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Targeting influence in a harmonic opinion model

Zachary M. Boyd, Nicolas Fraiman, Jeremy L. Marzuola, Peter J. Mucha, Braxton Osting

TL;DR

The paper studies how to strategically influence a network when multiple extreme opinions compete, by modeling opinions as a vector-valued harmonic field on a graph with Dirichlet zealots. It proves the targeting problem is NP-hard but the objective is monotone and submodular, enabling a $(1-1/e)$-approximation via greedy optimization, and it introduces a convex relaxation with explicit gradient and Hessian to compute approximate solutions efficiently. It also shows that symmetry in the graph preserves optimality in the relaxed problem and develops two computational approaches (greedy Schur-complement solves and relaxation-based optimization) whose performance is validated on grids and H-graphs, with an interactive game implementation. The results offer a principled, scalable framework for adversarial influence in networks, linking probabilistic hitting interpretations, energy minimization, and submodular optimization to practical strategies for influence targeting.

Abstract

Influence propagation in social networks is a central problem in modern social network analysis, with important societal applications in politics and advertising. A large body of work has focused on cascading models, viral marketing, and finite-horizon diffusion. There is, however, a need for more developed, mathematically principled \emph{adversarial models}, in which multiple, opposed actors strategically select nodes whose influence will maximally sway the crowd to their point of view. In the present work, we develop and analyze such a model based on harmonic functions and linear diffusion. We prove that our general problem is NP-hard and that the objective function is monotone and submodular; consequently, we can greedily approximate the solution within a constant factor. Introducing and analyzing a convex relaxation, we show that the problem can be approximately solved using smooth optimization methods. We illustrate the effectiveness of our approach on a variety of example networks.

Targeting influence in a harmonic opinion model

TL;DR

The paper studies how to strategically influence a network when multiple extreme opinions compete, by modeling opinions as a vector-valued harmonic field on a graph with Dirichlet zealots. It proves the targeting problem is NP-hard but the objective is monotone and submodular, enabling a -approximation via greedy optimization, and it introduces a convex relaxation with explicit gradient and Hessian to compute approximate solutions efficiently. It also shows that symmetry in the graph preserves optimality in the relaxed problem and develops two computational approaches (greedy Schur-complement solves and relaxation-based optimization) whose performance is validated on grids and H-graphs, with an interactive game implementation. The results offer a principled, scalable framework for adversarial influence in networks, linking probabilistic hitting interpretations, energy minimization, and submodular optimization to practical strategies for influence targeting.

Abstract

Influence propagation in social networks is a central problem in modern social network analysis, with important societal applications in politics and advertising. A large body of work has focused on cascading models, viral marketing, and finite-horizon diffusion. There is, however, a need for more developed, mathematically principled \emph{adversarial models}, in which multiple, opposed actors strategically select nodes whose influence will maximally sway the crowd to their point of view. In the present work, we develop and analyze such a model based on harmonic functions and linear diffusion. We prove that our general problem is NP-hard and that the objective function is monotone and submodular; consequently, we can greedily approximate the solution within a constant factor. Introducing and analyzing a convex relaxation, we show that the problem can be approximately solved using smooth optimization methods. We illustrate the effectiveness of our approach on a variety of example networks.
Paper Structure (23 sections, 10 theorems, 54 equations, 5 figures, 2 algorithms)

This paper contains 23 sections, 10 theorems, 54 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and results
  3. Previous work
  4. Opinion dynamics and targeting influence
  5. A model for opinions and influencers
  6. A harmonic model for opinion dynamics
  7. Measuring and targeting influence
  8. Reduction to a grouped opponent targeting influence problem
  9. Probabilistic interpretation
  10. Energy interpretation in the undirected case
  11. Hardness and approximation of the targeting influence problem
  12. Hardness of the targeting influence problem
  13. Approximation of the targeting influence problem
  14. A relaxation strategy for the targeting influence problem
  15. Targeting influence on symmetric graphs
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $Z \subset V$ and $f\colon Z \to \Delta_k$. Suppose $u \colon V \to \mathbb R^k$ solves Then $u(i) \in \Delta_k$ for every $i \in V$.

Figures (5)

  • Figure 1: On an $11\times 11$ square grid graph, we consider the case where first authority has chosen the center node and the second authority is choosing a vertex. (left) We plot the measure of influence $\tilde{\mathcal{I}}_2$ for the second authority for each available node. (right) We plot the $\tilde{\phi}$ value obtained as the solution to \ref{['alg:RelaxAlg']} with $\varepsilon = .15$. The $\tilde{\phi}$ values respect the underlying symmetry and correctly predict that the optimal single-node choice is to pick one of the 4 nearest neighbors. See \ref{['s:CompExp1']}.
  • Figure 2: We again consider an $11\times 11$ square grid graph, but now with a single edge missing. The left and right panels are as in \ref{['fig:square']}. See \ref{['s:CompExp2']}.
  • Figure 3: We consider an H-graph as described in \ref{['s:CompExp3']} and again consider the case where the first authority has chosen $Z_1$ and the second authority is choosing a vertex. (top) Here, $Z_1$ consists of a single node $(3,5)$ in the left subgraph. The left and right panels are as in \ref{['fig:square']}. (bottom) Here, $Z_1$ consists of a two nodes, one taken from the left and one taken from the right subgraphs. Again, The left and right panels are as in \ref{['fig:square']}. See \ref{['s:CompExp3']} for details.
  • Figure 4: As in \ref{['fig:hgraph']}, we consider an H-graph, but this time we plot $\tilde{\phi}$ for the value $\varepsilon = .015$, which is a factor of 10 smaller than the previous value. Compare to the bottom right figure of \ref{['fig:hgraph']}. We observe that the algorithm correctly wants to cordon off the first authority opinion.
  • Figure 5: An illustration of the interactive game implementation of the targeting influence problem gameweb. Here, we have an random geometric graph on 50 nodes. Two human players have each chosen a single node and Player 2 is currently winning; see \ref{['s:CompExp4']}.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2
  • proof : Proof of \ref{['t:NP-hard']}
  • Lemma 3.3
  • ...and 14 more