Whom to Befriend to Influence People.

Alice wants to join a new social network, and influence its members to adopt a new product or idea. Each person v in the network has a certain threshold t(v) for activation, i.e. adoption of the product or idea. If v has at least t(v) activated neighbors, then v will also become activated. If Alice wants to activate the entire social network, whom should she befriend? We study the problem of finding the minimum number of links that Alice should form to people in the network, in order to activate the entire social network. This Minimum Links Problem has applications in viral marketing and the study of epidemics. We show that the solution can be quite different from the related and widely studied Target Set Selection problem. We prove that the Minimum Links problem is NP-complete, in fact it is hard to approximate to within an is an element of ln n factor for some constant is an element of, even for graphs with degree 3 and with threshold at most 2. In contrast, we give linear time algorithms to solve the problem for trees, cycles, and cliques, and give precise bounds on the number of links needed.