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.
Keywords: Learning, Local interactions, Coordination game, Strategic complementarity, Convergence time, Convergence rate, Markov chain.
JEL Codes: C73, D83