Biased random walks on multiplex networks

Biased random walks on complex networks are a particular type of walks whose motion is biased on properties of the destination node, such as its degree. In recent years they have been exploited to design efficient strategies to explore a network, for instance by constructing maximally mixing trajectories or by sampling homogeneously the nodes. In multiplex networks, the nodes are related through different types of links (layers or communication channels), and the presence of connections at different layers multiplies the number of possible paths in the graph. In this work we introduce biased random walks on multiplex networks and provide analytical solutions for their long-term properties such as the stationary distribution and the entropy rate. We focus on degree-biased walks and distinguish between two subclasses of random walks: extensive biased walks consider the properties of each node separately at each layer, intensive biased walks deal instead with intrinsically multiplex variables. We study the effect of different structural properties, including the number of layers, the presence and sign of inter-layer degree correlations, and the redundancy of edges across layers, on the steady-state behaviour of the walkers, and we investigate how to design an efficient exploration of the system. Finally, we apply our results to the case of a multidimensional social network and to a multimodal transportation system, showing how an appropriate tuning of the bias parameters towards nodes which are truly multiplex allows to obtain a good trade-off between a maximal entropy rate and a homogeneous sampling of the nodes of the network.