36,028 research outputs found
The Power of Two Choices in Distributed Voting
Distributed voting is a fundamental topic in distributed computing. In pull
voting, in each step every vertex chooses a neighbour uniformly at random, and
adopts its opinion. The voting is completed when all vertices hold the same
opinion. On many graph classes including regular graphs, pull voting requires
expected steps to complete, even if initially there are only two
distinct opinions.
In this paper we consider a related process which we call two-sample voting:
every vertex chooses two random neighbours in each step. If the opinions of
these neighbours coincide, then the vertex revises its opinion according to the
chosen sample. Otherwise, it keeps its own opinion. We consider the performance
of this process in the case where two different opinions reside on vertices of
some (arbitrary) sets and , respectively. Here, is the
number of vertices of the graph.
We show that there is a constant such that if the initial imbalance
between the two opinions is ?, then with high probability two sample voting completes in a random
regular graph in steps and the initial majority opinion wins. We
also show the same performance for any regular graph, if where is the second largest eigenvalue of the transition
matrix. In the graphs we consider, standard pull voting requires
steps, and the minority can still win with probability .Comment: 22 page
Local majority dynamics on preferential attachment graphs
Suppose in a graph vertices can be either red or blue. Let be odd. At
each time step, each vertex in polls random neighbours and takes
the majority colour. If it doesn't have neighbours, it simply polls all of
them, or all less one if the degree of is even. We study this protocol on
the preferential attachment model of Albert and Barab\'asi, which gives rise to
a degree distribution that has roughly power-law ,
as well as generalisations which give exponents larger than . The setting is
as follows: Initially each vertex of is red independently with probability
, and is otherwise blue. We show that if is
sufficiently biased away from , then with high probability,
consensus is reached on the initial global majority within
steps. Here is the number of vertices and is the minimum of
and (or if is even), being the number of edges each new
vertex adds in the preferential attachment generative process. Additionally,
our analysis reduces the required bias of for graphs of a given degree
sequence studied by the first author (which includes, e.g., random regular
graphs)
An evolutionary game model for behavioral gambit of loyalists: Global awareness and risk-aversion
We study the phase diagram of a minority game where three classes of agents
are present. Two types of agents play a risk-loving game that we model by the
standard Snowdrift Game. The behaviour of the third type of agents is coded by
{\em indifference} w.r.t. the game at all: their dynamics is designed to
account for risk-aversion as an innovative behavioral gambit. From this point
of view, the choice of this solitary strategy is enhanced when innovation
starts, while is depressed when it becomes the majority option. This implies
that the payoff matrix of the game becomes dependent on the global awareness of
the agents measured by the relevance of the population of the indifferent
players. The resulting dynamics is non-trivial with different kinds of phase
transition depending on a few model parameters. The phase diagram is studied on
regular as well as complex networks
Dynamics of opinion formation in a small-world network
The dynamical process of opinion formation within a model using a local
majority opinion updating rule is studied numerically in networks with the
small-world geometrical property. The network is one in which shortcuts are
added to randomly chosen pairs of nodes in an underlying regular lattice. The
presence of a small number of shortcuts is found to shorten the time to reach a
consensus significantly. The effects of having shortcuts in a lattice of fixed
spatial dimension are shown to be analogous to that of increasing the spatial
dimension in regular lattices. The shortening of the consensus time is shown to
be related to the shortening of the mean shortest path as shortcuts are added.
Results can also be translated into that of the dynamics of a spin system in a
small-world network.Comment: 10 pages, 5 figure
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