We introduce the stochastic pseudo-star degree centrality problem, which focuses on a novel probabilistic group-based centrality metric. The goal is to identify a feasible induced pseudo-star, which is defined as a collection of nodes forming a star network with a certain probability, such that it maximizes the sum of the individual probabilities of unique assignments between the star and its open neighborhood. The feasibility is measured as the product of the existence probabilities of edges between the center node and leaf nodes and the product of one minus the existence probabilities of edges among the leaf nodes. First, the problem is shown to be $\mathcal{NP}$-complete. We then propose a non-linear binary optimization model subsequently linearized via McCormick inequalities. We test both classical and modern Benders Decomposition algorithms together with both two- and three-phase decomposition frameworks. Logic-based-Benders cuts are examined as alternative feasibility cuts when needed. The performance of our implementations is tested on small-world (SW) graphs and a real-world protein-protein interaction network. The SW networks resemble large-scale protein-protein interaction networks for which the deterministic star degree centrality has been shown to be an efficient centrality metric to detect essential proteins. Our computational results indicate that Benders implementations outperforms solving the model directly via a commercial solver in terms of both the solution time and the solution quality in every test instance. More importantly, we show that this new centrality metric plays an important role in the identification of essential proteins in real-world networks.

我们介绍了随机伪星度中心性问题，该问题侧重于一种新颖的基于概率组的中心性度量。目标是确定一个可行的诱导伪星，它被定义为以一定概率形成星形网络的节点集合，使得它最大化星与其开放邻域之间唯一分配的个体概率之和。可行性衡量为中心节点和叶节点之间边的存在概率与叶节点之间边的存在概率减去的乘积。首先，问题显示为$\mathcal{NP}$-完全的。然后，我们提出了一个非线性二元优化模型，随后通过 McCormick 不等式将其线性化。我们测试经典和现代 Benders 分解算法以及两相和三相分解框架。在需要时，将基于逻辑的 Benders 削减作为替代可行性削减进行检查。我们实施的性能在小世界 (SW) 图和真实世界的蛋白质-蛋白质相互作用网络上进行了测试。SW 网络类似于大规模蛋白质-蛋白质相互作用网络，其中确定性星度中心性已被证明是检测必需蛋白质的有效中心性指标。我们的计算结果表明，就每个测试实例的求解时间和求解质量而言，Benders 实施优于直接通过商业求解器求解模型。更重要的是，我们表明这种新的中心性指标在识别现实世界网络中的必需蛋白质方面发挥着重要作用。