Community Detection in NK Landscapes - An Empirical Study of Complexity Transitions in Interactive Networks

Abstract

NK-landscapes have been used as models of gene interaction in biology, but also to understand the influence of interaction between variables on the controllability and optimality organizations as a whole. As such, instead of gene networks they could also be models of economical systems or social networks. In NK-landscapes a fitness function is computed as a sum of N trait values. Each trait depends on the variable directly associated with the trait and K other variables – so-called epistatic variables. With increasing number of interactions the ruggedness of the function increases (local optima). The transition in complexity resembles phase transitions, and for $K \ge 2$ the problem to find global optima becomes NP hard. This is not the case if epistatic variables are local, meaning that all interactions are local.

In this research we study NK-landscapes from the perspective of communities and community detection. We will not look at communities of epistatic links but instead focus on links due to correlation between phenotypic traits. To this end we view a single trait as an individual agent which strives to maximize its contributed value to the net value of a community. If the value of a single trait is high whenever that of another trait is low we regard these traits as being conflicting. If high values of one trait coincide with high values of other traits we regard the traits as supporting each other. Finally, if the value of two traits is uncorrelated, we view their relationship as being neutral. We study what happens to the system of traits when the NK-landscape undergoes a critical transition to a more complex model via the increment of k. In particular, we study the effect of locality of interaction on the shape and number of the emerging communities of traits and show that the number of communities reaches its lowest point for medium values of k and not, as might be expected, for a fully connected epistatic matrix (case $k=N-1$).