Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjacency matrices for which we show the detectability limit vanishes as O(L −1/2 ) with increasing number of layers, L. Importantly, we find a similar scaling behavior when the summation is thresholded at an optimal value, providing insight into the common-but not well understoodpractice of thresholding pairwise-interaction data to obtain sparse network representations. PACS numbers: 89.75.Hc, 02.70.Hm, 64.60.aq The analysis of complex networks [1] has far-reaching applications ranging from social systems [2] to the brain [3] . Often, a natural representation is that of a multilayer network (see reviews [4, 5] ), whereby network layers encode different classes of interactions, such as categorical social ties [6] , types of critical infrastructure [7] , or a network at different instances in time [8] . In principle, the multilayer framework offers a more comprehensive representation of a data set or system, as compared to an aggregation of network layers that produces a simplified model but does so at the cost of information loss. For example, neglecting the layered structure can lead to severe and unintended consequences regarding structure [9] and dynamics [10] [11] [12] , which can fundamentally differ between single-layer and multilayer networks [13, 14] . However, layer aggregation also implements an information processing that can yield beneficial results. Network layers are often correlated with one another and can encode redundant information [15] . In some cases a multilayer representation is an over-modeling, which can negatively impact the computational and memory requirements for storage and analysis. In such situations, it is beneficial to seek a more concise representation in which certain layers are aggregated [16, 17] . Identifying sets of repetitive layers amounts to a clustering problem, and it is closely related to the topic of clustering networks in an ensemble of networks [17, 18] . Much remains to be studied regarding when layer aggregation is appropriate and how it should be implemented. We study here the effect of layer aggregation on community structure in multilayer networks in which each layer is drawn from a common stochastic block model (SBM). SBMs are a paradigmatic model [19] for complex structure in networks and are particularly useful for studying limitations on detectability-that is, if the community structure is too weak, it cannot be found upon inspection of the network [20] [21] [22] [23] [24] [25] . Recently, the detectability limit has been explored for networks with degree heterogeneity [26] and hierarchical structure [27, 28] , for temporal networks [29] , and for the detection of communities using multi-resolution methods [30] . Despite growing interest in multilayer SBMs [31-35] (which we note, focus on multiplex networks in which nodes are identical in every layer and edges are restricted to connecting nodes in the same layer [4, 5] ), the effect of layer aggregation on detectability limitations has yet to be explored outside the infinite layer limit [35] . To this end, we study detectability limitations for multilayer SBMs with layers following from identical SBM parameters and find that the method of aggregation significantly influences detectability. When the aggregate network corresponds to the summation of the adjacency matrices encoding the network layers, aggregation always improves detectability. In particular, the detectability limit vanishes with increasing number of layers, L, and decays as O(L −1/2 ). Because the summation of L adjacency matrices can often yield a weighted and dense network-which increases the complexity of community detection [36]-we also study binary adjacency matrices obtained by thresholding this summation at some valuẽ L. We find that the detectability limit is very sensitive to the choice ofL; however, we also find that there exist thresholds (e.g., mean edge probability for homogeneous communities) that are optimal in that the detectability limit also decays as O(L −1/2 ). These results provide insight into the use of thresholding pairwise-interaction data so as to produce sparse networks-a practice that is commonplace but for which the effects are not well understood. We begin by describing the multilayer SBM. We consider N nodes divided into K communities, and we denote by c i ∈ {1, . . . , K} the community index for each node i ∈ {1, . . . , N }. The multiplex network is defined by L layers encoded by a set of adjacency matrices, {A (l) }, where A (l) ij = 1 if (i, j) is an edge in layer l and A (l) ij = 0 otherwise. The probability of edge (i, j) in layer l is given by Π cicj ∈ [0, 1], where Π is a K × K matrix. The detectability of community structure relates to the ability to recover the nodes' community labels {c i }. To connect arXiv:1511.05271v3 [physics.soc-ph]
Under the assumption of two equally-sized clusters, the authors in REF show that if layer aggregation is used and if each layer is an independent realization of a common SBM, the inferential limit for cluster detectability decays at rate O(L −   1 2 ), where L is the number of layers.
Enhanced detectability of community structure in multilayer networks through layer aggregation
ArXiv
ArXiv
Physics Medicine Computer Science Mathematics