The directed L-distance minimal dominating set (MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erd$\acute{o}$s R$\acute{e}$nyi (ER) random graph and regular random graph, but this work can be extended to any type of networks. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation algorithm does not converge when the inverse temperature exceeds a threshold on either an ER random network or regular random network. Second, the entropy density of replica symmetric theory has a transition point at a finite inverse temperature on a regular random graph when the node degree exceeds 4 and on an ER random graph when the node degree exceeds 6.6; there is no entropy transition point (or $\beta=\infty$) in other circumstances. Third, the results of the replica symmetry (RS) theory are in perfect agreement with those of belief propagation (BP) algorithm while the results of the belief propagation decimation (BPD) algorithm are better than those of the greedy heuristic algorithm.

中文翻译：

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有向 2 距离最小支配集问题的统计力学

有向 L 距离最小支配集 (MDS) 问题在计算机科学和通信网络领域具有广泛的实际应用。在这里，我们从纯理论兴趣的角度来研究这个问题。我们只给出 Erd$\acute{o}$s R$\acute{e}$nyi (ER) 随机图和常规随机图的结果，但这项工作可以扩展到任何类型的网络。我们开发了自旋玻璃理论来研究有向的 2 距离 MDS 问题。首先，我们发现当逆温度超过 ER 随机网络或常规随机网络的阈值时，置信传播算法不会收敛。第二，复制对称理论的熵密度在节点度超过4时在规则随机图上和当节点度超过6.6时在ER随机图上在有限逆温度处有一个转变点；在其他情况下没有熵转换点（或 $\beta=\infty$）。第三，副本对称（RS）理论的结果与信念传播（BP）算法的结果完全一致，而信念传播抽取（BPD）算法的结果优于贪婪启发式算法的结果。