Random Multi-Hopper Model. Super-Fast Random Walks on Graphs

@article{Estrada2016RandomMM,
  title={Random Multi-Hopper Model. Super-Fast Random Walks on Graphs},
  author={Ernesto Estrada and Jean-Charles Delvenne and Naomichi Hatano and Jos{\'e} L. Mateos and Ralf Metzler and Alejandro P. Riascos and Michael T. Schaub},
  journal={J. Complex Networks},
  year={2016},
  volume={6},
  pages={382-403}
}
We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multi-hopper's hitting times between any… 

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References

SHOWING 1-10 OF 71 REFERENCES

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between

Random walks on complex networks.

The random walk centrality C is introduced, which is the ratio between its coordination number and a characteristic relaxation time, and it is shown that it determines essentially the mean first-passage time (MFPT) between two nodes.

Random Walks on Lattices. II

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary

Path Laplacian matrices: Introduction and application to the analysis of consensus in networks

Long-Range Navigation on Complex Networks using Lévy Random Walks

  • A. P. RiascosJ. Mateos
  • Computer Science, Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
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

Resistance distance in graphs and random walks

We study the resistance distance on connected undirected graphs, linking this concept to the fruitful area of random walks on graphs. We provide two short proofs of a general lower bound for the

Lattice walks by long jumps

Diffusion on surfaces has in the past been modeled as a random walk in continuous time between nearest‐neighbor sites on a lattice. In order to allow tests for the possible participation of long

The electrical resistance of a graph captures its commute and cover times

Known relations between random walks and electrical networks are extended by showing that resistance in this network is intimately connected with the lengths of random walks on the graph, and bounds on cover time obtained are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix.

Random walks and electric networks

The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.
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