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#### Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks

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arXiv:1804.07223.pdf

(Preprint), 918KB

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##### Citation

Cruciani, E., Natale, E., Nusser, A., & Scornavacca, G. (2018). Phase Transition of the 2-Choices Dynamics on Core-Periphery Networks. Retrieved from http://arxiv.org/abs/1804.07223.

Cite as: https://hdl.handle.net/21.11116/0000-0002-A446-6

##### Abstract

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.

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.