We discuss the design of interlayer edges in a multiplex network, under a limited budget, with the goal of improving its overall performance. We analyze the following three problems separately; first, we maximize the smallest nonzero eigenvalue, also known as the algebraic connectivity; second, we minimize the largest eigenvalue, also known as the spectral radius; and finally, we minimize the spectral width. Maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value. However, for larger budgets, the optimal weights are generally nonuniform. The dual formulation transforms the problem into a graph realization (embedding) problem that allows us to give a fuller picture. Namely, before the threshold budget, the optimal realization is one-dimensional with nodes in the same layer embedded to a single point, while beyond the threshold, the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity. Finally, for extremely large budgets the embeddings again revert to lower dimensions. Minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector. Before a threshold, the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector, and the optimal largest eigenvalue of the Laplacian remains constant. For larger budgets, the weight distribution tends to be almost uniform. In the dual picture, the optimal graph embedding is one-dimensional and nonhomogeneous at first, with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector, while the other layer is embedded at the origin. Beyond this threshold, the optimal embedding expands to be multidimensional, and for larger values of the budget, the two layers fill the embedding space. Finally, we show how these two problems are connected to minimizing the spectral width.

中文翻译：

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针对某些拉普拉斯光谱特性设计最佳的多路复用网络。

我们讨论在有限的预算下，在Multiplex网络中设计层间边缘的方法，目的是提高其整体性能。我们分别分析以下三个问题；首先，我们最大化最小的非零特征值，也称为代数连通性；第二，我们最小化最大特征值，也就是光谱半径。最后，我们将光谱宽度最小化。最大化代数连接性需要在中间层边缘具有相同的权重，以使预算小于阈值。但是，对于较大的预算，最佳权重通常是不一致的。对偶公式化将问题转换为图形实现（嵌入）问题，使我们可以提供更完整的图像。也就是说，在阈值预算之前，最佳实现是一维的，同一层中的节点嵌入到单个点，而超出阈值时，最佳嵌入通常展开到空间中，该空间的大小受代数连通性的限制。最后，对于非常大的预算，嵌入再次恢复为较小的尺寸。最小化最大特征值是由各个网络的光谱半径及其对应的特征向量决定的。在阈值之前，将总预算分配到与该特征向量的节点线相对应的层间边缘之间，并且拉普拉斯算子的最佳最大特征值保持恒定。对于较大的预算，权重分布趋于几乎均匀。在双图片中，最佳图形嵌入首先是一维且不均匀的，对应于具有最大光谱半径的层的节点根据其特征向量分布在一条线上，而另一层嵌入在原点。超过此阈值，最佳嵌入将扩展为多维，并且对于较大的预算值，两层将填充嵌入空间。最后，我们展示了如何将这两个问题与最小化光谱宽度联系起来。