Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks |
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Authors: | Zhongzhi Zhang Shuyang Gao |
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Institution: | (1) Department of Informatics, National Research Center, Giza, Egypt;(2) Department of Electronics and Communications Engineering, Cairo University, Giza, Egypt |
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Abstract: | Random walks on complex networks, especially scale-free
networks, have attracted considerable interest in the past few
years. A lot of previous work showed that the average receiving time
(ART), i.e., the average of mean first-passage time (MFPT) for
random walks to a given hub node (node with maximum degree) averaged
over all starting points in scale-free small-world networks exhibits
a sublinear or linear dependence on network order N (number of
nodes), which indicates that hub nodes are very efficient in
receiving information if one looks upon the random walker as an
information messenger. Thus far, the efficiency of a hub node
sending information on scale-free small-world networks has not been
addressed yet. In this paper, we study random walks on the class of
Koch networks with scale-free behavior and small-world effect. We
derive some basic properties for random walks on the Koch network
family, based on which we calculate analytically the average sending
time (AST) defined as the average of MFPTs from a hub node to all
other nodes, excluding the hub itself. The obtained closed-form
expression displays that in large networks the AST grows with
network order as N ln N, which is larger than the linear scaling
of ART to the hub from other nodes. On the other hand, we also
address the case with the information sender distributed uniformly
among the Koch networks, and derive analytically the global mean
first-passage time, namely, the average of MFPTs between all couples
of nodes, the leading scaling of which is identical to that of AST.
From the obtained results, we present that although hub nodes are
more efficient for receiving information than other nodes, they
display a qualitatively similar speed for sending information as
non-hub nodes. Moreover, we show that that AST from a starting point
(sender) to all possible targets is not sensitively affected by the
sender’s location. The present findings are helpful for better
understanding random walks performed on scale-free small-world
networks. |
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