随机块模型图中社团检测问题及其相关变迁的算法独立界
Algorithm independent bounds on community detection problems and associated transitions in stochastic block model graphs
摘要：
我们推导出严格的界限，定义良好的社区结构在复杂网络的随机块模型（SBM）基准。特别是，我们分析了社区间的“噪音”（社区间的边缘）对任何“社区检测”算法的能力，正确地分配到一个种植分区，一个问题，已被证明是NP完全的标准再现组节点的影响。我们的结果不依赖于任何一个特定的算法的使用，也不依赖于推理的局限性的分析。相反，我们把它的头上的问题和工作向后检查时，在第一位，定义良好的结构可能存在于SBMs。我们在这里介绍的方法可以潜在地应用到其他计算问题。社区检测算法的目标是将给定的网络划分为最优不相交的子图（或社区）。类似于k-SAT和其他组合优化问题，“社区检测”表现出不同的阶段。处于“不可解阶段”的网络缺乏明确的结构，因此没有有意义的划分。可解系统分裂成两个完全不同的阶段：“困难”阶段和“容易”阶段。正如它的名字一样，在简单阶段，分区很容易通过已知的算法实现。当网络处于硬阶段时，它仍然具有底层结构，但要找到一个可以在多项式时间内检查的有意义的分区，需要进行详尽的计算工作，并且随着图的大小而迅速增加。当考虑在一起，（i）严格的结果，我们在这里报告的图形时，有一个基本的结构和（ii）最近的结果有关的限制，而一般的算法，建议硬相的界限。
英文摘要：
We derive rigorous bounds for well-defined community structure in complex networks for a stochastic block model (SBM) benchmark. In particular, we analyze the effect of inter-community "noise" (inter-community edges) on any "community detection" algorithm's ability to correctly group nodes assigned to a planted partition, a problem which has been proven to be NP complete in a standard rendition. Our result does not rely on the use of any one particular algorithm nor on the analysis of the limitations of inference. Rather, we turn the problem on its head and work backwards to examine when, in the first place, well defined structure may exist in SBMs.The method that we introduce here could potentially be applied to other computational problems. The objective of community detection algorithms is to partition a given network into optimally disjoint subgraphs (or communities). Similar to k-SAT and other combinatorial optimization problems, "community detection" exhibits different phases. Networks that lie in the "unsolvable phase" lack well-defined structure and thus have no partition that is meaningful. Solvable systems splinter into two disparate phases: those in the "hard" phase and those in the "easy" phase. As befits its name, within the easy phase, a partition is easy to achieve by known algorithms. When a network lies in the hard phase, it still has an underlying structure yet finding a meaningful partition which can be checked in polynomial time requires an exhaustive computational effort that rapidly increases with the size of the graph. When taken together, (i) the rigorous results that we report here on when graphs have an underlying structure and (ii) recent results concerning the limits of rather general algorithms, suggest bounds on the hard phase.
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