Modularity measure of networks with overlapping communities

In this paper we introduce a non-fuzzy measure which has been designed to rank the partitions of a network's nodes into overlapping communities. Such a measure can be useful for both quantifying clusters detected by various methods and during finding the overlapping community-structure by optimization methods. The theoretical problem referring to the separation of overlapping modules is discussed, and an example for possible applications is given as well

A network-specific approach to percolation in networks with bidirectional links

Methods for determining the percolation threshold usually study the behavior of network ensembles and are often restricted to a particular type of probabilistic node/link removal strategy. We propose a network-specific method to determine the connectivity of nodes below the percolation threshold and offer an estimate to the percolation threshold in networks with bidirectional links. Our analysis does not require the assumption that a network belongs to a specific ensemble and can at the same time easily handle arbitrary removal strategies (previously an open problem for undirected networks). In validating our analysis, we find that it predicts the effects of many known complex structures (e.g., degree correlations) and may be used to study both probabilistic and deterministic attacks.Comment: 6 pages, 8 figure