Opinion formation under bounded confidence via gossip algorithms

This paper investigates two bounded confidence gossip algorithms, one with constant confidence threshold and the other with increasing one, for effective communicating between agents in a network among whom some opinion formation forms. Each agent in the network keeps a real value presenting its opinion about some matter. The opinions of agents will be updated time by time according to an iterative procedure. At each time, (i) two arbitrary agents are chosen randomly, (ii) they exchange their opinions, and (iii) if the distance between the opinions does not exceed some given confidence threshold, they update their opinions as the average of the two. It is shown that the algorithms almost surely drive any initial opinion profile to some opinion profile in which any two opinions either are the same or differ more than the confidence threshold. Moreover, the second algorithm can help achieving a prescribed number of different opinions on the convergence opinion profiles.