46 research outputs found
On dynamic monopolies of graphs with general thresholds
Let be a graph and be an
assignment of thresholds to the vertices of . A subset of vertices is
said to be dynamic monopoly (or simply dynamo) if the vertices of can be
partitioned into subsets such that and for any
each vertex in has at least neighbors in
. Dynamic monopolies are in fact modeling the irreversible
spread of influence such as disease or belief in social networks. We denote the
smallest size of any dynamic monopoly of , with a given threshold
assignment, by . In this paper we first define the concept of a
resistant subgraph and show its relationship with dynamic monopolies. Then we
obtain some lower and upper bounds for the smallest size of dynamic monopolies
in graphs with different types of thresholds. Next we introduce
dynamo-unbounded families of graphs and prove some related results. We also
define the concept of a homogenious society that is a graph with probabilistic
thresholds satisfying some conditions and obtain a bound for the smallest size
of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain
some bounds for their sizes and determine the exact values in some special
cases
On dynamic monopolies of graphs: the average and strict majority thresholds
Let be a graph and
be an assignment of thresholds to the vertices of . A subset of vertices
is said to be a dynamic monopoly corresponding to if the vertices
of can be partitioned into subsets such that
and for any , each vertex in has at least
neighbors in . Dynamic monopolies are in fact
modeling the irreversible spread of influence in social networks. In this paper
we first obtain a lower bound for the smallest size of any dynamic monopoly in
terms of the average threshold and the order of graph. Also we obtain an upper
bound in terms of the minimum vertex cover of graphs. Then we derive the upper
bound for the smallest size of any dynamic monopoly when the graph
contains at least one odd vertex, where the threshold of any vertex is set
as (i.e. strict majority threshold). This bound
improves the best known bound for strict majority threshold. We show that the
latter bound can be achieved by a polynomial time algorithm. We also show that
is an upper bound for the size of strict majority dynamic
monopoly, where stands for the matching number of . Finally, we
obtain a basic upper bound for the smallest size of any dynamic monopoly, in
terms of the average threshold and vertex degrees. Using this bound we derive
some other upper bounds
Majority Dynamics and Aggregation of Information in Social Networks
Consider n individuals who, by popular vote, choose among q >= 2
alternatives, one of which is "better" than the others. Assume that each
individual votes independently at random, and that the probability of voting
for the better alternative is larger than the probability of voting for any
other. It follows from the law of large numbers that a plurality vote among the
n individuals would result in the correct outcome, with probability approaching
one exponentially quickly as n tends to infinity. Our interest in this paper is
in a variant of the process above where, after forming their initial opinions,
the voters update their decisions based on some interaction with their
neighbors in a social network. Our main example is "majority dynamics", in
which each voter adopts the most popular opinion among its friends. The
interaction repeats for some number of rounds and is then followed by a
population-wide plurality vote.
The question we tackle is that of "efficient aggregation of information": in
which cases is the better alternative chosen with probability approaching one
as n tends to infinity? Conversely, for which sequences of growing graphs does
aggregation fail, so that the wrong alternative gets chosen with probability
bounded away from zero? We construct a family of examples in which interaction
prevents efficient aggregation of information, and give a condition on the
social network which ensures that aggregation occurs. For the case of majority
dynamics we also investigate the question of unanimity in the limit. In
particular, if the voters' social network is an expander graph, we show that if
the initial population is sufficiently biased towards a particular alternative
then that alternative will eventually become the unanimous preference of the
entire population.Comment: 22 page
Absorption Time of the Moran Process
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.Comment: minor change