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机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出划分空间映射（PSM）工具，结合规范标记和粒度-模块度二维投影，可视化社区检测的划分空间，揭示模块度简并结构，为算法评估与搜索引导提供精确参考。
AI 中文摘要
社区检测方法在网络的可能划分空间中进行搜索，但这一划分空间的结构很少被直接考察。本文引入了两种互补的工具，使$\mathcal{P}$在分析和视觉上变得可访问。首先，采用基于受限增长序列（RGS）的规范标记方案，为每个划分分配一个唯一的、置换不变的标识符，从而消除了对成对相似性度量（如归一化互信息）的需求。其次，定义了粒度$\Gamma$，并将其与模块度$Q$结合，产生一个二维投影：划分空间映射（PSM）。在$(\Gamma,Q)$平面内，只有$\Gamma$的离散整数集是可达到的，并且$Q$的上下界直接从偏差矩阵导出。这些工具通过对小型基准图的$\mathcal{P}$完全枚举进行演示，揭示了模块度简并的精细结构和塑造可行区域的组合约束。该框架为评估启发式社区检测算法提供了精确、无假设的参考，并为在更大网络中实现划分空间的几何感知探索奠定了基础。
英文摘要
Community-detection methods search over possible partitions of a network, but the structure of this partition space is rarely examined directly. This paper introduces two complementary tools that make~$\mathcal{P}$ analytically and visually accessible. First, a canonical labelling scheme based on the Restricted Growth Sequence~(RGS) is adopted, assigning each partition a unique, permutation-invariant identifier and eliminating the need for pairwise similarity measures such as the Normalized Mutual Information. Second, granularity~$Γ$ is defined and combined with modularity~$Q$ to produce a two-dimensional projection: the Partition Space Map~(PSM). Within the $(Γ,Q)$ plane, only a discrete set of integer values of~$Γ$ is attainable, and upper and lower bounds on~$Q$ are derived directly from the deviation matrix. These tools are demonstrated through complete enumeration of~$\mathcal{P}$ for small benchmark graphs, revealing the fine structure of modularity degeneracy and the combinatorial constraints that shape the feasible region. The framework provides an exact, assumption-free reference for evaluating heuristic community detection algorithms and lays the groundwork for geometry-aware exploration of partition space in larger networks.
