A myriad of approaches have been proposed to characterise the mesoscale structure of networks - most often as a partition based on patterns variously called communities, blocks, or clusters. Clearly, distinct methods designed to detect different types of patterns may provide a variety of answers to the network's mesoscale structure. Yet, even multiple runs of a given method can sometimes yield diverse and conflicting results, producing entire landscapes of partitions which potentially include multiple (locally optimal) mesoscale explanations of the network. Such ambiguity motivates a closer look at the ability of these methods to find multiple qualitatively different 'ground truth' partitions in a network. Here, we propose the stochastic cross-block model (SCBM), a generative model which allows for two distinct partitions to be built into the mesoscale structure of a single benchmark network. We demonstrate a use case of the benchmark model by appraising the power of stochastic block models (SBMs) to detect implicitly planted coexisting bi-community and core-periphery structures of different strengths. Given our model design and experimental set-up, we find that the ability to detect the two partitions individually varies by SBM variant and that coexistence of both partitions is recovered only in a very limited number of cases. Our findings suggest that in most instances only one - in some way dominating - structure can be detected, even in the presence of other partitions. They underline the need for considering entire landscapes of partitions when different competing explanations exist and motivate future research to advance partition coexistence detection methods. Our model also contributes to the field of benchmark networks more generally by enabling further exploration of the ability of new and existing methods to detect ambiguity in the mesoscale structure of networks.

翻译：已有大量方法用于刻画网络的中尺度结构——通常基于社区、区块或簇等模式形成划分。显然，针对不同类型模式设计的多种方法会提供关于网络中尺度结构的多样化答案。然而，即使对同一方法进行多次运行，有时也会产生多样且相互矛盾的结果，形成包含多个（局部最优）中尺度解释的整体划分图谱。这种模糊性促使我们更深入审视这些方法在同一网络中寻找多个本质不同的"真实"划分的能力。本文提出随机跨块模型（SCBM），这是一种生成模型，允许在单个基准网络的中尺度结构中内嵌两个独立划分。我们通过评估随机块模型（SBM）检测隐含植入的双重社区与核心-边缘结构（具有不同强度）的能力，展示了该基准模型的应用案例。根据模型设计与实验设置，发现两种划分的独立检测能力因SBM变体而异，且仅在极少数案例中能同时恢复两种划分。结果表明：即使存在其他划分，大多数情况下仅能检测到某种主导性结构。这凸显了在存在多种竞争解释时需考虑整体划分图谱的必要性，并推动未来研究发展划分共存检测方法。本模型通过支持进一步探索新方法与现有方法检测网络中尺度结构模糊性的能力，为基准网络领域做出更广泛的贡献。