Consensus Using a Network of Finite Memory Pólya Urns

We introduce a
finite memory
interacting Pólya urn process over a connected network which models consensus dynamics for interacting individuals. More specifically, each urn (individual) in the network is initially equipped with some red and black balls, with the fraction corresponding to the individual’s opinion (or belief) on a certain color. At each time instant and for each urn, a ball is drawn from a “super-urn”, which consists of all balls present in that urn and its neighboring urns; then reinforcing balls of the color just drawn are added to the urn for a limited period of
$M$
future time instants, where
$M$
denotes the memory parameter. Additionally, and important for our objective, as of time
$t=M+1$
, we remove the balls which were present in the urns initially. By examining the structure of the resulting underlying reducible Markov process, we show that individuals eventually reach consensus in the sense that they all achieve identical probabilities of drawing a red ball. Moreover, when the network has homogeneous reinforcement parameters, we construct a class of linear dynamical systems with time delay whose trajectory gives the probability of drawing a red ball for each node
$i$
at a time instant
$t$
. We examine the asymptotic behavior of such a network and exactly determine its consensus value. Our simulation confirms our theoretical findings by demonstrating the asymptotic behavior of draw variables of the network in some case studies.