# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 1 Introduction
Moreover, maximizing modularity is one of the few approaches for community detection in networks that has been extended in a principled way [16] to multilayer networks [19], though we also call attention to the multilayer extension [20] of Infomap [21] and recent developments extending stochastic block models [SBMs] to multilayer networks (see [22; 23; 24] and, for a general update of developments in SBMs, [4]). While multilayer modularity provides a means for community detection in multilayer networks using many of the same heuristics and applying some of the same conventional wisdom developed for single-layer networks, the generalization admits at least one more parameter to control the contribution of interlayer connections to modularity relative to that from intralayer connections, e.g., the interlayer coupling \(\omega\) in [16]. The same multilayer modularity framework can be applied generally to include multiple interlayer coupling parameters controlling the relative contributions of different parts of the multilayer structure, e.g., for data that is both temporal and multiplex. As such, multilayer modularity requires exploring a two-dimensional parameter space in its simplest setting, and higher dimensions in more general cases. For present purposes, we will here only explicitly consider the case of a single interlayer coupling parameter \(\omega\) in Sections 2.1 and 3.4; but this does not put constraints on the coupling topology or relative values, only that there is some selected interlayer coupling tensor that is multiplied by \(\omega\). Meanwhile, the approach we develop here can be naturally generalized to higher dimensions. Identifying appropriate values for parameter \(\gamma\) (and in the multilayer setting, \(\omega\)) involves running one or more heuristics at various parameter values and comparing the results. Because identifying globally optimal community structure is computationally intractable (both for modularity and most other approaches), these algorithms are usually run stochastically or with random initial conditions to account for entrapment in local extrema. The possibly different community structures found by computational heuristics at a particular \(\gamma\) parameter point [\((\gamma,\omega)\) for multilayer networks] are then typically assessed only at that point before moving on to generate results at other parameter values. For instance, one might select the partition with greatest modularity found at that specific value of \(\gamma\) or measure some statistic over the partitions that were generated at that \(\gamma\) (see, e.g., [25]). In order to determine whether the obtained community structures are \"robust\" to the \(\gamma\) selection in any sense, one might look for stable plateaus in the number of communities (see, e.g., [11; 26; 27; 28]), consider another metric such as significance [29], directly visualize the different community assignments across parameters (as in [28; 30]), or compare obtained communities with other generally-acceptable labels by some measure such as pairwise counting scores (see, e.g., the discussion in [31]) or information-theoretic measures like Variation of Information [32] and Normalized Mutual Information [33]. A more computationally-demanding approach that directly attacks the problem that there is no _a priori_ notion of what constitutes a \"good\" value of modularity is to compare the obtained best modularity at each \(\gamma\) with the distribution of modularities obtained by running community detection across some selected random-graph model, either on realizations from a model or from permutations of the data, repeating this process at different \(\gamma\) to identify parameter values where the obtained communities are strongest relative to the random cases [34]. Additionally, one may use a given set of partitions to generate a new partition by ensemble learning [35] or consensus clustering [36; 34].