Contagion in social networks: On contagion thresholds

Abstract

Consider a story initiated by a set S of nodes belonging to a network. We assume that all the remaining nodes in the network have the same adoption threshold q such that each of them will accept the story if and only if the story has been accepted by at least a fraction q of its neighbors. Consequently, there is a maximum threshold q at and below which a full contagion (i.e. acceptance of the story by all nodes in the network) can occur from S, called the contagion threshold of S. We study how it is influenced by the neighborhood of S or by the global topology of the network. Firstly, we establish two algorithms to compute the contagion threshold of any given set of nodes in a network. Secondly, we show that the contagion threshold of a connected set in a tree is completely determined by the maximum degree of the nodes not in the set. Next, we derive bounds for the contagion threshold, while indicating when it preserves the neighborhood-inclusion pre-order and when the preservation may fail. Finally, we look at the structural characterization of the sets of nodes with high contagion thresholds.