Properties of systems with majority voting rules have been exhaustingly studied. In this work we focus on the randomized case - where the system is initialized by randomized initial set of seeds. Our main aim is to give an asymptotic estimate for sampling probability, such that the initial set of seeds is (is not) a dynamic monopoly almost surely. After presenting some trivial examples, we present exhaustive results for toroidal mesh and random 4-regular graph under simple majority scenario.
Idea of majority voting is commonly used to resolve many problems related to achieving consistency between different parts of distributed computation. For example, majority voting is used to preserve data consistency when updating copies of the same data. Also, it is quite common to use majority voting to resolve inconsistencies in distributed database management. Majority based systems were also successfully used by Agur to examine the plasticity and precision of the immune response in [17].
The model for the system is as follows: Let G = (V, E) be simple undirected graph of size n. Every vertex has its color which is either black or white, and represents the state of the node (for example black = faulty, white = not faulty). By S we shall denote the set of black vertices at the beginning of the process. These vertices are also called seeds. By evolution of such system (or, the coloring process) we shall mean the synchronous process where in each step each vertex adjusts its color according to colors of its neighbors and its internal contamination and decontamination rules. (De)contamination rules determine under what configuration of its neighbors the white (black) vertex turns black (white). In this work we focus on the case when there is no decontamination rule, and the contamination rule is the simple majority rule. This means that the white vertex turns black if at least half of its neighbors are black and there is no possibility for black vertex to turn white. Set of seeds S is called dynamic monopoly, (or dynamo for short) if the corresponding coloring process leads to monochromatic black graph. For more rigorous definition, see section 2.1.
A significant attention was given to this model, and many interesting results were obtained. Probably the most basic (but certainly not trivial) question asks for determining the minimal cardinality of a dynamo on a fixed graph G [4,5]. Another interesting parameter of a dynamo besides its cardinality is the time that is needed for contamination to spread. This was analyzed for the first time in the literature in [4,5]. Several works are related to more advanced topics such 1 as decontamination of the system by external agents [7]. Finally, [9] defines the terminology of immune subgraphs and asks how the immune subgraph of a certain graph looks like.
All these tasks were solved for small class of graphs. Close attention is given to the ring and its several modifications [4], as well as to the torus and its modifications [5] (toroidal mesh, torus cordalis, torus serpentinus). Many results exist also for hypercube and binary tree.
Other possible questions ask about the minimal cardinality of a dynamo on arbitrary graphs. It is proved that on general directed graph the minimal dynamo has at most 0.727|V | vertices. It is likely that this result will be improved. On the other hand, in the case of general undirected graphs the minimal dynamo consists of at most V /2 (+1) vertices (depending on the exact specification of contaminating rules), this result is proved to be sharp [18].
There exist some interesting results about such systems even in the “most general” case, where contamination rule is that the vertex is contaminated if at least fraction α of its neighbors are black [20,21,22]. These works ask under what condition at least some fraction δ of all vertices is turned black.
Finally, we mention several works that we find most related to the paper. For a randomly chosen set of initial black vertices, Gleeson and Cahalane [21] gave an exact formula for the expected fraction of black vertices at the end in tree-like graphs. In [20,23] authors gave their estimate for minimal number of black vertices needed for re-coloring of at least fraction δ of all vertices on Erdős-Rényi random graph. For more thorough survey, see [10,8].
Motivation for studying coloring process induced by random set of seeds is quite straightforward. For example, considering vertices as computing nodes, each node can fail with probability p, independent on failure of other nodes. Although this looks like a very common scenario, there exists only few results about systems initialized with random initial coloring. Given graph G We consider random initial set of seeds S p = S p (G) as the set containing any vertex of G with probability p (independently on other vertices). Naturally, for every fixed graph G this gives us some probability that the random set of seeds is a dynamo. Determining these probabilities analytically is quite hard (if not impossible) and moreover, it can be done numerically with sufficient accuracy. Therefore, we shall try to obtain asymptotic results of the form: assuming 4-regular graph with n vertices, random set of seeds S 0.12 is dynamo with high probability (w.h.p.). On the other hand, S 0.10 is not a dynamo (w.h.p.).
In many situations, to determine the minimal value of p that S p is (w.h.p) dynamo is trivial. For example, assuming Erdős-Rényi random graph G(n
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