Detecting Overlapping Communities Based on Influence-Spreading Matrix and Local Maxima of a Quality Function

Community detection is a widely studied topic in network structure analysis. We propose a community detection method based on the search for the local maxima of an objective function. This objective function reflects the quality of candidate communities in the network structure. The objective function can be constructed from a probability matrix that describes interactions in a network. Different models, such as network structure models and network flow models, can be used to build the probability matrix, and it acts as a link between network models and community detection models. In our influence-spreading model, the probability matrix is called an influence-spreading matrix, which describes the directed influence between all pairs of nodes in the network. By using the local maxima of an objective function, our method can standardise and help in comparing different definitions and approaches of community detection. Our proposed approach can detect overlapping and hierarchical communities and their building blocks within a network. To compare different structures in the network, we define a cohesion measure. The objective function can be expressed as a sum of these cohesion measures. We also discuss the probability of community formation to analyse a different aspect of group behaviour in a network. It is essential to recognise that this concept is separate from the notion of community cohesion, which emphasises the need for varying objective functions in different applications. Furthermore, we demonstrate that normalising objective functions by the size of detected communities can alter their rankings.