Research Article
Topology Identification of Coupling Map Lattice under
Sparsity Condition
Jiangni Yu,
1,2
Lixiang Li,
1,2
and Yixian Yang
1,2
1
Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of
Posts and Telecommunications, Beijing 100876, China
2
National Engineering Laboratory for Disaster Backup and Recovery, Beijing University of Posts and Telecommunications,
Beijing 100876, China
Correspondence should be addressed to Lixiang Li; li
lixiang@.com
Received July ; Accepted September
Academic Editor: Florin Pop
Copyright © Jiangni Yu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Coupling map lattice is an ecient mathematical model for studying complex systems. is paper studies the topology identication
of coupled map lattice (CML) under the sparsity condition. We convert the identication problem into the problem of solving the
underdetermined linear equations. e
1
norm method is used to solve the underdetermined equations. e requirement of data
characters and sampling times are discussed in detail. We nd that the high entropy and small coupling coecient data are suitable
for the identication. When the measurement time is more than . times sparsity, the accuracy of identication can reach an
acceptable level. And when the measurement time reaches times sparsity, we can receive a fairly good accuracy.
1. Introduction
e coupled map lattice with nonlocally coupling chaotic
characteristic is widely observed and highly involved in many
elds, which ranges from complex network [–]toneural
network, from biological system to ecological system, and
from physics to computer science [–]. Driven by some
practical applications which benet from the better control-
ling of CML, a great deal of current research of CML has
focusedondynamicalanalysis,control,andmodeling.How-
ever, the behavior of CML is largely inuenced by the topol-
ogy of the network which generally is invisible to us. us, its
identication usually becomes the promise of application.
Up to now, various researches focus on identifying the
patterns of coupled map lattice models [, ], such as the
study of the formation and evolution of spatiotemporal
patterns based on a reference model [], and the identication
of CML based on the wavelet []. However, few researches
have discussed the topology identication of CML. ere
are two problems of topology identication: how we should
get the result with less measurement time ()andwhat
precondition the measured data need.
In this paper, as CML is discrete, for the facilitation of
identication, we transform the CML equation so that the
identication problem can be converted into a problem of
solving the linear equation =Φ.e
1
method is
introduced to solve the topology identication of CML. In
the process of identication, the relationship between the
measurement time ()andthesparsity()of is taken
into consideration. In the experiment, through the analysis
of entropy, we discuss the coupling coecient and the
solvability of the equation =Φ.Wendthatsmallercou-
pling coecient benets the identication. rough further
research, we study the inuence of the measurement time on
the identication precision; that is, when the measurement
time ≥2.86, we can achieve a decent identication result,
and when ≥4, the identication result is very good.
2. Analysis of Identification Process
Considering a CML with elements is as follows:
𝑡+1
(
)
=
(
1−
)
𝑡
(
)
+
2
𝑊
𝑗=1
𝑖𝑗
𝑖
,,
()
Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 303454, 6 pages
http://dx.doi.org/10.1155/2015/303454