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Towards influence centrality: where to not add an edge in the network?

Aashi Shrinate, Twinkle Tripathy

TL;DR

The paper addresses how to steer influence centrality in a strongly connected social network governed by the Friedkin-Johnsen model when two stubborn agents compete for influence. It develops a signal flow graph (SFG) framework, combined with index-residue reduction, to connect edge modifications to changes in the influence centrality and derives topology-based, sufficient conditions for when edge changes are redundant or beneficial, independent of edge weights. The key contributions include (i) a rigorous characterization of redundant edge modifications via level-set analysis, (ii) sufficient conditions under which edge modifications tilt influence toward one stubborn agent, and (iii) an illustrative example demonstrating both redundant and tilt-inducing modifications. The approach provides a practical, topology-driven alternative to combinatorial optimization for influence management in networks and points to future work with multiple stubborn agents and broader network classes.

Abstract

In this work, we consider a strongly connected group of individuals involved in decision-making. The opinions of the individuals evolve using the Friedkin-Johnsen (FJ) model. We consider that there are two competing `influencers' (stubborn agents) vying for control over the final opinion of the group. We investigate the impact of modifying the network interactions on their respective control over the final opinions (influence centrality). We use signal flow graphs (SFG) to relate the network interactions with the influence that each `influencer' exerts on others. We present the sufficient conditions on the edge modifications which lead to the increase of the influence of an `influencer' at the expense of the other. Interestingly, the analysis also reveals the existence of redundant edge modifications that result in no change in the influence centrality of the network. We present several numerical examples to illustrate these results.

Towards influence centrality: where to not add an edge in the network?

TL;DR

The paper addresses how to steer influence centrality in a strongly connected social network governed by the Friedkin-Johnsen model when two stubborn agents compete for influence. It develops a signal flow graph (SFG) framework, combined with index-residue reduction, to connect edge modifications to changes in the influence centrality and derives topology-based, sufficient conditions for when edge changes are redundant or beneficial, independent of edge weights. The key contributions include (i) a rigorous characterization of redundant edge modifications via level-set analysis, (ii) sufficient conditions under which edge modifications tilt influence toward one stubborn agent, and (iii) an illustrative example demonstrating both redundant and tilt-inducing modifications. The approach provides a practical, topology-driven alternative to combinatorial optimization for influence management in networks and points to future work with multiple stubborn agents and broader network classes.

Abstract

In this work, we consider a strongly connected group of individuals involved in decision-making. The opinions of the individuals evolve using the Friedkin-Johnsen (FJ) model. We consider that there are two competing `influencers' (stubborn agents) vying for control over the final opinion of the group. We investigate the impact of modifying the network interactions on their respective control over the final opinions (influence centrality). We use signal flow graphs (SFG) to relate the network interactions with the influence that each `influencer' exerts on others. We present the sufficient conditions on the edge modifications which lead to the increase of the influence of an `influencer' at the expense of the other. Interestingly, the analysis also reveals the existence of redundant edge modifications that result in no change in the influence centrality of the network. We present several numerical examples to illustrate these results.
Paper Structure (15 sections, 4 theorems, 12 equations, 9 figures)

This paper contains 15 sections, 4 theorems, 12 equations, 9 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Graph Preliminaries
  4. FJ model
  5. Signal Flow Graphs and Index Residue Reduction
  6. Problem Statement
  7. Implications of edge modifications in social networks
  8. Primary objectives
  9. Influence and signal flow graphs
  10. Impact of edge modification in $\mathcal{G}$ on $\mathcal{G}_s$
  11. Index nodes and the permissible edge modifications
  12. Impact of edge Modifications on Influence
  13. An Illustrative example
  14. CONCLUSIONS
  15. Index Residue Reduction of the SFG

Key Result

Lemma 1

FJ_Model When the underlying network $\mathcal{G}=(\mathcal{V},\mathcal{E})$ is strongly connected and $\beta_i \in [0,1)$ such that $\beta_i>0$ for at least one $i \in \mathcal{V}$, the following conditions hold:

Figures (9)

  • Figure 1: The SFG in Fig. \ref{['fig:1_index_node']} has nodes $p,q,r$ and source $S$. Splitting node $r$ into two nodes $r_1$ and $r_2$ eliminates all the loops in $\mathcal{G}_s$ passing through the node as observed in Fig. \ref{['fig:2_index_node']}.
  • Figure 2: Effect of the constant in-degree condition.
  • Figure 3: Effect of the edge modification $(1,3,4)$ in $\mathcal{G}$ on its corresponding SFG $\mathcal{G}_s$
  • Figure 4: The distribution of non-residual nodes in level sets.
  • Figure 5: The role of index nodes in the level sets' construction.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Lemma 1
  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Remark 1
  • Example 3
  • Lemma 2
  • proof
  • Example 4
  • ...and 4 more