Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Eliminating Majority Illusion is Easy

Jack Dippel, Max Dupré la Tour, April Niu, Sanjukta Roy, Adrian Vetta

TL;DR

This work studies majority illusion in graphs by formalizing the Illusion Elimination Problem and its three edge-edit variants, MIAE, MIRE, and MIE. It develops polynomial-time, LP-based algorithms for fully eliminating illusion via edge edits, leveraging a TI $b'$-matching–style formulation and a separation oracle to ensure integrality. The authors generalize to $p$-Illusion, showing polynomial-time solvability at $p=1/2$ but NP-hardness for most other rational $p$, with reductions from XSAT establishing hardness. The results showcase a striking separation: while complete elimination is tractable, partial elimination exhibits hardness, and they open avenues for weighted-edge extensions and finer complexity characterizations at special $p$ values.

Abstract

Majority Illusion is a phenomenon in social networks wherein the decision by the majority of the network is not the same as one's personal social circle's majority, leading to an incorrect perception of the majority in a large network. In this paper, we present polynomial-time algorithms which can eliminate majority illusion in a network by altering as few connections as possible. Additionally, we prove that the more general problem of ensuring all neighbourhoods in the network are at least a $p$-fraction of the majority is NP-hard for most values of $p$.

Eliminating Majority Illusion is Easy

TL;DR

This work studies majority illusion in graphs by formalizing the Illusion Elimination Problem and its three edge-edit variants, MIAE, MIRE, and MIE. It develops polynomial-time, LP-based algorithms for fully eliminating illusion via edge edits, leveraging a TI -matching–style formulation and a separation oracle to ensure integrality. The authors generalize to -Illusion, showing polynomial-time solvability at but NP-hardness for most other rational , with reductions from XSAT establishing hardness. The results showcase a striking separation: while complete elimination is tractable, partial elimination exhibits hardness, and they open avenues for weighted-edge extensions and finer complexity characterizations at special values.

Abstract

Majority Illusion is a phenomenon in social networks wherein the decision by the majority of the network is not the same as one's personal social circle's majority, leading to an incorrect perception of the majority in a large network. In this paper, we present polynomial-time algorithms which can eliminate majority illusion in a network by altering as few connections as possible. Additionally, we prove that the more general problem of ensuring all neighbourhoods in the network are at least a -fraction of the majority is NP-hard for most values of .
Paper Structure (14 sections, 27 theorems, 37 equations, 6 figures)

This paper contains 14 sections, 27 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Illusion Elimination Problem
  3. Our Contributions.
  4. Related Work.
  5. Preliminaries
  6. Majority Illusion Addition Elimination (MIAE)
  7. A First Attempt
  8. A Second Attempt
  9. A TDI System
  10. A Polynomial Time Algorithm
  11. Majority Illusion Removal Elimination (MIRE)
  12. Majority Illusion Elimination (MIE)
  13. Hardness
  14. Conclusion

Key Result

Lemma 1

In an optimal solution $E'$ to $(V,E,f,k)$ of MIAE, there are exactly $\max(r(v)-b(v),0)$ edges of $E'\setminus E$ incident to each red node $v$.

Figures (6)

  • Figure 1: In network (a) there are two nodes, $v_1$ and $v_2$, under majority illusion; in network (b) every node suffers illusion!
  • Figure 2: In (a) only $v_1$ and $v_2$ are under majority illusion. An optimal solution to the MIAE problem, adding just $2$ edges, is shown in (b); an optimal solution to the MIRE problem, removing $5$ edges, is shown in (c); an optimal solution to the MIE problem, adding $1$ edge and removing $1$ edge, is shown in (d).
  • Figure 3: In (a) every node is under majority illusion. An optimal solution to the MIAE problem, adding $11$ edges, is shown in (b); an optimal solution to the MIRE problem, requiring the removal of all $12$ edges, is shown in (c); an optimal solution to the MIE problem, adding $3$ edges and removing $3$ edges, is shown in (d).
  • Figure 4: A non-integral optimal solution to the linear program relaxation of ( IP-1a).
  • Figure 5: A violated constraint: $1-y_{v_3v_4} + 1-y_{v_3v_2} + 1-y_{v_2v_4} \leq \lfloor \frac{3}{2} \rfloor$
  • ...and 1 more figures

Theorems & Definitions (57)

  • Definition : Illusion
  • proof
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • ...and 47 more