Dense Subgraphs in Random Graphs

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest to $\frac{2}{\alpha(\gamma,p)}\big(\log n-\log\log n+\log\frac{e\alpha(\gamma,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap

Algorithms for Computing Edge-Connected Subgraphs

This thesis concentrates on algorithms for finding all the maximal k-edge-connected components in a given graph G = (V, E) where V and E represent the set of vertices and the set of edges, respectively, which are further used to develop a scale reduction procedure for the maximum clique problem. The proposed scale-reduction approach is based on the observation that a subset C of k + 1 vertices is a clique if and only if one needs to remove at least k edges in order to disconnect the corresponding induced subgraph G[C] (that is, G[C] is k-edge-connected). Thus, any clique consisting of k + 1 or more vertices must be a subset of a single k-edge connected component of the graph. This motivates us to look for subgraphs with edge connectivity at least k in a given graph G, for an appropriately selected k value. We employ the method based on the concept of the auxiliary graph, previously proposed in the literature, for finding all maximal k-edge-connected subgraphs. This method processes the input graph G to construct a tree-like graphic structure A, which stores the information of the edge connectivity between each pair of vertices of the graph G. Moreover, this method could provide us the maximal k-edge-connected components for all possible k and it shares the same vertex set V with the graph G. With the information from the auxiliary graph, we implement the scale reduction procedure for the maximum clique problem on sparse graphs based on the k-edge-connected subgraphs with appropriately selected values of k. Furthermore, we performed computational experiments to evaluate the performance of the proposed scale reduction and compare it to the previously used k-core method. The comparison results present the advancement of the scale reduction with k-edge-connected subgraphs. Even though our scale reduction algorithm based has higher time complexity, it is still of interest and deserves further investigation

The triangle k-club problem

Graph models have long been used in social network analysis and other social and natural sciences to render the analysis of complex systems easier. In applied studies, to understand the behaviour of social networks and the interactions that command that behaviour, it is often necessary to identify sets of elements which form cohesive groups, i.e., groups of actors that are strongly interrelated. The clique concept is a suitable representation for groups of actors that are all directly related pair-wise. However, many social relationships are established not only face-to-face but also through intermediaries, and the clique concept misses all the latter. To deal with these cases, it is necessary to adopt approaches that relax the clique concept. In this paper we introduce a new clique relaxation—the triangle k-club—and its associated maximization problem—the maximum triangle k-club problem. We propose integer programming formulations for the problem, stated in different variable spaces, and derive valid inequalities to strengthen their linear programming relaxations. Computational results on randomly generated and real-world graphs, with k = 2 and k = 3, are reported.info:eu-repo/semantics/publishedVersio

Algorithms for Computing Edge-Connected Subgraphs

This thesis concentrates on algorithms for finding all the maximal k-edge-connected components in a given graph G = (V, E) where V and E represent the set of vertices and the set of edges, respectively, which are further used to develop a scale reduction procedure for the maximum clique problem. The proposed scale-reduction approach is based on the observation that a subset C of k + 1 vertices is a clique if and only if one needs to remove at least k edges in order to disconnect the corresponding induced subgraph G[C] (that is, G[C] is k-edge-connected). Thus, any clique consisting of k + 1 or more vertices must be a subset of a single k-edge connected component of the graph. This motivates us to look for subgraphs with edge connectivity at least k in a given graph G, for an appropriately selected k value. We employ the method based on the concept of the auxiliary graph, previously proposed in the literature, for finding all maximal k-edge-connected subgraphs. This method processes the input graph G to construct a tree-like graphic structure A, which stores the information of the edge connectivity between each pair of vertices of the graph G. Moreover, this method could provide us the maximal k-edge-connected components for all possible k and it shares the same vertex set V with the graph G. With the information from the auxiliary graph, we implement the scale reduction procedure for the maximum clique problem on sparse graphs based on the k-edge-connected subgraphs with appropriately selected values of k. Furthermore, we performed computational experiments to evaluate the performance of the proposed scale reduction and compare it to the previously used k-core method. The comparison results present the advancement of the scale reduction with k-edge-connected subgraphs. Even though our scale reduction algorithm based has higher time complexity, it is still of interest and deserves further investigation

Low-Diameter Clusters in Network Analysis

In this dissertation, we introduce several novel tools for cluster-based analysis of complex systems and design solution approaches to solve the corresponding optimization problems. Cluster-based analysis is a subfield of network analysis which utilizes a graph representation of a system to yield meaningful insight into the system structure and functions. Clusters with low diameter are commonly used to characterize cohesive groups in applications for which easy reachability between group members is of high importance. Low-diameter clusters can be mathematically formalized using a clique and an s-club (with relatively small values of s), two concepts from graph theory. A clique is a subset of vertices adjacent to each other and an s-club is a subset of vertices inducing a subgraph with a diameter of at most s. A clique is actually a special case of an s-club with s = 1, hence, having the shortest possible diameter. Two topics of this dissertation focus on graphs prone to uncertainty and disruptions, and introduce several extensions of low-diameter models. First, we introduce a robust clique model in graphs where edges may fail with a certain probability and robustness is enforced using appropriate risk measures. With regard to its ability to capture underlying system uncertainties, finding the largest robust clique is a better alternative to the problem of finding the largest clique. Moreover, it is also a hard combinatorial optimization problem, requiring some effective solution techniques. To this aim, we design several heuristic approaches for detection of large robust cliques and compare their performance. Next, we consider graphs for which uncertainty is not explicitly defined, studying connectivity properties of 2-clubs. We notice that a 2-club can be very vulnerable to disruptions, so we enhance it by reinforcing additional requirements on connectivity and introduce a biconnected 2-club concept. Additionally, we look at the weak 2-club counterpart which we call a fragile 2-club (defined as a 2-club that is not biconnected). The size of the largest biconnected 2-club in a graph can help measure overall system reachability and connectivity, whereas the largest fragile 2-club can identify vulnerable parts of the graph. We show that the problem of finding the largest fragile 2-club is polynomially solvable whereas the problem of finding the largest biconnected 2-club is NP-hard. Furthermore, for the former, we design a polynomial time algorithm and for the latter - combinatorial branch-and-bound and branch-and-cut algorithms. Lastly, we once again consider the s-club concept but shift our focus from finding the largest s-club in a graph to the problem of partitioning the graph into the smallest number of non-overlapping s-clubs. This problem cannot only be applied to derive communities in the graph, but also to reduce the size of the graph and derive its hierarchical structure. The problem of finding the minimum s-club partitioning is a hard combinatorial optimization problem with proven complexity results and is also very hard to solve in practice. We design a branch-and-bound combinatorial optimization algorithm and test it on the problem of minimum 2-club partitioning