Laplacian spectrum and coherence analysis of weighted hypercube network

Hypercube network is one of the most important and attractive network topologies so far. In this paper, we consider the scaling for first- and second-order network coherence on the hypercube network controlled by a weight factor. Our objective is to quantify the robustness of algorithms to stochastic disturbances at the nodes by using a quantity called network coherence which can be characterized as Laplacian spectrum. Network coherence can capture how well a network maintains its formation in the face of stochastic external disturbances. Firstly, we deduce the recursive relationships of its eigenvalues at two successive generations of Laplacian matrix. Then, we obtain the Laplacian spectrum of Laplacian matrix. Finally, we calculate the first- and second-order network coherence quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues by using Squeeze Theorem. The obtained results show that the network coherence depends on generation number and weight factor. Meanwhile, the scalings of the first- and second-order network coherence of weighted hypercube decrease with the increasing of weight factor r, when 0 < r < 1.