A Vector Threshold Model for the Simultaneous Spread of Correlated Influence

Spread of influence is one of the most widely studied propagation processes in the literature on complex networks. Examples include the rise of collective action to join a riot and diffusion of beliefs, norms, and cultural fads, to name a few. Most existing works on modeling influence propagation consider a single content (e.g., an opinion, decision, product, political view, etc.) spreading over a network independent from everything else. However, most real-life examples involve multiple correlated contents spreading simultaneously and exhibiting positive (e.g., opinions on same-sex marriage and gun control) or negative (e.g., opinions on universal health care and tax-relief for the "rich") correlation. To accommodate these cases, this paper proposes the vector threshold model, as an extension of the widely used Watts threshold model for complex contagions. Here, the state of a node is represented by a binary vector representing their opinion on a number of content items. Nodes switch their states based on the influence they receive from their neighbors in the network. The influence is represented by a vector containing the proportion of neighbors who support each content; both positively and negatively correlated contents can be captured in this formulation by using different rules for switching node states. Our main result is concerned with the expected size of global cascades, i.e., cases where a randomly chosen node can initiate a propagation that eventually reaches a positive fraction of the whole population. We also derive conditions on network structure for global cascades to be possible. Analytic results are supported by a numerical study.