Mean first-passage time on a family of small-world treelike networks

2014 ◽  
Vol 25 (03) ◽  
pp. 1350097 ◽  
Author(s):  
Long Li ◽  
Weigang Sun ◽  
Guixiang Wang ◽  
Guanghui Xu
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Small World ◽  
Passage Time ◽  
Network Size ◽  
Mean First Passage Time ◽  
First Passage ◽  
Initial Node ◽  
Laplacian Spectra ◽  
Two Parameters

In this paper, we obtain exact scalings of mean first-passage time (MFPT) of random walks on a family of small-world treelike networks formed by two parameters, which includes three kinds. First, we determine the MFPT for a trapping problem with an immobile trap located at the initial node, which is defined as the average of the first-passage times (FPTs) to the trap node over all possible starting nodes, and it scales linearly with network size N in large networks. We then analytically obtain the partial MFPT (PMFPT) which is the mean of FPTs from the trap node to all other nodes and show that it increases with N as N ln N. Finally we establish the global MFPT (GMFPT), which is the average of FPTs over all pairs of nodes. It also grows with N as N ln N in the large limit of N. For these three kinds of random walks, we all obtain the analytical expressions of the MFPT and they all increase with network parameters. In addition, our method for calculating the MFPT is based on the self-similar structure of the considered networks and avoids the calculations of the Laplacian spectra.

Calculations of first passage time of delayed tree-like networks

2015 ◽  
Vol 29 (28) ◽  
pp. 1550200
Author(s):  
Shuai Wang ◽  
Weigang Sun ◽  
Song Zheng
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
The Novel ◽  
Mean First Passage Time ◽  
First Passage ◽  
Initial Node ◽  
Network Parameters ◽  
Self Similar ◽  
The Mean

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.

Determining entire mean first-passage time for Cayley networks

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009 ◽  
Author(s):  
Xiaoqian Wang ◽  
Meifeng Dai ◽  
Yufei Chen ◽  
Yue Zong ◽  
Yu Sun ◽  
...  
Keyword(s):  
First Passage Time ◽  
Polymer Networks ◽  
Laplacian Matrix ◽  
Constant Term ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage ◽  
Laplacian Spectra ◽  
Self Similar ◽  
The Relationship

In this paper, we consider the entire mean first-passage time (EMFPT) with random walks for Cayley networks. We use Laplacian spectra to calculate the EMFPT. Firstly, we calculate the constant term and monomial coefficient of characteristic polynomial. By using the Vieta theorem, we then obtain the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix. Finally, we obtain the scaling of the EMFPT for Cayley networks by using the relationship between the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix and the EMFPT. We expect that our method can be adapted to other types of self-similar networks, such as vicsek networks, polymer networks.

scholarly journals Mean first-passage time for random walks on the T-graph

2009 ◽  
Vol 11 (10) ◽  
pp. 103043 ◽  
Author(s):  
Zhongzhi Zhang ◽  
Yuan Lin ◽  
Shuigeng Zhou ◽  
Bin Wu ◽  
Jihong Guan
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage

Mean first passage time on a family of weighted directed hierarchical scale-free networks

2019 ◽  
Vol 33 (16) ◽  
pp. 1950179 ◽  
Author(s):  
Yu Gao ◽  
Zikai Wu
Keyword(s):  
Random Walks ◽  
Numerical Solutions ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage ◽  
Scale Free ◽  
Weight Parameter ◽  
Scale Free Networks ◽  
Hierarchical Scale

Random walks on binary scale-free networks have been widely studied. However, many networks in real life are weighted and directed, the dynamic processes of which are less understood. In this paper, we firstly present a family of directed weighted hierarchical scale-free networks, which is obtained by introducing a weight parameter [Formula: see text] into the binary (1, 3)-flowers. Besides, each pair of nodes is linked by two edges with opposite direction. Secondly, we deduce the mean first passage time (MFPT) with a given target as a measure of trapping efficiency. The exact expression of the MFPT shows that both its prefactor and its leading behavior are dependent on the weight parameter [Formula: see text]. In more detail, the MFPT can grow sublinearly, linearly and superlinearly with varied [Formula: see text]. Last but not least, we introduce a delay parameter p to modify the transition probability governing random walk. Under this new scenario, we also derive the exact solution of the MFPT with the given target, the result of which illustrates that the delay parameter p can only change the coefficient of the MFPT and leave the leading behavior of MFPT unchanged. Both the analytical solutions of MFPT in two distinct scenarios mentioned above agree well with the corresponding numerical solutions. The analytical results imply that we can get desired transport efficiency by tuning weight parameter [Formula: see text] and delay parameter p. This work may help to advance the understanding of random walks in general directed weighted scale-free networks.

scholarly journals Mean first-passage time for random walks on undirected networks

2011 ◽  
Vol 84 (4) ◽  
pp. 691-697 ◽  
Author(s):  
Zhongzhi Zhang ◽  
Alafate Julaiti ◽  
Baoyu Hou ◽  
Hongjuan Zhang ◽  
Guanrong Chen
Keyword(s):  
Random Walks ◽  
First Passage Time ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage

FIRST-ORDER NETWORK COHERENCE AND EIGENTIME IDENTITY ON THE WEIGHTED CAYLEY NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750049 ◽  
Author(s):  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YUE ZONG ◽  
JIAHUI ZOU ◽  
YUFEI CHEN ◽  
...  
Keyword(s):  
First Passage Time ◽  
Analytical Formula ◽  
Passage Time ◽  
Small Degree ◽  
Network Size ◽  
Mean First Passage Time ◽  
Weight Factor ◽  
First Passage ◽  
First Order ◽  
Definition Of

In this paper, we first study the first-order network coherence, characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, on the weighted Cayley networks with the weight factor. The analytical formula of the EMFPT is obtained by the definition of the EMFPT. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor. Then, we study eigentime identity quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on the weighted Cayley networks with the weight factor. We show that all their eigenvalues can be obtained by calculating the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity on the weighted Cayley networks obey two laws along with the range of the weight factor.

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