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The echo-chamber effect is a common term in opinion dynamic modeling todescribe how a person's opinion might be artificially enhanced as it isreflected back at her through social interactions. Here, we study the existenceof this effect in statistical mechanics models, which are commonly used tostudy opinion dynamics. We show that the Ising model does not exhibitecho-chambers, but this result is a consequence of a special symmetry. We thendistinguish between three types of models: (i) those with a strong echo-chambersymmetry, that have no echo-chambers at all; (ii) those with a weakecho-chamber symmetry that can exhibit echo-chambers but only if there areexternal fields in the system, and (iii) models without echo-chamber symmetrythat generically have echo-chambers. We use these results to construct anefficient algorithm to efficiently and precisely calculate magnetization inarbitrary tree networks. Finally, We apply this algorithm to study two systems:phase transitions in the random field Ising model on a Bethe lattice and theinfluence optimization problem in social networks.

Echo chambers in the Ising model and implications on the mean magnetization