Mathematical Programming Formulations for the Collapsed k-Core Problem

TL;DR

This paper introduces new mathematical programming formulations for the Collapsed k-Core Problem in social networks, aiming to identify the most critical users whose removal minimizes the network's cohesiveness.

Contribution

It is the first to formulate both the k-core and Collapsed k-Core problems using mathematical programming, including bilevel, single-level, and nonlinear models, with computational comparisons.

Findings

Proposed formulations outperform existing methods on benchmark instances.

Bilevel models can be effectively reformulated as single-level problems.

Pre-processing and valid inequalities improve computational efficiency.

Abstract

In social network analysis, the size of the k-core, i.e., the maximal induced subgraph of the network with minimum degree at least k, is frequently adopted as a typical metric to evaluate the cohesiveness of a community. We address the Collapsed k-Core Problem, which seeks to find a subset of $b$ users, namely the most critical users of the network, the removal of which results in the smallest possible k-core. For the first time, both the problem of finding the k-core of a network and the Collapsed k-Core Problem are formulated using mathematical programming. On the one hand, we model the Collapsed k-Core Problem as a natural deletion-round-indexed Integer Linear formulation. On the other hand, we provide two bilevel programs for the problem, which differ in the way in which the k-core identification problem is formulated at the lower level. The first bilevel formulation is reformulated as a single-level sparse model, exploiting a Benders-like decomposition approach. To derive the second bilevel model, we provide a linear formulation for finding the k-core and use it to state the lower-level problem. We then dualize the lower level and obtain a compact Mixed-Integer Nonlinear single-level problem reformulation. We additionally derive a combinatorial lower bound on the value of the optimal solution and describe some pre-processing procedures and valid inequalities for the three formulations. The performance of the proposed formulations is compared on a set of benchmarking instances with the existing state-of-the-art solver for mixed-integer bilevel problems proposed in (Fischetti et al., A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs, Operations Research 65(6), 2017).