Random walks on graphs: ideas, techniques and results

Abstract

Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and new ideas have been introduced, which can be fruitfully extended to different areas and disciplines. Here we aim at giving a brief but comprehensive perspective of these progresses, with a particular emphasis on physical aspects.

Random walks on deterministic scale-free networks: Exact results

“…Therefore, for large structures we have t O g ∼ (3/2) g , which is the same leading behavior found for t B g and t H g . This result is also consistent with Kac formula [27] according to which [19]: using Eq. 6 and Eq.…”

“…We also notice that for finite g,F g (1) = 1, that is the hub is a recurrent point, as expected for any point on finite graphs [19]. Interestingly, from Eq.…”

“…Once the thermodynamic limit is taken, a divergence inP g (z) for z → 1 would imply that the main hub is recurrent [19]. Recurrence can alternatively be proven by exploiting the connection between random walks an electric networks.…”

Non symmetric random walk on infinite graph

“…The quantity p 00 (1) is equal to the average number of visits at the starting point (we refer the reader to [1] in case of a random walk on graphs, or to [7] in general case).…”

Target annihilation by diffusing particles in inhomogeneous geometries

“…spectral dimension d, which is the natural generalization of the Euclidean dimension for dynamical processes [9]:…”