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

Abstract

International audienceIn this paper, we propose a new algorithm that computes the radius and the diameter of a weakly connected digraph G=(V,E), by finding bounds through heuristics and improving them until they are validated. Although the worst-case running time is O(|V||E|), we will experimentally show that it performs much better in the case of real-world networks, finding the radius and diameter values after 10–100 BFSs instead of |V| BFSs (independently of the value of |V|), and thus having running time O(|E|) in practice. As far as we know, this is the first algorithm able to compute the diameter of weakly connected digraphs, apart from the naive algorithm, which runs in time Ω(|V||E|) performing a BFS from each node. In the particular cases of strongly connected directed or connected undirected graphs, we will compare our algorithm with known approaches by performing experiments on a dataset composed by several real-world networks of different kinds. These experiments will show that, despite its generality, the new algorithm outperforms all previous methods, both in the radius and in the diameter computation, both in the directed and in the undirected case, both in average running time and in robustness. Finally, as an application example, we will use the new algorithm to determine the solvability over time of the “Six Degrees of Kevin Bacon” game, and of the “Six Degrees of Wikipedia” game. As a consequence, we will compute for the first time the exact value of the radius and the diameter of the whole Wikipedia digraph

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

“…These results explain the surprisingly small running time on most graphs, and the reason why the SumS algorithm is usually faster on "hard" instances, as observed in [13,14]. It is interesting to note that all the running times for β > 3 depend on the same constant C, which we prove to be close to the radius: in this case, in all regimes, the running time is almost linear, confirming the results in [14], where it is shown that the algorithm needed at most 10 BFSes on all inputs but one, and in the last input it needed 18 BFSes. The other two algorithms analyzed are the BCM algorithm, to compute the k most central vertices according to closeness centrality [15], and the distance oracle AIY in [4].…”

“…SumS [13,14] n 1+o(1) n 1+o(1) ≤ mn ance is unbounded, while if β > 3 also the variance is finite. Furthermore, all the results with β > 3 can be easily generalized to any degree distribution with finite variance, but the results become more cumbersome and dependent on specific characteristics of the distribution, such as the maximum degree of a vertex (for this reason, we focus on the power law case).…”

“…9 The SumSweep Heuristic. The SumS heuristic [13,14] provides a lower bound on the eccentricity of all vertices, by performing some BFSes from vertices t 1 , . .…”

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

“…In directed graphs, depending on the application, there may be multiple definitions for "closeness": a node can be close in the sense that it has short paths to other nodes ("source"), from other nodes ("target"), or even to and then back from other nodes ("roundtrip"). That is, there are several natural definitions of both radius and diameter for directed graphs; all of them are well-studied [28,40,27,36,6,30,26,35,11,12,57,58,23,38,54,47,25,2,17] (and many others). Even estimating the diameter and radius of a network efficiently is useful in practical applications (e.g.…”

A note on the complexity of computing the number of reachable vertices in a digraph

“…In this work, we consider the following problem: given a digraph G = (V, E), for each vertex v, we want to compute the number of vertices reachable from v. An efficient solution of this problem could have many applications: to name a few, there are algorithms that need to compute (or estimate) these values [6], the number of reachable vertices is used in the definition of other measures, like closeness centrality [10,14,11], and it can be useful in the analysis of the transitive closure of a graph (indeed, the out-degree of a vertex v in the transitive closure is the number of vertices reachable from v).…”