  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.
