Long-range social pressure and the evolution of cooperation in multiplex networks

Social pressure -- the awareness of being observed by others -- is a fundamental driver of prosocial behavior in human societies.

Yet it is typically assumed that only direct neighbors exert vigilance pressure on an individual, despite empirical evidence that social influence persists to at least three degrees of separation.

Here we show that extending the reach of social vigilance beyond direct neighbors substantially promotes cooperation.

We couple a Prisoner's Dilemma on one layer of a multiplex network to a vigilance cascade on the other, with influence decaying geometrically with network distance.

Extending vigilance to just the second circle of influence shifts the critical temptation for defection by nearly 30\% in sparse networks.

Extending to four circles raises this threshold by over 50\%.

The $L=1\to2$ transition already accounts for most of the gain, consistent with the decay coefficients of social influence reported in controlled experiments.

The effect is strongest in sparse topologies, requires that the vigilance and game layers be aligned, and reproduces directly on a real social network of physicians; in dense, hub-dominated networks the gain instead depends sharply on how fast influence decays with distance, switching between weak and strong cooperation as the decay rate crosses a threshold.

Our results strongly suggest that even modest expansions of social awareness -- such as those enabled by online social platforms -- can substantially reshape the landscape of cooperative behavior in human populations.