The art of interconnections: Achieving maximum algebraic connectivity in multilayer networks

The second smallest eigenvalue of the Laplacian matrix, known as algebraic connectivity, determines many network properties. This paper investigates the optimal design of interconnections that maximizes algebraic connectivity in multilayer networks. We identify an upper bound for maximum algebraic connectivity for total weight below a threshold, independent of interconnections pattern, and only attainable with a particular regularity condition. For efficient numerical approaches in regions of no analytical solution, we cast the problem into a convex framework and an equivalent graph embedding problem associated with the optimum diffusion phases in the multilayer. Allowing more general settings for interconnections entails regions of multiple transitions, giving more diverse diffusion phases than the more studied one-toone interconnection case. When there is no restriction on the interconnection pattern, we derive several analytical results characterizing the optimal weights using individual Fiedler vectors. We use the ratio of algebraic connectivity and layer sizes to explain the results. Finally, we study the placement of a limited number of interlinks heuristically, guided by each layer's Fiedler vector components.