Abstract
Networked discrete dynamical systems are commonly used as formal models to study the spread of contagions over networks. Fixed points of such dynamical systems represent configurations to which systems may converge. In the case of undesirable contagions (such as epidemics, fake news and rumors) spreading over social networks (where nodes represent a population), convergence to configurations with a small number of affected nodes is highly desirable. Motivated by such considerations, we formulate a novel optimization problem for discrete dynamical systems, where the state of each node is from {0, 1} and state 1 indicates a node that has been infected by the contagion. Specifically, the goal of the problem is to find a fixed point of such a system where the number of nodes in state 1 is a minimum. We also require such a fixed point to be nontrivial (to avoid the trivial case where all nodes are in state 0). Our results are for dynamical systems where the behavior of each node is specified by a threshold function. These functions capture simple as well as complex contagions and have also been studied in game theory. We show that, unless P = NP, there is no polynomial time algorithm for approximating the solution to within the factor n^{1 - epsilon} for any constant \epsilon > 0, even when the underlying network is bipartite. We then explore several approaches to cope with the computational intractability of the problem. First, we identify several special cases (e.g., dynamical systems over directed acyclic graphs, progressive contagions where nodes cannot transition from state 1 to state 0) for which the problem can be solved efficiently. We also identify a version that is fixed parameter tractable with respect to a structural parameter associated with the problem. Next, we develop an integer linear programming (ILP) formulation for the problem so that existing solvers such as Gurobi can be used to solve the problem in practice for networks of reasonable size. For larger networks, we propose a general heuristic framework for the problem. We present extensive experimental results using real-world networks to demonstrate the effectiveness of the proposed heuristic framework.
Keywords
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