33 research outputs found
Exact solution of mean geodesic distance for Vicsek fractals
The Vicsek fractals are one of the most interesting classes of fractals and
the study of their structural properties is important. In this paper, the exact
formula for the mean geodesic distance of Vicsek fractals is found. The
quantity is computed precisely through the recurrence relations derived from
the self-similar structure of the fractals considered. The obtained exact
solution exhibits that the mean geodesic distance approximately increases as an
exponential function of the number of nodes, with the exponent equal to the
reciprocal of the fractal dimension. The closed-form solution is confirmed by
extensive numerical calculations.Comment: 4 pages, 3 figure
Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect
A vast variety of real-life networks display the ubiquitous presence of
scale-free phenomenon and small-world effect, both of which play a significant
role in the dynamical processes running on networks. Although various dynamical
processes have been investigated in scale-free small-world networks, analytical
research about random walks on such networks is much less. In this paper, we
will study analytically the scaling of the mean first-passage time (MFPT) for
random walks on scale-free small-world networks. To this end, we first map the
classical Koch fractal to a network, called Koch network. According to this
proposed mapping, we present an iterative algorithm for generating the Koch
network, based on which we derive closed-form expressions for the relevant
topological features, such as degree distribution, clustering coefficient,
average path length, and degree correlations. The obtained solutions show that
the Koch network exhibits scale-free behavior and small-world effect. Then, we
investigate the standard random walks and trapping issue on the Koch network.
Through the recurrence relations derived from the structure of the Koch
network, we obtain the exact scaling for the MFPT. We show that in the infinite
network order limit, the MFPT grows linearly with the number of all nodes in
the network. The obtained analytical results are corroborated by direct
extensive numerical calculations. In addition, we also determine the scaling
efficiency exponents characterizing random walks on the Koch network.Comment: 12 pages, 8 figures. Definitive version published in Physical Review
Determining mean first-passage time on a class of treelike regular fractals
Relatively general techniques for computing mean first-passage time (MFPT) of
random walks on networks with a specific property are very useful, since a
universal method for calculating MFPT on general graphs is not available
because of their complexity and diversity. In this paper, we present techniques
for explicitly determining the partial mean first-passage time (PMFPT), i.e.,
the average of MFPTs to a given target averaged over all possible starting
positions, and the entire mean first-passage time (EMFPT), which is the average
of MFPTs over all pairs of nodes on regular treelike fractals. We describe the
processes with a family of regular fractals with treelike structure. The
proposed fractals include the fractal and the Peano basin fractal as their
special cases. We provide a formula for MFPT between two directly connected
nodes in general trees on the basis of which we derive an exact expression for
PMFPT to the central node in the fractals. Moreover, we give a technique for
calculating EMFPT, which is based on the relationship between characteristic
polynomials of the fractals at different generations and avoids the computation
of eigenvalues of the characteristic polynomials. Making use of the proposed
methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on
the fractals and show how they scale with the number of nodes. In addition, to
exhibit the generality of our methods, we also apply them to the Vicsek
fractals and the iterative scale-free fractal tree and recover the results
previously obtained.Comment: Definitive version published in Physical Review
Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices
The family of Vicsek fractals is one of the most important and
frequently-studied regular fractal classes, and it is of considerable interest
to understand the dynamical processes on this treelike fractal family. In this
paper, we investigate discrete random walks on the Vicsek fractals, with the
aim to obtain the exact solutions to the global mean first-passage time
(GMFPT), defined as the average of first-passage time (FPT) between two nodes
over the whole family of fractals. Based on the known connections between FPTs,
effective resistance, and the eigenvalues of graph Laplacian, we determine
implicitly the GMFPT of the Vicsek fractals, which is corroborated by numerical
results. The obtained closed-form solution shows that the GMFPT approximately
grows as a power-law function with system size (number of all nodes), with the
exponent lies between 1 and 2. We then provide both the upper bound and lower
bound for GMFPT of general trees, and show that leading behavior of the upper
bound is the square of system size and the dominating scaling of the lower
bound varies linearly with system size. We also show that the upper bound can
be achieved in linear chains and the lower bound can be reached in star graphs.
This study provides a comprehensive understanding of random walks on the Vicsek
fractals and general treelike networks.Comment: Definitive version accepted for publication in Physical Review