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.
The methods for constructing graphs with a given degree distribution are primarily either reductions to perfect matchings or sequential sampling methods. There are two popular perfect matching methods. The first is the @cite_15 @cite_48 : @math mini-vertices are created for each degree @math vertex, and all the mini-vertices are connected. Any perfect matching in the configuration graph corresponds to a graph with the correct degree distribution by merging all of the identified mini-vertices. This allows multiple edges and self-loops, which are often undesirable. See Figure . The second approach, the , prevents multi-edges and self-loops by creating a gadget for each vertex. If @math has degree @math , then it is replaced with a complete bipartite graph @math with @math and @math . Exactly one node in each @math is connected to each other @math , representing edge @math @cite_6 . Any perfect matching in this model corresponds exactly to a simple graph by using the edges in the matching that correspond with edges connecting any @math to any @math . We use a natural extension of the first configuration model to the joint degree distribution problem.
@cite_48 @cite_15 @cite_6
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We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called e-far from k-vertex-connectivity when at least @math edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is e-far from k-vertex-connectivity with a probability of at least 2 3. The algorithm runs in @math time (c>1 is a constant) for (k−1)-vertex-connected graphs, and in @math time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4. In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let @math be an undirected and connected graph with @math vertices and @math edges. The coalescence time, @math , is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs and prove that @math . Here @math is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from, e.g., lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if required. The value of @math is given by @math , where @math is the degree of vertex @math , and @math is the average degree. The parame... We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph @math with @math vertices, and define (for @math ) @math . Our aim is to estimate @math within a multiplicative error of @math (for a given approximation parameter @math ) in sublinear time. We consider the sparse graph model that allows access to: uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of @math (the average degree), @math queries suffice for any constant @math (Feige, SICOMP 06 and Goldreich-Ron, RSA 08). Gonen-Ron-Shavitt (SIDMA 11) extended this result to all integral @math , by designing an algorithms that performs @math queries. We design a new, significantly simpler algorithm for this problem. In the worst-case, it exactly matches the bounds of Gonen-Ron-Shavitt, and has a much simpler proof. More importantly, the running time of this algorithm is connected to the degeneracy of @math . This is (essentially) the maximum density of an induced subgraph. For the family of graphs with degeneracy at most @math , it has a query complexity of @math . Thus, for the class of bounded degeneracy graphs (which includes all minor closed families and preferential attachment graphs), we can estimate the average degree in @math queries, and can estimate the variance of the degree distribution in @math queries. This is a major improvement over the previous worst-case bounds. Our key insight is in designing an estimator for @math that has low variance when @math does not have large dense subgraphs. Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study @math -edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices @math and @math are @math -edge-connected if there are two edge-disjoint paths from @math to @math and two edge-disjoint paths from @math to @math . This relation partitions the vertices into blocks such that all vertices in the same block are @math -edge-connected. Differently from the undirected case, those blocks do not correspond to the @math -edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are @math -edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has @math edges and maintains the same @math -edge-connected blocks as the input graph. 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. A graph @math is called @math -full if @math contains a subgraph @math of @math edges such that, for any non-empty @math , @math holds. Here, @math denotes the set of vertices incident to @math . It is known that the family of edge sets of @math -full graphs forms a family of matroid, known as the sparsity matroid of @math . In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for @math -fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property @math or far from @math ). Depending on the values of @math and @math , it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for @math -edge-connected-orientability in the bounded-degree model, where an undirected graph @math is called @math -edge-connected-orientable if there exists an orientation @math of @math with a vertex @math such that @math contains @math arc-disjoint dipaths from @math to each vertex @math and @math arc-disjoint dipaths from each vertex @math to @math . A tester is called a one-sided error tester for @math if it always accepts a graph satisfying @math . We show, for @math and (proper) @math , any one-sided error tester for @math -fullness and @math -edge-connected-orientability requires @math queries. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . A theorem of A. Schrijver asserts that a d-regular bipartite graph on 2n vertices has at least @math perfect matchings. L. Gurvits gave an extension of Schrijver’s theorem for matchings of density p. In this paper we give a stronger version of Gurvits’s theorem in the case of vertex-transitive bipartite graphs. This stronger version in particular implies that for every positive integer k, there exists a positive constant c(k) such that if a d-regular vertex-transitive bipartite graph on 2n vertices contains a cycle of length at most k, then it has at least @math perfect matchings. We study constant-time approximation algorithms for bounded-degree graphs, which run in time independent of the number of vertices @math . We present an algorithm that decides whether a vertex is contained in a some fixed maximal independent set with expected query complexity @math , where @math is the degree bound. Using this algorithm, we show constant-time approximation algorithms with certain multiplicative error and additive error @math for many other problems, e.g., the maximum matching problem, the minimum vertex cover problem, and the minimum set cover problem, that run exponentially faster than existing algorithms with respect to @math and @math . Our approximation algorithm for the maximum matching problem can be transformed to a two-sided error tester for the property of having a perfect matching. On the contrary, we show that every one-sided error tester for the property requires at least @math queries. We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate @math ( @math ) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent @math , then, as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of @math and @math . The picture is similar with multiple starting points for the infections. 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 consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesnik (Acta Fac. Rerum Nat. Univ. Comen. Math. 30:71–93, 1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesnik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v)≤deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least @math , which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree. For the set of graphs with a given degree sequence, consisting of any number of @math and @math , and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of @math -matchings. We find the expected value of the number of @math -matchings of @math -regular bipartite graphs on @math vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for @math -matchings in @math -regular bipartite graphs on @math vertices, and their asymptotic versions for infinite @math -regular bipartite graphs. We prove these conjectures for @math -regular bipartite graphs and for @math -matchings with @math . Consider an Erdos---Renyi random graph in which each edge is present independently with probability @math 1 2, except for a subset @math CN of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of (ANALCO. SIAM, pp 67---75, 2011) provably identifies the clique @math CN in linear time, provided @math |CN|?1.261N. Spectral methods can be shown to fail on cliques smaller than @math N. In this paper we describe a nearly linear-time algorithm that succeeds with high probability for @math |CN|?(1+?)N e for any @math ?>0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on @math N vertices). For large-girth regular graphs of degree @math (Δ+1) we prove that so-called local algorithms succeed if @math |CN|?(1+?)N eΔ and fail if @math |CN|≤(1-?)N eΔ. Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set 1,...,w , and given a parameter @math , estimates in time @math the weight of the minimum spanning tree (MST) of G with a relative error of at most @math . Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of @math on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time @math the number of connected components of an unweighted graph to within an additive error of @math . (This becomes @math for @math .) The time bound is shown to be tight up to within the @math factor. Our connected-components algorithm picks @math vertices in the graph and then grows "local spanning trees" whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. Friedland's Lower Matching Conjecture asserts that if @math is a @math --regular bipartite graph on @math vertices, and @math denotes the number of matchings of size @math , then @math where @math . When @math , this conjecture reduces to a theorem of Schrijver which says that a @math --regular bipartite graph on @math vertices has at least @math perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that @math In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra @math factor compared to the conjecture if @math is separated away from @math and @math , and is tight up to a constant factor if @math is separated away from @math . We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to @math --biregular bipartite graphs. We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as @math -free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time. For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is @math , where @math is the number of vertices in the graph, @math is the average degree of its vertices, and @math is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most @math .