This paper models the dynamic process through which a large society may succeed in building up its “social capital” by establishing a stable and dense pattern of interaction among its members. In the model, agents interact according to a collection of infinitely repeated Prisoner’s Dilemmas played on the current social network. This network not only specifies the playing partners but, crucially, also determines how relevant strategic information diffuses or new cooperation opportunities are found. Over time, the underlying payoffs randomly change, i.e. display some “volatility”, which leads agents to react by creating new links and removing others. The process is ergodic, so we use numerical simulations to “compute” its long-run invariant behavior and obtain the following conclusions: (a) Only if payoff volatility is not too high can the society sustain a dense social network. (b) The social architecture endogenously responds to increased volatility by becoming more cohesive. (c) Network-based strategic effects are an essential buffer that preclude the abrupt collapse of the social network in the face of growing volatility. These conclusions, largely in tune with those of the social-capital literature, are further studied analytically in a companion paper through the use of mean-field techniques.