The robustness of democracy in Markov chain models

A sequence of irreducible Markov chains with increasing state cardinality is called democratic if the sequence of corresponding invariant probabilities converges to 0 uniformly. Democracy is a relevant property which naturally shows up when we deal with opinion dynamic models and cooperative algorithms over a network: it says that each agent measure/opinion is going to play a negligeable role in the asymptotic behavior of the global system. In this paper we prove a general result which says that, under some technical assumptions, perturbing the transition probabilities from a finite number of vertices of a time-reversible democratic sequence of chains, democracy is preserved. We want to stress the fact that the local perturbation in general breaks the time-reversibility of the chains. The main technical assumption needed in our result is the irreducibility of the limit Markov chains and we show with an example that this assumption is indeed necessary.