# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
###### Abstract
Multiresolution methods seemed to outperform other methods only because the community sizes used in previous tests were \"too close to one another,\" spanning less than one order of magnitude [42]. When the community sizes vary over up to two orders of magnitude, as in many real-world networks [20; 43], existing multiresolution methods also fail to recover the expected community structures [42]. The second limit of modularity optimization method is that finding an optimal division for any given network is normally infeasible. It has been recognized that the modularity landscape of a network often includes an exponentially growing (with system size) numbers of local maxima [42; 44]. These local maxima may all be very close to the global maximum in terms of modularity, but the corresponding divisions of the network can be topologically utterly different from one another [44]. This implies that not only an exactly optimal division for a network is intrinsically unreachable [42], but the available solutions in practice can be largely unreliable. This problem is even more serious for multiresolution approaches: communities detected in \"blurring\" resolution scales are often incomprehensible and, for all practical purposes, uninformative. Although it has been argued that inquiring which is the \"best\" or \"most relevant\" scale of resolution is an ill posed question [5], many methods still manage to find out the most stable communities that can be detected within a persistent range of resolution. The existence of such stable communities is an observed fact [5]: they form strong \"plateaus\" in the diagram of community numbers versus the resolution scale [35]. It has become popular in previous literature to rank the confidence of detection results by their strengths of plateaus; community structures suggested by strong plateaus have been demonstrated to be frequently consistent with the a priori knowledge about the network [5; 30; 35]. Nevertheless, existing methods using the stability of plateaus are not yet satisfying. On the one hand, a plateau was originally expected to reflect the multiple times of discovery of exactly the same communities at different resolutions. However, no comparison on the community structure represented by each data point within the same plateau has been explicitly conducted in previous literature, leaving a doubt that the same plateau may not represent the same community structure at all. Some methods, as in [5; 45; 46], simply define plateaus by the numbers of communities, some by the values of modularity [30], while some others, as in [35], execute quantitative comparisons among the communities detected at fixed resolutions--yet communities detected at different resolutions are still not compared. As discussed in [44], none of such definitions can guarantee each plateau as defined represents an identical community structure. On the other hand, evaluating the stability of plateaus by their lengths is, by far, not sufficient. Although large plateaus are almost always stable [5; 30; 35], stabilities of small plateaus are uncertain: some of them can be stable and informative, but some others just emerge due to randomness (we will show some examples later). Previous literature either ignores all small plateaus, or artificially selects their preferred ones. Due to these facts, we propose a strict definition for the term \"plateau,\" as well as an effective strategy to evaluate the stabilities of plateaus, as both are urgently needed. In this paper, we propose a new approach for multiresolution community detection. We adopt a modified community fitness function [30] and a heuristic Louvain algorithm [47] to find multilevel community structures for complex networks.