Local Coordination and the Geometry of Social Networks

TL;DR

The paper establishes a precise link between network geometry and local coordination efficiency by introducing amenability as the key property. It shows that amenable graphs admit near-efficient leader equilibria under local messaging, while non-amenable graphs render high correlations and low inefficiency impossible, even with private communication. The authors develop the Shapley influence distribution as a contraction tool and employ grand couplings to connect local information flow with global partitioning, yielding both sufficiency and a partial necessity for efficient coordination. They also analyze cycle graphs to illustrate near-optimality of leader equilibria in a concrete setting and discuss broader implications for distributed processes on networks.

Abstract

We study agents playing a pure coordination game on a large social network. Agents are restricted to coordinate locally, without access to a global communication device, and so different regions of the network will converge to different actions, precluding perfect coordination. We show that the extent of this inefficiency depends on the network geometry: on some networks, near-perfect efficiency is achievable, while on others welfare is strictly bounded away from the optimum. We provide a geometric condition on the network structure that characterizes when near-efficiency is attainable. On networks in which it is unattainable, our results more generally preclude high correlations between outcomes in a large spectrum of dynamic games.

Figures (4)

• Figure 1: $\mathbb{Z}$, amenable. Communities have length $c=2r+1$. Miscoordination occurs only along the red edges, which form a small fraction of the edges.
• Figure 2: $\mathbb{Z}^2$, amenable. Communities are $(r+1)\times(r+1)$ squares. Miscoordination occurs only along the red edges, which, as on the line, form a small fraction of the edges.
• Figure 3: Binary tree, non-amenable.
• Figure 4: In the equilibrium constructed in the proof of Theorem \ref{['thm:amenable-transitive']}, agents choose at random which tiles should be selected. Communities are the intersections of the selected tiles. Agents outside the selected tiles form singleton communities.

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• proof : Proof of Theorem \ref{['thm:amenable']}
• Theorem 2
• Proposition 1
• proof : Proof of Proposition \ref{['prop:stable']}
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 2