Triangles are an essential structure in complex network analysis representing metrics such as clustering coefficient and transitivity. In this work, we employ the recently introduced “Triangle Centrality” in exploring cohesive sub-networks of interest within the maximum $k -$core of complex networks. Motivated by the recent interest in higher-order connectivity patterns to understand fundamental structures that control behaviour of complex systems, we employ triangles as “higher-order” structural units in complex systems. Each triangle in the network is attributed with a numeric score, the so called $k -$count, to indicate its “influence” calculated based on the number of other triangles it is associated with. We demonstrate our proposed method to perform visual exploration of cohesive sub-networks of interest on a well-studied benchmark social network “Lusseau’s Dolphin Social Network”.

Triangles are an essential part of network analysis, representing metrics such as transitivity ratio and clustering coefficient Because of its diverse applications, enumeration and counting of triangles in large networks has been extensively studied, and continues to draw much interest from many different fields. This has only increased with the introduction of approximate counting, parallel and distributed implementations, and restricted and streaming data access scenarios. We propose a compact and efficient representation of network data based on the intersection of edge labels, and use sparse matrix data structures for its computer implementation. We then present a scalable algorithm that uses this structure to count triangles. On a set of large (the largest with more that 3.6 billion edges) real-world and synthetic networks, our algorithm performs significantly better than the reference implementation miniTri [1].

Several authors have quantified the modularity of software systems in terms of coupling and cohesion metrics. Most of these approaches focus on functional and procedural dependencies in the system. Although highly relevant at the design phase, these static dependencies alone do not account for how a software product evolves over time. Instead, this is also dictated by logical and hidden dependencies between system files. To a large extent, the co-change (co-commit) relation captures these different types of dependencies. In this paper, we define two measures of co-change-modularity of a software product based on a weighted design structure matrix (DSM). The first metric, called the weighted propagation cost, uses matrix exponential to measure how changes to one system file potentially affect the whole product. The second metric, called the weighted clustering cost, uses the output of the first metric to measure the partitionability of the system based on the co-change relation. In addition, we provide a visual representation of how the co-change structure of a system evolves over time. We discuss the theoretical foundation of our work and highlight its advantages over existing methodologies. We apply our approach to GNU Octave and show the findings to be consistent with the available literature on the evolution of Octave. Our analysis is extensible and applicable to a range of scenarios including open source systems.