布尔八卦网络
Boolean Gossip Networks
摘要：
本文提出并研究了一个布尔流言模型，它是一个简化但非平凡的概率布尔网络。与积极的节点相互作用，鉴于标准的理论，从马尔可夫链，我们证明了节点状态渐近收敛到一个协议，在一个二进制随机变量，其分布的特点是大规模网络的平均场近似。使用组合分析，我们还成功地计算了通信类的数量的正布尔网络显式的底层交互图的拓扑结构，其中显着微小的变化，在本地结构可以大大改变网络通信类的数量。与一般的布尔相互作用规则，吸收网络布尔动力学的出现被证明是由网络结构的必要和充分条件建立时，布尔八卦过程定义吸收马尔可夫链。特别是，它示出，对于大多数的布尔相互作用规则，除了九个总的$2^{16}-1$可能的非空的二进制布尔函数集，无论是诱导链吸收无关的拓扑结构的底层相互作用图，只要连接假设。这些结果说明了布尔网络的{相关动力学}属性的潜在相互作用的图形属性的可能性。
英文摘要：
This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total $2^{16}-1$ possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of {relating dynamical} properties of Boolean networks to graphical properties of the underlying interactions.
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