Limiting the Neighborhood: De-Small-World Network for Outbreak Prevention

In this work, we study a basic and practically important strategy to help prevent and/or delay an outbreak in the context of network: limiting the contact between individuals. In this paper, we introduce the average neighborhood size as a new measure for the degree of being small-world and utilize it to formally define the desmall- world network problem. We also prove the NP-hardness of the general reachable pair cut problem and propose a greedy edge betweenness based approach as the benchmark in selecting the candidate edges for solving our problem. Furthermore, we transform the de-small-world network problem as an OR-AND Boolean function maximization problem, which is also an NP-hardness problem. In addition, we develop a numerical relaxation approach to solve the Boolean function maximization and the de-small-world problem. Also, we introduce the short-betweenness, which measures the edge importance in terms of all short paths with distance no greater than a certain threshold, and utilize it to speed up our numerical relaxation approach. The experimental evaluation demonstrates the effectiveness and efficiency of our approaches.

💡 Research Summary

The paper addresses a fundamental yet under‑explored strategy for mitigating the spread of epidemics, computer viruses, spam, or misinformation: limiting contacts by removing edges rather than quarantining vertices. Recognizing that the classic small‑world measure—average shortest‑path length—fails to uniformly handle both connected and fragmented graphs, the authors introduce a new metric, the average neighborhood size. For a user‑specified propagation horizon k (e.g., 2–4 hops), the neighborhood N_k(v) of a vertex v is the set of vertices reachable within k steps; the average of |N_k(v)| over all vertices quantifies how many individuals could be infected within that time window. The “de‑small‑world network problem” is then defined as selecting a set E_r of exactly L edges to delete so that this average neighborhood size is minimized.

The authors reformulate the problem as a “reachable‑pair cut”: a pair (u, v) is locally reachable if their distance ≤ k in the original graph; removing edges that increase this distance beyond k constitutes a cut. Maximizing the number of such cut pairs under a budget L is shown to be NP‑hard (Theorem 1). Moreover, the set function counting cut pairs is neither sub‑ nor super‑modular, precluding the direct use of greedy submodular optimization techniques common in influence maximization or misinformation containment.

Two baseline algorithms are proposed based on edge betweenness. Global betweenness B(e) counts the fraction of all shortest paths that traverse e, while local betweenness LB(e) restricts the sum to vertex pairs whose shortest‑path length ≤ k. By iteratively removing the highest‑betweenness edges, a simple greedy heuristic is obtained; its computational cost is O(n m) per iteration, where n and m are the numbers of vertices and edges.

To move beyond greedy heuristics, the problem is expressed as an OR‑AND Boolean maximization. Each edge e is associated with a binary variable x_e (1 = removed). For each locally reachable pair (u, v), the clause ∨{e∈P{uv}} x_e captures the requirement that at least one edge on a shortest‑path set P_{uv} be removed. The overall objective is the AND over all such clauses, i.e., maximize the number of satisfied clauses. This formulation is also NP‑hard. The authors relax the binary variables to continuous values y_e∈