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We consider a class of opinion dynamics on networks where at each time-step,each node in the network disregards the opinions of a certain number of itsmost extreme neighbors and updates its own opinion as a weighted average of theremaining opinions. When all nodes disregard the same number of extremeneighbors, previous work has shown that consensus will be reached if and onlyif the network satisfies certain topological properties. In this paper, weconsider the implications of allowing each node to have a personal thresholdfor the number of extreme neighbors to ignore. We provide graph conditionsunder which consensus is guaranteed for such dynamics. We then study randomnetworks where each node's threshold is drawn from a certain distribution, andprovide conditions on that distribution, together with conditions on the edgeformation probability, that guarantee that consensus will be reachedasymptotically almost surely.

Ignoring Extreme Opinions in Complex Networks: The Impact of Heterogeneous Thresholds