A Simple Model of Influence

Abstract

We propose a simple model of influence in a network, based on edge density. In the model vertices (people) follow the opinion of the group they belong to. The opinion percolates down from an active vertex, the influencer, at the head of the group. Groups can merge, based on interactions between influencers (i.e., interactions along ‘active edges’ of the network), so that the number of opinions is reduced. Eventually no active edges remain, and the groups and their opinions become static.

Our analysis is for G(n, m) as m increases from zero to $N={n \atopwithdelims ()2}$. Initially every vertex is active, and finally G is a clique, and with only one active vertex. For $m\leqslant N/\omega$, where $\omega = \omega (n)$ grows to infinity, but arbitrarily slowly, we prove that the number of active vertices a(m) is concentrated and we give w.h.p. results for this quantity. For larger values of m our results give an upper bound on $\mathbb {E}\,a(m)$.

We make an equivalent analysis for the same network when there are two types of influencers. Independent ones as described above, and stubborn vertices (dictators) who accept followers, but never follow. This leads to a reduction in the number of independent influencers as the network density increases. In the deterministic approximation (obtained by solving the deterministic recurrence corresponding to the formula for the expected change in one step), when $m=cN$, a single stubborn vertex reduces the number of influencers by a factor of $\sqrt{1-c}$, i.e., from a(m) to $(\sqrt{1-c})\,a(m)$. If the number of stubborn vertices tends to infinity slowly with n, then no independent influencers remain, even if $m=N/\omega$.

Finally we analyse the size of the largest influence group which is of order $(n/k) \log k$ when there are k active vertices, and remark that in the limit the size distribution of groups is equivalent to a continuous stick breaking process.