The Layer Paradox

Coordination games on single-layer networks have well-understood dynamics: agents adjust their opinions to align with neighbors, and whether consensus emerges depends on the network structure. The spectral gap of the network determines the convergence rate. Add a second layer — the same agents connected differently — and the dynamics change in both directions simultaneously.

On multiplex networks, multilayer interactions can induce global consensus even when no single layer achieves it alone. Two network structures that individually fail to coordinate can, when their agents are coupled, produce agreement that neither could sustain in isolation. The coupling between layers provides pathways for information flow that don’t exist within either layer.

But the reverse also holds: individually coordinated layers may lose consensus once interconnected. Stable single-layer equilibria can become unstable when agents must also coordinate across a second set of relationships with different structure. The coupling that helps in one direction hurts in the other.

Spectral analysis of the merged and switching dynamics establishes the conditions for each outcome — when coupling helps and when it destroys. Periodically switching between layers (attending to one network, then the other) produces different stability properties than merging layers into a weighted aggregate.

The through-claim: the whole is not the sum of the parts, and the direction of the discrepancy is unpredictable from the parts alone. Two failing networks can succeed together; two succeeding networks can fail together. This is not merely emergence — it’s a qualitative inversion of the single-layer prediction. Any analysis of coordination that treats social layers independently (professional network, friend network, family network) and aggregates their predictions will be wrong in both directions.