Majority Illusion in Social Networks

• Majority illusion is a network phenomenon where agents perceive a local majority that contradicts the true global majority due to structural biases.
• Structural factors such as degree heterogeneity, assortativity, and attribute–degree correlation amplify misperceptions in various network types.
• This phenomenon impacts social contagion dynamics and informs targeted interventions aimed at correcting pervasive social misperceptions.

Majority illusion is a network-theoretic phenomenon wherein agents’ local perceptions of majority opinion in their personal neighborhoods systematically misalign with the true global majority in the overall population. This bias arises from the interplay of binary node attributes (such as behavioral adoption, voting decisions) and the graph structure of social networks, resulting in widespread misperception about “what most people think” even in the absence of misinformation. The effect is accentuated by structural factors such as degree heterogeneity, attribute–degree correlation, and network assortativity, and it has significant implications for contagion dynamics, perception formation, and interventions targeting social misperceptions.

1. Formal Definitions and Theoretical Framework

Consider an undirected simple graph $G=(V,E)$ modeling a social network, where each node $v\in V$ is assigned a binary label $f(v)\in\{B,R\}$ (e.g., “blue” and “red”, or $0$/$1$ for opinions) (Lerman et al., 2015, Los et al., 2023, Dippel et al., 29 Jul 2024, Fioravantes et al., 20 Feb 2025, Grandi et al., 2022). Nodes perceive majority opinion via their open neighborhood $N(v)$. Define $b(v)=|\{u\in N(v):f(u)=B\}|$ and $r(v)=|\{u\in N(v):f(u)=R\}|$. The global strict majority is the color with $|B|>|R|$ or vice versa.

A node $v$ is under majority illusion if its local majority is the global minority: $$\text{Majority illusion at } v \longleftrightarrow \begin{cases} b(v) < r(v), & \text{if } |B| > |R| \ r(v) < b(v), & \text{if } |R| > |B| \end{cases}$$ This is equivalently $f(v)$ experiences $b(v)/|N(v)| < 1/2$ while $|B|/|V| > 1/2$, or vice versa (Dippel et al., 29 Jul 2024).

The strict majority illusion requires both the local and global majorities to be strict and disagree; weak majority illusion counts cases where local and global majorities differ, regardless of ties (Los et al., 2023). The fraction or absolute number of illusioned nodes can be parameterized, leading to $q$–majority illusions (at least $q$-fraction of nodes are misled) (Grandi et al., 2022).

For random networks, the mathematical mean fraction of illusioned nodes is

$$P_{\text{illusion}} = \sum_k p(k) \Bigg[\sum_{n > k/2}^k \binom{k}{n} [P(x'=1|k)]^n [1-P(x'=1|k)]^{k-n}\Bigg]$$

where $p(k)$ is the degree distribution, and $P(x'=1|k)$ is the probability a neighbor of degree $k$ is “active” (Lerman et al., 2015).

2. Structural and Statistical Origins

Majority illusion is intimately tied to the friendship paradox: in networks with heterogeneous degree, the average neighbor has higher degree (and typically, more “active” or rare attributes) than a random person. This occurs because neighbor samples over-represent high-degree nodes, and if degree correlates positively with holding a minority attribute, many agents will see local majorities of that attribute—even if it is globally sparse (Lerman et al., 2015).

Assortativity ($r$), the Pearson correlation between the degrees of linked nodes, plays a key role: disassortative networks ($r<0$) expose low-degree nodes disproportionately to high-degree nodes, amplifying the illusion. Similarly, degree–attribute correlation ($\rho_{kx}$) modulates the bias, with stronger correlation producing sharper illusions.

Graph families exhibit diverse behaviors: bipartite graphs with odd size, and graphs with all nodes of odd degree, structurally support strict illusions; cycles and complete graphs forbid them. In scale-free (power-law) networks, majority illusion is pronounced due to extreme heterogeneity (Los et al., 2023, Lerman et al., 2015).

3. Computational Complexity and Algorithmic Methods

The main algorithmic questions concern both (i) detection—does a network with given coloring admit a (possibly majority) illusion for at least some fraction $q$ of the agents, and (ii) elimination—can the network be minimally modified (e.g., by edge addition, deletion, or vertex relabeling) to remove illusions (Grandi et al., 2022, Dippel et al., 29 Jul 2024, Fioravantes et al., 20 Feb 2025).

Decision Problem Complexity

Detecting whether a network admits a $q$–majority illusion for fixed $q\in(\tfrac{1}{2},1]$ is NP-complete, via reductions from SAT variants (e.g., 3-SAT) using variable, clause, and balance gadgets (Grandi et al., 2022). Editing the network (edge-addition, removal, or both) to eliminate illusions is also NP-complete for nontrivial $q$ in general graphs.

Polynomial-Time and FPT Results

Despite this, specific variants admit efficient (even polytime) algorithms under strong restrictions:

• Full elimination (q = 0): Given fixed coloring, finding a minimum-size set of edge edits (addition/removal) to eliminate all majority illusions (i.e., make every neighborhood reflect the global majority) is in P via LP-based methods (Dippel et al., 29 Jul 2024). These exploit the total dual integrality of a parity-constrained b-matching LP, solved using b-matching separation oracles in $O(n^4)$ time.
• Vertex Relabeling: The minimum relabeling problem to eliminate all illusions is W[2]-hard parameterized by $k$ (number of allowed relabels), and NP-hard even for planar bipartite bounded-degree graphs, but FPT parameterized by vertex cover number or by both treewidth and solution size (Fioravantes et al., 20 Feb 2025).
• Planar Networks: A PTAS exists for elimination in planar graphs using Baker's method and dynamic programming on bounded treewidth decompositions (Fioravantes et al., 20 Feb 2025).

Table: Algorithmic Tractability by Problem Variant

Variant	General Graphs	Planar Graphs	Bounded VC / TW
Detecting $q$-illusion	NP-complete ($q>\frac12$) (Grandi et al., 2022)	–	Linear/XP in special cases (Fioravantes et al., 20 Feb 2025)
Fixing (edge edit) $q>0$	NP-complete	NP-complete	FPT for small VC/tw + $k$
Fixing (edge edit) $q=0$	P (LP/ILP methods) (Dippel et al., 29 Jul 2024)	PTAS (Fioravantes et al., 20 Feb 2025)	FPT (Fioravantes et al., 20 Feb 2025)

4. Network Topology and Illusion Susceptibility

Network structure determines the extent and robustness of the majority illusion:

• Degree heterogeneity (e.g., in scale-free graphs) dramatically amplifies the illusion, often resulting in situations where a small minority attribute becomes a perceived majority for up to 60%–80% of agents (Lerman et al., 2015).
• Disassortativity increases exposure of low-degree agents to high-degree, minority-opinion holders, maximizing illusion prevalence.
• Bipartite networks (on odd $n$) require vertices of one partition to see only the opposite color, enabling strict illusions for more than half the graph (Los et al., 2023).
• Regular graphs possess combinatorial constraints; parameters $n,k$ (size, regularity) determine whether strict illusions are possible (Los et al., 2023).
• Cycles and complete graphs: immune to strict majority illusions due to combinatorial infeasibility.

5. Statistical Modeling and Empirical Observation

Lerman et al. (Lerman et al., 2015) present a statistical framework expressing the expected fraction of illusioned nodes in terms of the degree distribution $p(k)$, assortativity $r$, and degree–attribute correlation $\rho_{kx}$. The analytic estimate matches empirical simulation in synthetic networks (e.g., Erdős–Rényi, configuration model) and observed social networks (co-authorship, Digg, political blogs).

Key findings include:

• In heavy-tailed networks ($\alpha=2.1$), strong $\rho_{kx}$, and $r<0$, up to 80% of nodes perceive a local majority reversal.
• In random graphs, the effect is diminished but present, especially as active fraction or attribute-degree correlation increases.
• Real-world data confirm that substantial majorities of agents can experience the illusion when minority opinions are associated with highly connected individuals.

6. Interventions: Elimination, Hardness, and Feasibility

From a systems perspective, minimal interventions can dismantle the majority illusion. For the “full elimination” scenario ($q=0$), edge-editing models—specifically, the Majority Illusion Addition Elimination (MIAE), Removal Elimination (MIRE), or combined Elimination (MIE)—can be solved exactly via LP methods (Dippel et al., 29 Jul 2024). For more general “p-Illusion” assignments (where every neighborhood must see a $p$-fraction of majority, $p \neq \{1/3,1/2,2/3\}$), the problem is NP-hard by reduction from exact-one-SAT (Dippel et al., 29 Jul 2024).

For vertex relabeling, NP-completeness persists even under severe topological restrictions (planar, bipartite, low diameter); FPT is achievable only in graphs of bounded vertex cover or in combination with bounded solution size and treewidth (Fioravantes et al., 20 Feb 2025). Planar graphs admit a PTAS for illusion elimination (Fioravantes et al., 20 Feb 2025).

7. Real-World Implications and Social Contagion

Majority illusion has profound consequences for the dynamics of social contagion and perception cascades:

• In threshold models, the illusion can trigger global adoption of rare behaviors if local perceptions cross activation thresholds (Lerman et al., 2015).
• Social campaigns or malicious actors can exploit network design to “bandwagon” rare opinions, or to mitigate systematic misperceptions via targeted rewiring or influencer interventions (Dippel et al., 29 Jul 2024, Fioravantes et al., 20 Feb 2025).
• The universal existence of weak majority illusions (regardless of graph structure) implies that strictly structural defenses against local perception bias are impossible without trivializing network connectivity (Los et al., 2023).

Theoretical results indicate that robust algorithmic defenses require focus on structural parameters (vertex cover, treewidth), and that the trade-off between effort (number of edits/influencers) and topological complexity governs real-world feasibility (Fioravantes et al., 20 Feb 2025). Policy recommendations emphasize that correcting few critical links in heterogenous structure can disproportionately rebalance misperceptions (Grandi et al., 2022). In summary, majority illusion is a structurally emergent, algorithmically rich, and sociologically impactful property of networked systems.