Multigraph reconstruction via nonlinear random walk

Over the last few years, network science has proved to be useful in modeling a variety of complex systems, composed of a large number of interconnected units. The intricate pattern of interactions often allows the system to achieve complex tasks, such as synchronization or collective motions. In this regard, the interplay between network structure and dynamics has long been recognized as a cornerstone of network science. Among dynamical processes, random walks are undoubtedly among the most studied stochastic processes. While traditionally, the random walkers are assumed to be independent, this assumption breaks down if nodes are endowed with a finite carrying capacity, a feature shared by many real-life systems. Recently, a class of nonlinear diffusion processes accounting for the finite carrying capacities of the nodes was introduced. The stationary nodes densities were shown to be nonlinearly correlated with the nodes degrees, allowing to uncover the network structure by performing a few measurements of the stationary density at the level of a single arbitrary node and by solving an inverse problem. In this work, we extend this class of nonlinear diffusion processes to the case of multigraphs, in which links between nodes carry distinct attributes. Assuming the knowledge of the pattern of interactions associated with one type of links, we show how the degree distribution of the whole multigraph can be reconstructed. The effectiveness of the reconstruction algorithm is demonstrated through simulations on various multigraph topologies.