Mean field systems on networks, with singular interaction through hitting times
Mean field systems on networks, with singular interaction through hitting times
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DOI：
10.1214/19-aop1403
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发表时间：
2018-07
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影响因子：
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通讯作者：
S. Nadtochiy;Mykhaylo Shkolnikov
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作者：
S. Nadtochiy;Mykhaylo Shkolnikov
在[DIRT 15 a]，[DIRT 15 b]，[NS 17 a]，[DT 17]，[HLS 18]，[HS 18]的基础上，我们继续研究了具有奇异相互作用的粒子系统的碰撞时间。与以前的研究相比，我们（i）考虑非常一般的驱动过程和相互作用函数，（ii）允许非均匀的连接结构，（iii）分析一个游戏，在这个游戏中，粒子战略性地确定它们的连接。在此，我们揭示了两个全新的现象。首先，我们描述了这种系统的“脆弱时期”（例如，当群体的宏观部分同时违约或被感染时，或者当神经元细胞“同步”时的时间）明确地根据驱动过程的动力学、粒子值的当前分布以及底层网络的拓扑结构（由其Perron-Frobenius特征值表示）。其次，我们使用这样的系统来描述一个动态的信用网络游戏，并表明，在平衡，系统正则化：即，脆弱的时刻永远不会发生，因为粒子通过策略性地调整它们的连接来避免它们。两个辅助的数学结果，在他们自己的权利，发现在我们的调查：推广Schauder的不动点定理的Skorokhod空间与M1拓扑，和应用的最大代数的平衡版本的网络流问题。
Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures, and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values, and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes: i.e., the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of the max-plus algebra to the equilibrium version of the network flow problem.