基于彩色边缘的社区发现
Community Detection with Colored Edges
摘要：
在本文中，我们证明了一个尖锐的限制的社区发现问题的彩色边缘。我们假设两个大小相等的社区，有$m$不同类型的边缘。如果两个顶点在同一社区中，则对于$1\leq i \leq m$，边的分布遵循$p_i=α_i\log{n}/n$，否则对于$1\leq i \leq m$，边的分布为$q_i=β_i\log{n}/n$，其中$α_i$和$β_i$是正常数，$n$是顶点总数。在这些假设下，一个基本的限制社区发现的特点是使用两个分布之间的海林格距离。如果是$\sum_{i=1}^{m} {(\sqrt{α_i} - \sqrt{β_i})}^2 >2$，则通过最大似然（ML）估计器的社区发现以高概率是可能的。如果为$\sum_{i=1}^m {(\sqrt{α_i} - \sqrt{β_i})}^2 < 2$，则ML估计器未能检测到社区的概率不会变为零。
英文摘要：
In this paper, we prove a sharp limit on the community detection problem with colored edges. We assume two equal-sized communities and there are $m$ different types of edges. If two vertices are in the same community, the distribution of edges follows $p_i=α_i\log{n}/n$ for $1\leq i \leq m$, otherwise the distribution of edges is $q_i=β_i\log{n}/n$ for $1\leq i \leq m$, where $α_i$ and $β_i$ are positive constants and $n$ is the total number of vertices. Under these assumptions, a fundamental limit on community detection is characterized using the Hellinger distance between the two distributions. If $\sum_{i=1}^{m} {(\sqrt{α_i} - \sqrt{β_i})}^2 >2$, then the community detection via maximum likelihood (ML) estimator is possible with high probability. If $\sum_{i=1}^m {(\sqrt{α_i} - \sqrt{β_i})}^2 < 2$, the probability that the ML estimator fails to detect the communities does not go to zero.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
