The Dynamic Graph Clustering Problem - ILP-Based Approaches Balancing Optimality and the Mental Map
چکیده
Clustering is an established tool for the analysis of networks or network-like data. The partitioning of the graph of a network into so-called clusters is meant to yield insights into its function, and to reveal common properties amongst nodes, as well as properties of individual nodes. A cluster is understood to be a subset of the nodes of a network with large density of links amongst them and sparse connectivity to other clusters. Though lots of natural networks are subject to changes, the clustering of dynamic graphs (i.e. graphs, whose set of nodes and set of edges change over time) has not been studied in detail yet. In this thesis, a dynamic graph is understood to consist of a sequence of ordinary graphs {G1, . . . , Gn}. In brief, the Dynamic Graph Clustering Problem is to find a clustering CGi for each of those graphs Gi, such that the clustering’s quality is maximized and the distance to the clustering CGi−1 of the previous graph Gi−1 is minimized. The structural proximity to its predecessor is meant to conserve the mental map of the clustering, and thus to allow for an analysis of the structural changes in the dynamic clustering (i.e. a sequence of clusterings) over time. The quality of a clustering is examined by the application of the well known clustering indices modularity and significance performance. The structural distance between consecutive clusterings is evaluated by also well known distance measures for clusterings. During the course of this thesis, an existing integer linear program (ILP) formulation for finding optimal modularity clusterings is analyzed and optimized, in order to allow for the faster processing of larger graphs. Furthermore, an ILP formulation of the significance performance index is developed and compared to the modularity version. It is shown that both rank clusterings of a graph identically and thus result in the same optimal results. A bi-criterial ILP for solving the Dynamic Graph Clustering Problem is designed, which allows for the optimization of a weighted combination of a quality index and a distance measure at the same time. Both criteria are considered explicitly. In order to study the capabilities of the approach, a proof of concept employing the modularity index and the Rand distance measure is formulated and tested with a real dynamic graph. The influence of different weighting factors and different numbers of atomary modifications in the graph’s structure is examined. The relationship between a clustering’s quality and the distance to its predecessor is shown, as well as the user’s ability to balance their optimization. In addition to that, a complementary partial ILP is designed, which explicitly optimizes the clusterings’ quality while minimizing the distance between consecutive clusterings implicitly. The mathematical program is based on the assumption that minimal changes in the graph affect its semantics primarily locally. As a consequence of the modifications between two graphs Gi−1 and Gi, a neighborhood of nodes is extracted from the clusters of CGi−1 and re-assigned by the partial ILP according to the new structure of the graph. The preserved clusters stay unchanged. With respect to the remaining parts of the previous clustering and the modularity index, the result is optimal. The complexity of the constructed ILP is mostly independent