One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, i.e. the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this paper and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov Chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via end point switches. We empirically evaluate the mixing time of this Markov Chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov Chain mixes quickly on real graphs, allowing for utilization of our techniques in practice.
Another vein of related work is that of who introduce the concept of @math -series @cite_21 @cite_17 . In this model, @math refers to the dimension of the distribution and @math is the joint degree distribution. They propose a heuristic for generating random @math -graphs for a fixed @math distribution via edge rewirings. However, their method can get stuck if there exists a degree in the graph for which there is only 1 node with that degree. This is because the state space is not connected. We provide a theoretically sound method of doing this. Finally, Newman also studies the problem of fixing an assortativity value, finding a with that value, and then sampling a random graph with that distribution using Markov Chains @cite_50 @cite_36 . His Markov Chain starts at any graph with the correct degree distribution and converges to a pseudograph with the correct joint remaining degree distribution. By contrast, our work provides a theoretically sound way of constructing a simple graph with a given joint degree distribution first, and our Markov Chain only has simple graphs with the same joint degree distribution as its state space.
@cite_36 @cite_21 @cite_50 @cite_17
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One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes @math . To prove fast mixing, we first prove a general factorization theorem similar to the Martin--Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and ... The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting. We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree @math which satisfies @math , where @math is the sum of the degrees. The mixing time bound obtained is only a factor @math larger than that established in the regular case, where @math is the number of vertices. Our result covers a wide range of degree sequences, including power-law graphs with parameter @math and sufficiently many edges. For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by @math , where @math is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of @math larger than that established in the regular case. We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the SLEM, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, @math ) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say, 100,000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis--Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution. In network modeling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a joint degree matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tet ali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence and conjectured that it mixes rapidly (in poly-time) for arbitrary degree sequences. While the conjecture is still open, it was proven for special degree sequences, in particular, for those of undirected and directed regular simple graphs, of half-regular bipartite graphs, and of graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite splitted degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs. In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math . Sampling from combinatorial families can be difficult. However, complicated families can often be embedded within larger, simpler ones, for which easy sampling algorithms are known. We take advantage of such a relationship to describe a sampling algorithm for the smaller family, via a Markov chain started at a random sample of the larger family. The utility of the method is demonstrated via several examples, with particular emphasis on sampling labelled graphs with given degree sequence, a well-studied problem for which existing algorithms leave much room for improvement. For graphs with given degrees, with maximum degree @math where @math is the number of edges, we obtain an asymptotically uniform sample in @math steps, which substantially improves upon existing algorithms. In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on @math vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in @math in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method. We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius @math around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit. The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree @math , where @math is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock [ Annals of Applied Probability 1991], and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill [ Combinatorics, Probability and Computing 2007], but with significant extension required. We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter @math , the survival time of the process is larger than a stretched exponential function of the number of vertices, @math . We obtain sharp bounds for the typical density of infected sites in the graph, as @math is kept fixed and @math tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law. By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in @math O(Δ) rounds, independently of @math n; here @math Δ is the maximum degree of the graph and @math n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in @math o(Δ) rounds, independently of @math n. Our work gives the first linear-in- @math Δ lower bound for a natural graph problem in the standard @math LOCAL model of distributed computing--prior lower bounds for a wide range of graph problems have been at best logarithmic in @math Δ. We show that any randomised Monte Carlo distributed algorithm for the Lov 'asz local lemma requires @math communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of @math , where @math is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lov 'asz local lemma with a running time of @math rounds in bounded-degree graphs, and the best lower bound before our work was @math rounds [ 2014]. We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, clustering coefficient, and diameter relate to the threshold value and weight distribution. We give bounds on the threshold value guaranteeing the presence or absence of a giant component, connectivity and disconnectivity of the graph, and small diameter. Finally, we consider the clustering coefficient for nodes with a given degree l, finding that its scaling is very close to 1 l when the node weights are exponentially distributed. Distributed voting is a fundamental topic in distributed computing. In the standard model of pull voting, at each step every vertex chooses a neighbour uniformly at random and adopts its opinion. The voting is completed when all vertices hold the same opinion. In the simplest case, each vertex initially holds one of two different opinions. This partitions the vertices into arbitrary sets A and B. For many graphs, including regular graphs and irrespective of their expansion properties, if both A and B are sufficiently large sets, then pull voting requires @math expected steps, where n is the number of vertices of the graph. In this paper we consider a related class of voting processes based on sampling two opinions. In the simplest case, every vertex v chooses two random neighbours at each step. If both these neighbours have the same opinion, then v adopts this opinion. Otherwise, v keeps its own opinion. Let G be a connected graph with n vertices and m edges. Let P be the transition matrix of a simple random walk on G with second largest eigenvalue @math . We show that if the initial imbalance in degree between the two opinions satisfies @math , then with high probability voting completes in @math steps, and the opinion with the larger initial degree wins. The condition that @math , or only a bound on the conductance of the graph is known, the sampling process can be modified so that voting still provably completes in @math steps with high probability. The modification uses two sampling based on probing to a fixed depth @math from any vertex. In its most general form our voting process allows vertices to bias their sampling of opinions among their neighbours to achieve a desired outcome. This is done by allocating weights to edges. Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: @math , with @math . We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity. 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. Breadth First Search (BFS) and other graph traversal techniques are widely used for measuring large unknown graphs, such as online social networks. It has been empirically observed that incomplete BFS is biased toward high degree nodes. In contrast to more studied sampling techniques, such as random walks, the bias of BFS has not been characterized to date. In this paper, we quantify the degree bias of BFS sampling. In particular, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction of covered nodes, in a random graph RG(p k ) with a given (and arbitrary) degree distribution p k . Furthermore, we also show that, for RG(p k ), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, and Snowball Sampling) lead to the same bias, and we show how to correct for this bias. To give a broader perspective, we compare this class of exploration techniques to random walks that are well-studied and easier to analyze. Next, we study by simulation the effect of graph properties not captured directly by our model. We find that the bias gets amplified in graphs with strong positive assortativity. Finally, we demonstrate the above results by sampling the Facebook social network, and we provide some practical guidelines for graph sampling in practice. Let X_ 1 ,X_ 2 , be independent identically distributed random variables taking values in a finite set X and consider the conditional joint distribution of the first m elements of the sample X_ 1 , , X_ n on the condition that X_ 1 =x_ 1 and the sliding block sample average of a function h( , ) defined on X^ 2 exceeds a threshold > Eh(X_ 1 , X_ 2 ) . For m fixed and n , this conditional joint distribution is shown to converge m the m -step joint distribution of a Markov chain started in x_ 1 which is closest to X_ l , X_ 2 , in Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distribution P( , ) satisfies P(x, y)h(x, y) , provided some distribution P on X_ 2 having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when X_ 1 , X_ 2 , is an arbitrary finite-order Markov chain and more general conditioning is allowed. We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of @math nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus @math scales as @math , where @math is the @math th moment of the degree distribution. For a power-law degree distribution @math , @math thus scales as @math for @math , as @math for @math , as @math for @math , as @math for @math , and as @math for @math . These results agree with simulation data for networks with both uncorrelated and correlated node degrees.