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# 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.
(Impact Factor: 2.29). 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.

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• "For instance, in the upper panel L = 7 and αL = 5. to investigate the problem of diffusion in finite comb lattices and we will especially focus on first-passage quantities such as the hitting time H (i, f ) from i to f (i.e. the mean time for a random walker to first reach site f starting from i), the mean first-passage time MFPT f to f (i.e., the mean time needed to first reach the vertex f , averaged over the starting site), and the global mean-first passage time GMFPT (i.e., the mean time to go from a random vertex to a second random vertex). These quantities have been extensively studied in the past, also due to the number of different applications in several research areas: pharmacokinetics [20], reactiondiffusion processes [21], excitation transport in photosystems [22], target search processes [23], disease spreading [24] and many other physical problems [25] [26] [27] [28] [29] [30] [31] [32] [33]. arXiv:1412.5883v1 "
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ABSTRACT: In this work we consider a simple random walk embedded in a generic branched structure and we find a close-form formula to calculate the hitting time $H\left(i,f\right)$ between two arbitrary nodes $i$ and $j$. We then use this formula to obtain the set of hitting times $\left\{ H\left(i,f\right)\right\}$ for combs and their expectation values, namely the mean-first passage time ($\mbox{MFPT}_{f})$, where the average is performed over the initial node while the final node $f$ is given, and the global mean-first passage time (GMFPT), where the average is performed over both the initial and the final node. Finally, we discuss applications in the context of reaction-diffusion problems.
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• "In the last decade network theory has attracted an increasing interest and an impressive number of results, analytical and/or numerical, is nowadays available. Most of them are concerned with very popular models, like scalefree networks, random graphsà la Erdös-Rényi, smallworld networks, transfractals [4] [5] [6]. These models have proved to be very effective in describing superstructures, namely artificial structures such as the World Web Web, Internet, social networks, etc. "
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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.
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ABSTRACT: Based on the Koch network constructed using Koch fractals, we proposed a class of expanded Koch networks in this paper. The original triangle is replaced by r-polygon, and each node generates m sub r-polygons by every step, which makes the Koch network more general. We studied the structure and properties of the networks. The exact analytical result of the degree distribution, clustering coefficient and average path length were obtained. When parameters m and r satisfy some certain conditions, the networks follow a power-law distribution and have a small average path length. Finally, we introduced the random walk on the network. Our discussions focused on the trapping problem, particularly the calculation and derivation of mean first passage time (MFPT) and global mean first passage time (GMFPT). In addition, we also gave the relationship between the above results and the network size.
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