I. Rapaport
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K. Suchan
[email protected]
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I. Todinca
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J. Verstraete
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K. Suchan Faculty of Applied Mathematics, AGH-University of Science and Technology
, Cracow,
Poland
1
K. Suchan Facultad de Ingeniera y Ciencias,
Universidad Adolfo Ibaez
,
Santiago, Chile
2
I. Rapaport Departamento de Ingeniera Matemtica and Centro de Modelamiento Matemtico, Universidad de Chile
,
Santiago, Chile
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J. Verstraete University of California
,
San Diego, CA, USA
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I. Todinca ( ) LIFO,
Universit d'Orlans
, Orlans,
France
We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. This process is a particular case of bootstrap percolation. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if p < 12 . We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.
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Authors acknowledge the support of CONICYT via Anillo en Redes ACT08 (I.R., K.S.), Fondecyt
1090156 (I.R.), ECOS-CONICYT (I.R., I.T.), Fondap on Applied Mathematics (I.R.), French ANR
projects STAL-DEC-OPT and ALADDIN (I.T.) and an Alfred P. Sloan Fellowship (J.V.).
1 Introduction
Let G = (V , E) be a simple undirected graph. A configuration C of G is a
function that assigns to every vertex in V a value in {0, 1}. The value 1 means that the
corresponding vertex is active while the value 0 represents passive vertices.
We investigate the natural situation in which a vertex v needs a strong majority
of its neighbours, namely strictly more than 12 d (v) neighbours, to be active in order
to become an active vertex. Therefore, consider the following rule of dissemination
that acts on configurations: a passive vertex v whose strict majority of neighbours
are active becomes active; once active, a vertex never changes its state. The initial
configuration of a dissemination process is called an insemination. Since the set of
active vertices grows monotonically in a finite set V , a fixed point has to be reached
after a finite number of steps. If the fixed point is such that all vertices have become
active, then we say that the initial configuration overruns the graph G. A community
[12] in G is a subset of nodes X V each of which has at least as many neighbours
in X as in V \ X, i.e. for every v X, |N (v) X| |N (v) (V \ X)|. Notice that a
configuration overruns G if and only if it contains no community of passive vertices.
Dissemination has been intensively studied in the literature, using various
dissemination rules (see e.g. [18] for a survey). Among other types of rules we can cite
models in which a vertex becomes active if the total weight of its active neighbours
exceeds a fixed value [16], or symmetric majority voting rules, for which an active
vertex may also become passive if the number of passive neighbours outweights the
number of active neighbours [18]. One of the main questions for each of these
models is to find small sets of active vertices which overrun the network. Several authors
considered the problem of finding small communities in arbitrary graphs or special
graph classes [7, 9, 12, 13].
In this work we consider a probabilistic framework. A random configuration
in which each vertex is active with probability p and passive with probability
1 p is called a p-insemination. We are interested in the probability p(G) that
a p-insemination overruns G. It is clear that p(G) is a monotonic increasing
function of p. We investigate the majority dissemination process starting with a
p-insemination for various graph classes. Such random dissemination processes, with
different types of dissemination rules, have been studied in the literature in the
context of cellular automata or in bootstrap percolation [14].
One of the basic questions is to determine the ratio of active vertices (in other
words, the critical value of p) one needs in order to overrun the whole graph with
high probability. Without any restriction on the structure of the underlying graph,
it appears to be difficult to determine this ratio. It is therefore more instructive to
consider whole classes of graphs. If G is a class of graphs, let G = (Gn)nN denote
a generic sequence of graphs Gn G such that |V (Gn)| < |V (Gn+1)| for all n N.
We define dissemination half-thresholds pc+ and pc of class G by
In words, for p < pc and any increasing sequence G in G, the probability that a
random p-insemination overruns the graph tends to zero.
For example, for the class K of all complete graphs, it is straightforward to see that
pc+(K) = pc(K) = 21 . If for a class G the two half-thresholds are equal, we say that
pc(G) = pc+ (...truncated)