Democracy in Markov chains and its preservation under local perturbations

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 distributed algorithms like consensus or with opinion dynamic models: it says that each agent measure/opinion is going to play a negligible role in the asymptotic behavior of the global system. Simple random walks on undirected graphs of bounded degree and increasing cardinality are one of the simplest examples of democratic chains. Similar examples can be built considering more general time-reversible chains. 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.