The second smallest eigenvalue of the Laplacian matrix is determinative in characterizing many network properties and is known as algebraic connectivity. In this paper, we investigate the problem of maximizing algebraic connectivity in multilayer networks by allocating interlink weights subject to a budget while allowing arbitrary interconnections. For budgets below a threshold, we identify an upper-bound for maximum algebraic connectivity which is independent of interconnections pattern and is reachable with satisfying a certain regularity condition. For efficient numerical approaches in regions of no analytical solution, we cast the problem into a convex framework that explores the problem from several perspectives and, particularly, transforms into a graph embedding problem that is easier to interpret and related to the optimum diffusion phase. Allowing arbitrary interconnections entails regions of multiple transitions, giving more diverse diffusion phases with respect to one-to-one interconnection case. When there is no limitation on the interconnections pattern, we derive several analytical results characterizing the optimal weights by individual Fiedler vectors. We use the ratio of algebraic connectivity and the layer sizes to explain the results. Finally, we study the placement of a limited number of interlinks by greedy heuristics, using the Fiedler vector components of each layer.

中文翻译：

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最大化具有任意互连的多层网络中的代数连通性

拉普拉斯矩阵的第二小特征值在表征许多网络属性方面具有决定性作用，被称为代数连通性。在本文中，我们通过在允许任意互连的同时分配受预算约束的互连权重来研究最大化多层网络中代数连通性的问题。对于低于阈值的预算，我们确定了最大代数连通性的上限，该上限与互连模式无关，并且在满足一定规律性条件的情况下可达到。对于无解析解区域中的有效数值方法，我们将问题转换为凸框架，从多个角度探索问题，特别是转换为更易于解释并与最佳扩散阶段相关的图嵌入问题。允许任意互连需要多个过渡区域，从而在一对一互连情况下提供更多不同的扩散阶段。当对互连模式没有限制时，我们得出几个分析结果，通过单个 Fiedler 向量表征最佳权重。我们使用代数连通性和层大小的比率来解释结果。最后，我们使用每一层的 Fiedler 向量分量，通过贪婪启发式方法研究有限数量的互连的放置。我们推导出了几个分析结果，表征了各个 Fiedler 向量的最佳权重。我们使用代数连通性和层大小的比率来解释结果。最后，我们使用每一层的 Fiedler 向量分量，通过贪婪启发式方法研究有限数量的互连的放置。我们推导出了几个分析结果，表征了各个 Fiedler 向量的最佳权重。我们使用代数连通性和层大小的比率来解释结果。最后，我们使用每一层的 Fiedler 向量分量，通过贪婪启发式方法研究有限数量的互连的放置。