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A Dynamic Model of Social Network Formation

Brian Skyrms, Robin Pemantle

TL;DR

The paper tackles how social networks form and evolve when agents engage in repeated games, with pairings driven by a stochastically evolving network. It develops several reinforcement-based dynamics, from uniform reinforcement to asymmetric and symmetric weight updates, to study how network structure co-evolves with strategy. The results reveal that even simple baseline reinforcement can generate emergent structures such as sparse pairings and symmetric weight patterns, while more complex dynamics (discounting, noise, and nontrivial strategies) produce rich behaviors including pairing, star formations, and coordination outcomes in games like Rousseau's Stag Hunt. The findings underscore that incorporating structural dynamics into game-theoretic models can yield more realistic predictions and a broader range of equilibria, with implications for understanding cooperation and coordination in real networks.

Abstract

We consider a dynamic social network model in which agents play repeated games in pairings determined by a stochastically evolving social network. Individual agents begin to interact at random, with the interactions modeled as games. The game payoffs determine which interactions are reinforced, and the network structure emerges as a consequence of the dynamics of the agents' learning behavior. We study this in a variety of game-theoretic conditions and show that the behavior is complex and sometimes dissimilar to behavior in the absence of structural dynamics. We argue that modeling network structure as dynamic increases realism without rendering the problem of analysis intractable.

A Dynamic Model of Social Network Formation

TL;DR

The paper tackles how social networks form and evolve when agents engage in repeated games, with pairings driven by a stochastically evolving network. It develops several reinforcement-based dynamics, from uniform reinforcement to asymmetric and symmetric weight updates, to study how network structure co-evolves with strategy. The results reveal that even simple baseline reinforcement can generate emergent structures such as sparse pairings and symmetric weight patterns, while more complex dynamics (discounting, noise, and nontrivial strategies) produce rich behaviors including pairing, star formations, and coordination outcomes in games like Rousseau's Stag Hunt. The findings underscore that incorporating structural dynamics into game-theoretic models can yield more realistic predictions and a broader range of equilibria, with implications for understanding cooperation and coordination in real networks.

Abstract

We consider a dynamic social network model in which agents play repeated games in pairings determined by a stochastically evolving social network. Individual agents begin to interact at random, with the interactions modeled as games. The game payoffs determine which interactions are reinforced, and the network structure emerges as a consequence of the dynamics of the agents' learning behavior. We study this in a variety of game-theoretic conditions and show that the behavior is complex and sometimes dissimilar to behavior in the absence of structural dynamics. We argue that modeling network structure as dynamic increases realism without rendering the problem of analysis intractable.

Paper Structure

This paper contains 19 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Making friends: a baseline model of uniform reinforcement
  3. Friends I: asymmetric weights
  4. Friends II: symmetrized reinforcement
  5. Analysis of Friends I and II
  6. Analysis of Friends I and II
  7. Making enemies
  8. The transfer model
  9. The resistance model
  10. A better model?
  11. Perturbations of the models
  12. Discounting the past
  13. Analysis of discounting the past
  14. Introduction of noise
  15. Noise and discounting
  16. ...and 4 more sections

Key Result

Theorem 1

The probability matrix for Friends I with $n$ players will converge to a random limit $p$ as time goes to infinity. The distribution of the limit is that the rows of $p$ are independent, each having Dirichlet distribution (ignoring the zero entry on the diagonal) whose parameters are $n-1$ ones.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6