Abstract The analysis of social and biological networks often involves modeling clusters of interest as cliques or their graph‐theoretic generalizations. The ‐club model, which relaxes the requirement of pairwise adjacency in a clique to length‐bounded paths inside the cluster, has been used to model cohesive subgroups in social networks and functional modules or complexes in biological networks. However, if the graphs are time‐varying, or if they change under different conditions, we may be interested in clusters that preserve their property over time or under changes in conditions. To model such clusters that are conserved in a collection of graphs, we consider a cross‐graph ‐club model, a subset of nodes that forms a ‐club in every graph in the collection. In this article, we consider the canonical optimization problem of finding a cross‐graph ‐club of maximum cardinality in a graph collection. We develop integer programming approaches to solve this problem. Specifically, we introduce strengthened formulations, valid inequalities, and branch‐and‐cut algorithms based on delayed constraint generation. The results of our computational study indicate the significant benefits of using the approaches we introduce.

Cliques and their generalizations are frequently used to model “tightly knit” clusters in graphs and identifying such clusters is a popular technique used in graph-based data mining. One such model is the s-club, which is a vertex subset that induces a subgraph of diameter at most s. This model has found use in a variety of fields because low-diameter clusters have practical significance in many applications. As this property is not hereditary on vertex-induced subgraphs, the diameter of a subgraph could increase upon the removal of some vertices and the subgraph could even become disconnected. For example, star graphs have diameter two but can be disconnected by removing the central vertex. The pursuit of a fault-tolerant extension of the s-club model has spawned two variants that we study in this article: robust s-clubs and hereditary s-clubs. We analyze the complexity of the verification and optimization problems associated with these variants. Then, we propose cut-like integer programming formulations for both variants whenever possible and investigate the separation complexity of the cut-like constraints. We demonstrate through our extensive computational experiments that the algorithmic ideas we introduce enable us to solve the problems to optimality on benchmark instances with several thousand vertices. This work lays the foundations for effective mathematical programming approaches for finding fault-tolerant s-clubs in large-scale networks. History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications. Funding: The computing for this project was performed at the High Performance Computing Center at Oklahoma State University supported in part through the National Science Foundation [Grant OAC-1531128]. This material is based upon work supported by the National Science Foundation under [Grants 1662757 and 1942065]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1231 .

A γ-quasi-clique in a simple undirected graph refers to a subset of vertices that induces a subgraph with edge density at least γ. When γ equals one, this definition corresponds to a classical clique. When γ is less than one, it relaxes the requirement of all possible edges by the clique definition. Quasi-clique detection has been used in graph-based data mining to find dense clusters, especially in large-scale error-prone data sets in which the clique model can be overly restrictive. The maximum γ-quasi-clique problem, seeking a γ-quasi-clique of maximum cardinality in the given graph, can be formulated as an optimization problem with a linear objective function and a single quadratic constraint in binary variables. This article investigates the Lagrangian dual of this formulation and develops an upper-bounding technique using the geometry of ellipsoids to bound the Lagrangian dual. The tightness of the upper bound is compared with those obtained from multiple mixed-integer programming formulations of the problem via experiments on benchmark instances.

A k ‐plex is a clique relaxation introduced in social network analysis to model cohesive social subgroups that allows for a limited number of nonadjacent vertices (strangers) inside the cohesive subgroup. Several exact algorithms and heuristic approaches to find a maximum‐size k ‐plex in the graph have been developed recently for this NP‐hard problem. This article develops a greedy randomized adaptive search procedure (GRASP) for the maximum k ‐plex problem. We offer a key improvement in the design of the construction procedure that alleviates a drawback observed in multiple past studies. In existing construction heuristics, k ‐plexes found for smaller values of parameter k are sometimes not found for larger k even though they are feasible; instead inferior solutions are found. We identify the reasons behind this behavior and address these in our new construction procedure. We then show that an existing exact algorithm for solving this problem on power‐law graphs can be considerably enhanced by using GRASP. The overall approach is able to solve the problem to optimality on massive social networks, including some with several million vertices and edges. These are orders of magnitude larger than the largest real‐life social networks on which this problem has been solved to optimality in the current literature. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 388–407 2017

This article introduces the minimum spanning k-core problem that seeks to find a spanning subgraph with a minimum degree of at least k (also known as a k-core) that minimizes the total cost of the edges in the subgraph. The concept of k-cores was introduced in social network analysis to identify denser portions of a social network. We exploit the graph-theoretic properties of this model to introduce a new approach to survivable interhub network design via spanning k-cores; the model preserves connectivity and diameter under limited edge failures. The deterministic version of the problem is polynomial-time solvable due to its equivalence to generalized graph matching. We propose two conditional value-at-risk (CVaR) constrained optimization models to obtain risk-averse solutions for the minimum spanning k-core problem under probabilistic edge failures. We present polyhedral reformulations of the convex piecewise linear loss functions used in these models that enable Benders-like decomposition approaches. A decomposition and branch-and-cut approach is then developed to solve the scenario-based approximation of the CVaR-constrained minimum spanning k-core problem for the aforementioned loss functions. The computational performance of the algorithm is investigated via numerical experiments.