Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors, and are closer to criticality when chaotic dynamics would be expected, than classical RBNs.
Bastolla and Parisi @cite_3 studied modularity within classical RBNs, i.e. functionally independent clusters, but not topological modularity.
@cite_3
2952040885 2087471103 2337910323 2040858714
Modularity is one of the most widely used quality measures for graph clusterings. Maximizing modularity is NP-hard, and the runtime of exact algorithms is prohibitive for large graphs. A simple and effective class of heuristics coarsens the graph by iteratively merging clusters (starting from singletons), and optionally refines the resulting clustering by iteratively moving individual vertices between clusters. Several heuristics of this type have been proposed in the literature, but little is known about their relative performance. This paper experimentally compares existing and new coarsening- and refinement-based heuristics with respect to their effectiveness (achieved modularity) and efficiency (runtime). Concerning coarsening, it turns out that the most widely used criterion for merging clusters (modularity increase) is outperformed by other simple criteria, and that a recent algorithm by Schuetz and Caflisch is no improvement over simple greedy coarsening for these criteria. Concerning refinement, a new multi-level algorithm is shown to produce significantly better clusterings than conventional single-level algorithms. A comparison with published benchmark results and algorithm implementations shows that combinations of coarsening and multi-level refinement are competitive with the best algorithms in the literature. This is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of Random Boolean Networks. Here we study the functionally independent clusters in which the relevant elements, introduced and studied in our first paper [3], are subdivided. We show that the phase transition in Random Boolean Networks can also be described as a percolation transition. The statistical properties of the clusters of relevant elements (that we call modules) give an insight on the scaling behavior of the attractors of the critical networks that, according to Kauffman, have a biological analogy as a model of genetic regulatory systems. We consider a large class of random geometric graphs constructed from samples @math of independent, identically distributed observations of an underlying probability measure @math on a bounded domain @math . The popular modularity' clustering method specifies a partition @math of the set @math as the solution of an optimization problem. In this paper, under conditions on @math and @math , we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions @math converge in a certain sense to a continuum partition @math of the underlying domain @math , characterized as the solution of a type of Kelvin's shape optimization problem. Modularity is widely used to effectively measure the strength of the disjoint community structure found by community detection algorithms. Although several overlapping extensions of modularity were proposed to measure the quality of overlapping community structure, there is lack of systematic comparison of different extensions. To fill this gap, we overview overlapping extensions of modularity to select the best. In addition, we extend the Modularity Density metric to enable its usage for overlapping communities. The experimental results on four real networks using overlapping extensions of modularity, overlapping modularity density, and six other community quality metrics show that the best results are obtained when the product of the belonging coefficients of two nodes is used as the belonging function. Moreover, our experiments indicate that overlapping modularity density is a better measure of the quality of overlapping community structure than other metrics considered.