4 research outputs found
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks
Consider the following process on a network: Each agent initially holds
either opinion blue or red; then, in each round, each agent looks at two random
neighbors and, if the two have the same opinion, the agent adopts it. This
process is known as the 2-Choices dynamics and is arguably the most basic
non-trivial opinion dynamics modeling voting behavior on social networks.
Despite its apparent simplicity, 2-Choices has been analytically characterized
only on networks with a strong expansion property -- under assumptions on the
initial configuration that establish it as a fast majority consensus protocol.
In this work, we aim at contributing to the understanding of the 2-Choices
dynamics by considering its behavior on a class of networks with core-periphery
structure, a well-known topological assumption in social networks. In a
nutshell, assume that a densely-connected subset of agents, the core, holds a
different opinion from the rest of the network, the periphery. Then, depending
on the strength of the cut between the core and the periphery, a
phase-transition phenomenon occurs: Either the core's opinion rapidly spreads
among the rest of the network, or a metastability phase takes place, in which
both opinions coexist in the network for superpolynomial time. The interest of
our result is twofold. On the one hand, by looking at the 2-Choices dynamics as
a simplistic model of competition among opinions in social networks, our
theorem sheds light on the influence of the core on the rest of the network, as
a function of the core's connectivity towards the latter. On the other hand, to
the best of our knowledge, we provide the first analytical result which shows a
heterogeneous behavior of a simple dynamics as a function of structural
parameters of the network. Finally, we validate our theoretical predictions
with extensive experiments on real networks