150,013 research outputs found

    Ollivier curvature, betweenness centrality and average distance

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    We give a new upper bound for the average graph distance in terms of the average Ollivier curvature. Here, the average Ollivier curvature is weighted with the edge betweenness centrality. Moreover, we prove that equality is attained precisely for the reflective graphs which have been classified as Cartesian products of cocktail party graphs, Johnson graphs, halved cubes, Schl\"afli graphs, and Gosset graphs

    Shortest Distances as Enumeration Problem

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    We investigate the single source shortest distance (SSSD) and all pairs shortest distance (APSD) problems as enumeration problems (on unweighted and integer weighted graphs), meaning that the elements (u,v,d(u,v))(u, v, d(u, v)) -- where uu and vv are vertices with shortest distance d(u,v)d(u, v) -- are produced and listed one by one without repetition. The performance is measured in the RAM model of computation with respect to preprocessing time and delay, i.e., the maximum time that elapses between two consecutive outputs. This point of view reveals that specific types of output (e.g., excluding the non-reachable pairs (u,v,)(u, v, \infty), or excluding the self-distances (u,u,0)(u, u, 0)) and the order of enumeration (e.g., sorted by distance, sorted row-wise with respect to the distance matrix) have a huge impact on the complexity of APSD while they appear to have no effect on SSSD. In particular, we show for APSD that enumeration without output restrictions is possible with delay in the order of the average degree. Excluding non-reachable pairs, or requesting the output to be sorted by distance, increases this delay to the order of the maximum degree. Further, for weighted graphs, a delay in the order of the average degree is also not possible without preprocessing or considering self-distances as output. In contrast, for SSSD we find that a delay in the order of the maximum degree without preprocessing is attainable and unavoidable for any of these requirements.Comment: Updated version adds the study of space complexit

    Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees

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    The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the distance relation via weighted sampling. Based on that, we present highly practical algorithms with strong statistical guarantees for fundamental problems. We show that the average distance (and hence the centrality) for all nodes in a graph can be estimated using O(ϵ2)O(\epsilon^{-2}) single-source distance computations. For a set VV of nn points in a metric space, we show that after preprocessing which uses O(n)O(n) distance computations we can compute a weighted sample SVS\subset V of size O(ϵ2)O(\epsilon^{-2}) such that the average distance from any query point vv to VV can be estimated from the distances from vv to SS. Finally, we show that for a set of points VV in a metric space, we can estimate the average pairwise distance using O(n+ϵ2)O(n+\epsilon^{-2}) distance computations. The estimate is based on a weighted sample of O(ϵ2)O(\epsilon^{-2}) pairs of points, which is computed using O(n)O(n) distance computations. Our estimates are unbiased with normalized mean square error (NRMSE) of at most ϵ\epsilon. Increasing the sample size by a O(logn)O(\log n) factor ensures that the probability that the relative error exceeds ϵ\epsilon is polynomially small.Comment: 21 pages, will appear in the Proceedings of RANDOM 201

    Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs

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    We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension dd. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d=2d=2. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the AA^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra's algorithm. Specifically, if the nn corresponding points of a weighted unit-disk graph GG are picked from a unit square uniformly at random, and the connectivity radius is r(0,1)r\in (0,1), AA^* finds a shortest path in GG in O(n)O(n) expected time when r=Ω(logn/n)r=\Omega(\sqrt{\log n/n}), even though GG has Θ((nr)2)\Theta((nr)^2) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to AA^*. Specifically, we show that, if we are able to report the set of all kk points of GG from an arbitrary rectangular region of the plane in O(k+t(n))O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r2+t(n))O(1/r^2 + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r=Ω(logn/n)r=\Omega(\sqrt{\log n/n}) and O(n)O(\sqrt{n}) expected bound for r=Ω(n1/4)r=\Omega(n^{-1/4}) after only near-linear preprocessing of the point set.Comment: Full version of a SoCG'21 paper. Abstract truncated to meet arxiv requirement

    An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification

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    We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if GG is an nn-node weighted undirected graph of average combinatorial degree dd (that is, GG has dn/2dn/2 edges) and girth g>2d1/8+1g> 2d^{1/8}+1, and if λ1λ2λn\lambda_1 \leq \lambda_2 \leq \cdots \lambda_n are the eigenvalues of the (non-normalized) Laplacian of GG, then λnλ21+4dO(1d58) \frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O \left( \frac 1{d^{\frac 58} }\right) (The Alon-Boppana theorem implies that if GG is unweighted and dd-regular, then λnλ21+4dO(1d)\frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O\left( \frac 1 d \right) if the diameter is at least d1.5d^{1.5}.) Our result implies a lower bound for spectral sparsifiers. A graph HH is a spectral ϵ\epsilon-sparsifier of a graph GG if L(G)L(H)(1+ϵ)L(G) L(G) \preceq L(H) \preceq (1+\epsilon) L(G) where L(G)L(G) is the Laplacian matrix of GG and L(H)L(H) is the Laplacian matrix of HH. Batson, Spielman and Srivastava proved that for every GG there is an ϵ\epsilon-sparsifier HH of average degree dd where ϵ42d\epsilon \approx \frac {4\sqrt 2}{\sqrt d} and the edges of HH are a (weighted) subset of the edges of GG. Batson, Spielman and Srivastava also show that the bound on ϵ\epsilon cannot be reduced below 2d\approx \frac 2{\sqrt d} when GG is a clique; our Alon-Boppana-type result implies that ϵ\epsilon cannot be reduced below 4d\approx \frac 4{\sqrt d} when GG comes from a family of expanders of super-constant degree and super-constant girth. The method of Batson, Spielman and Srivastava proves a more general result, about sparsifying sums of rank-one matrices, and their method applies to an "online" setting. We show that for the online matrix setting the 42/d4\sqrt 2 / \sqrt d bound is tight, up to lower order terms
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