modify the clauset newman and moore cnm community detection algorithm to maximize weighted modularity and we show that the resulting algorithm has greatly improved accuracy in experiments with an emerging community standard benchmark we find that our simple cnm variant is competitive with the most accurate community detection methods yet proposed

lp harmonic analysis and radon transforms on the heisenberg group in this paper harmonic analysis in the heisenberg group heisn cn r with group law z t z t z z t t 12im z z is taken to mean the joint spectral theory of the operators l heisenberg laplacian and t t the spectral decomposition of e l2 is given explicitly as e 0 e d where e 1 k 0 1 2 and e z t n 2 n 1 n 2k n 1 exp ie tn 2k exp z 24 n 2k lkn 1 z 22 n 2k where lkn 1 denotes the laguerre polynomial the eigenvalues are it e e n 2k e and l k e k e this decomposition is essentially well known but has not previously been described in this light it is also the decomposition into irreducibles of the representation of the heisenberg motion group semi direct product of heisn and u n on l2 heisn the main result is a summability in lp of the decomposition for e lp 1 p limr 1 rk 0 ed for p 1 only a weaker substitute is available and for p the decomposition is false there is also a plancherel formula for l2 2 2 k e n 2k 0 k e z 0 2dzd a number of explicit examples of the decomposition are computed including part of the schrodinger propagator eisl the heisenberg radon transform is defined by r z t n z t w 0 dw this operator was studied by geller and stein it is shown to be bounded from lp to lq if and only if p 2n 2 2n 1 and q 2n 2 some generalizations of these results to the free step two nilpotent lie groups are given but the results are not as complete

comparisons of karci and shannon entropies and their effects on centrality of social networks abstract in order to measure the amount of different information in a system entropy concept can be used graph entropy measures nodes contribution to the entropy of the graph by this way the influential actors can be identified due to this case a new entropy based method was proposed to identify the influential actors karci entropy was applied to the social networks first time the alpha parameter allowed us to combine many different conditions together when measuring in the network the other important contribution of this paper is to predict the value of alpha parameter of karci entropy by using fuzzy logic after that karci and shannon entropies were compared based on experimental results moreover karci entropy was compared to traditional centrality measures if karci entropy definition is considered as a set of entropies shannon entropy can be regarded as an element of this set accordingly it can be concluded that karci entropy is superior to shannon entropy

structural information and dynamical complexity of networks in 1953 shannon proposed the question of quantification of structural information to analyze communication systems the question has become one of the longest great challenges in information science and computer science here we propose the first metric for structural information given a graph g we define the k dimensional structural information of g or structure entropy of g denoted by mathcal h k g to be the minimum overall number of bits required to determine the k dimensional code of the node that is accessible from random walk in g the k dimensional structural information provides the principle for completely detecting the natural or true structure which consists of the rules regulations and orders of the graphs for fully distinguishing the order from disorder in structured noisy data and for analyzing communication systems solving the shannon s problem and opening up new directions the k dimensional structural information is also the first metric of dynamical complexity of networks measuring the complexity of interactions communications operations and even evolution of networks the metric satisfies a number of fundamental properties including additivity locality robustness local and incremental computability and so on we establish the fundamental theorems of the one and two dimensional structural information of networks including both lower and upper bounds of the metrics of classic data structures general graphs the networks of models and the networks of natural evolution we propose algorithms to approximate the k dimensional structural information of graphs by finding the k dimensional structure of the graphs that minimizes the k dimensional structure entropy we find that the k dimensional structure entropy minimization is the principle for detecting the natural or true structures in real world networks consequently our structural information provides the foundation for knowledge discovering from noisy data we establish a black hole principle by using the two dimensional structure information of graphs we propose the natural rank of locally listing algorithms by the structure entropy minimization principle providing the basis for a next generation search engine

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

approximate von neumann entropy for directed graphs in this paper we develop an entropy measure for assessing the structural complexity of directed graphs although there are many existing alternative measures for quantifying the structural properties of undirected graphs there are relatively few corresponding measures for directed graphs to fill this gap in the literature we explore an alternative technique that is applicable to directed graphs we commence by using chung s generalization of the laplacian of a directed graph to extend the computation of von neumann entropy from undirected to directed graphs we provide a simplified form of the entropy which can be expressed in terms of simple node in degree and out degree statistics moreover we find approximate forms of the von neumann entropy that apply to both weakly and strongly directed graphs and that can be used to characterize network structure we illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real world data sets including structures from protein databases and high energy physics theory citation networks

network ensemble comparisons with stochastic rewiring and von neumann entropy assessing whether a given network is typical or atypical for a random network ensemble i e network ensemble comparison has widespread applications ranging from null model selection and hypothesis testing to clustering and classifying networks we develop a framework for network ensemble comparison by subjecting the network to stochastic rewiring we study two rewiring processes uniform and degree preserved rewiring which yield random network ensembles that converge to the erdos renyi and configuration model ensembles respectively we study convergence through von neumann entropy vne a network summary statistic measuring information content based on the spectra of a laplacian matrix and develop a perturbation analysis for the expected effect of rewiring on vne our analysis yields an estimate for how many rewires are required for a given network to resemble a typical network from an ensemble offering a computationally efficient quantity for network ensemble comparison that does not require

a restriction theorem for metivier groups abstract in the spirit of an earlier result of d muller on the heisenberg group we prove a restriction theorem on a certain class of two step nilpotent lie groups our result extends that of muller also in the framework of the heisenberg group

riemannian geometric entropy for measuring network complexity a central issue in the science of complex systems is the quantitative characterization of complexity in the present work we address this issue by resorting to information geometry actually we propose a constructive way to associate with a in principle any network a differentiable object a riemannian manifold whose volume is used to define the entropy the effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale free networks as well as of characterizing small exponential random graphs configuration models and real networks

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

a note on the von neumann entropy of random graphs abstract in this note we consider the von neumann entropy of a density matrix obtained by normalizing the combinatorial laplacian of a graph by its degree sum we prove that the von neumann entropy of the typical erdos renyi random graph saturates its upper bound since connected regular graphs saturate this bound as well our result highlights a connection between randomness and regularity a general interpretation of the von neumann entropy of a graph is an open problem

entropy of stochastic blockmodel ensembles stochastic blockmodels are generative network models where the vertices are separated into discrete groups and the probability of an edge existing between two vertices is determined solely by their group membership in this paper we derive expressions for the entropy of stochastic blockmodel ensembles we consider several ensemble variants including the traditional model as well as the newly introduced degree corrected version karrer et al phys rev e 83 016107 2011 which imposes a degree sequence on the vertices in addition to the block structure the imposed degree sequence is implemented both as soft constraints where only the expected degrees are imposed and as hard constraints where they are required to be the same on all samples of the ensemble we also consider generalizations to multigraphs and directed graphs we illustrate one of many applications of this measure by directly deriving a log likelihood function from the entropy expression and using it to infer latent block structure in observed data due to the general nature of the ensembles considered the method works well for ensembles with intrinsic degree correlations i e with entropic origin as well as extrinsic degree correlations which go beyond the block structure

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

gibbs entropy of network ensembles by cavity methods the gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints this quantity characterizes the level of order and randomness encoded in features of a given real network here we show how to relate this entropy to large deviations of conjugated canonical ensembles we derive exact expression for this correspondence using the cavity methods for the configuration model for the ensembles with constraint degree sequence and community structure and for the ensemble with constraint degree sequence and number of links at a given distance

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

analysis of heat kernel highlights the strongly modular and heat preserving structure of proteins abstract in this paper we study the structure and dynamical properties of protein contact networks with respect to other biological networks together with simulated archetypal models acting as probes we consider both classical topological descriptors such as modularity and statistics of the shortest paths and different interpretations in terms of diffusion provided by the discrete heat kernel which is elaborated from the normalized graph laplacians a principal component analysis shows high discrimination among the network types by considering both the topological and heat kernel based vector characterizations furthermore a canonical correlation analysis demonstrates the strong agreement among those two characterizations providing thus an important justification in terms of interpretability for the heat kernel finally and most importantly the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey notably the heat trace decay of an ensemble of varying size proteins denotes subdiffusion a peculiar property of proteins

local harmonic analysis on spheres abstract a continuous expansion x x h d is established for functions or distributions supported on a sufficiently small set k in the sphere sn valid in a neighborhood of k where x are eigenfunctions of the laplacian on with eigenvalue n 1 2 2 2 the function h and the operators p that give from are given by explicit formulas when n is odd the function x can be uniquely characterized by analyticity and growth conditions in of paley wiener type the expansion is applied to study functions of the laplacian and operators that commute with the laplacian

mean value properties of the laplacian via spectral theory soit z 2 une fonction entiere paire de type exponentiel temperee l une realisation auto adjointe de c x ou est l operateur de laplace beltrami sur une variete de riemann et l l operateur donne par la theorie spectrale on demontre un theoreme de paley wiener sur le support de l et on l utilise pour montrer que lu u sur un bon domaine implique l u u

the spherical transform of a schwartz function on the heisenberg group suppose thatk u n is a compact lie group acting on the 2n 1 dimensional heisenberg grouphn we say that k hn is a gelfand pair if the convolution algebral1k hn of integrablek invariant functions onhnis commutative in this case the gelfand space k hn is equipped with the godement plancherel measure and the spherical transform l2k hn l2 k hn is an isometry the main result in this paper provides a complete characterization of the set sk hn f f sk hn of spherical transforms ofk invariant schwartz functions onhn we show that a functionfon k hn belongs to sk hn if and only if the functions obtained fromfvia application of certain derivatives and difference operators satisfy decay conditions we also consider spherical series expansions fork invariant schwartz functions onhnmodulo its center

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

duality between equilibrium and growing networks in statistical physics any given system can be either at an equilibrium or away from it networks are not an exception most network models can be classified as either equilibrium or growing here we show that under certain conditions there exists an equilibrium formulation for any growing network model and vice versa the equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically but even for any finite system size the required conditions are satisfied in random geometric graphs in general and causal sets in particular and to a large extent in some real networks

growth and arrest of topological cycles in small physical networks the chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodes formula see text is below a critical value formula see text for covalent polymer networks increasing the network size as measured by the number of cross link nodes beyond formula see text results in a crossover to a new regime in which the characteristic size of the chordless cycles formula see text no longer increases from this result the onset and intensity of finite size effects can be predicted from measurement of formula see text in large networks although such information is largely inaccessible with experiments the agreement of simulation results from molecular dynamics metropolis monte carlo and kinetic monte carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation these results show random graphs as a promising model to capture structural differences in confined physical 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

fourier restriction theorem and characterization of weak l2 eigenfunctions of the laplace beltrami operator abstract in this paper we prove that for r 0 the poisson transform p f of f l 2 k m belongs to l 2 g k this complements an earlier result of lohoue and rychener using this we then prove an analogue of tomas stein restriction theorem for riemannian symmetric spaces of noncompact type with real rank one by using this estimate of the poisson transform we also characterize all weak l 2 eigenfunction of the laplace beltrami operator of riemannian symmetric spaces of noncompact type with real rank one and eigenvalue 2 2 for r 0

a new perspective on functional integration the core of this article is a general theorem with a large number of specializations given a manifold n and a finite number of one parameter groups of point transformations on n with generators y x 1 x d we obtain via functional integration over spaces of pointed paths on n paths with one fixed point a one parameter group of functional operators acting on tensor or spinor fields on n the generator of this group is a quadratic form in the lie derivatives lx in the x direction plus a term linear in ly the basic functional integral is over l2 1 paths x t n continuous paths with square integrable first derivative although the integrator is invariant under time translation the integral is powerful enough to be used for systems which are not time translation invariant seven nontrivial applications of the basic formula are given and the semiclassical expansion is computed the methods of proof are rigorous and combine albeverio ho egh krohn oscillatory integrals with elworthy s parametr

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

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

entropy of tournament digraphs abstract the renyi entropy h of complete antisymmetric directed graphs i e tournaments is explored we optimize h when 2 and 3 and find that as increases h s sensitivity to what we refer to as regularity increases as well a regular tournament on n vertices is one with each vertex having out degree n 1 2 but there is a lot of diversity in terms of structure among the regular tournaments for example a regular tournament may be such that each vertex s out set induces a regular tournament a doubly regular tournament or a transitive tournament a rotational tournament as increases on the set of regular tournaments h has maximum value on doubly regular tournaments and minimum value on rotational tournaments the more regular the higher the entropy we show however that among all tournaments on a fixed number of vertices h 2 and h 3 are maximized by any regular tournament on that number of vertices we also provide a calculation that is equivalent to the von neumann entropy but may be applied to any directed or undirected graph and shows that the von neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles

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

community detection in complex networks by density based clustering abstract we proposed a method to find the community structure in a complex network by density based clustering physical topological distance is introduced in density based clustering for determining a distance function of specific influence functions according to the distribution of the data the community structures are uncovered the method keeps a better connection mode of the community structure than the existing algorithms in terms of modularity which can be viewed as a basic characteristic of community detection in the future moreover experimental results indicate that the proposed method is efficient and effective to be used for community detection of medium and large networks

entropic origin of disassortativity in complex networks why are most empirical networks with the prominent exception of social ones generically degree degree anticorrelated to answer this long standing question we define the ensemble of correlated networks and obtain the associated shannon entropy maximum entropy can correspond to either assortative correlated or disassortative anticorrelated configurations but in the case of highly heterogeneous scale free networks a certain disassortativity is predicted offering a parsimonious explanation for the question above our approach provides a neutral model from which in the absence of further knowledge regarding network evolution one can obtain the expected value of correlations when empirical observations deviate from the neutral predictions as happens for social networks one can then infer that there are specific correlating mechanisms at work

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

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

sparse maximum entropy random graphs with a given power law degree distribution even though power law or close to power law degree distributions are ubiquitously observed in a great variety of large real networks the mathematically satisfactory treatment of random power law graphs satisfying basic statistical requirements of realism is still lacking these requirements are sparsity exchangeability projectivity and unbiasedness the last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints here we prove that the hypersoft configuration model belonging to the class of random graphs with latent hyperparameters also known as inhomogeneous random graphs or w random graphs is an ensemble of random power law graphs that are sparse unbiased and either exchangeable or projective the proof of their unbiasedness relies on generalized graphons and on mapping the problem of maximization of the normalized gibbs entropy of a random graph ensemble to the graphon entropy maximization problem showing that the two entropies converge to each other in the large graph limit

on riesz means of eigenfunction expansions for the kohn laplacian in a recent article c sogge improved some classical results by l hormander about the l p convergence of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in this paper we shall deal with the analogous problem for the model operator of a non elliptic but subelliptic differential operator the kohn laplacian on the heisenberg group

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

a restriction theorem for the heisenberg group since the discovery by e m stein see fefferman 5 that the fourier transform of an lp function on r has a well defined restriction to the unit sphere s which is square integrable on sn 1 if p is close enough to 1 various new restriction theorems have been proved see e g 10 22 27 29 the importance of such theorems in harmonic analysis see e g 5 6 24 28 as well as in the theory of partial differential equations 18 19 25 has become evident it has turned out 29 that the crucial step in proving the stein tomas restriction theorem is the following let a denote the surface measure of s n 1 and let k denote the fourier transform of a then the operator af f k is bounded from lp to lp if p is sufficiently close to 1 9 can be interpreted in terms of the laplacian a on rn let a fox dex be the spectral decomposition of a then formally 9 const f106 1 x dex where 6 denotes dirac s point measure therefore it is natural to study analogues for 9 where a is replaced by a more general positive differential operator l in the case of second order elliptic operators on compact manifolds this has been done by sogge 25

quantitative graph theory in this paper we describe some highlights of the new branch quantitative graph theory and explain its significant different features compared to classical graph theory the main goal of quantitative graph theory is the structural quantification of information contained in complex networks by employing a measurement approach based on numerical invariants and comparisons furthermore the methods as well as the networks do not need to be deterministic but can be statistic as such this complements the field of classical graph theory which is descriptive and deterministic in nature we provide examples of how quantitative graph theory can be used for novel applications in the context of the overarching concept network science

eigenspaces of the laplacian on hyperbolic spaces composition series and integral transforms abstract let x be a projective real complex or quaternion hyperbolic space realized as the pseudo riemannian symmetric space x g h with g o p q u p q or sp p q these are the classical isotropic symmetric spaces let be the g invariant laplace beltrami operator on x a complete description by k types for each c of all closed g invariant subspaces of the eigenspace f c x f f is given the eigenspace representations are compared with principal series representations using poisson transformations similar results are obtained also for the exceptional isotropic symmetric space the langlands parameters of the spherical discrete series representations are determined