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Past-aware game-theoretic centrality in complex contagion dynamics

Francesco Zigliotto

TL;DR

This work introduces past-aware game-theoretic centrality (PAGTC), a framework that conditions node centrality on a predefined set of already-active collaborators to capture both future potential and past activations in networks. It provides a general, computable approach for PAGTC across group centralities, including a closed-form for the $K$-complex contagion centrality $\nu_K$, enabling scalable algorithms for seed selection and influence maximization. The authors demonstrate that PAGTC-based heuristics outperform standard greedy methods in many non-submodular settings, especially as $K$ grows, and validate the approach on multiple realistic networks, including dynamic targeting variants. The results highlight practical impacts for efficient diffusion control, information spread, and strategy design in complex contagion scenarios.

Abstract

In this paper, we introduce past-aware game-theoretic centrality, a class of centrality measures that captures the collaborative contribution of nodes in a network, accounting for both uncertain and certain collaborators. A general framework for computing standard game-theoretic centrality is extended to the past-aware case. As an application, we develop a new heuristic for different versions of the influence maximization problem in complex contagion, which models processes requiring reinforcement from multiple neighbors to spread. A computationally efficient explicit formula for the corresponding past-aware centrality score is derived, leading to scalable algorithms for identifying the most influential nodes, which in most cases outperform the standard greedy approach in both efficiency and solution quality.

Past-aware game-theoretic centrality in complex contagion dynamics

TL;DR

This work introduces past-aware game-theoretic centrality (PAGTC), a framework that conditions node centrality on a predefined set of already-active collaborators to capture both future potential and past activations in networks. It provides a general, computable approach for PAGTC across group centralities, including a closed-form for the -complex contagion centrality , enabling scalable algorithms for seed selection and influence maximization. The authors demonstrate that PAGTC-based heuristics outperform standard greedy methods in many non-submodular settings, especially as grows, and validate the approach on multiple realistic networks, including dynamic targeting variants. The results highlight practical impacts for efficient diffusion control, information spread, and strategy design in complex contagion scenarios.

Abstract

In this paper, we introduce past-aware game-theoretic centrality, a class of centrality measures that captures the collaborative contribution of nodes in a network, accounting for both uncertain and certain collaborators. A general framework for computing standard game-theoretic centrality is extended to the past-aware case. As an application, we develop a new heuristic for different versions of the influence maximization problem in complex contagion, which models processes requiring reinforcement from multiple neighbors to spread. A computationally efficient explicit formula for the corresponding past-aware centrality score is derived, leading to scalable algorithms for identifying the most influential nodes, which in most cases outperform the standard greedy approach in both efficiency and solution quality.

Paper Structure

This paper contains 17 sections, 2 theorems, 37 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction and related works
  2. Notation
  3. Past-aware game-theoretic centrality
  4. Probabilistic interpretation of GTC
  5. Definition of PAGTC
  6. Application to set-function maximization problem
  7. Computation of PAGTC
  8. Application to complex contagion
  9. Complex contagion dynamics
  10. PAGTC for complex contagion
  11. One-round influence maximization
  12. Influence maximization
  13. Complex contagion with dynamic targeting
  14. Numerical experiments
  15. One-round and full influence maximization
  16. ...and 2 more sections

Key Result

Theorem 1

Given a graph $G=(V,E)$, a set of nodes $S_0$, and a group centrality $\nu$ of the form e:efficient_nu, the semivalue PAGTC of $u\in V$ can be computed as follows: where $C_\beta$ is given by e:c_beta and

Figures (5)

  • Figure 1: Violation of submodularity of $\nu_K$ and $\nu_K^*$, for $K=3$.
  • Figure 2: Maximization of $\nu_K(S)$, with $K=3$ and $r=7$, according to different algorithms. The solution set $S$ is depicted with black nodes, and the numbers represents the order in which they joined the solution set. The nodes outside $S$ that have at least $K$ neighbors in $S$ are marked with $\times$.
  • Figure 3: Comparison between the greedy* and pagtc$_\delta$ algorithms for influence maximization on a family of small-world graphs of increasing size, each with an average degree above $7$. The parameter $K$ is set to $5$, while $r$ corresponds to approximately $10\%$ of the total number of nodes.
  • Figure 4: Comparison between greedy* and pagtc$_{\delta}$ for influence maximization on the les-miserables graph, using various values of $K$ and $r$. Each plot shows the relative influence $\nu_K^*(S)/|V|$ of the obtained solution $S$.
  • Figure 5: Comparison of semivalue-based dynamic targeting strategies for complex contagion on the les-miserables graph.

Theorems & Definitions (15)

  • Definition 1
  • Example 1
  • Definition 2
  • Remark 1
  • Example 2
  • Definition 3
  • Remark 2
  • Theorem 1
  • proof
  • Definition 4
  • ...and 5 more