Finding Nontrivial Minimum Fixed Points in Discrete Dynamical Systems

TL;DR

This paper investigates the computational challenge of finding minimal fixed points in discrete dynamical systems, especially in network contagion models, and proposes algorithms and heuristics for practical solutions.

Contribution

It formulates a new optimization problem for minimal fixed points, proves its computational hardness, and offers efficient solutions for special cases and heuristic methods for larger networks.

Findings

Hardness of approximation established unless P=NP

Efficient solutions for special cases identified

Heuristics show effectiveness on real-world networks

Abstract

Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the dissemination of undesirable contagions (such as rumors and misinformation), convergence to fixed points with a small number of affected nodes is a desirable goal. Motivated by such considerations, we formulate a novel optimization problem of finding a nontrivial fixed point of the system with the minimum number of affected nodes. We establish that, unless P = NP, there is no polynomial time algorithm for approximating a solution to this problem to within the factor n^1-\epsilon for any constant epsilon > 0. To cope with this computational intractability, we identify several special cases for which the problem can be solved efficiently. Further, we introduce an integer linear program to address the problem for networks of reasonable sizes. For solving the problem on larger networks, we propose a general heuristic framework along with greedy selection methods. Extensive experimental results on real-world networks demonstrate the effectiveness of the proposed heuristics.