Consumer referral in a small world network

It is demonstrated that contrary to anticipation, the firm’s profit smoothly approaches the profit in the complete network as the rewiring probability increases, but that the per consumer profit is saturated to a finite non-zero value when the reWiring probability is near zero, implying a small world transition.

Abstract

We consider a firm's profit maximization problem when the demand for the product is created by referrals between consumers connected in a small world network. We generate a small world network by applying the rewiring algorithm of Watts and Strogatz and incorporate it into the profit maximization problem. We demonstrate (i) that contrary to our anticipation, the firm's profit smoothly (not rapidly) approaches the profit in the complete network as the rewiring probability increases, but (ii) that the per consumer profit is saturated to a finite non-zero value when the rewiring probability is near zero, implying a small world transition.