Majority Dynamics and the Retention of Information

Abstract

We consider a group of agents connected by a social network who participate in majority dynamics: each agent starts with an opinion in {-1,+1} and repeatedly updates it to match the opinion of the majority of its neighbors. We assume that one of {-1,+1} is the "correct" opinion S, and consider a setting in which the initial opinions are independent conditioned on S, and biased towards it. They hence contain enough information to reconstruct S with high probability. We ask whether it is still possible to reconstruct S from the agents' opinions after many rounds of updates. While this is not the case in general, we show that indeed, for a large family of bounded degree graphs, information on S is retained by the process of majority dynamics. Our proof technique yields novel combinatorial results on majority dynamics on both finite and infinite graphs, with applications to zero temperature Ising models.Comment: 19 pages, accepted for publication in the Israel Journal of Mathematic

Scaling limits of the Schelling model

“…In Section 3.1 we prove that the continuum Schelling model describes the discrete Schelling model well for small times and large w. In Section 3.2 we conclude the proof of the scaling limit result for the one-dimensional Schelling model. We also prove (proceeding similarly as in [TT14]) that the opinion of each node converges a.s. for any N ∈ N and M ≥ 2, and we include a lemma which might be related to the typical cluster size for the limiting opinions in higher dimensions. We conclude the paper with a list of open questions in Section 4.…”

“…Choose the w ij 's such that (i,j)∈E w ij < ∞. The existence of appropriate w ij satisfying these properties follows by [TT14, Proposition 3.4], since the degree of each node is bounded by (2w + 1) N , and since our graph satisfies the growth criterion considered in [TT14]. For each i ∈ Z N and t ≥ 0 define J i t by J i t := j∈N(i) w ij 1 X(j,t) =X(i,t) − j∈N(i) w ij 1 X(j,t − ) =X(i,t − ) .…”

Percolation in majority dynamics

“…They prove that, provided the graph does not grow very fast, it presents the period-two property, that says each vertex eventually has an orbit of period at most two. In Tamuz and Tessler [18], this result is strengthened, proving that, in the asynchronous model, almost surely each vertex eventually fixates and, besides, that it changes opinion only a bounded number of times. This last result is actually combinatorial: If no two clocks ring at the same time, then, for any initial condition, each site eventually fixates.…”

“…We remark that their proof also works for Z 2 , provided p is not close to 1 2 . As a consequence of [18], majority dynamics on Z 2 fixates. This allows us to define the limiting configuration η ∞ as the pointwise limit of η t .…”

Majority dynamics with one nonconformist

“…Other facets of majority dynamics have been studied, such as the question of whether a bias in the initial opinions tends to be preserved by this process (Tamuz and Tessler, [15]), and the threshold of initial bias which results in consensus on infinite trees (Kanoria and Montanari, [8]). Probabilistic versions of majority dynamics have been studied on highly-structured graphs.…”