  Multilayer networks are used to represent the interdependence between the
relational data of individuals interacting with each other via different types
of relationships. To study the information-theoretic phase transitions in
detecting the presence of planted partition among the nodes of a multi-layer
network with additional nodewise covariate information and diverging average
degree, Ma and Nandy (2023) introduced Multi-Layer Contextual Stochastic Block
Model. In this paper, we consider the problem of detecting planted partitions
in the Multi-Layer Contextual Stochastic Block Model, when the average node
degrees for each network is greater than $1$. We establish the sharp phase
transition threshold for detecting the planted bi-partition. Above the
phase-transition threshold testing the presence of a bi-partition is possible,
whereas below the threshold no procedure to identify the planted bi-partition
can perform better than random guessing. We further establish that the derived
detection threshold coincides with the threshold for weak recovery of the
partition and provide a quasi-polynomial time algorithm to estimate it.
