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

Maximizing the algebraic connectivity in multilayer networks with arbitrary interconnections