A Clustering Coefficient Network Formation Game

For the most up-to-date version please visit http://www.cis.upenn.edu/~brautbar/ccgame.pdf

💡 Research Summary

The paper introduces a novel network formation game in which each agent’s utility is driven by the local clustering coefficient of the node they control, balanced against a linear cost for maintaining each incident edge. Formally, for a graph G = (V,E) with n agents, the utility of agent i is defined as U_i(G) = α·C_i(G) − β·deg_i(G), where C_i(G) is the standard local clustering coefficient (the fraction of possible triangles among i’s neighbors that actually exist), deg_i(G) is i’s degree, and α, β > 0 are parameters that weight the benefit of being in many triangles against the expense of forming links.

The authors first prove that this game is an exact potential game. By defining the global potential Φ(G) = ∑_{i∈V}U_i(G), they show that any unilateral edge addition or deletion by a single player changes Φ by exactly the same amount as it changes that player’s utility. Consequently, best‑response dynamics are guaranteed to converge to a pure Nash equilibrium (NE) in a finite number of steps, establishing the existence of equilibrium for any choice of α and β.

A central contribution is the structural characterization of equilibria as a function of the ratio r = α/β. Three regimes emerge:

1. High clustering incentive (r ≫ 1). When the benefit of triangles dominates edge cost, the unique equilibrium is the complete graph K_n. Every possible edge is present, giving each node a clustering coefficient of 1 and maximizing the α·C_i term despite the high degree cost.

2. Balanced regime (r around 1). Here the equilibrium consists of a dense “core” clique of size k < n together with peripheral nodes that connect only to the core. The core maximizes intra‑core triangles, while peripheral nodes obtain a modest clustering boost by linking to many core members without incurring the cost of forming many edges among themselves.

3. Low clustering incentive (r ≪ 1). When edge cost outweighs the triangle benefit, the equilibrium collapses to a star or a very sparse tree. The central hub may have degree n − 1, but its clustering coefficient is near zero; the overall structure minimizes total edge cost at the expense of clustering.

The paper then evaluates the social welfare W(G) = ∑_i U_i(G) and compares the welfare of equilibria to that of the socially optimal graph (SO). By analyzing the three regimes, the authors derive upper bounds on the Price of Anarchy (PoA):

• For r ≥ 2, the complete graph is both NE and SO, giving PoA = 1.
• For 0.5 < r < 2, PoA ≤ 2·(1 + r), which smoothly interpolates between the two extremes.
• For r ≤ 0.5, the star is the worst‑case NE and its welfare is roughly half that of the optimal dense graph, yielding PoA ≈ 2.

The Price of Stability (PoS) is shown to be close to 1 in all regimes because a socially optimal configuration can always be supported as a Nash equilibrium (e.g., a suitably sized core‑clique with peripheral attachments).

Algorithmically, the authors implement a best‑response dynamics simulation. In each round a randomly selected agent evaluates all possible edge additions and deletions, chooses the move that maximizes its utility, and updates the graph. Empirical results indicate convergence within O(n log n) steps on average, regardless of the initial graph (empty, complete, or random). The final structures observed match the theoretical predictions for the corresponding r values, confirming the robustness of the analytical results.

Finally, the paper discusses extensions. It sketches how the model can be generalized to higher‑order clustering (e.g., counting squares or larger cycles), to heterogeneous edge costs (different β_i per player), and to dynamic settings where α and β evolve over time. In each case, the authors outline conditions under which the potential‑game property is preserved, suggesting that the framework can be adapted to more realistic social‑network scenarios where users have personalized preferences and the environment changes.

Overall, the work provides a rigorous foundation for studying network formation when agents care about local cohesiveness. By linking clustering incentives to game‑theoretic equilibrium analysis, it bridges a gap between network science (where clustering is a key structural metric) and algorithmic game theory (where strategic link formation is traditionally modeled by distance or cost alone). The results have practical implications for the design of online platforms, peer‑to‑peer systems, and any setting where encouraging tightly knit communities while controlling link overhead is desirable.