how community structure influences epidemic spread in social networks abstract two key features of social networks are the community structure and the high clustering coefficient for understanding their influences on dynamical processes we present a model with both an adjustable clustering coefficient and an adjustable degree of community this model has an invariant degree distribution when its clustering coefficient is being adjusted we find that the efficiency of epidemic spreading in this model depends mainly on the degree of community and decreases with increase of the degree of community for a fixed degree of community the efficiency will decrease with increase of the clustering coefficient numerical simulations have confirmed the theoretic analysis

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

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

an evolving model of online bipartite networks abstract understanding the structure and evolution of online bipartite networks is a significant task since they play a crucial role in various e commerce services nowadays recently various attempts have been tried to propose different models resulting in either power law or exponential degree distributions however many empirical results show that the user degree distribution actually follows a shifted power law distribution the so called mandelbrot s law which cannot be fully described by previous models in this paper we propose an evolving model considering two different user behaviors random and preferential attachment extensive empirical results on two real bipartite networks delicious and citeulike show that the theoretical model can well characterize the structure of real networks for both user and object degree distributions in addition we introduce a structural parameter p to demonstrate that the hybrid user behavior leads to the shifted power law degree distribution and the region of power law tail will increase with the increment of p the proposed model might shed some lights in understanding the underlying laws governing the structure of real online bipartite networks

convergence analysis of a projection semi implicit coupling scheme for fluid structure interaction problems in this paper we provide a convergence analysis of a projection semi implicit scheme for the simulation of fluid structure systems involving an incompressible viscous fluid the error analysis is performed on a fully discretized linear coupled problem a finite element approximation and a semi implicit time stepping strategy are respectively used for space and time discretization the fluid is described by the stokes equations the structure by the classical linear elastodynamics equations linearized elasticity plate or shell models and all changes of geometry are neglected we derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least sqrt delta t finally some numerical experiments that confirm the theoretical analysis are presented

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

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

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

a variant of k theory and topological t duality for real circle bundles for a space with involutive action there is a variant of k theory motivated by t duality in type ii orbifold string theory we establish that a twisted version of the variant enjoys a topological t duality for real circle bundles i e circle bundles with real structure

line graphs as social networks abstract it was demonstrated recently that the line graphs are clustered and assortative these topological features are known to characterize some social networks m e j newman y park why social networks are different from other types of networks phys rev e 68 2003 036122 it was argued that this similarity reveals their cliquey character in the model proposed here a social network is the line graph of an initial network of families communities interest groups school classes and small companies these groups play the role of nodes and individuals are represented by links between these nodes the picture is supported by the data on the livejournal network of about 8 106 people

analysis of a fluid structure interaction problem recast in an optimal control setting fluid structure interaction fsi simulation presents many computational difficulties particularly when the densities of the fluid and structure are close a previous report p kuberry and h lee comput methods appl mech engrg 267 2013 pp 594 605 has suggested that recasting the fsi problem in the context of optimal control may significantly reduce computation time this paper introduces a neumann type control along with detailed analysis for the stability and existence of an optimal solution for a given time step the existence of lagrange multipliers is proved and an optimality system is derived a gradient based optimization algorithm is then presented with numerical results confirming its effectiveness in computing an accurate solution

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

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

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

generating multi scaling networks with two types of vertices abstract a variety of scale free networks have been created since the pioneer work by barabasi and albert science 286 1999 509 most of these models are homogeneous since they are composed of the same kind of nodes in the realistic world however elements nodes or vertices in the network may play different roles or have different functions in this work we develop an alternative way of vertex classification other than the ordinary modularity method by introducing two types of vertices the interaction between two neighbor vertices is dependent on their types it is found that the vertex degree exhibits a multi scaling law distribution with the scaling exponent of each types of vertex adjustable this network model may exhibit some interesting properties concerning the dynamical processes on it

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

organization of modular networks we examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts char22 communities or modules we develop an analytical approach allowing us to obtain the statistics of connected components and the intervertex distance distribution in these modular networks and to describe their global organization and structure in particular we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks we demonstrate that even a relatively small number of shortcuts unite the networks into one in more precise terms if the number of interlinks is any finite fraction of the total number of connections then the intervertex distance distribution approaches a ensuremath delta function peaked form and so the network is united

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

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

transitivity and degree assortativity explained the bipartite structure of social networks dynamical processes such as the diffusion of knowledge opinions pathogens fake news innovation and others are highly dependent on the structure of the social network in which they occur however questions on why most social networks present some particular structural features namely high levels of transitivity and degree assortativity when compared to other types of networks remain open first we argue that every one mode network can be regarded as a projection of a bipartite network and we show that this is the case using two simple examples solved with the generating functions formalism second using synthetic and empirical data we reveal how the combination of the degree distribution of both sets of nodes of the bipartite network together with the presence of cycles of lengths four and six explain the observed values of transitivity and degree assortativity coefficients in the one mode projected network bipartite networks with top node degrees that display a more right skewed distribution than the bottom nodes result in highly transitive and degree assortative projections especially if a large number of small cycles are present in the bipartite structure

good wannier bases in hilbert modules associated to topological insulators for a large class of physically relevant operators on a manifold with discrete group action we prove general results on the non existence of a basis of well localized wannier functions for their spectral subspaces this turns out to be equivalent to the freeness of a certain hilbert module over the group c algebra canonically associated with the spectral subspace this brings into play k theoretic methods and justifies their importance as invariants of topological insulators in physics

interpersonal interactions and human dynamics in a large social network we study a large social network consisting of over 10 6 individuals who form an internet community and organize themselves in groups of different sizes on the basis of the users list of friends and other data registered in the database we investigate the structure and time development of the network the structure of this friendship network is very similar to the structure of different social networks however here a degree distribution exhibiting two scaling regimes power law for low connectivity and exponential for large connectivity was found the groups size distribution and distribution of number of groups of an individual have power law form we found very interesting scaling laws concerning human dynamics our research has shown how long people are interested in a single task

finding local community structure in networks although the inference of global community structure in networks has recently become a topic of great interest in the physics community all such algorithms require that the graph be completely known here we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time this algorithm runs in time o k2d for general graphs when d is the mean degree and k is the number of vertices to be explored for graphs where exploring a new vertex is time consuming the running time is linear o k we show that on computer generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge as an application we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer

exploring the assortativity clustering space of a network s degree sequence nowadays there is a multitude of measures designed to capture different aspects of network structure to be able to say if a measured value is expected or not one needs to compare it with a refere

analysis and approximation of the cross model for quasi newtonian flows with defective boundary conditions a numerical method for a generalized newtonian fluid with flow rate boundary conditions is considered the defect boundary condition problem is formulated as a constrained optimal control problem where a flow balance is forced on the inflow and outflow boundaries using a neumann control the control problem is analyzed for an existence result and the lagrange multiplier rule a decoupling solution algorithm is presented and numerical experiments are provided to validate robustness of the algorithm

efficient sampling of complex network with modified random walk strategies abstract we present two novel random walk strategies choosing seed node csn random walk and no retracing nr random walk different from the classical random walk sampling the csn and nr strategies focus on the influences of the seed node choice and path overlap respectively three random walk samplings are applied in the erdos renyi er barabasi albert ba watts strogatz ws and the weighted usair networks respectively then the major properties of sampled subnets such as sampling efficiency degree distributions average degree and average clustering coefficient are studied the similar conclusions can be reached with these three random walk strategies firstly the networks with small scales and simple structures are conducive to the sampling secondly the average degree and the average clustering coefficient of the sampled subnet tend to the corresponding values of original networks with limited steps and thirdly all the degree distributions of the subnets are slightly biased to the high degree side however the nr strategy performs better for the average clustering coefficient of the subnet in the real weighted usair networks some obvious characters like the larger clustering coefficient and the fluctuation of degree distribution are reproduced well by these random walk strategies

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

complex network study of brazilian soccer players departamento de f isica e inform atica instituto de f isica de s ao carlos universidade de s ao paulo c p 369 13560 970 s ao carlos sp brazilalthough being a very popular sport in many countries soccer has not received much attentionfrom the scienti c community in this paper we study soccer from a complex network point of view first we consider a bipartite network with two kinds of vertices or nodes the soccer players andthe clubs real data were gathered from the 32 editions of the brazilian soccer championship in atotal of 13 411 soccer players and 127 clubs we nd a lot of interesting and perhaps unsuspectedresults the probability that a brazilian soccer player has worked at n clubs or played m gamesshows an exponential decay while the probability that he has scored g goals is power law now iftwo soccer players who have worked at the same club at the same time are connected by an edge then a new type of network arises composed exclusively by soccer players nodes our analysisshows that for this network the degree distribution decays exponentially we determine the exactvalues of the clustering coe cient the assortativity coe cient and the average shortest path lengthand compare them with those of the erd os r enyi and con guration model the time evolution ofthese quantities are calculated and the corresponding results discussed

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

small world properties emerge in highly compartmentalized networks with intermediate group sizes and numbers many recent studies have focused on two statistical properties observed in diverse real world networks the small world property and compartmentalization d j watts and s h strogatz nature 393 440 1998 m girvan and m e j newman proc natl acad sci 99 7821 2002 models that include group affiliations have been shown to produce networks with high clustering coefficients a necessary condition for small world properties m e j newman phys rev e 68 026121 2003 m e j newman and j park phys rev e 68 036122 2003 however the consequences of varying the number and size of groups in a network are not well understood in order to investigate the consequences of group organization we examined sets of networks that varied simultaneously in the size and number of groups while maintaining the same overall size and average degree here we show that the small world property arises in maximally compartmentalized and clustered networks that occur in the intermediate region between few very large groups and many very small groups

two walks degree assortativity in graphs and networks degree assortativity is the tendency for nodes of high degree resp low degree in a graph to be connected to high degree nodes resp to low degree ones it is usually quantified by the pearson correlation coefficient of the degreedegree correlation here we extend this concept to account for the effect of second neighbours to a given node in a graph that is we consider the two walks degree of a node as the sum of all the degrees of its adjacent nodes the two walks degree assortativity of a graph is then the pearson correlation coefficient of the two walks degreedegree correlation we found here analytical expression for this two walks degree assortativity index as a function of contributing subgraphs we then study all the 261 000 connected graphs with 9 nodes and observe the existence of assortativeassortative and disassortativedisassortative graphs according to degree and two walks degree respectively more surprisingly we observe a class of graphs which are degree disassortative and two walks degree assortative we explain the existence of some of these graphs due to the presence of certain topological features such as a node of low degree connected to high degree ones more importantly we study a series of 49 real world networks where we observe the existence of the disassortativeassortative class in several of them in particular all biological networks studied here were in this class we also conclude that no graphs networks are possible with assortativedisassortative structure

dynamics and approximations of a velocity tracking problem for the navier stokes flows with piecewise distributed controls we study the dynamics of a piecewise in time distributed optimal control problem for the navier stokes equations which models velocity tracking over time we also study the dynamics of semidiscrete and fully discrete approximations of this velocity tracking problem we prove that the rates of velocity tracking are exponential some computational results are presented which reinforces the theoretical results derived

network formation determined by the diffusion process of random walkers we studied the diffusion process of random walkers in networks formed by their traces this model considers the rise and fall of links determined by the frequency of transports of random walkers in order to examine the relation between the formed network and the diffusion process a situation in which multiple random walkers start from the same vertex is investigated the difference in diffusion rate of random walkers according to the difference in dimension of the initial lattice is very important for determining the time evolution of the networks for example complete subgraphs can be formed on a one dimensional lattice while a graph with a power law vertex degree distribution is formed on a two dimensional lattice we derived some formulae for predicting network changes for the 1d case such as the time evolution of the size of nearly complete subgraphs and conditions for their collapse the networks formed on the 2d lattice are characterized by the existence of clusters of highly connected vertices and their life time as the life time of such clusters tends to be small the exponent of the power law distribution changes from 1 2 to 3

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

complex network properties of chinese natural science basic research in this paper we studied the research areas of chinese natural science basic research from a point view of complex network two research areas are considered to be connected if they appear in one fund proposal the explicit network of such connections using data from 1999 to 2004 is constructed the analysis of the real data shows that the degree distribution of the bf research areas network ran may be better fitted by the exponential distribution it displays small world effect in which randomly chosen pairs of research areas are typically separated by only a short path of intermediate research areas the average distance of ran decreases with time while the average clustering coefficient increases with time which indicates that the scientific study would like to be integrated together in terms of the studied areas the relationship between the clustering coefficient c k and the degree k indicates that there is no hierarchical organization in ran

revealing the efficiency of information diffusion in online social networks of microblog abstract online social networks osns such as microblogs enable people to simultaneously share information with a large number of peers and revealing the properties of the formed network structure and the corresponding impacts on information diffusion are important in this article we empirically study 10 millions user profiles from the largest chinese mircoblog sina weibo and 41 7 millions profiles from twitter by studying followers topology we find an invariant characteristic that the followers count of users obeys a power law distribution with exponent near 2 we then investigate the reason from the theoretical view and find this network structure achieves the optimal information diffusion in terms of inducing a minimal propagation times our study thus reveals the efficiency of information diffusion in osns of microblogs which is validated by experimental examination theoretical analysis and numerical simulation

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

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

evolving networks with distance preferences we study evolving networks where new nodes when attached to the network form links with other nodes of preferred distances a particular case is where always the shortest distances are selected make friends with the friends of your present friends we present simulation results for network parameters like the first eigenvalue of the graph laplacian synchronizability clustering coefficients average distances and degree distributions for different distance preferences and compare them with the parameter values for random and scale free networks we find that for the shortest distance rule we obtain a power law degree distribution as in scale free networks while the other parameters are significantly different especially the clustering coefficient

epidemic outbreaks in growing scale free networks with local structure abstract the class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in ba model most of these models assume that only one node per time step joins the network in this paper we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment this successfully generates key features observed in social networks these include power law degree distribution p k k 3 where n 1 m is a tuning parameter defined as the modularity strength of the network nontrivial clustering assortative mixing and modular structure moreover all these features are dependent in a similar way on the parameter we then study the susceptible infected epidemics on this network with identical infectivity and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure especially the modularity strength the modularity of the network makes the spreading velocity much lower than that of the ba model on the other hand increasing the modularity strength will accelerate the propagation velocity

diversity of graphs with highly variable connectivity a popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large scale connectivity while it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent the relationship between competing descriptions is not entirely understood this paper lends perspective on this problem by showing how the degree sequence and other constraints e g connectedness no self loops or parallel edges on a particular graph play a primary role in dictating many features including its correlation structure building on recent work we show how a simple structural metric characterizes key differences between graphs having the same degree sequence more broadly we show how the often implicit choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions

convergence of a fluid structure interaction problem decoupled by a neumann control over a single time step abstract building off of previous analytical results for recasting fluid structure interaction into an optimal control setting an a priori error estimate is given for the optimality system by means of brr theory the convergence of the steepest descent method is proven in a discrete setting for a sufficiently small time step and mesh size a numerical study is included supporting the theoretical rate of convergence over a single time step additional results demonstrate optimal convergence in space and time over several time steps

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

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

a decoupling algorithm for fluid structure interaction problems based on optimization abstract simulating fluid structure interactions is challenging due to the tight coupling between the fluid and solid substructures explicit and implicit decoupling methods often either fail or require relaxation when densities of the two materials are close in this paper a fluid structure interaction problem is formulated as a least squares problem where the jump in velocities of the two substructures is minimized by a neumann control enforcing the continuity of stress on the interface a decoupling optimization algorithm is discussed which requires few nonlinear solves at each time step and numerical results are presented

smooth dense subalgebras of reduced group c algebras schwartz cohomology of groups and cyclic cohomology abstract we introduce the notion of schwartz co homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group we prove that if the length function is chosen to be a word length on a group of polynomial growth then its schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of connes and moscovici and that there are no nontrivial idempotents in the group c algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of connes

subcubic equivalences between path matrix and triangle problems we say an algorithm on n n matrices with integer entries in l m mr or n node graphs with edge weights from l m m is truly subcubic if it runs in o n3 poly log m time for some 0 we define a notion of subcubic reducibility and show that many important problems on graphs and matrices solvable in o n3 time are equivalent under subcubic reductions namely the following weighted problems either all have truly subcubic algorithms or none of them do the all pairs shortest paths problem on weighted digraphs apsp detecting if a weighted graph has a triangle of negative total edge weight listing up to n2 99 negative triangles in an edge weighted graph finding a minimum weight cycle in a graph of non negative edge weights the replacement paths problem on weighted digraphs finding the second shortest simple path between two nodes in a weighted digraph checking whether a given matrix defines a metric verifying the correctness of a matrix product over the min semiring finding a maximum subarray in a given matrix therefore if apsp cannot be solved in n3 e time for any e 0 then many other problems also need essentially cubic time in fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization verifying a matrix product over the same structure and corresponding triangle detection problems over the structure these equivalences simplify prior work on subcubic algorithms for all pairs path problems since it now suffices to give appropriate subcubic triangle detection algorithms other consequences of our work are new combinatorial approaches to boolean matrix multiplication over the or and semiring abbreviated as bmm we show that practical advances in triangle detection would imply practical bmm algorithms among other results building on our techniques we give two improved bmm algorithms a derandomization of the combinatorial bmm algorithm of bansal and williams focs 09 and an improved quantum algorithm for bmm

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

main memory triangle computations for very large sparse power law graphs finding counting and or listing triangles three vertices with three edges in massive graphs are natural fundamental problems which have recently received much attention because of their importance in complex network analysis here we provide a detailed survey of proposed main memory solutions to these problems in a unified way we note that previous authors have paid surprisingly little attention to space complexity of main memory solutions despite its both fundamental and practical interest we therefore detail space complexities of known algorithms and discuss their implications we also present new algorithms which are time optimal for triangle listing and beats previous algorithms concerning space needs they have the additional advantage of performing better on power law graphs which we also detail we finally show with an experimental study that these two algorithms perform very well in practice allowing us to handle cases which were previously out of reach

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

cost effective campaigning in social networks campaigners are increasingly using online social networking platforms for promoting products ideas and information a popular method of promoting a product or even an idea is incentivizing individuals to evangelize the idea vigorously by providing them with referral rewards in the form of discounts cash backs or social recognition due to budget constraints on scarce resources such as money and manpower it may not be possible to provide incentives for the entire population and hence incentives need to be allocated judiciously to appropriate individuals for ensuring the highest possible outreach size we aim to do the same by formulating and solving an optimization problem using percolation theory in particular we compute the set of individuals that are provided incentives for minimizing the expected cost while ensuring a given outreach size we also solve the problem of computing the set of individuals to be incentivized for maximizing the outreach size for given cost budget the optimization problem turns out to be non trivial it involves quantities that need to be computed by numerically solving a fixed point equation our primary contribution is that for a fairly general cost structure we show that the optimization problems can be solved by solving a simple linear program we believe that our approach of using percolation theory to formulate an optimization problem is the first of its kind c 2016 elsevier b v all rights reserved

clustering coefficients of growing networks abstract in this paper we develop a general analytical method to compute clustering coefficients of growing networks this method can be applied to any network as long as we can construct and solve the dynamic equation for the degree of any node we also verify the accuracy of the method through simulation

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

a gradient method approach to optimization based multidisciplinary simulations and nonoverlapping domain decomposition algorithms it has been shown recently that optimization based nonoverlapping domain decomposition algorithms are connected to many well known algorithms using a gradient type iterative strategy for the optimization problem we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain in particular the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter

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

scaling behavior in probabilistic neuronal cellular automata we study a neural network model of interacting stochastic discrete two state cellular automata on a regular lattice the system is externally tuned to a critical point which varies with the degree of stochasticity or the effective temperature there are avalanches of neuronal activity namely spatially and temporally contiguous sites of activity a detailed numerical study of these activity avalanches is presented and single joint and marginal probability distributions are computed at the critical point we find that the scaling exponents for the variables are in good agreement with a mean field theory

stationary state structure of a random copying mechanism over a complex network abstract an analytical approach to network dynamics is used to show that when agents copy their state randomly the network arrives to a stationary regime in which the distribution of states is independent of the degree the effects of network topology on the process are characterized introducing a quantity called influence and studying its behavior for scale free and random networks we show that for this model degree averaged quantities are constant in time regardless of the number of states involved

revising the simple measures of assortativity in complex networks we find that traditional statistics for measuring degree mixing are strongly affected by superrich nodes to counteract and measure the effect of superrich nodes we propose a paradigm to quantify the mixing pattern of a real network in which different mixing patterns may appear among low degree nodes and among high degree nodes this paradigm and the simple revised measure uncover the true complex degree mixing patterns of complex networks with superrich nodes the alternate method indicates that some networks show a false disassortative mixing induced by superrich nodes and have no tendency to be genuinely disassortative our results also show that the previously observed fragility of scale free networks is actually greatly exacerbated by the presence of even a very small number of superrich nodes

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

stability and geometric conservation laws for ale formulations the aim of this paper is to investigate a model ale scheme with respect to various possible choices of time discretizations for each time scheme we investigate the relationships between stability and the so called geometric conservation laws gcl we shall see that gcl condition proves neither necessary nor sufficient for stability in doing so we review some known theoretical results and we prove some new stability results for space time ale discretizations some new error estimates are also presented several numerical experiments confirm the theory

a family network model for wealth distribution in societies a model based on first degree family relations network is used to describe the wealth distribution in societies the network structure is not a priori introduced in the model it is generated in parallel with the wealth values through simple and realistic dynamical rules the model has two main parameters governing the wealth exchange in the network choosing their values realistically leads to wealth distributions in good agreement with measured data the cumulative wealth distribution function has an exponential behavior in the low and medium wealth limit and shows the pareto like power law tail for the upper 5 of the society the obtained pareto indexes are in good agreement with the measured ones the generated family networks also converge to a statistically stable topology with a simple poissonian degree distribution on this family network many interesting correlations are studied and the main factors leading to wealth diversification and the formation of the pareto law are identified

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

degree landscapes in scale free networks we generalize the degree organizational view of real world networks with broad degree distributions in a landscape analog with mountains high degree nodes and valleys low degree nodes for exam

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

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

optimization based nonoverlapping domain decomposition algorithms and their convergence we present some optimization based domain decomposition algorithms with nonoverlapping subdomains a number of existing algorithms as well as new algorithms may be derived using this approach both continuous problems and their finite element discretizations are considered convergence properties are examined

percolation in hierarchical scale free nets we study the percolation phase transition in hierarchical scale free nets depending on the method of construction the nets can be fractal or small world the diameter grows either algebraically or logarithmically with the net size assortative or disassortative a measure of the tendency of like degree nodes to be connected to one another or possess various degrees of clustering the percolation phase transition can be analyzed exactly in all these cases due to the self similar structure of the hierarchical nets we find different types of criticality illustrating the crucial effect of other structural properties aside from the scale free degree distribution of the nets