24,230 research outputs found

    The Minimum Wiener Connector

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    The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph G=(V,E)G=(V,E) and a set QVQ\subseteq V of query vertices, find a subgraph of GG that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless P=NP\mathbf{P} = \mathbf{NP}. Our main contribution is a constant-factor approximation algorithm running in time O~(QE)\widetilde{O}(|Q||E|). A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set QQ a small number of important vertices (i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Dat

    Distributed Community Detection in Dynamic Graphs

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    Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied \emph{Planted Bisection Model} \sdG(n,p,q) where the node set [n][n] of the network is partitioned into two unknown communities and, at every time step, each possible edge (u,v)(u,v) is active with probability pp if both nodes belong to the same community, while it is active with probability qq (with q<<pq<<p) otherwise. We also consider a time-Markovian generalization of this model. We propose a distributed protocol based on the popular \emph{Label Propagation Algorithm} and prove that, when the ratio p/qp/q is larger than nbn^{b} (for an arbitrarily small constant b>0b>0), the protocol finds the right "planted" partition in O(logn)O(\log n) time even when the snapshots of the dynamic graph are sparse and disconnected (i.e. in the case p=Θ(1/n)p=\Theta(1/n)).Comment: Version I
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