A Scalable Distributed Dynamical Systems Approach to Learn the Strongly Connected Components and Diameter of 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(Dd_{\rm{in-degree}}^{\max })$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> is the (finite) diameter of the network and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d_{\rm{in-degree}}^{\max }$</tex-math></inline-formula> is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D+1$</tex-math></inline-formula> iterations, which allows us to retrieve the diameter of the network. We illustrate the performance of our algorithm on several random networks, including Erdö-Rényi, Barabási-Albert, and Watts-Strogatz networks.