Article

Exact calculations of first-passage quantities on recursive networks.

Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Case Courrier 121, Université Paris 6, 4 Place Jussieu, FR-75255 Paris Cedex, France.
Physical Review E (Impact Factor: 2.31). 02/2012; 85(2 Pt 2):026113. DOI: 10.1103/PhysRevE.85.026113
Source: PubMed

ABSTRACT We present general methods to exactly calculate mean first-passage quantities on self-similar networks defined recursively. In particular, we calculate the mean first-passage time and the splitting probabilities associated to a source and one or several targets; averaged quantities over a given set of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities highlights the dependency of first-passage processes with respect to the source-target distance, which has recently revealed to be a key parameter in characterizing transport in complex media. We explicitly perform calculations for different classes of recursive networks [finitely ramified fractals, scale-free (trans)fractals, nonfractals, mixtures between fractals and nonfractals, nondecimable hierarchical graphs] of arbitrary size. Our approach unifies and significantly extends the available results in the field.

0 Bookmarks
 · 
138 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider degree-biased random walkers whose probability to move from a node to one of its neighbours of degree k is proportional to k^{\alpha}, where \alpha is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to pass from a given node, and iii) the time it takes to visit all the nodes of the network. We consider a large database of real-world networks and we show that the value of \alpha which minimizes the three characteristic times is different from the value \alpha_{min}=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, assortative networks have preferentially a value of \alpha_{min} in the range [-1,-0.5], while disassortative networks have \alpha_{min} in the range [-0.5, 0]. When only local information is available, degree-biased random walks can guarantee smaller characteristic times by means of an appropriate tuning of the motion bias.
    Physical Review E 01/2014; 89:012803. · 2.31 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Efficiently controlling the diffusion process is crucial in the study of diffusion problem in complex systems. In the sense of random walks with a single trap, mean trapping time (MTT) and mean diffusing time (MDT) are good measures of trapping efficiency and diffusion efficiency, respectively. They both vary with the location of the node. In this paper, we analyze the effects of node's location on trapping efficiency and diffusion efficiency of T-fractals measured by MTT and MDT. First, we provide methods to calculate the MTT for any target node and the MDT for any source node of T-fractals. The methods can also be used to calculate the mean first-passage time between any pair of nodes. Then, using the MTT and the MDT as the measure of trapping efficiency and diffusion efficiency, respectively, we compare the trapping efficiency and diffusion efficiency among all nodes of T-fractal and find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that the hub node of T-fractal is the best trapping site, but it is also the worst diffusing site; and that the three boundary nodes are the worst trapping sites, but they are also the best diffusing sites. Comparing the maximum of MTT and MDT with their minimums, we find that the maximum of MTT is almost 6 times of the minimum of MTT and the maximum of MDT is almost equal to the minimum for MDT. Thus, the location of target node has large effect on the trapping efficiency, but the location of source node almost has no effect on diffusion efficiency. We also simulate random walks on T-fractals, whose results are consistent with the derived results.
    The Journal of Chemical Physics 04/2014; 140(13):134102. · 3.12 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: On infinite homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. However, on general inhomogeneous structures this property does not hold and, although a single random walker will certainly return to its starting point, two moving particles may never meet. This striking property has been shown to hold, for instance, on infinite combs. Due to the huge variety of natural phenomena which can be modeled in terms of encounters between two (or more) particles diffusing in comb-like structures, it is fundamental to investigate if and, if so, to what extent similar effects may take place in finite structures. By means of numerical simulations we evidence that, indeed, even on finite structures, the topological inhomogeneity can qualitatively affect the two-particle problem. In particular, the mean encounter time can be polynomially larger than the time expected from the related one particle problem.
    05/2014;

Full-text (2 Sources)

View
47 Downloads
Available from
May 27, 2014