9 research outputs found
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
Distributed Community Detection via Metastability of the 2-Choices Dynamics
We investigate the behavior of a simple majority dynamics on networks of
agents whose interaction topology exhibits a community structure. By leveraging
recent advancements in the analysis of dynamics, we prove that, when the states
of the nodes are randomly initialized, the system rapidly and stably converges
to a configuration in which the communities maintain internal consensus on
different states. This is the first analytical result on the behavior of
dynamics for non-consensus problems on non-complete topologies, based on the
first symmetry-breaking analysis in such setting. Our result has several
implications in different contexts in which dynamics are adopted for
computational and biological modeling purposes. In the context of Label
Propagation Algorithms, a class of widely used heuristics for community
detection, it represents the first theoretical result on the behavior of a
distributed label propagation algorithm with quasi-linear message complexity.
In the context of evolutionary biology, dynamics such as the Moran process have
been used to model the spread of mutations in genetic populations [Lieberman,
Hauert, and Nowak 2005]; our result shows that, when the probability of
adoption of a given mutation by a node of the evolutionary graph depends
super-linearly on the frequency of the mutation in the neighborhood of the node
and the underlying evolutionary graph exhibits a community structure, there is
a non-negligible probability for species differentiation to occur.Comment: Full version of paper appeared in AAAI-1
The Power of Small Coalitions under Two-Tier Majority on Regular Graphs
In this paper, we study the following problem. Consider a setting where a
proposal is offered to the vertices of a given network , and the vertices
must conduct a vote and decide whether to accept the proposal or reject it.
Each vertex has its own valuation of the proposal; we say that is
``happy'' if its valuation is positive (i.e., it expects to gain from adopting
the proposal) and ``sad'' if its valuation is negative. However, vertices do
not base their vote merely on their own valuation. Rather, a vertex is a
\emph{proponent} of the proposal if the majority of its neighbors are happy
with it and an \emph{opponent} in the opposite case. At the end of the vote,
the network collectively accepts the proposal whenever the majority of its
vertices are proponents. We study this problem for regular graphs with loops.
Specifically, we consider the class of -regular graphs
of odd order with all loops and happy vertices. We are interested
in establishing necessary and sufficient conditions for the class
to contain a labeled graph accepting the proposal, as
well as conditions to contain a graph rejecting the proposal. We also discuss
connections to the existing literature, including that on majority domination,
and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied
Mathematic
Find Your Place: Simple Distributed Algorithms for Community Detection
International audienceGiven an underlying graph, we consider the following dynamics: Initially, each node locally chooses a value in {−1, 1}, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node. We prove that the process resulting from this dynamics produces a clustering that exactly or approximately (depending on the graph) reects the underlying cut in logarithmic time, under various graph models that exhibit a sparse balanced cut, including the stochastic block model. We also prove that a natural extension of this dynamics performs community detection on a regularized version of the stochastic block model with multiple communities. Rather surprisingly, our results provide rigorous evidence for the ability of an extremely simple and natural dynamics which is non-trivial even in a centralized setting
Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks
International audienc
Phase transition of the 2-Choices dynamics on core–periphery networks
International audienceAbstract The 2-Choices dynamics is a process that models voting behavior on networks and works as follows: 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. We study its behavior on a class of networks with core–periphery structure. Assume that a densely-connected subset of agents, the core , holds a different opinion from the rest of the network, the periphery . We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core’s opinion rapidly spreads across the network, or a metastability phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the influence of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network
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