Controlling a Social Network of Individuals with Coevolving Actions and Opinions

TL;DR

The paper addresses controlling a social network where actions and opinions coevolve by injecting a committed minority to flip the population consensus. It develops a monotone, convergent controlled dynamic, proves NP-hardness of minimal-control-set identification, and provides a polynomial-time algorithm to certify effectiveness, plus a Markov-chain heuristic for finding minimal control sets in general networks. A complete-graph analysis yields explicit feasibility conditions for different control modes, and a real-world Malawi case study demonstrates practical gains from joint control and the effectiveness of the proposed heuristics. These results offer rigorous tools for designing robust social-change interventions and for assessing system vulnerability to targeted manipulation.

Abstract

In this paper, we consider a population of individuals who have actions and opinions, which coevolve, mutually influencing one another on a complex network structure. In particular, we formulate a control problem for this social network, in which we assume that we can inject into the network a committed minority -- a set of stubborn nodes -- with the objective of steering the population, initially at a consensus, to a different consensus state. Our study focuses on two main objectives: i) determining the conditions under which the committed minority succeeds in its goal, and ii) identifying the optimal placement for such a committed minority. After deriving general monotone convergence result for the controlled dynamics, we leverage these results to build a computationally-efficient algorithm to solve the first problem and an effective heuristics for the second problem, which we prove to be NP-complete. For both algorithms, we establish theoretical guarantees. The proposed methodology is illustrated though academic examples, and demonstrated on a real-world case study.

Figures (6)

• Figure 1: Schematic of the update rule for a generic individual $i\in\mathcal{R}(t)$.
• Figure 2: Example of a setting that satisfies Assumption \ref{['a:initial_condition']}. The top layer represents individuals' action (white for $-1$, green for $+1$), the bottom layer represents individuals' opinion (shades from white for $-1$ to violet for $+1$). Control sets are $\mathcal{C}^{X}=\{1,6\}$ and $\mathcal{C}^{Y}=\{1,2\}$.
• Figure 3: Results for a complete graph. The color intensity represents the cardinality of the minimal control set $\gamma=|\mathcal{C}|/(n-1)$ that solves Problem \ref{['problem2']}.
• Figure 4: Example of one iteration of the Markov chain from set $\mathcal{\bar{C}}$. In (a), we highlight with a blue circle the set of admissible control sets. The red circles highlight all the candidate control sets. The last control set is not admissible and so it is not considered. In (b), we illustrate the transitions of the chain, described in Eq. (\ref{['eq:MC_probabilities']}). Nodes in the control set are denoted in black, nodes not in the control set in white.
• Figure 5: The network used in Section \ref{['sec:case']}. Green nodes are those identified by our algorithm as control nodes, in the scenario of joint control.

Theorems & Definitions (28)

• Remark 1
• Proposition 1
• Remark 2
• Proposition 2
• Proposition 3: Proposition 3 from raineri2024
• Remark 3
• Remark 4
• Theorem 1
• Remark 5
• Lemma 1