Suppose there is a spreading process such as an infectious disease propagating on a graph. How would we reduce the number of affected nodes in the spreading process? This question appears in recent studies about implementing mobility interventions on mobility networks (Chang et al. (2021)). A practical algorithm to reduce infections on unweighted graphs is to remove edges with the highest edge centrality score (Tong et al. (2012)), which is the product of two adjacent nodes' eigenscores. However, mobility networks have weighted edges; Thus, an intervention measure would involve edge-weight reduction besides edge removal. Motivated by this example, we revisit the problem of minimizing top eigenvalue(s) on weighted graphs by decreasing edge weights up to a fixed budget. We observe that the edge centrality score of Tong et al. (2012) is equal to the gradient of the largest eigenvalue of $WW^{\top}$, where $W$ denotes the weight matrix of the graph. We then present generalized edge centrality scores as the gradient of the sum of the largest $r$ eigenvalues of $WW^{\top}$. With this generalization, we design an iterative algorithm to find the optimal edge-weight reduction to shrink the largest $r$ eigenvalues of $WW^{\top}$ under a given edge-weight reduction budget. We also extend our algorithm and its guarantee to time-varying graphs, whose weights evolve over time. We perform a detailed empirical study to validate our approach. Our algorithm significantly reduces the number of infections compared with existing methods on eleven weighted networks. Further, we illustrate several properties of our algorithm, including the benefit of choosing the rank $r$, fast convergence to global optimum, and an almost linear runtime per iteration.

翻译：假设存在一个传播过程，例如传染病在图结构上的扩散。我们如何减少该传播过程中受影响的节点数量？这一议题出现在近期关于在移动网络中实施流动性干预措施的研究中（Chang et al., 2021）。在无权图中降低感染率的一个实用算法是移除具有最高边中心性得分的边（Tong et al., 2012），该得分定义为相邻两节点特征向量中心性得分的乘积。然而，移动网络具有加权边，因此干预措施除移除边之外，还涉及边权重的降低。受此实例启发，我们通过减少边权重（在固定预算约束下）来最小化加权图中的最大特征值（集）。我们注意到，Tong等人（2012）提出的边中心性得分等于$WW^{\top}$最大特征值的梯度，其中$W$表示图的权重矩阵。进而，我们将广义边中心性得分定义为$WW^{\top}$最大$r$个特征值之和的梯度。基于这一推广，我们设计了一种迭代算法，在给定边权重削减预算下，通过最优边权重减少来缩小$WW^{\top}$的最大$r$个特征值。我们还将该算法及其理论保证扩展至权重随时间演化的时变图。通过详细的实证研究验证了方法的有效性。在十一个加权网络上，与现有方法相比，我们的算法显著减少了感染数量。此外，我们展示了算法的若干性质，包括选择秩$r$的优势、快速收敛至全局最优解以及近乎线性的每轮迭代时间复杂度。