摘要翻译：
我们将信念传播算法推广到具有任意模体（三角形、环等）分布的稀疏随机网络。这些网络中的每个顶点都属于一组给定的模体（配置模型的推广）。这些网络可以被视为稀疏的不相关超图，其中超边表示模体。这里的超图是图的推广，超边可以连接任意数量的顶点。这些不相关的超图是树状的（超树），它极大地简化了问题，并允许我们将信念传播算法应用于这些具有任意模体的环网络。作为自然例子，我们考虑有限环和团形式的模体。我们将信念传播算法应用于铁磁Ising模型，得到了随机网络。我们在以有限环或团为模体的网络上得到了该模型的精确解。我们找到了铁磁相变的一个精确临界温度，并证明了与普通树状复杂网络相比，随着聚类系数和环尺寸的增加，临界温度增加。我们的解决方案也给出了在这些环状网络中巨型连通组件的诞生点。
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英文标题：
《Belief-propagation algorithm and the Ising model on networks with
  arbitrary distributions of motifs》
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作者：
S. Yoon, A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes
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最新提交年份：
2012
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分类信息：

一级分类：Physics        物理学
二级分类：Disordered Systems and Neural Networks        无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Computer Science        计算机科学
二级分类：Artificial Intelligence        人工智能
分类描述：Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域，除了视觉、机器人、机器学习、多智能体系统以及计算和语言（自然语言处理），这些领域有独立的学科领域。特别地，包括专家系统，定理证明（尽管这可能与计算机科学中的逻辑重叠），知识表示，规划，和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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英文摘要：
  We generalize the belief-propagation algorithm to sparse random networks with arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in these networks belongs to a given set of motifs (generalization of the configuration model). These networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs. Here a hypergraph is a generalization of a graph, where a hyperedge can connect any number of vertices. These uncorrelated hypergraphs are tree-like (hypertrees), which crucially simplify the problem and allow us to apply the belief-propagation algorithm to these loopy networks with arbitrary motifs. As natural examples, we consider motifs in the form of finite loops and cliques. We apply the belief-propagation algorithm to the ferromagnetic Ising model on the resulting random networks. We obtain an exact solution of this model on networks with finite loops or cliques as motifs. We find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size, the critical temperature increases compared to ordinary tree-like complex networks. Our solution also gives the birth point of the giant connected component in these loopy networks.
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PDF链接：
https://arxiv.org/pdf/1106.4925