models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

semi classical properties of berezin toeplitz operators with c k symbol we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

spherical and hyperbolic embeddings of data many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects for many types of data these dissimilarities are not euclidean i e they do not represent the distances between points in a euclidean space and therefore cannot be isometrically embedded in a euclidean space examples include shape dissimilarities graph distances and mesh geodesic distances in this paper we provide a means of embedding such non euclidean data onto surfaces of constant curvature we aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data the space can be either of positive curvature spherical or of negative curvature hyperbolic we give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation for objects which do not reside exactly on the manifold we develop a optimisation based procedure for approximate embedding on a hyperspherical manifold we use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space this process is efficient enough to allow us to embed data sets of several thousand objects we apply our method to a variety of data including time warping functions shape similarities graph similarity and gesture similarity data in each case the embedding maintains the local structure of the data while placing the points in a metric space

on the asymptotic expansion of bergman kernel we study the asymptotic of the bergman kernel of the spin c dirac operator on high tensor powers of a line bundle

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

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

scale free network clustering in hyperbolic and other random graphs random graphs with power law degrees can model scale free networks as sparse topologies with strong degree heterogeneity mathematical analysis of such random graphs proved successful in explaining scale free network properties such as resilience navigability and small distances we introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees we apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale free networks with latent geometries and clustering we show that clustering in the hyperbolic model is non vanishing and self averaging so that a single random graph sample is a good representation in the large network limit we also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model

effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution in every network a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes and therefore one may speak of a ball its volume and how it grows as a function of the radius spatial networks tend to feature peculiar volume scaling functions as well as other topological features including clustering degree degree correlation clique complexes and heterogeneity here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function we show that such structures fall into the category of m colored random graphs and study the percolation transition by using this theory we prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function additionally we show that the size of the giant component and the cyclomatic number are not affected by volume scaling these findings may explain the surprisingly good performance of network models that neglect volume scaling even though this paper focuses on the implications of the volume growth the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications

the accumulative law and its probability model an extension of the pareto distribution and the log normal distribution the divergence between the pareto distribution and the log normal distribution has been observed persistently over the past couple of decades in complex network research economics and social scie

non vanishing elements in finite groups abstract we show that if a is an elementary abelian normal p subgroup of a finite group g and p is a sylow p subgroup of g then no irreducible character of g vanish on any element of z p a

search in power law networks many communication and social networks have power law link distributions containing a few nodes that have a very high degree and many with low degree the high connectivity nodes play the important role of hubs in communication and networking a fact that can be exploited when designing efficient search algorithms we introduce a number of local search strategies that utilize high degree nodes in power law graphs and that have costs scaling sublinearly with the size of the graph we also demonstrate the utility of these strategies on the gnutella peer to peer network

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

bootstrapping on undirected binary networks via statistical mechanics we propose a new method inspired from statistical mechanics for extracting geometric information from undirected binary networks and generating random networks that conform to this geometry in this method an undirected binary network is perceived as a thermodynamic system with a collection of permuted adjacency matrices as its states the task of extracting information from the network is then reformulated as a discrete combinatorial optimization problem of searching for its ground state to solve this problem we apply multiple ensembles of temperature regulated markov chains to establish an ultrametric geometry on the network this geometry is equipped with a tree hierarchy that captures the multiscale community structure of the network we translate this geometry into a parisi adjacency matrix which has a relative low energy level and is in the vicinity of the ground state the parisi adjacency matrix is then further optimized by making block permutations subject to the ultrametric geometry the optimal matrix corresponds to the macrostate of the original network an ensemble of random networks is then generated such that each of these networks conforms to this macrostate the corresponding algorithm also provides an estimate of the size of this ensemble by repeating this procedure at different scales of the ultrametric geometry of the network it is possible to compute its evolution entropy i e to estimate the evolution of its complexity as we move from a coarse to a fine description of its geometric structure we demonstrate the performance of this method on simulated as well as real data networks

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

navigation in a small world with local information it is commonly known that there exist short paths between vertices in a network showing the small world effect yet vertices for example the individuals living in society usually are not able to find the shortest paths due to the very serious limit of information to study this issue theoretically here the navigation process of launching messages toward designated targets is investigated on a variant of the one dimensional small world network swn in the network structure considered the probability of a shortcut falling between a pair of nodes is proportional to r alpha where r is the lattice distance between the nodes when alpha 0 it reduces to the swn model with random shortcuts the system shows the dynamic small world effect which is different from the well studied static sw effect we study the effective network diameter the path length as a function of the lattice distance and the dynamics they are controlled by multiple parameters and we use data collapse to show that the parameters are correlated the central finding is that in the one dimensional network studied the dynamic sw effect exists for 0 alpha 2 for each given value of alpha in this region the point where the dynamic sw effect arises is m l approximately 1 where m is the number of useful shortcuts and l is their average reduced effective length

what is the dimension of citation space abstract citation networks represent the flow of information between agents they are constrained in time and so form directed acyclic graphs which have a causal structure here we provide novel quantitative methods to characterise that structure by adapting methods used in the causal set approach to quantum gravity by considering the networks to be embedded in a minkowski spacetime and measuring its dimension using myrheim meyer and midpoint scaling estimates we illustrate these methods on citation networks from the arxiv supreme court judgements from the usa and patents and find that otherwise similar citation networks have measurably different dimensions we suggest that these differences can be interpreted in terms of the level of diversity or narrowness in citation behaviour

curvature and temperature of complex networks we show that heterogeneous degree distributions in observed scale free topologies of complex networks can emerge as a consequence of the exponential expansion of hidden hyperbolic space fermi dirac statistics provides a physical interpretation of hyperbolic distances as energies of links the hidden space curvature affects the heterogeneity of the degree distribution while clustering is a function of temperature we embed the internet into the hyperbolic plane and find a remarkable congruency between the embedding and our hyperbolic model besides proving our model realistic this embedding may be used for routing with only local information which holds significant promise for improving the performance of internet routing

identifying optimal targets of network attack by belief propagation for a network formed by nodes and undirected links between pairs of nodes the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components this problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin glass theory and an associated belief propagation guided decimation bpd algorithm zhou eur phys j b 86 455 2013 in the present work we apply the bpd algorithm which has approximately linear time complexity to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm morone and makse nature 524 65 2015 for different types of random networks and real world network instances the bpd guided attack scheme often induces an abrupt collapse of the whole network which may make it very difficult to defend

anomalous biased diffusion in networks we study diffusion with a bias toward a target node in networks this problem is relevant to efficient routing strategies in emerging communication networks like optical networks bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node we investigate the scaling of the mean first passage time mfpt with the size of the network we find by using theoretical analysis and computer simulations that for random regular rr and erd ifmmode mbox h o else h o fi s r enyi networks there exists a threshold probability p mathrm th such that for pl p mathrm th the mfpt scales anomalously as n ensuremath alpha where n is the number of nodes and ensuremath alpha depends on p for pg p mathrm th the mfpt scales logarithmically with n the threshold value p mathrm th of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes an exact solution for every value of p is given for the scaling of the mfpt in rr networks the regime transition is also observed for the second moment of the probability distribution function the standard deviation for the case of scale free sf networks we present analytical bounds and simulations results showing that the mfpt scales at most as mathrm ln n to a positive power for any finite bias which means that in sf networks even a very small bias is considerably more efficient in comparison to unbiased walk

hyperbolicity measures democracy in real world networks in this work we analyze the hyperbolicity of real world networks a geometric quantity that measures if a space is negatively curved we provide two improvements in our understanding of this quantity first of all in our interpretation a hyperbolic network is aristocratic since few elements connect the system while a non hyperbolic network has a more democratic structure with a larger number of crucial elements the second contribution is the introduction of the average hyperbolicity of the neighbors of a given node through this definition we outline an influence area for the vertices in the graph we show that in real networks the influence area of the highest degree vertex is small in what we define local networks i e social or peer to peer networks and large in global networks i e power grid metabolic networks or autonomous system networks

community structure and scale free collections of erdős renyi graphs community structure plays a significant role in the analysis of social networks and similar graphs yet this structure is little understood and not well captured by most models we formally define a community to be a subgraph that is internally highly connected and has no deeper substructure we use tools of combinatorics to show that any such community must contain a dense erd ifmmode mbox h o else h o fi s r enyi er subgraph based on mathematical arguments we hypothesize that any graph with a heavy tailed degree distribution and community structure must contain a scale free collection of dense er subgraphs these theoretical observations corroborate well with empirical evidence from this we propose the block two level erd ifmmode mbox h o else h o fi s r enyi bter model and demonstrate that it accurately captures the observable properties of many real world social networks

isolation and connectivity in random geometric graphs with self similar intensity measures random geometric graphs consist of randomly distributed nodes points with pairs of nodes within a given mutual distance linked in the usual model the distribution of nodes is uniform on a square and in the limit of infinitely many nodes and shrinking linking range the number of isolated nodes is poisson distributed and the probability of no isolated nodes is equal to the probability the whole graph is connected here we examine these properties for several self similar node distributions including smooth and fractal uniform and nonuniform and finitely ramified or otherwise we show that nonuniformity can break the poisson distribution property but it strengthens the link between isolation and connectivity it also stretches out the connectivity transition finite ramification is another mechanism for lack of connectivity the same considerations apply to fractal distributions as smooth with some technical differences in evaluation of the integrals and analytical arguments

exploring complex networks by walking on them we carry out a comparative study of the problem of a walker searching several typical complex networks the search efficiency is evaluated for various strategies having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes it is found that the best search strategy is the self avoiding random walk the preferentially self avoiding random walk does not help in improving the search efficiency further in return topological information of the underlying networks may be drawn by comparing the results of the different search strategies

crossing on hyperbolic lattices we divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p we consider the 7 3 heptagonal enhanced or extended binary tree ebt the ebt dual and the 5 5 pentagonal lattices we find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p l to the upper p u critical values we find bounds and estimates for the values of p l and p u for these lattices and identify the self duality point p corresponding to where the crossing probability equals 1 2 comparison is made with recent numerical and theoretical results

a random geometric graph built on a time varying riemannian manifold abstract the theory of random geometric graph enables the study of complex networks through geometry to analyze evolutionary networks time varying geometries are needed solutions of the generalized hyperbolic geometric flow are such geometries here we propose a scale free network model which is a random geometric graph on a two dimensional disc the metric of the disc is a ricci flat solution of the flow the model is used to physically simulate the growth and aggregation of a type of cancer cell

exploring complex networks via topological embedding on surfaces we demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate to characterize and to simulate networks with a broad range of properties any network can be embedded on a surface with sufficiently high genus and therefore the study of topologically embedded graphs is non restrictive we show that the local properties of the network are affected by the surface genus which determines the average degree which influences the degree distribution and which controls the clustering coefficient the global properties of the graph are also strongly affected by the surface genus which is constraining the degree of interwovenness changing the scaling properties of the network from large world kind small genus to small and ultrasmall world kind large genus two elementary moves allow the exploration of all networks embeddable on a given surface and naturally introduce a tool to develop a statistical mechanics description for these networks within such a framework we study the properties of topologically embedded graphs which dynamically tend to lower their energy towards a ground state with a given reference degree distribution we show that the cooling dynamics between high and low temperatures is strongly affected by the surface genus with the manifestation of a glass like transition occurring when the distance from the reference distribution is low we prove with examples that topologically embedded graphs can be built in a way to contain arbitrary complex networks as subgraphs this method opens a new avenue to build geometrically embedded networks on hyperbolic manifolds

perception of similarity a model for social network dynamics some properties of social networks e g the mixing patterns and the community structure appear deeply influenced by the individual perception of people in this work we map behaviors by considering similarity and popularity of people also assuming that each person has his her proper perception and interpretation of similarity although investigated in different ways depending on the specific scientific framework from a computational perspective similarity is typically calculated as a distance measure in accordance with this view to represent social network dynamics we developed an agent based model on top of a hyperbolic space on which individual distance measures are calculated simulations performed in accordance with the proposed model generate small world networks that exhibit a community structure we deem this model to be valuable for analyzing the relevant properties of real social networks

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

the spin bf c dirac operator on high tensor powers of a line bundle we study the asymptotic of the spectrum of the rm spin c dirac operator on high tensor powers of a line bundle as application we get a simple proof of the main result of guillemin uribe 13 theorem 2 which was originally proved by using the analysis of toeplitz operators of boutet de monvel and guillemin 10

superlinear and sublinear urban scaling in geographical networks modeling cities using a geographical scale free network to describe relations between people in a city we explain both superlinear and sublinear allometric scaling of urban indicators that quantify activities or performances of the city the urban indicator y n of a city with the population size n is analytically calculated by summing up all individual activities produced by person to person relationships our results show that the urban indicator scales superlinearly with the population namely y n n with 1 if y n represents a creative productivity and the indicator scales sublinearly 1 if y n is related to the degree of infrastructure development these results coincide with allometric scaling observed in real world urban indicators we also show how the scaling exponent depends on the strength of the geographical constraint in the network formation

berezin toeplitz quantization on kaehler manifolds we study the berezin toeplitz quantization on kaehler manifolds we explain first how to compute various associated asymptotic expansions then we compute explicitly the first terms of the expansion of the kernel of the berezin toeplitz operators and of the composition of two berezin toeplitz operators as application we estimate the norm of donaldson s q operator

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

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

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

bounds of percolation thresholds on hyperbolic lattices we analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative gaussian curvature the quantity of our main concern is p c2 the value of occupation probability where a unique unbounded cluster begins to emerge by applying the substitution method to known bounds of the order 5 pentagonal tiling we show that p c2 0 382508 for the order 5 square tiling p c2 0 472043 for its dual and p c2 0 275768 for the order 5 4 rhombille tiling

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

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

tuning clustering in random networks with arbitrary degree distributions we present a generator of random networks where both the degree dependent clustering coefficient and the degree distribution are tunable following the same philosophy as in the configuration model the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori the algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs the procedure unveils an universal relation among clustering and degree degree correlations for all networks where the level of assortativity establishes an upper limit to the level of clustering maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior correlation measures in real networks are seen to observe this structural bound

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

two remarks about non vanishing elements in finite groups let g be a finite group we prove that for x g we have x 0 for all irreducible characters of g iff the class sum of x in the group algebra over c is a unit from this we conclude that if g has a normal p subgroup v and a hall p subgroup then g has non vanishing elements different from 1 hence we get another proof that a finite solvable group always has non trivial non vanishing elements moreover we give an example for a finite solvable group g which has a non vanishing involution not contained in an abelian normal subgroup of g

fast community detection based on sector edge aggregation metric model in hyperbolic space by studying the edge aggregation characteristic of nodes in hyperbolic space sector edge aggregation metric seam model is proposed and theoretically proved in this paper in hyperbolic disk seam model determines the minimum angular range of a sector which possesses the maximal edge aggregation of nodes the set of nodes within such sector has dense internal links which corresponds with the characteristic of community structure based on seam model we propose a fast community detection algorithm called greedy optimization modularity algorithm goma which employs greedy optimization strategy and hyperbolic coordinates goma firstly divides initial communities according to the quantitative results of sector edge aggregation given by seam and the nodes hyperbolic coordinates then based on greedy optimization strategy only merges the two angular neighboring communities in hyperbolic disk to optimize the network modularity function and consequently obtains high quality community detection the strategies of initial community partition and merger in hyperbolic space greatly improve the speed of searching the most optimal modularity experimental results indicate that goma is able to detect out high quality community structure in synthetic and real networks and performs better when applied to the large scale and dense networks with strong clustering

berezin toeplitz operators a semi classical approach this article is devoted to the toeplitz operators 4 in the context of the geometric quantization 11 15 we propose an ansatz for their schwartz kernel from this we deduce the main known properties of the principal symbol of these operators and obtain new results we define their covariant and contravariant symbols which are full symbols and compute the product of these symbols in terms of the kahler metric this gives canonical star products on the kahlerian manifolds this ansatz is also useful to introduce the notion of microsupport

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

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

novel fake news spreading model with similarity on pso based networks abstract this paper proposes a fake news spreading model with similarity taken into account assuming that the similarity between individuals can affect the transmission rate simulations show that the similarity of two connected nodes and the product of their degrees are positively correlated when the network temperature is small and the similarity of two connected nodes decreases as the product of their degrees increases thus the transmission rate can be expressed as the function of their degrees in the proposed model the theoretic analysis demonstrates the critical threshold is related to both the influence coefficient and the similarity function simulation results show a smaller influence coefficient leads to a larger critical threshold and smaller final density of stiflers

network growth by copying we introduce a growing network model in which a new node attaches to a randomly selected node as well as to all ancestors of the target node this mechanism produces a sparse ultrasmall network where the average node degree grows logarithmically with network size while the network diameter equals 2 we determine basic geometrical network properties such as the size dependence of the number of links and the in and out degree distributions we also compare our predictions with real networks where the node degree also grows slowly with time the internet and the citation network of all physical review papers

greedy connectivity of geographically embedded graphs we introduce a measure of greedy connectivity for geographical networks graphs embedded in space and where the search for connecting paths relies only on local information such as a node s location and that of its neighbors constraints of this type are common in everyday life applications greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability greedy connectivity can be used as a criterion for optimal network design

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

complex networks in the euclidean space of communicability distances we study the properties of complex networks embedded in a euclidean space of communicability distances the communicability distance between two nodes is defined as the difference between the weighted sum of walks self returning to the nodes and the weighted sum of walks going from one node to the other we give some indications that the communicability distance identifies the least crowded routes in networks where simultaneous submission of packages is taking place we define an index q based on communicability and shortest path distances which allows reinterpreting the small world phenomenon as the region of minimum q in the watts strogatz model it also allows the classification and analysis of networks with different efficiency of spatial uses consequently the communicability distance displays unique features for the analysis of complex networks in different scenarios

mean field diffusive dynamics on weighted networks diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks here we introduce a general formalism that allows to easily write down mean field equations for any diffusive dynamics on weighted networks we also propose the concept of annealed weighted networks in which such equations become exact we show the validity of our approach addressing the problem of the random walk process pointing out a strong departure of the behavior observed in quenched real scale free networks from the mean field predictions additionally we show how to employ our formalism for more complex dynamics our work sheds light on mean field theory on weighted networks and on its range of validity and warns about the reliability of mean field results for complex dynamics

soft communities in similarity space the mathbb s 1 model has been central in the development of the field of network geometry it places nodes in a similarity space and connects them with a likelihood depending on an effective distance which combines similarity and popularity dimensions with popularity directly related to the degrees of the nodes the mathbb s 1 model has been mainly studied in its homogeneous regime in which angular coordinates are independently and uniformly scattered on the circle we now investigate if the model can generate networks with targeted topological features and soft communities that is inhomogeneous angular distributions to that end hidden degrees must depend on angular coordinates and we propose a method to estimate them we conclude that the model can be topologically invariant with respect to the soft community structure our results expand the scope of the model beyond the independent hidden variables limit and can have an important impact in the embedding of real world networks

latent geometry of bipartite networks despite the abundance of bipartite networked systems their organizing principles are less studied compared to unipartite networks bipartite networks are often analyzed after projecting them onto one of the two sets of nodes as a result of the projection nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network even though these projections allow one to study bipartite networks using tools developed for unipartite networks one mode projections lead to significant loss of information and artificial inflation of the projected network with fully connected subgraphs here we pursue a different approach for analyzing bipartite systems that is based on the observation that such systems have a latent metric structure network nodes are points in a latent metric space while connections are more likely to form between nodes separated by shorter distances this approach has been developed for unipartite networks and relatively little is known about its applicability to bipartite systems here we fully analyze a simple latent geometric model of bipartite networks and show that this model explains the peculiar structural properties of many real bipartite systems including the distributions of common neighbors and bipartite clustering we also analyze the geometric information loss in one mode projections in this model and propose an efficient method to infer the latent pairwise distances between nodes uncovering the latent geometry underlying real bipartite networks can find applications in diverse domains ranging from constructing efficient recommender systems to understanding cell metabolism