Being Corrupt Requires Being Clever, But Detecting Corruption Doesn't

TL;DR

This paper studies corruption detection in networks, establishing a combinatorial parameter that determines the minimal corruptions needed to hide truthful nodes, and provides efficient algorithms and hardness results based on graph properties.

Contribution

It introduces the parameter m(G) for corruption resistance, offers a linear-time algorithm for detection when corruptions are below m(G)/2, and proves NP-hardness of more extensive corruption prevention under the Small Set Expansion Hypothesis.

Findings

Efficient algorithm detects truthful nodes if corruptions < m(G)/2.

NP-hardness of preventing detection with corruptions up to α m(G).

Relation between corruptions needed and graph vertex separability.

Abstract

We consider a variation of the problem of corruption detection on networks posed by Alon, Mossel, and Pemantle '15. In this model, each vertex of a graph can be either truthful or corrupt. Each vertex reports about the types (truthful or corrupt) of all its neighbors to a central agency, where truthful nodes report the true types they see and corrupt nodes report adversarially. The central agency aggregates these reports and attempts to find a single truthful node. Inspired by real auditing networks, we pose our problem for arbitrary graphs and consider corruption through a computational lens. We identify a key combinatorial parameter of the graph $m(G)$, which is the minimal number of corrupted agents needed to prevent the central agency from identifying a single truthful node. We give an efficient (in fact, linear time) algorithm for the central agency to identify a truthful node that is successful whenever the number of corrupt nodes is less than $m(G)/2$. On the other hand, we prove that for any constant $\alpha > 1$, it is NP-hard to find a subset of nodes $S$ in $G$ such that corrupting $S$ prevents the central agency from finding one truthful node and $|S| \leq \alpha m(G)$, assuming the Small Set Expansion Hypothesis (Raghavendra and Steurer, STOC '10). We conclude that being corrupt requires being clever, while detecting corruption does not. Our main technical insight is a relation between the minimum number of corrupt nodes required to hide all truthful nodes and a certain notion of vertex separability for the underlying graph. Additionally, this insight lets us design an efficient algorithm for a corrupt party to decide which graphs require the fewest corrupted nodes, up to a multiplicative factor of $O(\log n)$.