61,780 research outputs found

    Local Fiedler vector centrality for detection of deep and overlapping communities in networks

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    Abstract—In this paper, a new centrality called local Fiedler vector centrality (LFVC) is proposed to analyze the connectivity structure of a graph. It is associated with the sensitivity of algebraic connectivity to node or edge removals and features distributed computations via the associated graph Laplacian matrix. We prove that LFVC can be related to a monotonic submodular set function that guarantees that greedy node or edge removals come within a factor 11=e of the optimal non-greedy batch removal strategy. Due to the close relationship between graph topology and community structure, we use LFVC to detect deep and overlapping communities on real-world social network datasets. The results offer new insights on community detection by discovering new significant communities and key members in the network. Notably, LFVC is also shown to significantly out-perform other well-known centralities for community detection. I

    Correlation Decay in Random Decision Networks

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    We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one

    Fast Distributed PageRank Computation

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    Over the last decade, PageRank has gained importance in a wide range of applications and domains, ever since it first proved to be effective in determining node importance in large graphs (and was a pioneering idea behind Google's search engine). In distributed computing alone, PageRank vector, or more generally random walk based quantities have been used for several different applications ranging from determining important nodes, load balancing, search, and identifying connectivity structures. Surprisingly, however, there has been little work towards designing provably efficient fully-distributed algorithms for computing PageRank. The difficulty is that traditional matrix-vector multiplication style iterative methods may not always adapt well to the distributed setting owing to communication bandwidth restrictions and convergence rates. In this paper, we present fast random walk-based distributed algorithms for computing PageRanks in general graphs and prove strong bounds on the round complexity. We first present a distributed algorithm that takes O\big(\log n/\eps \big) rounds with high probability on any graph (directed or undirected), where nn is the network size and \eps is the reset probability used in the PageRank computation (typically \eps is a fixed constant). We then present a faster algorithm that takes O\big(\sqrt{\log n}/\eps \big) rounds in undirected graphs. Both of the above algorithms are scalable, as each node sends only small (\polylog n) number of bits over each edge per round. To the best of our knowledge, these are the first fully distributed algorithms for computing PageRank vector with provably efficient running time.Comment: 14 page

    Distance matrices on the H-join of graphs: A general result and applications

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    Given a graph HH with vertices 1,,s1,\ldots ,s and a set of pairwise vertex disjoint graphs G1,,Gs,G_{1},\ldots ,G_{s}, the vertex ii of HH is assigned to Gi.G_{i}. Let GG be the graph obtained from the graphs G1,,GsG_{1},\ldots ,G_{s} and the edges connecting each vertex of GiG_{i} with all the vertices of GjG_{j} for all edge ijij of H.H. The graph GG is called the HjoinH-join of G1,,GsG_1,\ldots,G_s. Let M(G)M(G) be a matrix on a graph GG. A general result on the eigenvalues of M(G)M\left( G\right) , when the all ones vector is an eigenvector of M(Gi)M\left( G_{i}\right) for i=1,2,,si=1,2,\ldots ,s, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of GG when G1,,GsG_{1},\ldots ,G_{s} are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.publishe

    Algebraic Connectivity and Degree Sequences of Trees

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    We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector.Comment: 8 page

    On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks

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    We consider the multiple unicast problem with three source-terminal pairs over directed acyclic networks with unit-capacity edges. The three sitis_i-t_i pairs wish to communicate at unit-rate via network coding. The connectivity between the sitis_i - t_i pairs is quantified by means of a connectivity level vector, [k1k2k3][k_1 k_2 k_3] such that there exist kik_i edge-disjoint paths between sis_i and tit_i. In this work we attempt to classify networks based on the connectivity level. It can be observed that unit-rate transmission can be supported by routing if ki3k_i \geq 3, for all i=1,,3i = 1, \dots, 3. In this work, we consider, connectivity level vectors such that mini=1,,3ki<3\min_{i = 1, \dots, 3} k_i < 3. We present either a constructive linear network coding scheme or an instance of a network that cannot support the desired unit-rate requirement, for all such connectivity level vectors except the vector [1 2 4][1~2~4] (and its permutations). The benefits of our schemes extend to networks with higher and potentially different edge capacities. Specifically, our experimental results indicate that for networks where the different source-terminal paths have a significant overlap, our constructive unit-rate schemes can be packed along with routing to provide higher throughput as compared to a pure routing approach.Comment: To appear in the IEEE/ACM Transactions on Networkin
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