novel scaling limits for critical inhomogeneous random graphs we flnd scaling limits for the sizes of the largest components at criticality for the rank 1 inho mogeneous random graphs with power law degrees with exponent we investigate the case where 2 3 4 so that the degrees have flnite variance but inflnite third moment the sizes of the largest clusters rescaled by n i i2 i1 converge to hitting times of a thinned levy process this process is intimately connected to the general multiplicative coalescents studied in 1 and 2 in particular we use the results in 2 to show that when interpreting the location inside the critical window as time the limiting process is a multiplicative process with difiusion constant 0 and the entrance boundary describing the size of relative components in the i1 regime proportional to i i i1 i1 i 1 a crucial ingredient is the identiflcation of the scaling of the largest connected components in the barely subcritical regime our results should be contrasted to the case where the degree exponent satisfles 4 so that the third moment is flnite there instead we see that the sizes of the components rescaled by n i2 3 converge to the excursion lengths of an inhomogeneous brownian motion as proved in 1 for the erd os renyi random graph and extended to the present setting in 3 4 the limit again is a multiplicative coalescent the only difierence with the limit for 2 3 4 being the initial state corresponding to the barely subcritical regime

stable isomorphism of dual operator spaces we prove that two dual operator spaces x and y are stably isomorphic if and only if there exist completely isometric normal representations and of x and y respectively and ternary rings of operators m1 m2 such that x m2 y m1 w and y m2 x m1 w we prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic we apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic consequently we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are morita equivalent in our sense finally we provide examples motivated by the theory of csl algebras

local clustering in scale free networks with hidden variables we investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices the class rules out self loops and multiple edges we focus on the regime where the hidden variables follow a power law with exponent 2 3 so that the degrees have infinite variance the natural cutoff h c characterizes the largest degrees in the hidden variable models and a structural cutoff h s introduces negative degree correlations disassortative mixing due to the infinite variance degrees we show that local clustering decreases with the hidden variable or degree we also determine how the average clustering coefficient c scales with the network size n as a function of h s and h c for scale free networks with exponent 2 3 and the default choices h s n 1 2 and h c n 1 1 this gives c n 2 lnn for the universality class at hand we characterize the extremely slow decay of c when 2 and show that for 2 1 say clustering starts to vanish only for networks as large as n 10 9

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

a morita theorem for dual operator algebras abstract we prove that two dual operator algebras are weak morita equivalent in the sense of d p blecher u kashyap morita equivalence of dual operator algebras j pure appl algebra 212 2008 2401 2412 if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak continuous on appropriate morphism spaces moreover in a fashion similar to the operator algebra case we characterize such functors as the module normal haagerup tensor product with an appropriate weak morita equivalence bimodule we also develop the theory of the w dilation which connects the non selfadjoint dual operator algebra with the w algebraic framework in the case of weak morita equivalence this w dilation is a w module over a von neumann algebra generated by the non selfadjoint dual operator algebra the theory of the w dilation is a key part of the proof of our main theorem

asymptotic bifurcation results for coupled nonlinear wave equations with variable coefficients abstract this paper is concerned with the bifurcation phenomenon of large amplitude periodic solutions of coupled nonlinear wave equations with either constant or variable coefficients such model arises when two waves propagate simultaneously in nonisotropic media the change of degree argument is used to obtain necessary and sufficient conditions for the existence of asymptotic bifurcation points with respect to the matrix spectrum the results we get are also valid for uncoupled systems by changing one of our assumptions on the nonlinear term we also obtain the necessary and sufficient conditions for the existence of the standard bifurcation points

minimum vertex cover in generalized random graphs with power law degree distribution in this paper we study the approximability of the minimum vertex cover problem in power law graphs in particular we investigate the behavior of a standard 2 approximation algorithm together with a simple pre processing step when the input is a random sample from a generalized random graph model with power law degree distribution more precisely if the probability of a vertex of degree i to be present in the graph is c i where 2 and c is a normalizing constant the expected approximation ratio is 1 1 l i e where is the riemann zeta function of li is the polylogarithmic special function of and li 2 1 e 1

modularity based graph partitioning using conditional expected models modularity based partitioning methods divide networks into modules by comparing their structure against random networks conditioned to have the same number of nodes edges and degree distribution we propose a novel way to measure modularity and divide graphs based on conditional probabilities of the edge strength of random networks we provide closed form solutions for the expected strength of an edge when it is conditioned on the degrees of the two neighboring nodes or alternatively on the degrees of all nodes comprising the network we analytically compute the expected network under the assumptions of gaussian and bernoulli distributions when the gaussian distribution assumption is violated we prove that our expression is the best linear unbiased estimator finally we investigate the performance of our conditional expected model in partitioning simulated and real world 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

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

generating simple random graphs with prescribed degree distribution let f be a probability distribution with support on the non negative integers four methods for generating a simple undirected graph with approximate degree distribution f are described and compared two methods are based on the so called configuration model with modifications ensuring a simple graph one method is an extension of the classical erdos renyi graph where the edge probabilities are random variables and the last method starts with a directed random graph which is then modified to a simple undirected graph all methods are shown to give the correct distribution in the limit of large graph size but under different assumptions on the degree distribution f and also using different order of operations

the multiplicative coalescent inhomogeneous continuum random trees and new universality classes for critical random graphs one major open conjecture in the area of critical random graphs formulated by statistical physicists and supported by a large amount of numerical evidence over the last decade braunstein et al in phys rev lett 91 16 168701 2003 wu et al in phys rev lett 96 14 148702 2006 braunstein et al int j bifurc chaos 17 07 2215 2255 2007 chen et al in phys rev lett 96 6 068702 2006 is as follows for a wide array of random graph models with degree exponent tau in 3 4 distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n tau 3 tau 1 in this paper we study the metric space structure of maximal components of the multiplicative coalescent in the regime where the sizes converge to excursions of levy processes without replacement aldous and limic electron in j probab 3 3 59 1998 yielding a completely new class of limiting random metric spaces a by product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent tau in 3 4 where edges are rescaled by n tau 3 tau 1 yielding the first rigorous proof of the above conjecture the limits in this case are compact tree like random fractals with a dense collection of hubs infinite degree vertices a finite number of which are identified with leaves to form shortcuts in a special case we show that the minkowski dimension of the limiting spaces equal tau 2 tau 3 a s in stark contrast to the erd s renyi scaling limit whose minkowski dimension is 2 a s it is generally believed that dynamic versions of a number of fundamental random graph models as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent in work in progress the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent tau in 3 4 our proof makes crucial use of inhomogeneous continuum random trees icrt which have previously arisen in the study of the entrance boundary of the additive coalescent we show that tilted versions of the same objects using the associated mass measure describe connectivity properties of the multiplicative coalescent since convergence of height processes of corresponding approximating mathbf p trees is not known we use general methodology in athreya et al 2014 and develop novel techniques relying on first showing convergence in the gromov weak topology and then extending this to gromov hausdorff prokhorov convergence by proving a global lower mass bound

tuning the overlap and the cross layer correlations in two layer networks application to a susceptible infectious recovered model with awareness dissemination we study the properties of the potential overlap between two networks a b sharing the same set of n nodes a two layer network whose respective degree distributions p a k p b k are given defining the overlap coefficient as the jaccard index we prove that is very close to 0 when a and b are random and independently generated we derive an upper bound m for the maximum overlap coefficient permitted in terms of p a k p b k and n then we present an algorithm based on cross rewiring of links to obtain a two layer network with any prescribed inside the range 0 m a refined version of the algorithm allows us to minimize the cross layer correlations that unavoidably appear for values of beyond a critical overlap c m finally we present a very simple example of a susceptible infectious recovered epidemic model with information dissemination and use the algorithms to determine the impact of the overlap on the final outbreak size predicted by the model

triadic closure in configuration models with unbounded degree fluctuations the configuration model generates random graphs with any given degree distribution and thus serves as a null model for scale free networks with power law degrees and unbounded degree fluctuations for this setting we study the local clustering c k i e the probability that two neighbors of a degree k node are neighbors themselves we show that c k progressively falls off with k and the graph size n and eventually for k varomega sqrt n settles on a power law c k sim n 5 2 tau k 2 3 tau with tau in 2 3 the power law exponent of the degree distribution this fall off has been observed in the majority of real world networks and signals the presence of modular or hierarchical structure our results agree with recent results for the hidden variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi edges we show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting

ising critical exponents on random trees and graphs we study the critical behavior of the ferromagnetic ising model on random trees as well as so called locally tree like random graphs we pay special attention to trees and graphs with a power law offspring or degree distribution whose tail behavior is characterized by its power law exponent 2 we show that the critical inverse temperature of the ising model equals the hyperbolic arctangent of the reciprocal of the mean offspring or mean forward degree distribution in particular the critical inverse temperature equals zero when tau in 2 3 where this mean equals infinity

replica theory for learning curves for gaussian processes on random graphs we use a statistical physics approach to derive accurate predictions for the challenging problem of predicting the performance of gaussian process regression performance is quantified by the learning curve defined as the average error versus number of training examples we assume the gaussian process prior is defined by a random walk kernel inputs are vertices on a random graph and the outputs are noisy function values we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs after first rewriting the average error in terms of a graphical model conventionally the gaussian process kernel is only globally normalized so that the prior variance can differ between vertices as a more principled alternative we also consider local normalization where the prior variance is uniform the normalization constants for the prior then have to be defined as thermal averages in an unnormalized model and this requires the introduction of a second auxiliary set of replicas our results for both types of kernel normalization apply generically to all random graph ensembles constrained by a fixed but arbitrary degree distribution we compare with numerically simulated learning curves and find excellent agreement a significant improvement over existing approximations

strong morita equivalence of operator spaces abstract we introduce and examine the notions of strong equivalence and strong tro equivalence for operator spaces we show that they behave in an analogous way to how strong morita equivalence does for the category of c algebras in particular we prove that strong equivalence coincides with stable isomorphism under the expected countability hypothesis and that strongly tro equivalent operator spaces admit a correspondence between particular representations furthermore we show that strongly equivalent operator spaces have stably isomorphic second duals and strongly equivalent tro envelopes in the case of unital operator spaces strong equivalence implies stable isomorphism of the c envelopes

the structure of typical clusters in large sparse random configurations the initial purpose of this work is to provide a probabilistic explanation of recent results on a version of smoluchowski s coagulation equations in which the number of aggregations is limited the latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations under appropriate assumptions the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a galton watson process started from two ancestors

cluster tails for critical power law inhomogeneous random graphs recently the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank 1 type having finite variance but infinite third moment degrees was obtained in bhamidi et al ann probab 40 2299 2361 2012 it was proved that when the degrees obey a power law with exponent tau in 3 4 the sequence of clusters ordered in decreasing size and multiplied through by n tau 2 tau 1 converges as n rightarrow infty to a sequence of decreasing non degenerate random variables here we study the tails of the limit of the rescaled largest cluster i e the probability that the scaling limit of the largest cluster takes a large value u as a function of u this extends a related result of pittel j combin theory ser b 82 2 237 269 2001 for the erd s renyi random graph to the setting of rank 1 inhomogeneous random graphs with infinite third moment degrees we make use of delicate large deviations and weak convergence arguments

homotopy linearization and bifurcation on presente une prise en compte essentiellement auto contenue de la relation entre bifurcation locale et la determination a l aide de la linearisation de la non trivialite de certaines classes d homotopie d operateurs non lineaires

continuum limit of critical inhomogeneous random graphs the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real world systems in this study we analyze the maximal components at criticality of one famous class of such models the rank one inhomogeneous random graph model norros and reittu adv appl probab 38 1 59 75 2006 bollobas et al random struct algorithms 31 1 3 122 2007 section 16 4 viewing these components as measured random metric spaces under finite moment assumptions for the weight distribution we show that the components in the critical scaling window with distances scaled by n 1 3 converge in the gromov haussdorf prokhorov metric to rescaled versions of the limit objects identified for the erd s renyi random graph components at criticality in addario berry et al probab theory related fields 152 3 4 367 406 2012 a key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called mathbf p trees camarri and pitman electron j probab 5 2 1 18 2000 aldous et al probab theory related fields 129 2 182 218 2004 this is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics braunstein et al phys rev lett 91 16 168701 2003 for such graph models by asymptotic equivalence janson random struct algorithms 36 1 26 45 2010 the same results are true for the chung lu model chung and lu proc natl acad sci 99 25 15879 15882 2002 chung and lu ann combin 6 2 125 145 2002 chung and lu complex graphs and networks 2006 and the britton deijfen martin lof model britton et al j stat phys 124 6 1377 1397 2006 a crucial ingredient of the proof of independent interest are tail bounds for the height of mathbf p trees the techniques developed in this paper form the main technical bedrock for the general program developed in bhamidi et al scaling limits of random graph models at criticality universality and the basin of attraction of the erd s renyi random graph arxiv preprint 2014 for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels bollobas et al random struct algorithms 31 1 3 122 2007

universality for first passage percolation on sparse random graphs we consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density assuming that the degree distribution satises a uniform x 2 logx condition we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices as well the number of edges on this path or hopcount the hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous time branching process centered by a multiple of logn where the constant is the inverse of the malthusian rate of growth of the associated branching process the minimal weight converges in distribution the limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices and a gumbel random variable and thus shows a remarkably universal behavior the proofs rely on a rened coupling between the shortest path problems on these graphs and continuous time branching processes and on a poisson point process limit for the potential closing edges of shortest weight paths between the source and destination the results extend to a host of related random graph models ranging from random rregular graphs inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence

robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases this paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ode model defined on contact networks the model incorporates the spread of awareness among individuals and moreover a small inflow of imported cases these cases prevent stochastic extinctions when we simulate the epidemics and hence they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ode model stochastic simulations confirm the existence of sustained oscillations for different types of random networks with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values this abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the hopf bifurcation curve computed from the ode model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters

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

stably isomorphic dual operator algebras we prove that two unital dual operator algebras a b are stably isomorphic if and only if they are equivalent eleftherakis in j pure appl algebra arxiv math oa 0607489v4 2007 if and only if they have completely isometric normal representations on hilbert spaces h k respectively and there exists a ternary ring of operators mathcal m subset b h k such that alpha a mathcal m beta b mathcal m w and beta b mathcal m alpha a mathcal m w

bias in generation of random graphs we study the statistical properties of the generation of random graphs according to the configuration model in which one assigns randomly degrees to nodes this model is often used for example for the scale free degree distribution ensuremath sim phantom rule 0 16em 0ex d ensuremath ensuremath gamma for the efficient variant where nonfeasible edges are rejected and the construction of a graph continues there exists a bias which we calculate explicitly for a small sample ensemble we find that this bias does not disappear with growing system size this becomes visible for example also for scale free graphs when measuring quantities such as the graph diameter hence the efficient generation of general scale free graphs with a very broad distribution ensuremath gamma l2 remains an open problem

global results concerning bifurcation for fredholm maps of index zero with a transversality condition on considere l equation f x ax bx r x 0 ou a b est la derivee de frechet de f a l origine dans x et r un terme d ordre eleve x est un espace de banach et f x r y est un operateur non lineaire continu et y un espace de banach on etudie comment l ensemble solution varie avec le parametre reel

an operator space characterization of dual operator algebras this paper is devoted to dual algebras that is w closed algebras of bounded operators on hilbert space we give abstract characterizations of unital dual algebras resp dual algebras up to w homeomorphic unital complete isometry resp complete isomorphism the results we obtain are natural w versions of analogous characterizations due to blecher ruan sinclair and blecher respectively in the category of operator algebras

random walk kernels and learning curves for gaussian process regression on random graphs we consider learning on graphs guided by kernels that encode similarity between vertices our focus is on random walk kernels the analogues of squared exponential kernels in euclidean spaces we show that on large locally treelike graphs these have some counter intuitive properties specifically in the limit of large kernel lengthscales we consider using these kernels as covariance functions of gaussian processes in this situation one typically scales the prior globally to normalise the average of the prior variance across vertices we demonstrate that in contrast to the euclidean case this generically leads to significant variation in the prior variance across vertices which is undesirable from a probabilistic modelling point of view we suggest the random walk kernel should be normalised locally so that each vertex has the same prior variance and analyse the consequences of this by studying learning curves for gaussian process regression numerical calculations as well as novel theoretical predictions for the learning curves using belief propagation show that one obtains distinctly different probabilistic models depending on the choice of normalisation our method for predicting the learning curves using belief propagation is significantly more accurate than previous approximations and should become exact in the limit of large random graphs

a note on dynamical models on random graphs and fokker planck equations we address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model i e a model on the complete graph with a suitably renormalized interaction parameter examples include erdos renyi graphs with edge probability p n n is the number of vertices such that lim n to infty p n n infty the purpose of this note it twofold 1 to establish this proximity on finite time horizon by exploiting the fact that both systems are accurately described by a fokker planck pde or equivalently by a nonlinear diffusion process in the n infty limit 2 to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system

global bifurcation of a class of periodic boundary value problems abstract we prove that 0 is a global bifurcation point of the second order periodic boundary value problem p t x t x t 2 x t f t x t x t x t x 0 x 1 x 0 x 1 we study this equation under hypotheses for which it may be solved explicitly for x t however it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation thus we need to study the implicit equation with regard to global bifurcation

counting cliques and cycles in scale free inhomogeneous random graphs scale free networks contain many small cliques and cycles we model such networks as inhomogeneous random graphs with regularly varying infinite variance weights for these models the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis we obtain various asymptotic descriptions for how the average number of cliques and cycles of any size grow with the network size for the cycle asymptotics we invoke the theory of circulant matrices

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

growing networks with preferential deletion and addition of edges a preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed at each time step t 1 2 with probability 1 0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree with probability 2 a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex and with probability 3 1 1 2 1 3 the fraction pk decays exponentially at rate 1 2 2 3 there is hence a non trivial upper bound for how much deletion the network can incorporate without losing the power law behavior of the degree distribution the analytical results are supported by simulations

a characterization and a generalization of w modules we give a new banach module characterization of w modules also known as self dual hilbert c modules over a von neumann algebra this leads to a generalization of the notion and the theory of w modules to the setting where the operator algebras are weakly closed algebras of operators on a hilbert space that is we find the appropriate weak topology variant of our earlier notion of rigged modules and their theory which in turn generalizes the notions of a c module and a hilbert space successively our w rigged modules have canonical envelopes which are w modules indeed a w rigged module may be defined to be a subspace of a w module possessing certain properties

universality for distances in power law random graphs we survey the recent work on phase transition and distances in various random graph models with general degree sequences we focus on inhomogeneous random graphs the configuration model and affine preferential attachment models and pay special attention to the setting where these random graphs have a power law degree sequence this means that the proportion of vertices with degree k in large graphs is approximately proportional to k for some 1 since many real networks have been empirically shown to have power law degree sequences these random graphs can be seen as more realistic models for real complex networks than classical random graphs such as the erd s renyi random graph it is often suggested that the behavior of random graphs should have a large amount of universality meaning in this case that random graphs with similar degree sequences share similar behavior we survey the available results on graph distances in power law random graphs that are consistent with this prediction

universality for the distance in finite variance random graphs we generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs among others this class contains the poissonian random graph the expected degree random graph and the generalized random graph including the classical erdos renyi graph in the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes we consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two so that in particular the degrees have finite variance we prove that the graph distance grows like log nu n where the nu depends on the capacities and n denotes the size of the graph in addition the random fluctuations around this asymptotic mean log nu n are shown to be tight we also consider the case where the capacities are independent copies of a positive random lambda with p lambda x 3 againg resulting in graphs where the degrees have finite variance the method of proof of these results is to couple each member of the class to the poissonian random graph for which we then give the complete proof by adapting the arguments of van der hofstad et al