In the voter model, each node of a graph has an opinion, and in every round each node chooses independently a random neighbour and adopts its opinion. We are interested in the consensus time, which is the first point in time where all nodes have the same opinion. We consider dynamic graphs in which the edges are rewired in every round (by an adversary) giving rise to the graph sequence @math , where we assume that @math has conductance at least @math . We assume that the degrees of nodes don't change over time as one can show that the consensus time can become super-exponential otherwise. In the case of a sequence of @math -regular graphs, we obtain asymptotically tight results. Even for some static graphs, such as the cycle, our results improve the state of the art. Here we show that the expected number of rounds until all nodes have the same opinion is bounded by @math , for any graph with @math edges, conductance @math , and degrees at least @math . In addition, we consider a biased dynamic voter model, where each opinion @math is associated with a probability @math , and when a node chooses a neighbour with that opinion, it adopts opinion @math with probability @math (otherwise the node keeps its current opinion). We show for any regular dynamic graph, that if there is an @math difference between the highest and second highest opinion probabilities, and at least @math nodes have initially the opinion with the highest probability, then all nodes adopt w.h.p. that opinion. We obtain a bound on the convergences time, which becomes @math for static graphs.
There are several other models which are related to the voter model, most notably the Moran process and rumor spreading in the phone call model. In the case of the Moran process, a population resides on the vertices of a graph. The initial population consists of one mutant with fitness @math and the rest of the nodes are non-mutants with fitness 1. In every round, a node is chosen at random with probability proportional to its fitness. This node then reproduces by placing a copy of itself on a randomly chosen neighbour, replacing the individual that was there. The main quantities of interest are the probability that the mutant occupies the whole graph (fixation) or vanishes (extinction), together with the time before either of the two states is reached (absorption time). There are several publications considering the fixation probabilities @cite_14 @cite_10 @cite_7 .
@cite_14 @cite_10 @cite_7
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This work extends what is known so far for a basic model of evolutionary antagonism in undirected networks (graphs). More specifically, this work studies the generalized Moran process, as introduced by Lieberman, Hauert, and Nowak [Nature, 433:312-316, 2005], where the individuals of a population reside on the vertices of an undirected connected graph. The initial population has a single mutant of a fitness value r (typically r>1), residing at some vertex v of the graph, while every other vertex is initially occupied by an individual of fitness 1. At every step of this process, an individual (i.e. vertex) is randomly chosen for reproduction with probability proportional to its fitness, and then it places a copy of itself on a random neighbor, thus replacing the individual that was residing there. The main quantity of interest is the fixation probability, i.e. the probability that eventually the whole graph is occupied by descendants of the mutant. In this work we concentrate on the fixation probability when the mutant is initially on a specific vertex v, thus refining the older notion of which studied the fixation probability when the initial mutant is placed at a random vertex. We then aim at finding graphs that have many "strong starts" (or many "weak starts") for the mutant. Thus we introduce a parameterized notion of selective amplifiers (resp. selective suppressors) of evolution. We prove the existence of strong selective amplifiers (i.e. for h(n)=Θ(n) vertices v the fixation probability of v is at least @math for a function c(r) that depends only on r), and the existence of quite strong selective suppressors. Regarding the traditional notion of fixation probability from a random start, we provide strong upper and lower bounds: first we demonstrate the non-existence of "strong universal" amplifiers, and second we prove the Thermal Theorem which states that for any undirected graph, when the mutant starts at vertex v, the fixation probability at least @math . This theorem (which extends the "Isothermal Theorem" of for regular graphs) implies an almost tight lower bound for the usual notion of fixation probability. Our proof techniques are original and are based on new domination arguments which may be of general interest in Markov Processes that are of the general birth-death type. The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.