Constant Communities in Complex Networks

Abstract

Identifying community structure is a fundamental problem in network analysis. Most community detection algorithms are based on optimizing a combinatorial parameter, for example modularity. This optimization is generally NP-hard, thus merely changing the vertex order can alter their assignments to the community. However, there has been less study on how vertex ordering influences the results of the community detection algorithms. Here we identify and study the properties of invariant groups of vertices (constant communities) whose assignment to communities are, quite remarkably, not affected by vertex ordering. The percentage of constant communities can vary across different applications and based on empirical results we propose metrics to evaluate these communities. Using constant communities as a pre-processing step, one can significantly reduce the variation of the results. Finally, we present a case study on phoneme network and illustrate that constant communities, quite strikingly, form the core functional units of the larger communities.A fundamental problem in understanding the behavior of complex networks is the ability to correctly detect communities. Communities are groups of entities (represented as vertices) that are more connected to each other as opposed to other entities in the system. Mathematically, this question can be translated to a combinatorial optimization problem with the goal of optimizing a given metric of interrelation, such as modularity or conductance. The goodness of community detection algorithms (see 1,2 for a review) is often objectively measured according to how well they achieve the optimization.However, these algorithms can be applied to any network, regardless of whether it possesses a community structure or not. Furthermore when the optimization problem is NP-hard, as in the case of modularity 3 , the order in which vertices are processed as well as the heuristics can change the results. These inherent fluctuations of the results associated with modularity have long been a source of concern among researchers. Indeed the goodness of modularity as an indicator of community structure has also been questioned, and there exist examples 5 which demonstrate that high modularity does not always indicate the correct community structure. Other orthogonal graph-theoretic metrics (e.g., conductance 6 which is also NP-complete 4 ) have been proposed in the past literatures to judge the goodness of a community detection algorithm.Research in addressing the fluctuations in the results due to modularity maximization heuristics include identifying stability among communities from the consensus networks built from the successive iterations of a non-deterministic community detection algorithm (such as by Seifi et al. 7 ). Lancichinetti et al. 8 proposed consensus clustering by reweighting the edges based on how many times the pair of vertices were allocated to the same community, for different identification methods. Delvenne et al. 9 introduced the notion of the stability of a partition, a me...

GenPerm: A Unified Method for Detecting Non-Overlapping and Overlapping Communities

“…This can be more specifically measured in terms of number of invariant groups of vertices which stay in the same output community in spite of the fluctuation. These groups have been termed as "constant communities" [25]. In our earlier work, we showed that despite such fluctuations in the final outcome, there exist few such constant communities which always remain same across different vertex orderings.…”

On the permanence of vertices in network communities

“…Recently, Chakraborty et al [5] pointed out how vertex ordering influences the results of the community detection algorithms. They identify invariant groups of vertices (named as "constant communities") whose assignment to communities are not affected by vertex ordering.…”

“…Moreover, the optimal values of the scoring functions do not provide any insight as to whether a network actually possesses community structure or not. For example, the highest modularity in the Jazz network is 0.45 [5] and that of the Western USA power grid is 0.98 [5,13]. However, it has been observed [5,13], that Jazz has a much stronger community structure than the power grid.…”

“…To the best of our knowledge, we are the first to make this distinction between the total external connections and their distribution. Permanence of a vertex thus quantifies its propensity to remain in its assigned community and the extent to which it is "pulled" [5] by the neighboring communities.…”

Overlapping community detection using neighborhood ratio matrix

“…Each c i ∈ C is an induced sub-graph of G. Membership nodes are determined by using 3 NMF which can be defined as; let X be an n × m non-negative matrix then, NMF factorizes X into non-negative matrices W 4 and H , such that [15]; 5 X ≈ WH .…”