Eliminating Majority Illusions

TL;DR

This work formalizes Eliminating Illusion (EI), the problem of removing majority illusion in social networks by minimally flipping vertex labels, and reveals a deep connection to Total Vector Domination (TVD). It proves strong hardness results—NP-hardness on restricted planar bipartite graphs, W[2]-hardness by the number of flips, and XNLP-hardness by pathwidth—while also identifying tractable avenues: FPT algorithms parameterised by vertex cover, XP and FPT results tied to treewidth, and a PTAS for planar graphs. The approach leverages the EI–TVD equivalence to transfer algorithmic techniques and derives practical, scalable strategies for network interventions, notably Baker-style layering to achieve a $(1+\varepsilon)$-approximation in $n^{\mathcal{O}(1/\varepsilon)}$ time. Overall, the paper lays a theoretical foundation for understanding and mitigating majority illusion through principled graph-structural methods with potential real-world impact in information integrity and democratic processes.

Abstract

An opinion illusion refers to a phenomenon in social networks where agents may witness distributions of opinions among their neighbours that do not accurately reflect the true distribution of opinions in the population as a whole. A specific case of this occurs when there are only two possible choices, such as whether to receive the COVID-19 vaccine or vote on EU membership, which is commonly referred to as a majority illusion. In this work, we study the topological properties of social networks that lead to opinion illusions and focus on minimizing the number of agents that need to be influenced to eliminate these illusions. To do so, we propose an initial, but systematic study of the algorithmic behaviour of this problem. We show that the problem is NP-hard even for underlying topologies that are rather restrictive, being planar and of bounded diameter. We then look for exact algorithms that scale well as the input grows (FPT). We argue the in-existence of such algorithms even when the number of vertices that must be influenced is bounded, or when the social network is arranged in a ``path-like'' fashion (has bounded pathwidth). On the positive side, we present an FPT algorithm for networks with ``star-like'' structure (bounded vertex cover number). Finally, we construct an FPT algorithm for ``tree-like'' networks (bounded treewidth) when the number of vertices that must be influenced is bounded. This algorithm is then used to provide a PTAS for planar graphs.

Figures (5)

• Figure 1: In the leftmost figure, the vertex $v_7$ is under illusion. It may change its colour over time and subsequently influence the colour of $v_6$ followed by $v_5$ and $v_4$. The final configuration is presented in the rightmost figure.
• Figure 2: The graph $G_I$ constructed in the proof of Theorem \ref{['thm:W2-set-cover']}. The white vertices have label $0$, and the grey vertices have label $1$.
• Figure 3: The graph $G_\phi$ constructed in the proof of Theorem \ref{['thm:NP-3sat']}. The white vertices have label $0$, and the grey vertices have label $1$. The clauses are $c_1 = (X_1 \lor X_2 \lor X_3)$ and $c_2 = (\overline{X_2} \lor \overline{X_3} \lor \overline{X_k})$.
• Figure 4: A chessboard structure between vertices $u$ and $v$, used in the proof of Theorem \ref{['thm:treewidth_hard']}. The grey vertices have label $1$, and the others label $0$.
• Figure 5: The same structure as in Fig. \ref{['fig:maxmin-a']}. The grey vertices were relabelled with label $0$. This corresponds to the edge $uv\in E(G)$ being directed from $u$ to $v$ in $G^*$. The only other valid relabelling with size $2 w(u, v)$ is the reversed relabelling.

Theorems & Definitions (17)

• Lemma 1
• Corollary 1
• Corollary 2
• Lemma 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof