Discovering hidden layers in quantum graphs

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Abstract
Finding hidden layers in complex networks is an important and a non-trivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multi-layer system exist and if so then what is their extent, i.e. how many unknown layers there are. We assume that all information we have is the time evolution of a wave propagation on a single layer of a network and show that it is indeed possible to uncover that which is hidden by merely observing the dynamics. We present evidence on both synthetic and real world networks that the frequency spectrum of the wave dynamics can express distinct features in the form of additional frequency peaks. These peaks exhibit dependence on the number of layers taking part in the propagation and thus allowing for the extraction of said number. In fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare this approach to the work of Aziz et al. in which there has been established a wave packet signature method for discerning between various single layer graphs and we modify it for the purposes of multi-layer systems.

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