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

TL;DR

This paper introduces an axiomatic framework to analyze the efficiency of algorithms computing metric properties of complex networks, demonstrating their effectiveness on models of power law random graphs with practical subquadratic time complexities.

Contribution

It provides the first axiomatic and average-case analysis of algorithms for graph metric problems, showing their efficiency on probabilistic models of real-world networks.

Findings

Algorithms compute diameter and radius in subquadratic time.

Possible to find top central vertices efficiently.

Distance oracle can be built with sublinear query time.

Abstract

In recent years, researchers proposed several algorithms that compute metric quantities of real-world complex networks, and that are very efficient in practice, although there is no worst-case guarantee. In this work, we propose an axiomatic framework to analyze the performances of these algorithms, by proving that they are efficient on the class of graphs satisfying certain axioms. Furthermore, we prove that the axioms are verified asymptotically almost surely by several probabilistic models that generate power law random graphs, such as the In recent years, researchers proposed several algorithms that compute metric quantities of real-world complex networks, and that are very efficient in practice, although there is no worst-case guarantee. In this work, we propose an axiomatic framework to analyze the performances of these algorithms, by proving that they are efficient on the class of graphs satisfying certain properties. Furthermore, we prove that these properties are verified asymptotically almost surely by several probabilistic models that generate power law random graphs, such as the Configuration Model, the Chung-Lu model, and the Norros-Reittu model. Thus, our results imply average-case analyses in these models. For example, in our framework, existing algorithms can compute the diameter and the radius of a graph in subquadratic time, and sometimes even in time $n^{1+o(1)}$. Moreover, in some regimes, it is possible to compute the $k$ most central vertices according to closeness centrality in subquadratic time, and to design a distance oracle with sublinear query time and subquadratic space occupancy. In the worst case, it is impossible to obtain comparable results for any of these problems, unless widely-believed conjectures are false.