Eliminating Illusion in Directed Networks

2026-04-02

AI Summary

This study addresses the problem of p-illusion elimination in directed social networks, where—despite a global majority of blue nodes—certain nodes perceive an anomalously high fraction of red nodes among their out-neighbors. The goal is to eliminate all such p-illusions via a minimum number of recoloring operations. We provide the first systematic characterization of the computational complexity landscape for this problem, proving it to be NP-hard and even W[2]-hard on general directed graphs, grids, and bipartite DAGs. In contrast, we identify several sparse graph classes—including outerplanar graphs, trees, and cycles—that admit efficient polynomial-time algorithms. Furthermore, we develop fixed-parameter tractable algorithms parameterized by treewidth and the number of illusioned nodes, highlighting the critical role of structural sparsity in rendering the problem computationally tractable.

Abstract

We study illusion elimination problems on directed social networks where each vertex is colored either red or blue. A vertex is under \textit{majority illusion} if it has more red out-neighbors than blue out-neighbors when there are more blue vertices than red ones in the network. In a more general phenomenon of $p$-illusion, at least $p$ fraction of the out-neighbors (as opposed to $1/2$ for majority) of a vertex is red. In the directed illusion elimination problem, we recolor minimum number of vertices so that no vertex is under $p$-illusion, for $p\in (0,1)$. Unfortunately, the problem is NP-hard for $p =1/2$ even when the network is a grid. Moreover, the problem is NP-hard and W[2]-hard when parameterized by the number of recolorings for each $p \in (0,1)$ even on bipartite DAGs. Thus, we can neither get a polynomial time algorithm on DAGs, unless P=NP, nor we can get a FPT algorithm even by combining solution size and directed graph parameters that measure distance from acyclicity, unless FPT=W[2]. We show that the problem can be solved in polynomial time in structured, sparse networks such as outerplanar networks, outward grids, trees, and cycles. Finally, we show tractable algorithms parameterized by treewidth of the underlying undirected graph, and by the number of vertices under illusion.

Problem

• majority illusion
• p-illusion
• directed networks
• illusion elimination
• NP-hard

Innovation

• majority illusion
• p-illusion
• NP-hardness
• fixed-parameter tractability
• directed social networks