Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH ...Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH 〉 0＂, and also obtain the induced subgraph counts inside the threshold interval.展开更多
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 theo...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.展开更多
基金This research is supported by the National Natural Science of Foundation under Grant Nos.10531070 and 10721101 of China.
文摘Let Gn,d be a random d-regular graph with n vertices, where d = o（n）. Given a fixed graph H, YH denotes the number of induced copies of H in Gn d In this paper, the authors determine the threshold of the event ＂YH 〉 0＂, and also obtain the induced subgraph counts inside the threshold interval.
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)。
文摘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.
