Large-scale clique cover of real-world networks

The edge clique cover (ECC ) problem deals with discovering a set of (possibly overlapping) cliques in a given graph that covers each of the graph's edges. This problem finds applications ranging from social networks to compiler optimization and stringology. We consider several variants of the ECC problem, using classical quality measures (like the number of cliques) and new ones. We describe efficient heuristic algorithms, the fastest one taking O(mdG) time for a graph with m edges, degeneracy dG (also known as k-core number). For large real-world networks with millions of nodes, like social networks, an algorithm should have (almost) linear running time to be practical: Our algorithm for finding ECCs of large networks has linear-time performance in practice because dG is small, as our experiments show, on real-world networks with thousands to several million nodes

On the threshold-width of graphs

The GG-width of a class of graphs GG is defined as follows. A graph G has GG-width k if there are k independent sets N1,...,Nk in G such that G can be embedded into a graph H in GG such that for every edge e in H which is not an edge in G, there exists an i such that both endpoints of e are in Ni. For the class TH of threshold graphs we show that TH-width is NP-complete and we present fixed-parameter algorithms. We also show that for each k, graphs of TH-width at most k are characterized by a finite collection of forbidden induced subgraphs

Graph decomposition algorithms for analyzing social and large complex networks

Graphs are often used to model or represent large and sparse networks with billions of vertices and edges and store extensive amounts of structural and semantic information. Therefore, analyzing characteristics in networked data, such as graphs that can yield important information on the modelled structure, is challenging due to their linked nature and size. A common way to uncover this high-quality information is by analyzing subgraphs to get a deeper understanding of the data, which are helpful for classification, clustering, and knowledge discovery. This thesis proposes using a compact network data representation based on sparse matrix data structures. We will consider the enumeration of subgraphs (edge clique cover problem) with some ordering schemes. Finally, we benefit from the linear algebraic approach to graph algorithms for counting triangles, triangle enumeration, the k-count algorithm, and triangle centrality calculation. This thesis will present both serial and parallel algorithms for solving these problems