具有衰减置信度的意见动力学：在图中社区检测中的应用
Opinion Dynamics with Decaying Confidence: Application to Community Detection in Graphs
摘要：
我们研究了一类离散时间多智能体系统建模的意见动态与衰减的信心。我们考虑一个代理网络，每个代理都有自己的观点。在每一个时间步，代理与他们的邻居交换意见，并通过只考虑与他们自己的意见相差小于一些置信界限来更新它。这个置信区间正在衰减：一个代理人只对接近其观点足够快的邻居重复地给予信任。基本上，代理试图达成协议的约束，它必须接近不低于规定的收敛速度。在这种限制下，可能无法达成全球共识，只能达成地方协议。达成局部协议的代理在网络内部形成社区。在本文中，我们分析了这个意见动力学模型：我们表明，社区对应于网络的渐近连接的组件，并给出了一个代数表征的社区定义的集体动态矩阵的特征值。最后，我们应用我们的意见动力学模型来解决图中的社区检测问题。我们提出了一个新的配方的社区检测问题的基础上的特征值的归一化拉普拉斯矩阵的图，并表明这个问题可以解决我们的意见动力学模型。我们考虑三个网络的例子，并比较我们检测到的社区与现有算法的基础上模块化优化。我们表明，我们的意见动力学模型不仅提供了一个有吸引力的方法，社区检测，但它也是有效的。
英文摘要：
We study a class of discrete-time multi-agent systems modelling opinion dynamics with decaying confidence. We consider a network of agents where each agent has an opinion. At each time step, the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound. This confidence bound is decaying: an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion. Essentially, the agents try to reach an agreement with the constraint that it has to be approached no slower than a prescribed convergence rate. Under that constraint, global consensus may not be achieved and only local agreements may be reached. The agents reaching a local agreement form communities inside the network. In this paper, we analyze this opinion dynamics model: we show that communities correspond to asymptotically connected component of the network and give an algebraic characterization of communities in terms of eigenvalues of the matrix defining the collective dynamics. Finally, we apply our opinion dynamics model to address the problem of community detection in graphs. We propose a new formulation of the community detection problem based on eigenvalues of normalized Laplacian matrix of graphs and show that this problem can be solved using our opinion dynamics model. We consider three examples of networks, and compare the communities we detect with those obtained by existing algorithms based on modularity optimization. We show that our opinion dynamics model not only provides an appealing approach to community detection but that it is also effective.
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