Gajewski Łukasz G, Sienkiewicz Julian, Hołyst Janusz A
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland.
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland and ITMO University, Kronverkskiy Prospekt 49, St Petersburg, Russia 197101.
Phys Rev E. 2021 Sep;104(3-1):034311. doi: 10.1103/PhysRevE.104.034311.
Finding hidden layers in complex networks is an important and a nontrivial problem in modern science. We explore the framework of quantum graphs to determine whether concealed parts of a multilayer system exist and if so then what is their extent, i.e., how many unknown layers are there. Assuming that the only information available is the time evolution of a wave propagation on a single layer of a network 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. We show that, in fact, with sufficient observation time, one can fully reconstruct the row-normalized adjacency matrix spectrum. We compare our propositions to a machine learning approach using a wave packet signature method modified for the purposes of multilayer systems.
