An Axiomatic and an Average-Case Analysis of Algorithms and Heuristics for Metric Properties of Graphs

Borassi, Michele; Crescenzi, Pierluigi; Trevisan, Luca

doi:10.1137/1.9781611974782.58

Cited by 17 publications

(17 citation statements)

References 46 publications

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“…Our proof techniques have the merit of adopting a unified approach that simultaneously works in all models considered. These models well represent metric properties of real-world networks [14]: indeed, our results are confirmed by practical experiments.…”

Section: Our Contributionsupporting

confidence: 76%

“…Miller, Marvin (I) 0.004708 0.004859 0.005015 [9][10][11][12] de Còrdova, Arturo 0.004147 0.004299 0.004457 [9][10][11][12][13][14][15][16][17][18] Haas, Hugo (I) 0.003888 0.004039 0.004197 [9][10][11][12][13][14][15][16][17][18] Singh, Ram (I) 0.003854 0.004004 0.004160 9-18) Kamiyama, Sōjin 0.003848 0.003999 0.004155 [10][11][12][13][14][15][16][17][18] Sauli, Anneli 0.003827 0.003978 0.004135 [10][11][12][13][14][15][16][17][18] King, Walter Woolf 0.003774 0.003923 0.004078 [10][11][12][13][14][15][16]…”

Section: G2 the Results On The Wikipedia Graphunclassified

“…Flowers, Bess 0.003446 0.003625 0.003815 [7][8][9][10][11][12][13][14][15][16][17] Mitchell, Gordon (I) 0.003417 0.003596 0.003785 7-17) Sullivan, Elliott 0.003371 0.003551 0.003740 7-17) Rietty, Robert 0.003368 0.003547 0.003735 7-17)…”

Section: G2 the Results On The Wikipedia Graphmentioning

confidence: 99%

“…The next ingredient is used to bound the first integers l s , l t such that r ls (s), r lt (t) > n 1 2 + . Theorem 18 (Theorem 5.1 in [24] for the CM, Theorem 14.8 in [12] for IRG (see also [45,14])). The diameter of a graph generated through the aforementioned models is O(log n).…”

Section: E Balanced Bidirectional Bfs On Random Graphsmentioning

confidence: 99%

“…Rey, Fernando (I) 0.003234 0.003420 0.003617 [7][8][9][10][11][12][13][14][15] Smith, William (I) 0.003012 0.003199 0.003397 [8][9][10][11][12][13][14][15] Welles, Orson 0.002885 0.003072 0.003271 [10][11][12][13][14][15] Mitchell, Gordon (I) 0.002851 0.003036 0.003232 [10][11][12][13][14][15] Kinski, Klaus 0.002705 0.002890 0.003087 [10][11][12][13][14][15] Mitchell, Cameron (I) 0.002671 0.002858 0.003058 [10][11][12][13][14][15] Quinn, Anthony (I) 0.002640 0.002826 0.003026 Carradine, John 0.002943 0.003114 0.003296 [6][7][8][9][10][11] Chen, Sing 0.002662 0.002834 0.003018 [7]…”

Section: Number Of Iterationsmentioning

confidence: 99%

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KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

Borassi

Natale

2019

ACM J. Exp. Algorithmics

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We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest.The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E| 1 2 +o(1) with high probability, obtaining a significant speedup with respect to the Θ(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well.The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well. * This work was done while the authors were visiting the Simons Institute for the Theory of Computing. 1 As explained in see Section 2, to simplify notation we consider the normalized betweenness centrality.

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Section: Our Contributionsupporting

confidence: 76%

Section: G2 the Results On The Wikipedia Graphunclassified

Section: G2 the Results On The Wikipedia Graphmentioning

confidence: 99%

Section: E Balanced Bidirectional Bfs On Random Graphsmentioning

confidence: 99%

Section: Number Of Iterationsmentioning

confidence: 99%

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KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

Borassi

Natale

2019

ACM J. Exp. Algorithmics

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The diameter of AT‐free graphs

Ducoffe

2021

Journal of Graph Theory

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A graph algorithm is truly subquadratic if it runs in O(m b ) time on connected m-edge graphs, for some positive b < 2. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no truly subquadratic algorithm for computing the diameter of general graphs. In this work, we present positive and negative results on the existence of such algorithms for computing the diameter on some special graph classes. Specifically, three vertices in a graph form an asteroidal triple (AT) if between any two of them there exists a path that avoids the closed neighbourhood of the third one. We call a graph AT-free if it does not contain an AT. We first prove that for all m-edge AT-free graphs, one can compute all the eccentricities in truly subquadratic O(m 3/2 ) time. For the AT-free bipartite graphs, it can be improved to linear time. Then, we extend our study to several subclasses of chordal graphs -all of them generalizing interval graphs in various ways -, as an attempt to understand which of the properties of AT-free graphs, or natural generalizations of the latter, can help in the design of fast algorithms for the diameter problem on broader graph classes. For instance, for all chordal graphs with a dominating shortest path, there is a linear-time algorithm for computing a diametral pair if the diameter is at least four. However, already for split graphs with a dominating edge, under plausible complexity assumptions, there is no truly subquadratic algorithm for deciding whether the diameter is either 2 or 3.

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Fast Approximation of Centrality and Distances in Hyperbolic Graphs

Chepoi¹,

Dragan²,

Habib³

et al. 2018

Combinatorial Optimization and Applications

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We show that the eccentricities (and thus the centrality indices) of all vertices of a δhyperbolic graph G = (V, E) can be computed in linear time with an additive one-sided error of at most cδ, i.e., after a linear time preprocessing, for every vertex v of G one can compute in O(1) time an estimateê(v) of its eccentricity eccG(v) such that eccG(v) ≤ê(v) ≤ eccG(v) + cδ for a small constant c. We prove that every δ-hyperbolic graph G has a shortest path tree, constructible in linear time, such that for every vertexThese results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of G, the smaller its eccentricity is. We also show that the distance matrix of G with an additive one-sided error of at most c ′ δ can be computed in O(|V | 2 log 2 |V |) time, where c ′ < c is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

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An Axiomatic and an Average-Case Analysis of Algorithms and Heuristics for Metric Properties of Graphs

Cited by 17 publications

References 46 publications

KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

The diameter of AT‐free graphs

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

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