arXiv subjects

Junhao Peng

Publications and source records attributed to Junhao Peng.

24 records · Page 2Linked to original sources

arXiv2015

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and the reduced moment as the measures. Note that the FRT and FPT are deeply affected by the starting or target site. We don't intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e., node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, the mean, the variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with the increasing of the network order $N$ and tends to infinity while $N\rightarrow\infty$; but for the reduced moments of GFPT, it is almost a constant($\approx1.1605$) for large $N$. Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT is much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with self-similar structure, such as $(u, v)$ flowers and recursive scale-free trees.

cond-mat.stat-mech↗

arXiv2013

Analysis of diffusion and trapping efficiency for random walks on non-fractal scale-free trees

We study discrete random walks on the NFSFT and provide new methods to calculate the analytic solutions of the MFPT for any pair of nodes, the MTT for any target node and MDT for any source node. Further more, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we compare the trapping efficiency and diffusion efficiency for any two nodes of NFSFT and find the best (or worst) trapping sites and the best (or worst) diffusion sites. Our results show that: the two hubs of NFSFT is the best trapping site, but it is also the worst diffusion site, the nodes which are the farthest nodes from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maximum and minimum of MTT and MDT, we found that the ratio between the maximum and minimum of MTT grows logarithmically with network order, but the ratio between the maximum and minimum of MTT is almost equal to $1$. These results implie that the trap's position has great effect on the trapping efficiency, but the position of source node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.

cond-mat.stat-mech↗

arXiv2013

Efficiency analysis of diffusion on T-fractals in the sense of random walks

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 study random walks on T-fractal and provided general methods to calculate the MTT for any target node and the MDT for any source node. 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, the three boundary nodes are the worst trapping sites, but they are also the best diffusing sites. Comparing the minimum and maximum of MTT and MDT, we found that the maximum of MTT is almost $6$ times of the minimum for MTT and the maximum of MDT is almost equal to the minimum for MDT. These results show that the location of target node has big effect on the trapping efficiency, but the location of source node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.

cond-mat.stat-mech↗

arXiv2013

Effects of node position on diffusion and trapping efficiency for random walks on fractal scale-free trees

We study unbiased discrete random walks on the FSFT based on the its self-similar structure and the relations between random walks and electrical networks. First, we provide new methods to derive analytic solutions of the MFPT for any pair of nodes, the MTT for any target node and MDT for any starting node. And then, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we analyze the effect of trap's position on trapping efficiency and the effect of starting position on diffusion efficiency. Comparing the trapping efficiency and diffusion efficiency among all nodes of FSFT, we find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that: the node which is at the center of FSFT is the best trapping site, but it is also the worst diffusing site. The nodes which are the farthest nodes from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maximum and minimum of MTT and MDT, we found that the maximum of MTT is almost $\frac{20m^2+32m+12}{4m^2+4m+1}$ times of the minimum of MTT, but the the maximum of MDT is almost equal to the minimum of MDT. These results shows that the position of target node has big effect on trapping efficiency, but the position of starting node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.

cond-mat.stat-mech↗

arXiv2013

Tutte polynomial of pseudofractal scale-free web

The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the number of spanning forests, the number of acyclic orientations, the reliability polynomial, chromatic polynomial and flow polynomial. In this paper, we study and gain recursive formulas for the Tutte polynomial of pseudofractal scale-free web (PSW) which implies logarithmic complexity algorithm is obtained to calculate the Tutte polynomial of PSW although it is NP-hard for general graph. We also obtain the rigorous solution for the the number of spanning trees of PSW by solving the recurrence relations derived from Tutte polynomial, which give an alternative approach for explicitly determining the number of spanning trees of PSW. Further more, we analysis the all-terminal reliability of PSW and compare the results with that of Sierpinski gasket which has the same number of nodes and edges with PSW. In contrast with the well-known conclusion that scale-free networks are more robust against removal of nodes than homogeneous networks (e.g., exponential networks and regular networks). Our results show that Sierpinski gasket (which is a regular network) are more robust against random edge failures than PSW (which is a scale-free network). Whether it is true for any regular networks and scale-free networks, is still a unresolved problem.

math-ph↗

arXiv2011

Average path length for Sierpinski pentagon

In this paper,we investigate diameter and average path length(APL) of Sierpinski pentagon based on its recursive construction and self-similar structure.We find that the diameter of Sierpinski pentagon is just the shortest path lengths between two nodes of generation 0. Deriving and solving the linear homogenous recurrence relation the diameter satisfies, we obtain rigorous solution for the diameter. We also obtain approximate solution for APL of Sierpinski pentagon, both diameter and APL grow approximately as a power-law function of network order $N(t)$, with the exponent equals $\frac{\ln(1+\sqrt{3})}{\ln(5)}$. Although the solution for APL is approximate,it is trusted because we have calculated all items of APL accurately except for the compensation($Δ_{t}$) of total distances between non-adjacent branches($Λ_t^{1,3}$), which is obtained approximately by least-squares curve fitting. The compensation($Δ_{t}$) is only a small part of total distances between non-adjacent branches($Λ_t^{1,3}$) and has little effect on APL. Further more,using the data obtained by iteration to test the fitting results, we find the relative error for $Δ_{t}$ is less than $10^{-7}$, hence the approximate solution for average path length is almost accurate.

math-ph↗