有向网络上类投票者模型的守恒律
Conservation laws for voter-like models on directed networks
摘要：
我们研究了有向网络中节点和链路更新下的投票者模型，以及相关的入侵过程。我们实现了一个分析处理的热力学极限使用的非均匀平均场假设。从微观层次上的动力学规则出发，我们找到了具有任意度分布和度-度相关性的异构网络上给定状态下节点相对密度的演化方程。我们证明了保守量作为加权线性叠加的自旋状态存在的所有三个过程，对于不相关的有向网络，我们推导出其具体的表达式。我们还讨论了相对密度的时间演化，指数衰减到一个均匀的平稳值的守恒量。在热力学极限中得到的守恒定律决定了一个有限系统达到吸收态的概率，而这个系统在热力学极限中并不有序。每个度类对守恒量的贡献由局部性质决定。根据动态，最高贡献与到达大量传出邻居的有影响力的节点相关联，不太有影响力的节点与少量传入连接相关联，或者同时到达两者。
英文摘要：
We study the voter model, under node and link update, and the related invasion process on a single strongly connected component of a directed network. We implement an analytical treatment in the thermodynamic limit using the heterogeneous mean field assumption. From the dynamical rules at the microscopic level, we find the equations for the evolution of the relative densities of nodes in a given state on heterogeneous networks with arbitrary degree distribution and degree-degree correlations. We prove that conserved quantities as weighted linear superpositions of spin states exist for all three processes and, for uncorrelated directed networks, we derive their specific expressions. We also discuss the time evolution of the relative densities that decay exponentially to a homogeneous stationary value given by the conserved quantity. The conservation laws obtained in the thermodynamic limit for a system that does not order in that limit determine the probabilities of reaching the absorbing state for a finite system. The contribution of each degree class to the conserved quantity is determined by a local property. Depending on the dynamics, the highest contribution is associated to influential nodes reaching a large number of outgoing neighbors, not too influenceable ones with a low number of incoming connections, or both at the same time.
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