共识与分歧：量化行为在意见动态中的作用
Consensus and disagreement: the role of quantized behaviours in opinion dynamics
AI总结：
本文研究连续时间意见动态中连续意见与离散行为的相互作用，分析Carathéodory解的路径学问题，并通过Krasowskii解证明量化行为对共识的破坏作用，同时在特殊通信图结构下证明共识收敛。
摘要：
本文研究连续时间意见动态中连续意见与离散行为的相互作用。在模型中，个体意见仅受他人行为影响。研究发现，由于方程右端不连续，需采用广义解法：分析Carathéodory解的存在性和完备性，并指出其路径学问题，即解可能收敛至非共识点。通过Krasowskii解研究发现，所有解与共识的距离估计为二次于群体规模，证明量化行为可大幅破坏共识。在完全图或完全二分图通信结构下，可证明收敛至共识。
英文摘要：
This paper deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviours. In our model, the opinion of one individual is only influenced by the behaviours of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carathéodory and Krasowskii solutions. We first prove existence and completeness of Carathéodory solutions from every initial condition and we highlight a pathological behaviour of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to non-consensus configurations is very common. In order to cope with these pathological attractors, we then study Krasowskii solutions. We give an estimate of the asymptotic distance of all Krasowskii solutions from consensus and we prove its tightness via an example: this estimate is quadratic in the number of agents, implying that quantization can drastically destroy consensus. However, we are able to prove convergence to consensus in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph.
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