On the Influence of Committed Minorities on Social Consensus

Human behavior is profoundly affected by the influenceability of individuals and the social networks that link them together. In the sociological context, spread of ideas, ideologies and innovations is often studied to understand how individuals adopt new behaviors, opinions, or ideologies through the influence of others interacting with them either personally or through social media. Our work studies the evolution of opinions and the dynamics of its spread. We use the binary agreement variant of the model known as the Naming Game, in which individuals hold opinions and periodically a pair of randomly chosen nodes sharing an edge try to agree on their opinions. In our studies we allow just two opinions to exist. In an initial state, a given opinion B is shared by all individuals except for a fraction p<1 of the community committed to opinion A. In a generalization of this model, we consider also the initial state in which some of the individuals holding opinion B are also committed to it. Committed individuals are defined as those who are immune to influence but they can influence others through the usual prescribed rules for opinion change.

We show that under certain conditions the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed individuals. Our work defines the conditions necessary for such rapid reversal of opinions held in the community. We refer to the time needed to achieve such a reversal as a consensus time, Tc, and we study how the consensus time varies with the size of the committed fraction. Specifically, we show that when the committed fraction grows beyond a critical value, pc, (for large fully connected graphs, pc=9.79%), there is a dramatic change of network behavior. At pc, also referred to as a tipping-point, there is drastic decrease in the consensus time, Tc, that it takes the entire population to hold the committed opinion. Below this value, the consensus time is proportional to the exponential function of the network size, while above this value this time is proportional to the logarithm of the network size. We also discuss conditions under which the committed minority can rapidly reverse the influenceable majority even if the latter is supported by a small fraction of individuals committed to their opinion. Finally we study impact of the network structure on the critical value at which the tipping point exists. These results are relevant to understanding the network based mechanisms involved in the social movements and underlying the societal changes.