Finding Large Independent Sets in Networks Using Competitive Dynamics

TL;DR

This paper demonstrates that a biological-inspired competitive dynamical system on networks can approximate solutions to the computationally hard maximum independent set problem, with improved results under increased competitive pressure.

Contribution

It introduces a novel biologically inspired discrete algorithm for approximating maximum independent sets based on dynamical bifurcations and Katz centrality.

Findings

Dynamical systems can recover near-optimal independent sets.

Increasing competitive pressure improves solution optimality.

Bifurcation cascades relate to Katz centrality-based node removal.

Abstract

Many decision-making algorithms draw inspiration from the inner workings of individual biological systems. However, it remains unclear whether collective behavior among biological species can also lead to solutions for computational tasks. By studying the coexistence of species that interact through simple rules on a network, we demonstrate that the underlying dynamical system can recover near-optimal solutions to the maximum independent set problem -- a fundamental, computationally hard problem in graph theory. Furthermore, we observe that the optimality of these solutions is improved when the competitive pressure in the system is gradually increased. We explain this phenomenon by showing that the cascade of bifurcation points, which occurs with rising competitive pressure in our dynamical system, naturally gives rise to Katz centrality-based node removal in the network. By formalizing this connection, we propose a biologically inspired discrete algorithm for approximating the maximum independent set problem on a graph. Our results indicate that complex systems may collectively possess the capacity to perform non-trivial computations, with implications spanning biology, economics, and other fields.