Resolution Limits for Detecting Community Changes in Multilayer Networks

Multilayer networks capture pairwise relationships between the components of complex systems across multiple modes or scales of interactions. An important meso-scale feature of these networks is measured though their community structure, which defines groups of strongly connected nodes that exist within and across network layers. Because interlayer edges can describe relationships between different modalities, scales, or time points, it is essential to understand how communities change and evolve across layers. A popular method for detecting communities in multilayer networks consists of maximizing a quality function known as modularity. However, in the multilayer setting the modularity function depends on an interlayer coupling parameter, ω, and how this parameter affects community detection is not well understood. Here, we expose an upper bound for ω beyond which community changes across layers can not be detected. This upper bound has non-trivial, purely multilayer effects and acts as a resolution limit for detecting evolving communities. Further, we establish an explicit and previously undiscovered relationship between the single layer resolution parameter, γ, and interlayer coupling parameter, ω, that provides new understanding of the modularity parameter space. Our findings not only represent new theoretical considerations but also have important practical implications for choosing interlayer coupling values when using multilayer networks to model real-world systems whose communities change across time or modality. Multilayer networks are quickly becoming the modeling framework of choice to represent complex interactions in large, multi-modal datasets. A multilayer network is a rich generalization of a traditional network that captures interactions between nodes by separating each interaction type into its own layer together with interlayer coupling of nodes between layers [25, 17, 8]. Multilayer networks have found applications in a diverse range of settings such as neuroscience [32, 13, 4, 46], financial assets [7, 10], congressional voting similarity [30, 31], social networks [42, 9, 19], and spreading processes [11, 41, 14]. Recently there has been much interest in detecting communities in dynamic and multilayer settings [18, 3, 21, 15, 24, 16, 29, 22, 43, 12, 36, 28]. A community is a group of nodes with stronger connections to nodes within the group than to nodes external to the group, and the organization of the network into communities has strong implications for the function and structure of the system. Communities in multilayer networks represent a balance between the community structure in and between layers, and detecting multilayer communities can provide insight into the network structure which is hidden at the level of the individual layers [30]. Because multilayer communities can describe multiple interactions (throughout time, space, modality, etc.) between different layers of the network, it is especially important to understand how communities change and evolve across layers. A popular class of algorithms attempt to optimize a quality function that measures how well a given partition of the network matches the underlying community structure. Multiple quality functions have been developed from the perspective of network topology [34, 33], information theory [40], and statistical physics [38, 26], and the optimization 1 ar X iv :1 80 3. 03 59 7v 1 [ ph ys ic s. so cph ] 9 M ar 2 01 8 of different quality functions can return different community partitions. The first and most popular quality function is the modularity function [34, 33] which measures the number of internal community edges compared to a random network. However, Fortunato and Barthélemy exposed a fundamental problem with modularity maximization [20] by showing that in single layer networks, there is a resolution limit such that communities that are small relative to the network can not be detected. Later, Traag et al. [45] showed that any method that relies on optimizing a global quality function suffers from a resolution limit, thereby demonstrating that the resolution limit represents a fundamental challenge for a large class of algorithms. Other work relating statistical physics and modularity [38, 26] introduced a tunable multiresolution parameter, γ, to the modularity function that can be used to control the resolution of community detection. In fact, it has been shown [44, 45] that several quality functions [34, 38, 37, 2, 39] can all be realized as a specific formulation of a generalized multiresolution modularity function, and this multiresolution modularity function has been widely adopted to mitigate the resolution limit problem. Importantly, modularity maximization was one of the first methods to be extended to multilayer networks [30, 3] through a simple modification of the multiresolution modularity function. As such, it currently remains one of the most commonly used algorithms for performing community detection in the multilayer setting. The multilayer modularity function includes two tunable parameters: the resolution parameter, γ, and an interlayer coupling parameter, ω, that controls the strength of the interlayer coupling, i.e. the edges that run between layers. The interlayer coupling allows communities to span across layers, and the balance between detecting community structure within and between layers is controlled by ω. When ω is small, the community structure of each layer will be preferred and nodes can easily switch communities between layers. When ω is large, nodes will prefer to stay in one community across layers and will be less compelled by the community structure within the layers. While some work attempts to provide guidance on how to choose these parameters [47, 1], little is known about how the choice of parameter values influences community detection. Here, we expose a resolution limit on community detection in multilayer networks such that a change in community structure between two layers can not be detected. We show precisely how the interlayer coupling parameter, ω, controls the ability to detect communities in multilayer networks and give an upper bound on ω beyond which it can be guaranteed that certain cross-layer community changes can not be detected. We demonstrate an explicit relationship between our bound and the previously established single layer resolution limit and show how our bound has non-trivial and purely multilayer effects. Further, we show that ω is bounded above by a linear function in γ establishing an explicit and previously unknown relationship between these parameters. Multi-resolution Modularity in Single Layer Networks We first review modularity maximization in a traditional single layer network. Given a network with an adjacency matrix, A, and a randomized version of the network, R, the (multi-resolution) modularity function is