Unveiling Oligarchy in Influence Networks From Partial Information

In modern society, individuals' opinions on various topics evolve as the result of their continuous interactions and are shaped by interpersonal influences and individual social power. Friedkin's reflected appraisal theory reveals how social power evolves along discussion sequences as a consequence of direct and indirect interpersonal influence over issue outcomes. This reflected appraisal theory also suggests how, in enduring social groups, a form of concentrated power in an entrenched minority arises as a near iron law of influence network dynamics. Motivated by theoretical and empirical findings, this article studies oligarchic influence systems, i.e., systems in which the social power is accumulated within a small group of individuals. First, we propose a new mathematical model of the reflected appraisal mechanism and illustrate how, under appropriate conditions, this model leads the influence system to asymptotically converge to an oligarchy. Second and most important, we address the data-driven social power estimation problem, i.e., we propose algorithms to unveil oligarchies in influence networks along issue sequences. Our algorithmic approach is based upon 1) casting the problem of learning social power as a sparse recovery problem and 2) estimating individuals' social power directly from the observation of initial and final average opinions, without the burdensome intermediate step of social influence recovery. We prove that the social power estimation can be performed with partial sampling of initial opinions and we derive theoretical bounds on the estimation error in terms of the number of observations. We study sample complexity and computational requirements of the proposed methods. Finally, our findings are validated via numerical experiments.