Source-linked AI summary
Opinion dynamics in social networks with stubborn agents: an issue-based perspective
Ye Tian, Long Wang
TL;DR
The F-J model generally does not reach consensus on a single issue when agents are stubborn, although consensus may occur across issue sequences. This paper derives convergence and consensus conditions for the model over single issues and issue sequences, including a bounded-confidence setting. It shows that consensus can be achieved over path-dependent issue sequences and that the issue-sequence network has enhanced connectivity.
Problem
The F-J model generally does not reach consensus over a single issue when agents are stubborn, while empirical evidence suggests consensus may occur across issue sequences.
Source: §1 · p. 2, §7 · p. 15Method
The paper derives single-issue convergence conditions and analyzes issue-sequence consensus through connections between interpersonal influence and successive initial-opinion networks, including bounded confidence.
Source: §1 · p. 2, §7 · pp. 14–15Results
The F-J model may achieve consensus over each path-dependent issue sequence even though it generally does not achieve consensus over a single issue.
Source: §7 · p. 15Takeaways & Limitations
The issue-sequence analysis reveals enhanced connectivity relative to opinion formation over a single issue.
Source: §7 · p. 15The bounded-confidence analysis assumes opinions converge before the next issue arises, so its stated setting concerns sequential issues with that timing.
Source: §5 · p. 9
Abstract
from arXiv · showhide
Classic models on opinion dynamics usually focus on a group of agents forming their opinions interactively over single issue. Yet generally consensus can not be achieved over single issue when agents are not completely open to interpersonal influence. In this paper, opinion consensus in social networks with stubborn agents is considered over issue sequences. The social network with stubborn agents is described by the Friedkin-Johnsen (F-J) model where agents are stubborn to their initial opinions. Firstly, we propose some sufficient and necessary conditions both in terms of network topology and system matrix for convergence of the F-J model over single issue. Secondly, opinion consensus of the F-J model is investigated over issue sequences. Our analysis establishes connections between the interpersonal influence network and the network describing the relationship of agents' initial opinions for successive issues. Taking advantage of these connections, we derive the sufficient and necessary condition for the F-J model to achieve opinion consensus over issue sequences. Finally, we consider a more general scenario where each agent has bounded confidence in forming its initial opinion for every issue. By analyzing the evolution of agents' ultimate opinions for each issue over issue sequences, we prove that the connectivity of the state-dependent network is preserved in this setting. Then the conditions for agents to achieve opinion consensus over issue sequences are proposed. Simulation examples are provided to illustrate the effectiveness of our theoretical results.
1 Introduction
The paper addresses why stubborn agents may fail to reach consensus on a single issue, while consensus may emerge across interdependent or repeatedly arising issues. It studies the F-J model across issue sequences, including a bounded-confidence setting.
1 Introduction
- The study is motivated by the limited consensus of the F-J model on a single issue and observations that consensus may occur over issue sequences.The authors argue that interdependent issues are relevant to associations handling repeatedly arising issues.Source: §1 · p. 2
- Unlike prior work on self-appraisals across independent issues or opinions on unordered interdependent issues, this paper studies F-J opinion formation across interdependent issue and time sequences.The cited prior approaches are Mirtabatabaei et al. (2014), Jia et al. (2015), and Parsegov et al. (2017).Source: §1 · p. 2
- For a single issue, the paper proposes milder sufficient and necessary network-topology conditions for convergence than conditions requiring an undirected connected network or a strictly row-substochastic matrix.It also gives properties of agents’ ultimate opinions.Source: §1 · p. 2
- For interdependent issue sequences, the paper links interpersonal influence to the network of agents’ initial opinions across successive issues and derives conditions for consensus or clustering.The sequence model assumes each agent’s cognitive inertia equals its stubborn factor.Source: §1 · p. 2
- In the bounded-confidence setting, the paper analyzes connectivity preservation in the modified F-J model and derives conditions for consensus.Agents selectively weight information when forming initial opinions.Source: §1 · p. 2
- Across path-dependent issue sequences, consensus can emerge despite stubbornness when one partially stubborn agent’s opinion reaches the others, forming a star relationship network.The star describes the relationship among agents’ initial opinions across successive issues.Source: §1 · p. 3
2 Preliminaries
The preliminaries define directed-network concepts used to describe information flow and introduce the F-J model, which combines interpersonal influence with agents’ attachment to their initial opinions.
2.1 Basic concepts on graph theory
- A weighted directed graph represents information flow: an edge from i to j means j can receive information from i, without requiring reciprocal influence.The section also defines paths, strong connectivity, strongly connected components, spanning trees, and star subgraphs.Source: §2.1 · p. 3
2.2 Introduction of the F-J model
- The F-J model generalizes DeGroot by adding a positive diagonal matrix that quantifies agents’ susceptibility to interpersonal influence.Each agent’s stubbornness to its initial opinion is quantified by one minus its susceptibility.Source: §2.2 · p. 3
- The model can also be represented as a leader-following system, with agents’ initial opinions treated as stationary leaders.The augmented system is introduced as an equivalent representation of the F-J model.Source: §2.2 · p. 4
3 Convergence of the F-J model over a single issue
For a single issue, the F-J model converges exactly when each ISCC consisting only of non-stubborn agents is aperiodic. The section also characterizes convergence spectrally and describes how stubborn agents shape ultimate opinions.
3 Convergence of the F-J model over a single issue
- The same convergence criterion is that 1 is the only eigenvalue of ΞW with maximum modulus; its algebraic multiplicity may exceed one.Any other eigenvalue of modulus 1 makes the system non-convergent.Source: §3 · p. 4
- The graph condition is milder than requiring every ISCC to be aperiodic, since it applies only to components consisting entirely of non-stubborn agents.Earlier F-J convergence conditions commonly required an undirected connected network or a stubborn agent in every ISCC.Source: §3 · pp. 4–5
- Convergence holds if and only if each ISCC consisting only of non-stubborn agents is aperiodic, with an agent having two coprime-length cycles.This is the stated necessary and sufficient graph condition for convergence.Source: §3 · p. 4
- Each agent’s ultimate opinion is a convex combination of initial opinions, but a non-stubborn agent’s own initial opinion contributes zero when a stubborn agent can reach it.Otherwise, non-stubborn agents in ISCCs without stubborn agents can contribute to ultimate opinions.Source: §3 · pp. 5–6
- A non-stubborn agent can determine the group’s ultimate opinion only if it is not influenced by stubborn agents.The paper connects this condition to the role of leaders with greater communication power or external information.Source: §3 · p. 6
4 Opinion formation of the F-J model over issue sequences
The section models path-dependent issue sequences by linking agents’ initial opinions across issues to their within-issue F-J dynamics. It relates the resulting influence network to the original social network to characterize consensus and clustering.
4 Opinion formation of the F-J model over issue sequences
- The model treats successive issues as path-dependent: past decisions constrain later ones, and an agent’s cognitive-inertia factor is set equal to its stubbornness factor.Issue intervals may be any positive integer; x(s,ks) denotes agents’ initial opinions for issue s.Source: §4 · pp. 6–7
- I − ΞW is nonsingular if and only if every strongly connected component of G(W) contains at least one stubborn agent.This is Assumption 2, which the section uses to ensure the issue-sequence dynamics are defined.Source: §4 · p. 7
- When the time between issues is large enough, the next issue’s initial opinions equal the current issue’s limiting opinions, producing cognitive freezing.The paper relates this case to cognitive inertia reinforced into freezing.Source: §4 · p. 8
- Under Assumption 2, Ψ links agents’ initial opinions across issues: its positive entries reflect paths in G(W) through partially stubborn agents.Roots in G(Ψpp) connect to every other vertex, so a spanning tree there implies a star subgraph.Source: §4 · p. 8
- The Ψ block structure implies that fully stubborn agents never change their opinions, so the consensus theorem considers networks without them.The theorem’s criterion assumes no fully stubborn agents.Source: §4 · p. 9
- With no fully stubborn agents and Assumption 2, consensus occurs if and only if a partially stubborn agent has directed paths to every other partially stubborn agent.Paths may pass through partially stubborn or non-stubborn agents.Source: §4 · p. 9
- With no fully stubborn agents and Assumption 2, the system forms opinion clusters if and only if G(W) has more than one initial strongly connected component.The paper states this as Corollary 1.Source: §4 · p. 9
5 Opinion formation of the F-J model over issue sequences with bounded confidence
The section extends the F-J model to bounded-confidence initial opinions across issue sequences. Under stated connectivity and opinion-distribution conditions, the state-dependent network retains its links and the system can reach consensus, including with fully stubborn agents.
Bounded-confidence model
- Each next issue’s initial opinions are weighted averages of agents’ ultimate opinions from the prior issue that lie within confidence bound d.The model assumes opinions settle within each issue before the next begins; its update uses a symmetric modification of the H-K model.Source: §5 · pp. 9–10
Convergence argument
- If G(Φ′(s)) has a spanning tree, Φ1(s) is SIA; preserving the state-dependent network’s connectivity therefore supports the convergence argument across issues.The section connects connectivity preservation to Φ1(s) being SIA.Source: §5 · p. 10
Connectivity preservation
- Under Assumption 3, every edge initially in G(H(0)) remains in G(H(s)) for every s ∈ N, preserving the initial graph’s edges across issues.The stated bounds are 1/(2n) < h < 1/(n−1).Source: §5 · p. 10
Consensus conditions
- Consensus follows under Assumptions 2 and 3 if a partially stubborn agent reaches every other partially stubborn agent through partially or non-stubborn agents, G(H_ff(0)) is connected, and G(H(0)) links a partially and fully stubborn agent.These are sufficient conditions for consensus in system (7), (15).Source: §5 · pp. 11–12
- Theorem 3 establishes that the bounded-confidence model can achieve consensus even with fully stubborn agents, weakening the cognitive freezing associated with the unbounded-confidence model.The section attributes cognitive freezing under Assumption 2 to interpersonal-influence externalities.Source: §5 · p. 12
6 Simulations
Three simulations illustrate the paper’s convergence and consensus results: for a single issue, opinions settle; across issues, opinions reach consensus, both with and without bounded confidence.
Example 1: Single-issue convergence
- In the single-issue example, opinions tend to constant values as k increases; the non-stubborn ISCC {8, 9, 10} reaches consensus, while agent 7 has a different ultimate opinion.The fully stubborn agents’ opinions remain fixed.Source: §6 · p. 12
Example 2: Consensus over issue sequences
- By issue s = 30, the simulated system reaches consensus; within each issue opinions settle as k increases, and steady opinions approach a common value across issues.The example also shows a star topology among partially stubborn agents in G(Ψ2).Source: §6 · p. 13, Figure 3 · p. 13, Figure 4 · p. 14
Example 3: Bounded confidence
- In the bounded-confidence example, x(s, 0) and x(s, k) reach consensus across issues under the stated conditions.The simulation uses confidence bound d = 1 and h = 0.1.Source: §6 · p. 13, Figure 6 · p. 14
7 Conclusions
The conclusion synthesizes the paper’s analysis of the F-J model over issue sequences, including bounded confidence. It reports that consensus may emerge across path-dependent issues even when it generally does not on a single issue.
Contributions
- The paper studies single-issue convergence and ultimate opinions, then uses connections between interpersonal influence and successive initial-opinion networks to analyze consensus and clustering across issues.It also considers agents with bounded confidence when forming initial opinions.Source: §7 · pp. 14–15
Conclusions
- The F-J model may achieve consensus over path-dependent issue sequences, while generally not achieving consensus over a single issue.The paper links this difference to enhanced network connectivity over issue sequences.Source: §7 · p. 15
Implications and future work
- The authors identify theoretical explanations for observations such as decision-making in human society or the animal kingdom, and leave the evolution of stubbornness across issues for future work.Source: §7 · p. 15