Abstract
We describe a model for polarization in multi-agent systems based on Esteban and Ray’s standard measure of polarization from economics. Agents evolve by updating their beliefs (opinions) based on an underlying influence graph, as in the standard DeGroot model for social learning, but under a confirmation bias; i.e., a discounting of opinions of agents with dissimilar views. We show that even under this bias polarization eventually vanishes (converges to zero) if the influence graph is strongly-connected. If the influence graph is a regular symmetric circulation, we determine the unique belief value to which all agents converge. Our more insightful result establishes that, under some natural assumptions, if polarization does not eventually vanish then either there is a disconnected subgroup of agents, or some agent influences others more than she is influenced. We also show that polarization does not necessarily vanish in weakly-connected graphs under confirmation bias. We illustrate our model with a series of case studies and simulations, and show how it relates to the classic DeGroot model for social learning.
Mário S. Alvim and Bernardo Amorim were partially supported by CNPq, CAPES and FAPEMIG. Santiago Quintero and Frank Valencia were partially supported by the ECOS-NORD project FACTS (C19M03).
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Notes
- 1.
W.l.o.g. we can assume the values of \(\pi _i\) are all non-zero and add up to 1.
- 2.
Note that this assumption implies that an agent has an influence on himself, and hence cannot be used as a “puppet” who immediately assumes another’s agent’s belief.
- 3.
Recall from Definition 2 that our model allows arbitrary discretizations \(D_k\) –i.e., different number of bins, with not-necessarily uniform widths– depending on the scenario of interest.
- 4.
It is worthwhile to note that this discontinuity at borderline points matches real scenarios where each bin represents a sharp action an agent takes based on his current belief value. Even when two agents’ beliefs are asymptotically converging to a same borderline value from different sides, their discrete decisions will remain distinct. E.g., in the vaccine case of Example 1, even agents that are asymptotically converging to a common belief value of 0.5 will take different decisions on whether or not to vaccinate, depending on which side of 0.5 their belief falls. In this sense, although there is convergence in the underlying belief values, there remains polarization w.r.t. real-world actions taken by agents.
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Alvim, M.S., Amorim, B., Knight, S., Quintero, S., Valencia, F. (2021). A Multi-agent Model for Polarization Under Confirmation Bias in Social Networks. In: Peters, K., Willemse, T.A.C. (eds) Formal Techniques for Distributed Objects, Components, and Systems. FORTE 2021. Lecture Notes in Computer Science(), vol 12719. Springer, Cham. https://doi.org/10.1007/978-3-030-78089-0_2
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