Consider a graph G = (V, E) and an initial random coloring where each vertex v ∈ V is blue with probability P b and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random d-regular graph G n,d . It is shown that for all ǫ > 0, P b ≤ 1/2 − ǫ results in final complete occupancy by red in O(log d log n) rounds with high probability, provided that d ≥ c/ǫ 2 for a suitable constant c. Furthermore, we show that with high probability, G n,d is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can "take over" in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg [21].
Assume for a graph G = (V, E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the Erdős-Rényi random graph G n,p and regular expanders. First we consider the behavior of the majority model on G n,p with an initial random configuration, where each node is blue independently with probability p b and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely log n n . Furthermore, we say a graph G is λ-expander if the second-largest absolute eigenvalue of its adjacency matrix is λ. We prove that for a ∆-regular λ-expander graph if λ/∆ is sufficiently small, then the majority model by starting from ( 1 2 − δ)n blue nodes (for an arbitrarily small constant δ > 0) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an "efficient" and "fast" density classifier on regular expanders. As a by-product of our results, we show regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node ∆-regular Ramanujan graph if the initial number of blue nodes is s ≤ βn, the number of blue nodes in the next round is at most cs ∆ for some constants c, β > 0. This settles an open problem by Peleg [33]. Eligible for best student paper.
The dynamics of rumor spreading is investigated using a model with three kinds of agents who are, respectively, the Seeds, the Agnostics, and the Others. While Seeds are the ones who start spreading the rumor being adamantly convinced of its truth, Agnostics reject any kind of rumor and do not believe in conspiracy theories. In between, the Others constitute the main part of the community. While Seeds are always Believers and Agnostics are always Indifferents, Others can switch between being Believer and Indifferent depending on who they are discussing with. The underlying driving dynamics is implemented via local updates of randomly formed groups of agents. In each group, an Other turns into a Believer as soon as m or more Believers are present in the group. However, since some Believers may lose interest in the rumor as time passes by, we add a flipping fixed rate 0<d<1 from Believers into Indifferents. Rigorous analysis of the associated dynamics reveals that switching from m=1 to m≥2 triggers a drastic qualitative change in the spreading process. When m=1, even a small group of Believers may manage to convince a large part of the community very quickly. In contrast, for m≥2, even a substantial fraction of Believers does not prevent the rumor dying out after a few update rounds. Our results provide an explanation on why a given rumor spreads within a social group and not in another and also why some rumors will not spread in neither groups.
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