Learning and convergence in networks

We study the convergence properties of learning in social and economic networks. We characterize the effect of network structure on the long-run convergent behaviour and on the time of convergence to steady state. Agents play a repeated game governed by two underlying behavioural rules; they are myopic and boundedly rational. Under this setup, the long-run limit of the adoptive process converges to a unique equilibrium. We treat the dynamic process as a Markov chain and derive the bound for the convergence time of the chain in terms of the stationary distribution of the initial and steady state population configurations, and the spectrum of the transition matrix. We in turn differentiate between the two antagonistic effects of the topology of the interaction structure on the convergence time; that through the initial state and that through the asymptotic behaviour. The effect through the initial state favours local interactions and sparsely connected network structures, while that through the asymptotic behaviour favours densely and uniformly connected network structures. The main result is that the most efficient network topologies for faster convergence or learning are those where agents belong to subgroups in which the inter-subgroup interactions are "weaker" relative to within subgroup interactions. We further show that the inter-subgroup interactions or long-ties should not be "too weak" lest the convergence time be infinitely long and hence slow learning.