Moreira Carolina A, Schneider David M, de Aguiar Marcus A M
Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, São Paulo, Brazil.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042812. doi: 10.1103/PhysRevE.92.042812. Epub 2015 Oct 26.
We study a binary dynamical process that is a representation of the voter model with two candidates and opinion makers. The voters are represented by nodes of a network of social contacts with internal states labeled 0 or 1 and nodes that are connected can influence each other. The network is also perturbed by opinion makers, a set of external nodes whose states are frozen in 0 or 1 and that can influence all nodes of the network. The quantity of interest is the probability of finding m nodes in state 1 at time t. Here we study this process on star networks, which are simple representations of hubs found in complex systems, and compare the results with those obtained for networks that are fully connected. In both cases a transition from disordered to ordered equilibrium states is observed as the number of external nodes becomes small. For fully connected networks the probability distribution becomes uniform at the critical point. For star networks, on the other hand, we show that the equilibrium distribution splits in two peaks, reflecting the two possible states of the central node. We obtain approximate analytical solutions for the equilibrium distribution that clarify the role of the central node in the process. We show that the network topology also affects the time scale of oscillations in single realizations of the dynamics, which are much faster for the star network. Finally, extending the analysis to two stars we compare our results with simulations in simple scale-free networks.
我们研究一种二元动力学过程，它是具有两个候选者和舆论制造者的选民模型的一种表示。选民由具有标记为0或1的内部状态的社会联系网络的节点表示，并且相连的节点可以相互影响。该网络也受到舆论制造者的干扰，舆论制造者是一组外部节点，其状态固定为0或1，并且可以影响网络的所有节点。我们感兴趣的量是在时刻t找到m个处于状态1的节点的概率。在这里，我们在星型网络上研究这个过程，星型网络是复杂系统中发现的枢纽的简单表示，并将结果与在完全连接的网络上获得的结果进行比较。在这两种情况下，随着外部节点数量变小，都观察到从无序到有序平衡态的转变。对于完全连接的网络，概率分布在临界点变得均匀。另一方面，对于星型网络，我们表明平衡分布分裂为两个峰值，反映了中心节点的两种可能状态。我们获得了平衡分布的近似解析解，阐明了中心节点在该过程中的作用。我们表明网络拓扑也会影响动力学单个实现中的振荡时间尺度，对于星型网络来说振荡要快得多。最后，将分析扩展到两个星型网络，我们将我们的结果与简单无标度网络中的模拟结果进行比较。
