任意互连多层网络代数连通度的最大化
Maximizing the algebraic connectivity in multilayer networks with arbitrary interconnections
摘要：
拉普拉斯矩阵的第二个最小特征值在表征许多网络特性方面是决定性的，并且被称为代数连通性。在本文中，我们研究的问题，最大限度地提高代数连通性的多层网络分配互联权的预算，同时允许任意互连。对于低于阈值的预算，我们确定了一个最大代数连通性的上限，它是独立的互连模式，并满足一定的正则性条件是可达的。对于没有解析解的区域中的有效数值方法，我们将问题转换为凸框架，该框架从多个角度探索问题，特别是转换为更容易解释且与最佳扩散相相关的图嵌入问题。允许任意互连需要多个过渡区，相对于一对一互连的情况下，给更多样化的扩散阶段。当有没有限制的互连模式，我们得到几个分析结果表征的最佳权重由个别费德勒向量。我们使用代数连通率和层的大小来解释的结果。最后，我们研究了放置有限数量的互连贪婪算法，使用费德勒向量分量的每一层。
英文摘要：
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.
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