Random @math -intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random @math -intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least @math items (the graph is called a random intersection graph when @math ). A special case of particular interest is a uniform random @math -intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random @math -intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random @math -intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random @math -intersection graphs). In this paper, for binomial uniform random @math -intersection graphs, we present results related to @math -connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) @math -vertex-connectivity, (ii) @math -edge-connectivity and (iii) the property of minimum vertex degree being at least @math .
Bloznelis @cite_45 investigates the clustering coefficient of graph @math and show that the chance of two neighbors of a given vertex @math to be adjacent decays as @math , where @math is a positive constant and @math is the degree of vertex @math . Bloznelis @cite_13 study the correlation coefficient of degrees of adjacent vertices in @math . Bloznelis @cite_19 demonstrate that a connected component in @math with at at least a constant fraction of @math emerges as @math when the edge probability @math exceeds @math .
@cite_19 @cite_45 @cite_13
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We establish asymptotic vertex degree distribution and examine its relation to the clustering coecient in two popular random intersection graph models of Godehardt and Jaworski (2001). For sparse graphs with positive clustering coecient, we examine statistical dependence between the (local) clustering coecient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of (2011) that correlates negatively with degree." Moreover, they explain empirical results on k 1 scaling of the local clustering coecient of a vertex of degree k reported in Ravasz and Barab asi (2003). 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 . We consider sparse random intersection graphs with the property that the clustering coefficient does not vanish as the number of nodes tends to infinity. We find explicit asymptotic expressions for the correlation coefficient of degrees of adjacent nodes (called the assortativity coefficient), the expected number of common neighbours of adjacent nodes, and the expected degree of a neighbour of a node of a given degree k. These expressions are written in terms of the asymptotic degree distribution and, alternatively, in terms of the parameters defining the underlying random graph model. Newman’s measure for (dis)assortativity, the linear degree correlation coefficient ?D, is reformulated in terms of the total number Nk of walks in the graph with k hops. This reformulation allows us to derive a new formula from which a degree-preserving rewiring algorithm is deduced, that, in each rewiring step, either increases or decreases ?D conform our desired objective. Spectral metrics (eigenvalues of graph-related matrices), especially, the largest eigenvalue ?1 of the adjacency matrix and the algebraic connectivity ?N?1 (second-smallest eigenvalue of the Laplacian) are powerful characterizers of dynamic processes on networks such as virus spreading and synchronization processes. We present various lower bounds for the largest eigenvalue ?1 of the adjacency matrix and we show, apart from some classes of graphs such as regular graphs or bipartite graphs, that the lower bounds for ?1 increase with ?D. A new upper bound for the algebraic connectivity ?N?1 decreases with ?D. Applying the degree-preserving rewiring algorithm to various real-world networks illustrates that (a) assortative degree-preserving rewiring increases ?1, but decreases ?N?1, even leading to disconnectivity of the networks in many disjoint clusters and that (b) disassortative degree-preserving rewiring decreases ?1, but increases the algebraic connectivity, at least in the initial rewirings. In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS C, a key component of a practical successful graph partitioner ( in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS C, which computes the similarity measure used in Bubble-FOS C and is therefore the most crucial component. By relating FOS C to random walks, we obtain precise characterizations of the behavior of FOS C on tori and hypercubes. Besides leading to new knowledge on FOS C (and therefore also on Bubble-FOS C), these characterizations have been recently used for the analysis of load balancing algorithms ( in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS C, at least one of the two parts is connected. Using the aforementioned relation between FOS C and random walks, we prove that in vertex-transitive graphs both parts must be connected components. We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. N. (2002) cast the problem thus: given a graph G whose edges are labeled "+" (similar) or "-" (dissimilar), partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respect to the input labeling is maximized (resp. minimized). Complete graphs, where the classifier labels every edge, and general graphs, where some edges are not labeled, are both worth studying. We answer several questions left open by N. (2002) and provide a sound overview of clustering with qualitative information. We give a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs is shown by N. (2002); we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APX-hardness. We also prove the APX-hardness of minimization on complete graphs. We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. Consider the random graph on n vertices 1,...,n. Each vertex i is assigned a type x(i) with x(1),...,x(n) being independent identically distributed as a nonnegative random variable X. We assume that EX3 < infinity. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability min 1, x(i)x(j) n (1 + a n(1 3)) . We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n(-2 3) the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient root EXEX3 and drift a - EX3 EX s. In particular, we conclude that the size of the largest connected component is of order n(2 3). (c) 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486-539, 2013 (Less) We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: • We present a poly log n round deterministic algorithm for (2Δ−1)·(1+o(1))-edge-coloring, where Δ denotes the maximum degree. Modulo the 1 + o(1) factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of (2Δ − 1) · poly log Δ-edge-coloring in poly log n rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. • We show that sinkless orientation---i.e., orienting edges such that each node has at least one out-going edge---on Δ-regular graphs can be solved in O(logΔ log n) rounds randomized and in O(logΔ n) rounds deterministically. These prove the corresponding lower bounds by [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of for Δ-coloring Δ-regular trees. • We present a randomized O(log4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + e). This can be also turned into a decomposition into a(1 + e) forests when a = Ω(log n) and into a(1 + e) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin. We consider the following general correlation-clustering problem [N. Bansal, A. Blum, S. Chawla, Correlation clustering, in: Proc. 43rd Annu. IEEE Symp. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250]: given a graph with real nonnegative edge weights and a 〈+〉 〈-〉 edge labelling, partition the vertices into clusters to minimize the total weight of cut 〈+〉 edges and uncut 〈-〉 edges. Thus, 〈+〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈-〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be best possible by the problem definition.Correlation clustering was introduced by [Correlation clustering, in: Proc. 43rd Annu. IEEE Syrup. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250], motivated by both document clustering and agnostic learning. They proved NP-hardness and gave constant-factor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)-approximation algorithm for the general case based on a linear-programming rounding and the "region-growing" technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r3)-approximation algorithm for Kr, r-minor-free graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APX-hard and difficult to approximate better than Θ(log n). Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ|I|. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ< λc=(Δ-1)Δ-1 (Δ-2)Δ. This improves on the previously known general bound of λ ≤ (2 ‾ Δ-2). The new regime includes the interesting case of λ=1 (uniform weights) and Δ ≤ 5. The previous bound required Δ ≤ 4 for uniform approximate counting and there is evidence that for Δ ≥ 6 the problem is hard. Note that λc is the critical activity for uniqueness of the Gibbs measure on the regular tree of degree Δ, i.e., for λ ≤ λc the probability that the root is in the independent set is asymptotically independent of the configuration on the leaves far below. Indeed, our analysis is focused on establishing decay of correlations with distance in the above weighted distribution. We show that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree. This resolves an open conjecture in statistical physics. Our comparison of a general graph with the tree uses an algorithmic argument yielding the approximation scheme mentioned above. Also, by existing arguments, establishing decay of correlations for all graphs and λ<λc gives that the Glauber dynamics is rapidly mixing in this regime. However, the implication from decay of correlations to rapid mixing of the dynamics is only known to hold for graphs of subexponential growth, and hence, our result regarding the Glauber dynamics is limited to this class of graphs. We consider the problem of encoding graphs with n vertices and m edges compactly supporting adjacency, neighborhood and degree queries in constant time in the @Q(logn)-bit word RAM model. The adjacency query asks whether there is an edge between two vertices, the neighborhood query reports the neighbors of a given vertex in constant time per neighbor, and the degree query reports the number of incident edges to a given vertex. We study the problem in the context of succinctness, where the goal is to achieve the optimal space requirement as a function of n and m, to within lower order terms. We prove a lower bound in the cell probe model indicating it is impossible to achieve the information-theory lower bound up to lower order terms unless the graph is either too sparse (namely, m=o(n^@d) for any constant @d>0) or too dense (namely m=@w(n^2^-^@d) for any constant @d>0). Furthermore, we present a succinct encoding of graphs supporting aforementioned queries in constant time. The space requirement of the encoding is within a multiplicative 1+@e factor of the information-theory lower bound for any arbitrarily small constant @e>0. This is the best achievable space bound according to our lower bound where it applies. The space requirement of the representation achieves the information-theory lower bound tightly within lower order terms where the graph is very sparse (m=o(n^@d) for any constant @d>0), or very dense (m>n^2 lg^1^-^@dn for an arbitrarily small constant @d>0). We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters ( ) and ( ). The non-negative parameter ( ) measures the amount of loops, and ( ) plays the role of killing on vertices penalizing ( ( >0 )) or favoring ( ( <0 )) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed ( ) and large enough ( ), there exists an infinite cluster in the loop percolation on ( Z ^d ). In the present article, we show a non-trivial phase transition on the integer lattice ( Z ^d ) ( (d 3 )) for ( =0 ). More precisely, we show that there is no loop percolation for ( =0 ) and ( ) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of ( ), namely, for ( =0 ) and any sub-critical value of ( ), the probability of one-arm event decays at most polynomially. For (d 5 ), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For ( ) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For (d=3 ) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity ( ). Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan[2010] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target @math -clustering, the projection of a point @math onto the line joining its cluster center @math and some other center @math , is a large additive factor closer to @math than to @math . This additive factor can be roughly described as @math times the spectral norm of the matrix representing the differences between the given (known) dataset and the means of the (unknown) target clustering. Clearly, the proximity condition implies center separation -- the distance between any two centers must be as large as the above mentioned bound. In this paper we improve upon the work of Kumar and Kannan along several axes. First, we weaken the center separation bound by a factor of @math , and secondly we weaken the proximity condition by a factor of @math . Using these weaker bounds we still achieve the same guarantees when all points satisfy the proximity condition. We also achieve better guarantees when only @math -fraction of the points satisfy the weaker proximity condition. The bulk of our analysis relies only on center separation under which one can produce a clustering which (i) has low error, (ii) has low @math -means cost, and (iii) has centers very close to the target centers. Our improved separation condition allows us to match the results of the Planted Partition Model of McSherry[2001], improve upon the results of [2006], and improve separation results for mixture of Gaussian models in a particular setting. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is far from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k-connected (for k > 1), cycle-free and Eulerian. Our algorithms work in time polynomial in 1 e, always accept the graph when it has the tested property, and reject with high probability if the graph is s-far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N-vertex d-degree graph for which more than edN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(√N) on the query complexity of testing algorithms for the bipartite and expander properties. 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. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114. The networks considered here consist of sets of intercon- nected vertices, examples of which include social networks, technological networks, and biological networks. Two important issues are to measure the extent of proximity between vertices and to identify the community structure of a network. In this paper, the proximity index between two nearest-neighboring vertices of a network is measured by a biased Brown- ian particle which moves on the network. This proximity index integrates both the local and the global structural information of a given network, and it is used by an agglomerative hierarchical algorithm to identify the community structure of the network. This method is applied to several artificial or real-world networks and satisfying results are attained. Find- ing the proximity indices for all nearest-neighboring vertex pairs needs a computational time that scales as O(N 3 ), with N being the total number of vertices in the network. In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficiency and or effectiveness perspective. Recent research results show that in high dimensional space, the concept of proximity, distance or nearest neighbor may not even be qualitatively meaningful. In this paper, we view the dimensionality curse from the point of view of the distance metrics which are used to measure the similarity between objects. We specifically examine the behavior of the commonly used Lk norm and show that the problem of meaningfulness in high dimensionality is sensitive to the value of k. For example, this means that the Manhattan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more meaningful results both from the theoretical and empirical perspective. The results show that fractional distance metrics can significantly improve the effectiveness of standard clustering algorithms such as the k-means algorithm. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.