Statistical mechanics of the directed 2-distance minimal dominating set problem  

作　　者：Yusupjan Habibulla 

机构地区：[1]School of Physics and Technology,Xinjiang University,Sheng-Li Road 666,Urumqi 830046,China

出　　处：《Communications in Theoretical Physics》2020年第9期132-139,共8页理论物理通讯（英文版）

基　　金：supported by the doctoral startup fund of Xinjiang University of China (grant number 208-61357);the National Natural Science Foundation of China (grant number 11 465 019,11 664 040)。

摘　　要：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 Erdós Rényi(ER)random graph and regular random(RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation(BP) algorithm does not converge when the inverse temperatureβ exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds3.3;there is no entropy transition point(or β = ■) in other circumstances. Third, the results of the replica symmetry(RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.

关 键 词：directed 2-distance minimal dominating set belief propagation regular random graph ER random graph belief propagation decimation 

分 类 号：O157.5[理学—数学]