Fast computation of empirically tight bounds for the diameter of massive graphs

Abstract

The diameter of a graph is among its most basic parameters. Since a few years ago, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various real-world cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice. ACM Reference Format:Magnien, C., Latapy, M., and Habib, M. 2008. Fast computation of empirically tight bounds for the diameter of massive graphs. ACM J.Throughout this article, we consider a connected undirected unweighted graph G = (V , E) with n = |V | vertices and m = |E| edges. We denote d (u, v) the distance between u and v in G, ecc(v) = max u d (v, u) the eccentricity of v in G, and D = max u,v d (u, v) = max v ecc(v) the diameter of G.

Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs

“…This algorithm will not only be more general than all previous counterparts, but it will also outperform them on directed, strongly connected graphs or undirected graphs. It relates the sweep approach (i.e., a new visit of the graph depends on the previous one, as in [14,13,22,23]) with the techniques developed in [31,32]. It is based on a new heuristic, named SumSweep, which is able to compute very efficiently lower bounds on the diameter and upper bounds on the radius of a given graph.…”

“…For this reason, several papers dealt with the problem of appropriately choosing the vertices from which the BFSs have to be performed. For example, the so-called 2Sweep heuristic picks one of the farthest vertices x from a random vertex r and returns the distance of the farthest vertex from x [22], while the 4Sweep picks the vertex in the middle of the longest path computed by a 2Sweep execution and performs another 2Sweep from that vertex [13]. Both methods work quite well and very often provide tight bounds.…”

Determining the diameter of small world networks

“…Similarly, the maximum upper bound on the eccentricity over all nodes can be seen as an upper bound on the diameter. The upper bound can even be made more tight by observing that this bound can be at most twice as big as the smallest eccentricity upper bound over all nodes, as observed in [14]. These observations can be formalized as follows:…”

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

“…We used the NetworkX Python package [19] to generate our random intersection graphs (using the uniform random intersection graph method), and the SageMath software system [14] to compute the hyperbolicity [6,9,16], degeneracy [1,38] and diameter [10,11,27,45] of the generated graphs. The measurements of the p-centered coloring number (presented below) were executed using the implementation available in [37].…”