a practical guide to heavy tails statistical techniques and applications part 1 applications heavy tailed probability distributions in the world wide web m e crovella et al self similarity and heavy tails structural modelling of network traffic w willinger et al heavy tails in high frequency financial data u a muller et al stable paretian modelling in finance some empirical and theoretical aspects s mittnik et al risk management and quantile estimation f bassi et al part 2 time series analyzing stable time series r j adler et al inference for linear processes with stable noise m calder r a davis on estimating the intensity of long range dependence in finite and infinite variance time series m s taqqu v teverovsky why non linearities can ruin the heavy tailed modeller s day s i resnick periodogram estimates from heavy tailed data t mikosch bayesian inference for time series with infinite variance stable innovations n ravishanker z qiou part 3 heavy tail estimation hill bootstrap and jackknife estimators for heavy tails o v pictet et al characteristic function based estimation of stable distribution parameters s m kogan d b williams part 4 regression bootstrapping signs and permutations for regression with heavy tailed errors a robust resampling r lepage et al linear regression with stable disturbances j h mcculloch part 5 signal processing deviation from normality in statistical signal processing parameter estimation with alpha stable distributions p tsakalides c l nikias statistical modelling and receiver design for multi user communication networks g a tsihrintzis part 6 model structures subexponential distributions c m goldie c kluppelberg structure of stationary stable processes j rosinski tail behaviour of some shot noise processes g samorodnitsky part 7 numerical procedures numerical approximation of the symmetric stable distribution and density j h mcculloch table of the maximally skewed stable distributions j h mcculloch d b panton multivariate stable distributions approximation estimation simulation and identification j p nolan univariate stable distributions parametrizations and software j p nolan

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

power law statistics and universal scaling in the absence of criticality critical states are sometimes identified experimentally through power law statistics or universal scaling functions we show here that such features naturally emerge from networks in self sustained irregular regimes away from criticality in these regimes statistical physics theory of large interacting systems predict a regime where the nodes have independent and identically distributed dynamics we thus investigated the statistics of a system in which units are replaced by independent stochastic surrogates and found the same power law statistics indicating that these are not sufficient to establish criticality we rather suggest that these are universal features of large scale networks when considered macroscopically these results put caution on the interpretation of scaling laws found in nature

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

radial structure of the internet the structure of the internet at the autonomous system as level has been studied by the physics mathematics and computer science communities we extend this work to include features of the core and the periphery taking a radial perspective on as network structure new methods for plotting as data are described and they are used to analyse datasets that have been extended to contain edges missing from earlier collections the average distance from one vertex to the rest of the network is used as the baseline metric for investigating radial structure common vertex specific quantities are plotted against this metric to reveal distinctive characteristics of central and peripheral vertices two datasets are analysed

exponential random graph models for networks with community structure although the community structure organization is an important characteristic of real world networks most of the traditional network models fail to reproduce the feature therefore the models are useless as benchmark graphs for testing community detection algorithms they are also inadequate to predict various properties of real networks with this paper we intend to fill the gap we develop an exponential random graph approach to networks with community structure to this end we mainly built upon the idea of blockmodels we consider both the classical blockmodel and its degree corrected counterpart and study many of their properties analytically we show that in the degree corrected blockmodel node degrees display an interesting scaling property which is reminiscent of what is observed in real world fractal networks a short description of monte carlo simulations of the models is also given in the hope of being useful to others working in the field

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

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

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

on approximating the number of k cliques in sublinear time we study the problem of approximating the number of k cliques in a graph when given query access to the graph we consider the standard query model for general graphs via 1 degree queries 2

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

direct quadrature method of moments solution of fokker planck equations in aeroelasticity the direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the fokker planck equation the theoretical details of the solution procedure are first presented the method is then used to solve the fokker planck equations for both one and two dimensional processes that possess nonlinear stochastic differential equations higher order moments of the stationary solutions are computed and prove to be very accurate when compared with analytic one dimensional process and monte carlo two dimensional process solutions trends in the standard deviation and coefficient of kurtosis with respecttotheadditivenoiselevelandbifurcationparameterarereportedforwhatappearstobethe firsttimeforthe saddle node subcritical hopf bifucation problem the deterministic form of this problem exhibits hysteresis which is often a phenomenon present in more complex nonlinear aeroelastic systems finally statistical results are shown for a typical section airfoil with nonlinear stiffness subjected to random aerodynamic loading the constants for the nonlinear stiffness are chosen such that the deterministic behavior exhibits both a saddle node and subcritcal bifurcation standard deviation results computed using the direct quadrature method of moments for a reduced frequency below the deterministic saddle node bifurcation point compare well with monte carlo results

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

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

a scalable generative graph model with community structure network data is ubiquitous and growing yet we lack realistic generative network models that can be calibrated to match real world data the recently proposed block two level erdos renyi bter model can be tuned to capture two fundamental properties degree distribution and clustering coefficients the latter is particularly important for reproducing graphs with community structure such as social networks in this paper we compare bter to other scalable models and show that it gives a better fit to real data we provide a scalable implementation that requires only o d rm max storage where d rm max is the maximum number of neighbors for a single node the generator is trivially parallelizable and we show results for a hadoop mapreduce implementation for modeling a real world web graph with over 4 6 billion edges we propose that the bter model can be used as a graph generator for benchmarking purposes and provide idealized degree distributions and clustering coefficient profiles that can b

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

global renyi index of the distance matrix abstract in previous studies the heterogeneity of complex networks has been extensively studied in our study the heterogeneity of distance matrices is studied based on the renyi index of networks we define a new metric and name it global renyi index g r i and prove several properties in particular the g r i value of the distance matrix corresponding to the evenly distributed point set in the euclidean space is zero some model data were used to clarify the geometric meanings of g r i and then we studied the g r i value of financial data the results show that the g r i value in the real market changes drastically and is significantly different from the g r i value of the model generated data these results suggest that the proposed concept g r i is meaningful to the study distance matrix and provides a new perspective based on the network

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

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

reduction of su n loop tensors to trees we present a systematic method to express all su n invariant tensors in terms of forests i e products of tree tensors

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

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

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

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

evolution of a modified binomial random graph by agglomeration in the classical erd s renyi random graph g n p there are n vertices and each of the possible edges is independently present with probability p the random graph g n p is homogeneous in the sense that all vertices have the same characteristics on the other hand numerous real world networks are inhomogeneous in this respect such an inhomogeneity of vertices may influence the connection probability between pairs of vertices the purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the erd s renyi random graph g n p given a configuration of n vertices arranged in n subsets of vertices we call each subset a super vertex we define a random graph with n super vertices by letting two super vertices be connected if and only if there is at least one edge between them in g n p our main result concerns the threshold for connectedness we also analyze the phase transition for the emergence of the giant component and the degree distribution even though our model begins with g n p it assumes the existence of some community structure encoded in the configuration furthermore under certain conditions it exhibits a power law degree distribution both properties are important for real world applications

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

computer friendly d tensor identities for su n the identity of degree n 1 satisfied by the su n tensor dijk is derived and presented in a simple recursive form suitable for computation

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

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

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

stability in a system subject to noise with regulated periodicity the stability of a simple dynamical system subject to multiplicative one side pulse noise with hidden periodicity is investigated both analytically and numerically the stability analysis is based on the exact result for the characteristic functional of the renewal pulse process the influence of the memory effects on the stability condition is analyzed for two cases i the dead time distorted poissonian process and ii the renewal process with pareto distribution we show that for fixed noise intensity the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity

first passage time of nonlinear ship rolling in narrow band non stationary random seas the generalized extended stochastic central difference gescd method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non stationary random seas the gescd method is based on a combination of the extended stochastic central difference method with a statistical linearization technique modified adaptive time scheme and time coordinate transformation the extended stochastic central difference method is however an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non stationary random disturbances approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined it is concluded that the gescd method is very accurate simple and efficient to apply compared with monte carlo simulation the proposed method is applicable to cases with large nonlinearities and intensive random excitations the approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the monte carlo simulation it is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature

on the fractal characterization of paretian poisson processes abstract paretian poisson processes are poisson processes which are defined on the positive half line have maximal points and are quantified by power law intensities paretian poisson processes are elemental in statistical physics and are the bedrock of a host of power law statistics ranging from pareto s law to anomalous diffusion in this paper we establish evenness based fractal characterizations of paretian poisson processes considering an array of socioeconomic evenness based measures of statistical heterogeneity we show that amongst the realm of poisson processes which are defined on the positive half line and have maximal points paretian poisson processes are the unique class of fractal processes exhibiting scale invariance the results established in this paper are diametric to previous results asserting that the scale invariance of poisson processes with respect to physical randomness based measures of statistical heterogeneity is characterized by exponential poissonian intensities

a statistical nonlinearization technique for multi degrees of freedom nonlinear systems under white noise excitations a statistical nonlinearization snl technique is proposed for the solution of the joint probability density function of general multi degrees of freedom mdof nonlinear systems under stationary white noise excitations the nonlinearities associated with damping and restoring forces as well as the intensities of excitations are not necessarily small it is based on the application of the exact solution of the joint probability density function of a mdof nonlinear systems under stationary white noise excitations this exact solution is different from that of caughey in that in the present exact solution the ratios of damping coefficients to applied white noise excitations are not identical the exact solution is also different from those of cai and lin and zhu and huang in that the present exact solution is obtained directly from the theory of differential equations while that of cai and lin requires satisfaction of a relatively restrictive criterion furthermore the hamiltonian formulation is applied by zhu and huang their solution depends on the number of independent integrals of motion for example results by applying the proposed technique for a general two degrees of freedom nonlinear system are compared with those obtained by monte carlo simulation mcs it is concluded that the technique is accurate simple to implement and is applicable to mdof systems with both nonlinear damping and nonlinear restoring forces

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

understanding the determinants of volatility clustering in terms of stationary markovian processes abstract volatility is a key variable in the modeling of financial markets the most striking feature of volatility is that it is a long range correlated stochastic variable i e its autocorrelation function decays like a power law for large time lags in the present work we investigate the determinants of such feature starting from the empirical observation that the exponent of a certain stock s volatility is a linear function of the average correlation of such stock s volatility with all other volatilities we propose a simple approach consisting in diagonalizing the cross correlation matrix of volatilities and investigating whether or not the diagonalized volatilities still keep some of the original volatility stylized facts as a result the diagonalized volatilities result to share with the original volatilities either the power law decay of the probability density function and the power law decay of the autocorrelation function this would indicate that volatility clustering is already present in the diagonalized un correlated volatilities we therefore present a parsimonious univariate model based on a non linear langevin equation that well reproduces these two stylized facts of volatility the model helps us in understanding that the main source of volatility clustering once volatilities have been diagonalized is that the economic forces driving volatility can be modeled in terms of a smoluchowski potential with logarithmic tails

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

modeling statistics and kinetics of the natural aggregation structures and processes with the solution of generalized logistic equation abstract the generalized logistic equation is proposed to model kinetics and statistics of natural processes such as earthquakes forest fires floods landslides and many others this equation has the form d n a d a s 1 n a n a q a q 0 and a 0 is the size of an element of a structure and 0 the equation contains two exponents and q taking into account two important properties of elements of a system their fractal geometry and their ability to interact either to enhance or to damp the process of aggregation the function n a can be understood as an approximation to the number of elements the size of which is less than a the function d n a d a where n a is the general solution of this equation for q 1 is a product of an increasing bounded function and power law function with stretched exponential cut off the relation with tsallis non extensive statistics is demonstrated by solving the generalized logistic equation for q 0 in the case 0 q 1 the equation models super additive and the case q 1 it models sub additive structures the gutenberg richter g r formula results from interpretation of empirical data as a straight line in the area of stretched exponent with small the solution is applied for modeling distribution of foreshocks and aftershocks in the regions of napa valley 2014 and sumatra 2004 earthquakes fitting the observed data well both qualitatively and quantitatively

the emergence of scale free networks with a seceding mechanism in order to explore further the underlying mechanism of the scale free networks we study stochastic secession as a mechanism for the creation of complex networks in this evolution the network growth incorporates the addition of new links between existing nodes the deleting and rewiring of some existing links and the stochastic secession of nodes to random growing networks with preferential attachment the model yields scale free behavior for the degree distribution furthermore we get the analytical expression of the power law degree distribution with scaling exponent ranges from 1 1 to 9 the analytical expressions are in good agreement with the numerical simulation results

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

counting triangles in massive graphs with mapreduce graphs and networks are used to model interactions in a variety of contexts there is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure some of the most useful metrics are triangle based and give a measure of the connectedness of mutual friends this is often summarized in terms of clustering coefficients which measure the likelihood that two neighbors of a node are themselves connected computing these measures exactly for large scale networks is prohibitively expensive in both memory and time however a recent wedge sampling algorithm has proved successful in efficiently and accurately estimating clustering coefficients in this paper we describe how to implement this approach in mapreduce to deal with massive graphs we show results on publicly available networks the largest of which is 132m nodes and 4 7b edges as well as artificially generated networks using the graph500 benchmark the largest of which has 240m nodes and 8 5b edges

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

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

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

growing networks of overlapping communities with internal structure we introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks the model comprises two complementary mechanisms one mechanism accounts for the evolution of the internal link structure of a single community and the second mechanism coordinates the growth of multiple overlapping communities the first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates we demonstrate that this simple mechanism gives rise to an effective maximal degree within communities this observation is related to the anthropological theory known as dunbar s number i e the empirical observation of a maximal number of ties which an average individual can sustain within its social groups the second mechanism is based on a recently proposed generalization of preferential attachment to community structure appropriately called structural preferential attachment spa the combination of these two mechanisms into a single model spa allows us to reproduce a number of the global statistics of real networks the distribution of community sizes of node memberships and of degrees the spa model also predicts a three qualitative regimes for the degree distribution within overlapping communities and b strong correlations between the number of communities to which a node belongs and its number of connections within each community we present empirical evidence that support our findings in real complex networks

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

on the bias of traceroute sampling or power law degree distributions in regular graphs understanding the graph structure of the internet is a crucial step for building accurate network models and designing efficient algorithms for internet applications yet obtaining this graph structure can be a surprisingly difficult task as edges cannot be explicitly queried for instance empirical studies of the network of internet protocol ip addresses typically rely on indirect methods like traceroute to build what are approximately single source all destinations shortest path trees these trees only sample a fraction of the network s edges and a paper by lakhina et al 2003 found empirically that the resulting sample is intrinsically biased further in simulations they observed that the degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is poisson in this article we study the bias of traceroute sampling mathematically and for a very general class of underlying degree distributions explicitly calculate the distribution that will be observed as example applications of our machinery we prove that traceroute sampling finds power law degree distributions in both regular and poisson distributed random graphs thus our work puts the observations of lakhina et al on a rigorous footing and extends them to nearly arbitrary degree distributions

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

on characteristic equations trace identities and casimir operators of simple lie algebras two approaches are developed to exploit for simple complex or compact real lie algebras g the information that stems from the characteristic equations of representation matrices and casimir operators these approaches are selected so as to be viable not only for small lie algebras but also to be suitable for treatment by computer algebra a very large body of new results emerges in the forms of a identities of a tensorial nature involving structure constants etc of g b trace identities for powers of matrices of the adjoint and defining representations of g c expressions of nonprimitive casimir operators of g in terms of primitive ones the methods are sufficiently tractable to allow not only explicit proof by hand of the nonprimitive nature of the quartic casimir of g2 f4 e6 but also e g of that of the tenth order casimir of f4

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

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

direct quadrature method of moments solution of the fokker planck equation abstract the direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the fokker planck equation corresponding to stochastic nonlinear dynamical systems the theoretical details of the solution procedure are first presented the method is then used to solve fokker planck equations for both 1d and 2d noisy van der pol oscillator processes which possess nonlinear stochastic differential equations higher order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic 1d process and monte carlo 2d process solutions

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

multifractal network generator we introduce a new approach to constructing networks with realistic features our method in spite of its conceptual simplicity it has only two parameters is capable of generating a wide variety of network types with prescribed statistical properties e g with degree or clustering coefficient distributions of various very different forms in turn these graphs can be used to test hypotheses or as models of actual data the method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies the main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously a very unique feature of this construction is that with the increasing system size the generated graphs become topologically more structured we present analytic expressions derived from the parameters of the to be iterated initial generating measure for such major characteristics of graphs as their degree clustering coefficient and assortativity coefficient distributions the optimal parameters of the generating measure are determined from a simple simulated annealing process thus the present work provides a tool for researchers from a variety of fields such as biology computer science biology or complex systems enabling them to create a versatile model of their network data

power law distribution of degree degree distance a better representation of the scale free property of complex networks whether real world complex networks are scale free or not has long been controversial recently in broido and clauset a d broido a clauset nat commun 10 1017 2019 it was claimed that the degree distributions of real world networks are rarely power law under statistical tests here we attempt to address this issue by defining a fundamental property possessed by each link the degree degree distance the distribution of which also shows signs of being power law by our empirical study surprisingly although full range statistical tests show that degree distributions are not often power law in real world networks we find that in more than half of the cases the degree degree distance distributions can still be described by power laws to explain these findings we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted two way selection process the model does not always produce solid power law distributions but predicts that the degree degree distance distribution exhibits stronger power law behavior than the degree distribution of a finite size network especially when the network is dense we test the strength of our model and its predictive power by examining how real world networks evolve into an overly dense stage and how the corresponding distributions change we propose that being scale free is a property of a complex network that should be determined by its underlying mechanism e g preferential attachment rather than by apparent distribution statistics of finite size we thus conclude that the degree degree distance distribution better represents the scale free property of a complex network

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

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

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

multifractal formalism by enforcing the universal behavior of scaling functions a b s t r a c t despite its solid foundations multifractal analysis is still a challenging task the inversed singularity spectrum is a major pitfall in standard multifractal analyses especially for empirical signals to resolve this issue we identified the fan like convergent geometry of scaling functions yielding a limit value termed focus for all moments at the largest scale building on this behavior of scaling we introduce the novel concept of focus based multifractal formalism it relies on enforcing this universal behavior when the moment wise scaling exponents are assessed for the scaling functions besides developing focus based variants of the established multifractal detrended fluctuation analysis and the wavelet leader method we present a novel analytical tool of multifractal signal summation conversion all methods are extensively tested on exact multifractal signals synthesized by the generalized binomial multifractal model in terms of precision and incidence of inversed singularity spectra our focus based variants never yielded inversed spectra and their precision was found similar to that of standard methods our approach allowed computing a moment wise and a global error parameter describing the impact of finite size effect and degree of multifractality as compared to that of the fitted exact multifractal model we demonstrate that the standard approach to multifractal analyses contains a central element that is essentially monofractal due to its regression scheme assessing the scaling exponents for each and every moment separately hence these methods can yield reliable estimates only for ideally behaving multifractal signals in contrast our focus based variants due to their genuine multifractal model fitting always yield reliable estimates accompanied by goodness of fit statistics the presented novel multifractal tools offer means of dealing with the consequences of endogenous impurities of potentially multifractal empirical signals

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

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling

estimating heavy tail exponents through max self similarity in this paper a novel approach to the problem of estimating the heavy tail exponent 0 of a distribution is proposed it is based on the fact that block maxima of size m scale at a rate m1 for independent as well as for a number of dependent data this scaling rate can be captured well by the max spectrum plot of the data that leads to regression based estimators for consistency and asymptotic normality of these estimators is established for independent data under mild conditions on the behavior of the tail of the distribution the proposed estimators have an important computational advantage over existing methods namely they can be calculated and updated sequentially in an on line fashion without having to store the entire data set practical issues on the automatic selection of tuning parameters for the estimators and corresponding confidence intervals are also addressed extensive numerical simulations show that the proposed method is competitive for both small and large sample sizes robust to contaminants and continues to work under the presence of substantial amount of dependence the proposed estimators are used to illustrate the close connection between long range dependence and heavy tails over an internet traffic trace