Long-range navigation on complex networks using Lévy random walks

Abstract

We introduce a strategy of navigation in undirected networks, including regular, random, and complex networks, that is inspired by Lévy random walks, generalizing previous navigation rules. We obtained exact expressions for the stationary probability distribution, the occupation probability, the mean first passage time, and the average time to reach a node on the network. We found that the long-range navigation using the Lévy random walk strategy, compared with the normal random walk strategy, is more efficient at reducing the time to cover the network. The dynamical effect of using the Lévy walk strategy is to transform a large-world network into a small world. Our exact results provide a general framework that connects two important fields: Lévy navigation strategies and dynamics on complex networks.

Mean First Passage Time of Preferential Random Walks on Complex Networks with Applications

“…In fact, the above-mentioned various types of biased random walks in networks [24][25][26] can also be transformed into random walks on weighted networks equivalently in similar way. For example, a biased random walk in uncorrelated networks and a biased lions-lamb model were introduced in [24,25], respectively.…”

“…In addition, despite some studies of biased or preferential random walks [18,[24][25][26], a general framework for the scaling behaviour of the walks in networks with different topologies has not been available. That is to say, there is not a unified approach for understanding the behaviour of biased random walks.…”

“…We can attach edge weight = ( ) and thus revisit the biased random walks. Another example is the Lévy random walks in [26] which can be got by configuring general weight = ( ) − between node and node where denotes the shortest path length. Remark 1.…”

Fractional graph-based semi-supervised learning

“…Using this route, it is possible to plug "by hand" long-range transitions: this approach was explored in [7] and it relies on the construction of a transition matrix informed by the geodesic distances between nodes. In the first incarnation [7], the geodesic distance was taken into account weighted by a power β: therefore the transition probability from node i to j is imposed a priori as τ i→j = d…”

“…The dynamics of these 1 We thank CONACYT and Labex MILyon for the financial support. more volatile random walks has been shown to be more efficient/rapid at diffusing over irregular graph structures [7]- [10] and therefore may, in some circumstances, address some of the limitations of G-SSL. We start recalling in Sec.…”