AI 中文总结
该研究针对构造与给定拉普拉斯最接近且满足规定连通分量结构的图拉普拉斯问题，提出凸半定优化框架，证明解的存在唯一性，通过桑普森修道院网络示例验证了方法的有效性。
AI 中文摘要
我们研究在强制规定连通分量结构的条件下，构造与给定拉普拉斯矩阵最接近的图拉普拉斯矩阵的问题。设顶点集被划分为非空且互不相交的块 $C_1,\ldots,C_k$，令 $U=[u_1,\ldots,u_k]$ 为对应块指示向量构成的矩阵。约束条件 $MU=0$ 确保这些规定的指示向量位于优化后的拉普拉斯矩阵 $M^\star$ 的零空间中，因此关联图至少有 $k$ 个连通分量。为保证恰好为规定的分量，我们对主子块 $M_j=M[C_j,C_j]$ 施加额外的块连通性约束，这些约束确保每个规定的块诱导出一个连通的加权子图。所得问题是一个凸半定优化问题，具有严格凸的弗罗贝尼乌斯范数目标函数。我们证明了极小值解的存在性与唯一性，并表明优化后的拉普拉斯矩阵恰好具有规定的连通分量，其零空间为 $\operatorname{span}\{u_1,\ldots,u_k\}$。本工作提出的框架为量化将基于图的网络转换为具有规定群组分离结构所需的最小结构干预提供了一种原则性工具。数值示例包括桑普森修道院正情感网络，展示了最接近派系一致的加权重构以及实现规定派系结构所需的最小拉普拉斯扰动。
英文摘要
We study the problem of constructing the nearest graph Laplacian matrix to a given Laplacian while enforcing a prescribed connected-component structure. Let the vertex set be partitioned into nonempty disjoint blocks $C_1,\ldots,C_k$, and let $U=[u_1,\ldots,u_k]$ be the matrix of the corresponding block-indicator vectors. The constraint $MU=0$ ensures that these prescribed indicators lie in the nullspace of the optimized Laplacian $M^\star$, and hence the associated graph has at least $k$ connected components. To guarantee exactly the prescribed components, we impose additional block-connectivity constraints on the principal blocks $M_j=M[C_j,C_j]$. These constraints ensure that each prescribed block induces a connected weighted subgraph. The resulting problem is a convex semidefinite optimization problem with a strictly convex Frobenius-norm objective. We prove existence and uniqueness of the minimizer and show that the optimized Laplacian has exactly the prescribed connected components, with nullspace $\operatorname{span}\{u_1,\ldots,u_k\}$. The framework proposed in this work provides a principled tool for quantifying the minimum structural intervention required to transform a graph-based network into one having a prescribed group-separated structure. Numerical examples, including the Sampson monastery positive-affection network, illustrate the nearest faction-consistent weighted reconstruction and the minimum Laplacian perturbation required to realize the prescribed faction structure.
