A scalable distributed dynamical systems approach to compute the strongly connected components and diameter of networks

TL;DR

This paper introduces a scalable, distributed algorithm based on consensus principles for efficiently computing strongly connected components and the diameter of large directed networks, with proven convergence and applicability to various network models.

Contribution

It presents a novel distributed method for SCC and diameter computation with proven complexity and convergence guarantees, suitable for large-scale networks.

Findings

Algorithm terminates in D+1 iterations

Computational complexity is O(D d_in_max)

Effective on Erdős-Rényi, Barabási-Albert, Watts-Strogatz networks

Abstract

Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of discrete optimization problems, and subsequently, machine learning and control theory problems. On the one hand, SCCs are used in solving the 2-satisfiability problem, which has applications in clustering, scheduling, and visualization. On the other hand, the diameter has applications in network learning and discovery problems enabling efficient internet routing and searches, as well as identifying faults in the power grid. In this paper, we leverage consensus-based principles to find the SCCs in a scalable and distributed fashion with a computational complexity of $\mathcal{O}\left(Dd_{\text{in-degree}}^{\max}\right)$, where $D$ is the (finite) diameter of the network and $d_{\text{in-degree}}^{\max}$ is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in $D+1$ iterations, which allows us to retrieve the diameter of the network. We illustrate the performance of our algorithm on several random networks, including Erd\H{o}s-R\'enyi, Barab\'asi-Albert, and \mbox{Watts-Strogatz} networks.