81 research outputs found
Opinion fluctuations and disagreement in social networks
We study a tractable opinion dynamics model that generates long-run
disagreements and persistent opinion fluctuations. Our model involves an
inhomogeneous stochastic gossip process of continuous opinion dynamics in a
society consisting of two types of agents: regular agents, who update their
beliefs according to information that they receive from their social neighbors;
and stubborn agents, who never update their opinions. When the society contains
stubborn agents with different opinions, the belief dynamics never lead to a
consensus (among the regular agents). Instead, beliefs in the society fail to
converge almost surely, the belief profile keeps on fluctuating in an ergodic
fashion, and it converges in law to a non-degenerate random vector. The
structure of the network and the location of the stubborn agents within it
shape the opinion dynamics. The expected belief vector evolves according to an
ordinary differential equation coinciding with the Kolmogorov backward equation
of a continuous-time Markov chain with absorbing states corresponding to the
stubborn agents and converges to a harmonic vector, with every regular agent's
value being the weighted average of its neighbors' values, and boundary
conditions corresponding to the stubborn agents'. Expected cross-products of
the agents' beliefs allow for a similar characterization in terms of coupled
Markov chains on the network. We prove that, in large-scale societies which are
highly fluid, meaning that the product of the mixing time of the Markov chain
on the graph describing the social network and the relative size of the
linkages to stubborn agents vanishes as the population size grows large, a
condition of \emph{homogeneous influence} emerges, whereby the stationary
beliefs' marginal distributions of most of the regular agents have
approximately equal first and second moments.Comment: 33 pages, accepted for publication in Mathematics of Operation
Researc
Scale-free interpersonal influences on opinions in complex systems
An important side effect of the evolution of the human brain is an increased
capacity to form opinions in a very large domain of issues, which become points
of aggressive interpersonal disputes. Remarkably, such disputes are often no
less vigorous on small differences of opinion than large differences. Opinion
differences that may be measured on the real number line may not directly
correspond to the subjective importance of an issue and extent of resistance to
opinion change. This is a hard problem for field of opinion dynamics, a field
that has become increasingly prominent as it has attracted more contributions
to it from investigators in the natural and engineering sciences. The paper
contributes a scale-free approach to assessing the extents to which
individuals, with unknown heterogeneous resistances to influence, have been
influenced by the opinions of others
Effects of Network Communities and Topology Changes in Message-Passing Computation of Harmonic Influence in Social Networks
The harmonic influence is a measure of the importance of nodes in social
networks, which can be approximately computed by a distributed message-passing
algorithm. In this extended abstract we look at two open questions about this
algorithm. How does it perform on real social networks, which have complex
topologies structured in communities? How does it perform when the network
topology changes while the algorithm is running? We answer these two questions
by numerical experiments on a Facebook ego network and on synthetic networks,
respectively. We find out that communities can introduce artefacts in the final
approximation and cause the algorithm to overestimate the importance of "local
leaders" within communities. We also observe that the algorithm is able to
adapt smoothly to changes in the topology.Comment: 4 pages, 7 figures, submitted as conference extended abstrac
On a Modified DeGroot-Friedkin Model of Opinion Dynamics
This paper studies the opinion dynamics that result when individuals
consecutively discuss a sequence of issues. Specifically, we study how
individuals' self-confidence levels evolve via a reflected appraisal mechanism.
Motivated by the DeGroot-Friedkin model, we propose a Modified DeGroot-Friedkin
model which allows individuals to update their self-confidence levels by only
interacting with their neighbors and in particular, the modified model allows
the update of self-confidence levels to take place in finite time without
waiting for the opinion process to reach a consensus on any particular issue.
We study properties of this Modified DeGroot-Friedkin model and compare the
associated equilibria and stability with those of the original DeGroot-Friedkin
model. Specifically, for the case when the interaction matrix is doubly
stochastic, we show that for the modified model, the vector of individuals'
self-confidence levels asymptotically converges to a unique nontrivial
equilibrium which for each individual is equal to 1/n, where n is the number of
individuals. This implies that eventually, individuals reach a democratic
state
On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence
This paper introduces a class of non-linear and continuous-time opinion
dynamics model with additive noise and state dependent interaction rates
between agents. The model features interaction rates which are proportional to
a negative power of opinion distances. We establish a non-local partial
differential equation for the distribution of opinion distances and use Mellin
transforms to provide an explicit formula for the stationary solution of the
latter, when it exists. Our approach leads to new qualitative and quantitative
results on this type of dynamics. To the best of our knowledge these Mellin
transform results are the first quantitative results on the equilibria of
opinion dynamics with distance-dependent interaction rates. The closed form
expressions for this class of dynamics are obtained for the two agent case.
However the results can be used in mean-field models featuring several agents
whose interaction rates depend on the empirical average of their opinions. The
technique also applies to linear dynamics, namely with a constant interaction
rate, on an interaction graph
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