Identifying the Topology of a Coupled FitzHugh–Nagumo Neurobiological Network via a Pinning Mechanism
ABSTRACT Topology identification of a network has received great interest for the reason that the study on many key properties of a network assumes a special known topology. Different from recent similar works in which the evolution of all the nodes in a complex network need to be received, this brief presents a novel criterion to identify the topology of a coupled FitzHugh-Nagumo (FHN) neurobiological network by receiving the membrane potentials of only a fraction of the neurons. Meanwhile, although incomplete information is received, the evolution of all the neurons including membrane potentials and recovery variables are traced. Based on Schur complement and Lyapunov stability theory, the exact weight configuration matrix can be estimated by a simple adaptive feedback control. The effectiveness of the proposed approach is successfully verified by neural networks with fixed and switching topologies.
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ABSTRACT: The topological structure of a dynamical network plays a pivotal part in its properties, dynamics and control. Thus, understanding and modeling the structure of a network will lead to a better knowledge of its evolutionary mechanisms and to a better cottoning on its dynamical and functional behaviors. However, in many practical situations, the topological structure of a dynamical network is usually unknown or uncertain. Thus, exploring the underlying topological structure of a dynamical network is of great value. In recent years, there has been a growing interest in structure identification of dynamical networks. As a result, various methods for identifying the network structure have been proposed. However, in most of the previous work, few of them were discussed in the perspective of optimization. In this paper, an optimization algorithm based on the projected conjugate gradient method is proposed to identify a network structure. It is straightforward and applicable to networks with or without observation noise. Furthermore, the proposed algorithm is applicable to dynamical networks with partially observed component variables for each multidimensional node, as well as small-scale networks with time-varying structures. Numerical experiments are conducted to illustrate the good performance and universality of the new algorithm.Physica A: Statistical Mechanics and its Applications 02/2013; 392(4):1038–1049. · 1.68 Impact Factor - SourceAvailable from: Lixiang Li[Show abstract] [Hide abstract]
ABSTRACT: Nowadays, the topology of complex networks is essential in various fields as engineering, biology, physics, and other scientific fields. We know in some general cases that there may be some unknown structure parameters in a complex network. In order to identify those unknown structure parameters, a topology identification method is proposed based on a chaotic ant swarm algorithm in this paper. The problem of topology identification is converted into that of parameter optimization which can be solved by a chaotic ant algorithm. The proposed method enables us to identify the topology of the synchronization network effectively. Numerical simulations are also provided to show the effectiveness and feasibility of the proposed method.Mathematical Problems in Engineering 10/2013; 2013. · 1.38 Impact Factor - SourceAvailable from: Hamid Soleimani[Show abstract] [Hide abstract]
ABSTRACT: This paper presents a modified astrocyte model that allows a convenient digital implementation. This model is aimed at reproducing relevant biological astrocyte behaviors, which provide appropriate feedback control in regulating neuronal activities in the central nervous system. Accordingly, we investigate the feasibility of a digital implementation for a single astrocyte and a biological neuronal network model constructed by connecting two limit-cycle Hopf oscillators to an implementation of the proposed astrocyte model using oscillator-astrocyte interactions with weak coupling. Hardware synthesis, physical implementation on field-programmable gate array, and theoretical analysis confirm that the proposed astrocyte model, with considerably low hardware overhead, can mimic biological astrocyte model behaviors, resulting in desynchronization of the two coupled limit-cycle oscillators.IEEE transactions on neural networks and learning systems 03/2014; · 3.77 Impact Factor
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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 20091679
Identifying the Topology of a Coupled FitzHugh–Nagumo
Neurobiological Network via a Pinning Mechanism
Jin Zhou, Wenwu Yu, Xiumin Li, Michael Small, and Jun-an Lu
Abstract—Topology identification of a network has received great
interest for the reason that the study on many key properties of a network
assumes a special known topology. Different from recent similar works
in which the evolution of all the nodes in a complex network need to be
received, this brief presents a novel criterion to identify the topology of a
coupled FitzHugh–Nagumo (FHN) neurobiological network by receiving
the membrane potentials of only a fraction of the neurons. Meanwhile, al-
though incomplete information is received, the evolution of all the neurons
including membrane potentials and recovery variables are traced. Based
on Schur complement and Lyapunov stability theory, the exact weight
configuration matrix can be estimated by a simple adaptive feedback
control. The effectiveness of the proposed approach is successfully verified
by neural networks with fixed and switching topologies.
Index Terms—Complex network, neural network, pinning, topology
identification, weight couplings.
I. INTRODUCTION
Dynamical neural networks, consisting of a large number of neu-
rons, have been a fascinating and important subject of research [1],
[2]. These neurons are connected to each other by synapses, which are
the specialized junctions where a neuron communicates with a target
cell [3]. The interactions among neurons, namely, the couplings, have
a great influence on the dynamical characteristics of neurobiological
network [3].
The properties of a neural network with a certain coupling config-
uration have been extensively investigated [1]–[6]. Network topology
is critical for the understanding of geometry characteristics, synchro-
nization,andapplicationofanetwork[7]–[14].Therefore,asaninverse
problem incomplex networkandneurobiology, topologyidentification
of a neural network is very important.
Inthehumanbrain,thereisalargenumberofneuronswhichinteract
mutually to represent and process information. When measuring brain
activity by electroencephalography (EEG) [15], magnetoencephalog-
raphy (MEG) [16], functional magnetic resonance imaging (fMRI)
[17], or positron emitted tomography (PET) [18], the sensor informa-
tion reflects the dynamics of neurons mediated as a local field potential
(LFP). By receiving this information, it is possible to estimate the
topology of the network. From signal processing point of view, there
Manuscript received April 22, 2008; revised June 01, 2009; accepted July 16,
2009. First published August 21, 2009; current version published October 07,
2009. This work was supported by the University Grants Council of the Hong
Kong Government under the Competitive Earmarked Research Grant PolyU
5269/06E.TheworkofJ.Zhou andJ.-anLuwassupportedbythe NationalNat-
uralScienceFoundationofChinaunderGrants70771084and60574045andthe
National Basic Research Program of China under Grant 2007CB310805.
J. Zhou is with the School of Mathematics and Statistics, Wuhan University,
Wuhan 430072, China, and also with the Department of Electronic and Infor-
mation Engineering, Hong Kong Polytechnic University, Hong Kong (e-mail:
enjzhou@gmail.com).
W. Yu is with the Department of Electronic Engineering, City University of
Hong Kong, Hong Kong (e-mail: wenwuyu@gmail.com; wwyu@ee.cityu.edu.
hk).
X. Li and M. Small are with the Department of Electronic and Informa-
tion Engineering, Hong Kong Polytechnic University, Hong Kong (e-mail:
07901216r@eie.polyu.edu.hk; small@ieee.org).
J. Lu is with the School of Mathematics and Statistics, Wuhan University,
Wuhan 430072, China (e-mail: jalu@whu.edu.cn).
Color versions of one or more of the figures in this brief are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNN.2009.2029102
are many algorithms such as cross correlation and partial correlation
to identify the interactions among neurons in a network [19]–[21].
In addition, by exploiting information theory algorithms such as
transfer entropy [22] and mutual information [23], reconstructing the
functional connectivity among neurons is possible. While in the area
of mathematics and engineering, works on the rigorous theoretical
derivation for these techniques have been rare.
Very recently, topology identification of complex networks has been
intensively studied. First, Yu et al. estimated the adjacency matrix of a
linearly coupled complex network in 2006 [24]; then, in 2007, Tang et
al. modified Yu’s method and applied it into a neural network in which
the dynamics of each neuron is a Hindmarsh–Rose model [25]; in the
same year, Zhou and Lu recognized the topology of a general weighted
complex network even with different coupling nodes [26]; later, in
2008, Wu estimated the topology of a network with time-varying cou-
pling delay [27]. In the above methods, the states of all the nodes in the
network were monitored to achieve topology identification.
However, in many cases, only some of the couplings in several sub-
regions are unknown or uncertain, and the remainder is known [28].
Therefore, only some of the couplings need to be identified. The pin-
ning control mechanism, which was first applied to complex networks
by Wang et al. [30] and Li et al. [31], is a more economical and prac-
tical technique. Although the pinning mechanism requires more infor-
mation of the network to decide which nodes should be controlled, this
mechanism can reduce the number of controllers. Provided with some
known couplings, pinning mechanism is a possible way to estimate the
value of those which are uncertain.
Consider the original neural network as a drive network, and design
a response network which receives only the membrane potential evo-
lution of some of the neurons. Based on Schur complement [32], [33]
andLyapunovstabilitytheory[34],anovelcriterionispresentedtoesti-
mate the weight configuration matrix for a coupled FitzHugh–Nagumo
(FHN) neural network. In this paper, some simple adaptive feedback
controllers are used to identify the topology of this network by a pin-
ning mechanism.
The remainder of this brief is organized as follows. In Section II,
some preliminaries are briefly outlined. The mechanism of control-
ling the membrane potentials of a fraction of neurons in the response
network to reach topology estimation is discussed in Section III. In
Section IV, examples are simulated to illustrate the effectiveness of the
proposed approach. The main ideas and conclusions are summarized
in Section V.
II. PRELIMINARIES
A. FitzHugh–Nagumo Model
Since 1951, the quantitative study of electrically active cells has re-
ceived its principal impetus from the remarkable work by Hodgkin and
Huxley [35] on nerve conduction in the squid giant axon. The subject
of Hodgkin and Huxley’s work is the process by which the impulse
travelsalongtheaxoninthegiantaxonofthesquidusinga4-Dexpres-
sion [35]. Since the equations were too complicated to analyze com-
pletely, simpler systems were indeed necessary to aid in understanding
the properties of the Hodgkin–Huxley equations. Thus, by taking into
account the physiological background, FitzHugh (1961) and Nagumo
(1962), independently derived a 2-D system that provides a convenient
simplification of the 4-D Hodgkin–Huxley equations [35]–[38].
The FHN model is described by
?? ? ? ??
?? ? ??? ? ? ? ????
???? ? ? ???
Here, ? is the membrane potential, ? is the recovery variable, ??? is
the external stimulus current, and ????? are positive constants. Gener-
ally, ? ? ? ? ? makes ? as a fast variable and ? as a slow variable.
1045-9227/$26.00 © 2009 IEEE
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1680IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Fig. 1. Phase of ? versus phase of ? in FHN model with parameters ? ?
????, ? ? ???, ? ? ???, ?? ????????????, ? ??? ? ??, and ???? ? ?.
It is shown that the trajectories converge to a limit cycle. Thus, ? and ? are
bounded in a certain region ultimately and the ultimate boundary of ? can be
chosen as 2 here.
The parameter ? is one of the critical terms that can significantly in-
fluence the dynamics of the system. Choosing ? ? ????, ? ? ???,
? ? ???, ??? ? ????????????, ? ??? ? ??, and ???? ? ?, the state
variables of FHN model can be depicted by Fig. 1. Since the trajecto-
ries starting from arbitrary points will converge to a limit cycle, ? and
? are bounded in a certain region ultimately.
B. The Neurobiological Network Model
Ingeneral,consideraneuralnetworkwith? coupledneuronswhich
is described by FHN equations. The dynamics of the ?th neuron in the
network can then be formulated as
???? ????
???
?????????
?
????????
??????????????????
?
? ????????? ?
?
????????????
?
? ?????????
???? ????? ? ? ????
(1)
where ? ? ??? ? ?, ????????? ? ??? ???????
????????? ? ????? ? ? ????. As for couplings, ?????? is a non-
linearinner-couplingfunctionthatistheoutputofnode?,and???isthe
synaptic outer-coupling strength. The network topology is determined
by the weight configuration matrix ? ? ????????: if the ?th neuron
is a neighbor of the ?th neuron ?? ?? ??, then the weight ??? ?? ?;
otherwise, ??? ? ?. In addition, the matrix ? is diffusive satisfying
??? ? ?
For an FHN equation, the following facts are satisfied:
? ? ??? ???, and
???????????.
???? ????????????? ? ??????????
? ???? ??? ??? ????
????? ??? ??
?? ????? ??
?
? ???? ??????? ???
? ???? ????
? ??
?
??
?? ????? ??
?
? ???? ??????? ???
? ???
?
?
???? ????? ???? ??????? ???
and
???? ????????????? ? ??????????
? ????? ??????? ??? ? ?????? ????
? ????? ??????? ??? ? ?????? ????
for any two vectors ????????and ????????, where ? is a positive
constant representing the ultimate boundary of the first variable. For
example, when ? ? ????, ? ? ???, ? ? ???, ??? ? ????????????,
? ??? ? ??, and ???? ? ?, ? can be chosen as 2 from Fig. 1.
C. The Lemmas
In order toderive the main results, the followinglemmas areneeded.
Lemma 1 (Schur complement [32], [33]): See Appendix I.
Lemma 2: Assume that ? is a diagonal matrix whose ?th ?? ?
????????diagonal elements are ? and the others are 0, where ? ? ? is a
properconstantwhichislargeenough.??istheminormatrixofasym-
metric matrix ? by removing all the ?th ?? ? ???????? row–column
pairs of ?. Then, ? ? ? ? ? is equivalent to ??? ?.
The proof of Lemma 2 is presented in Appendix II.
III. THE CONTROLLING MECHANISM
To identify the topology of a complex network, usually the orig-
inal network is served as a drive network. Design a response network
through receiving the evolution of nodes; it is possible to estimate the
weight configuration matrix. In previous literature, however, all the
nodesintheresponsenetworkshouldbecontrolledtoachievetopology
identification.
Inspired by pinning control and provided with some known weight
couplings, it is possible to identify the topology of the neurobiological
network without controlling all the neurons in the response network.
Without loss of generality, assume that in neural network (1) the
couplings ????? ? ? ? ???? ? ? ? ????? ? ??? are uncertain. To
realize weight configuration matrix estimation, the following response
network is designed:
???? ? ???????? ??? ?
?
???? ?????????? ? ?????
?? ??? ???????? ????
?? ???? ??????????????
? ? ? ? ?
? ? ? ? ???? ? ? ? ??
(2)
where????? ?? are tracing state variables,??? ???? ? ?? is the error
variable, ? ??? is the estimation of ??? for ? ? ? ? ??, ? ? ? ? ??,
? ??? ? ??? for others, ?? is positive constant, and ?? is the adaptive
feedback gain satisfying
???? ?????
??? ??
??? ? ? ? ??
otherwise
(3)
where ???? ? ? ? ??? is positive constant.
In order to propose the main results, the following hypothesis for
inner-coupling function is introduced.
Hypothesis 1 (H1): Assume that ??????? ? ? ? ??? are linearly
independent [39], and their differentials satisfy ?? ? ???
where ??and ??are positive constants.
Denote ??? ???????? as the symmetric part of the matrix ?.
Let??beamodifiedmatrixof??whichisobtainedbychangingthedi-
agonalelements???into??????????for? ? ? ? ?,???betheminor
matrix of?? by removing the first ?? row–column pairs, and ???????
????? ? ??,
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Page 3
ZHOU et al.: IDENTIFYING THE TOPOLOGY OF A COUPLED FITZHUGH–NAGUMO NEUROBIOLOGICAL NETWORK VIA A PINNING MECHANISM1681
be the maximum eigenvalue of a symmetric matrix. Then, for identi-
fying topology of the coupled FHN network (1) in which the weight
couplings among a section of neurons are unknown, a criterion based
on pinning control is attained.
Theorem 1: Suppose that H1 holds. Then, the unknown elements
????? ? ? ? ???? ? ? ? ??? of the weight configuration matrix ?
in network (1) can be identified by the estimation ? ??? in the response
network(2)and(3),providedthat????????? ? ???????????????
??? ? ???????????.
Proof: Let? ??
?? ??? ??and ? ???
?. In addition, denote ?? as a variable between ?? and??? satisfying
??????? ? ?????? ? ??
didate as
?
?
? ? ??? ? ??? for ? ? ??? ?
?????????? ???. Then, consider a Lyapunov can-
? ??
?
?
???
???
? ?? ??
?
??
?
?
???
?
???
? ??
??
??
??
?
?
???
???? ???
??
where ? is a positive constant to be determined.
Taking the derivative of ? along the trajectories of (2) and (3) yields
?? ?
?
???
??? ???????? ??? ? ?????????
?? ?? ???????? ??? ? ?????????
?
?
???
?
???
??? ? ?????????? ? ????????? ?
?
???
?????
?
?
?
???
?
???
? ????????????? ?
?
???
???? ?????
?
?
?
???
? ???
?
???
? ? ?? ? ????????? ??? ? ??? ??
?
?
?
???
?
???
?????? ??????? ? ?????? ?
?
???
????
?
?
?
???
? ???
?
???
? ? ?? ? ????????? ??? ? ??? ??
?
?
?
???
????????
?????????
?
???
?
????????
????????
?????????
?
???
????
?
?
?
???
? ???
?
???
? ? ?? ? ????????? ??? ? ??? ??
?
?
?
???
????????
? ?
?
???
?
????????
??????????????? ?
?
???
????
?
???????
???
?
???? ? ? ?
? ???
?
???
? ? ?
?
???
? ? ?
?
??
?
?
???
??????
??
?
?
?
???????
????
where?? ? ??????????????????,?
? ? ??? ?????????? ?????
? ?
??
? ? ?
?
???
? ? ?
?
???
??????
?? ? ???? ? ? ? ?? ? ??????????, ??? is the ? ? ? identity
matrix, and ? is the diagonal matrix whose ?th ?? ? ? ? ??? elements
are?andothersare0.Duetothat????????? ? ???????????????
??????????????,one can choose an appropriate constant ? ? ? such
that ???????? ? ???? ? ???????? according to Lemma 2. It grows
??? ???? ? ????? ? ??? ? ??? ? ??? ? ????????? ? ?. As a result,
since ?????? ? ?, ? is negative definite according to Lemma 1.
The largest invariant set of ??? ? ?? is ? ? ? ? ???? ? ? and? ?? ?
??? ? ? ? ??. According to LaSalle’s invariance principle [34], all
the trajectories of systems (1)–(3) will converge to ? ? ? asymptotically
for any initial values. In this set, it is obvious that???? ? ?, and fur-
ther,
constants ???? ? ? ? ???, such that
have ? ??? ? ? for ? ? ? ? ???? ? ? ? ?? in the set ? ? ?. Therefore, we
get ???
? ?? ? ?for? ? ? ? ?.Asaconclusion,theunknowncoupling
strength in neural network (1) can be identified using the response net-
work (2) and (3). Thus, the proof is completed.
From this theorem, it is shown that using the pinning adaptive feed-
back control approach, the exact topology of model (1) can be esti-
mated. At the same time, it is obvious that although just the mem-
brane potentials of some of the neurons are received, all the evolution
of the neurons including membrane potentials and recovery variables
are traced. As there are many circumstances in which the connections
among some of nodes in a complex network are unknown, this mech-
anism is of great practical significance.
Remark 1: As a special case, if the couplings of the entire net-
work (1) are unknown, the proposed mechanism (2) and (3) guarantees
topology identification provided only with H1 holding.
Remark 2: For neurobiological networks whose weight configu-
ration matrix is switching, our method works well too as shown in
Section IV.
Remark 3: The proposed mechanism can also be applied to the
neural network in which dynamics of each neuron is something other
than FHN equations, and even to neural networks with different types
of neurons (dynamics of each neuron need not to be identical). Similar
mathematical or engineering conclusions can be derived.
?
???? ?????????? ? ?. Since H1 holds, there do not exist nonzero
?
???????????? ? ?. Thus, we
????? ??? ? ? for ? ? ? ? ??, ? ? ? ? ??, and ???
???
????
????
??? ? ?,
IV. NUMERICAL SIMULATIONS
Here, 100 neurons are considered to form a Barabási–Albert scale-
free (see Appendix III) neurobiological network ??? ? ? ? ?? in
whichthedynamicsofeachneuronisanFHNequationwithparameters
? ? ????, ? ? ???, ? ? ???, and ??? ? ????????????. Assume
that the adjacent matrix of this network is ?, the common weight is
? ? ??. Suppose that the weight coupling matrix is ? ? ??, where
??? ? ????, ??? ? ??, ??? ? ??, ??? ? ?, ??? ? ??, ??? ? ????,
??? ? ??, and ??? ? ? are unknown. In addition, ????? ? ? ?
??????? ? ? ? ???? are assumed to be inner-coupling function. It is
obvious that ?? ? ?? ? ?. Then, we have?? ? ??.
In view of ????? ? ? ? ??? ? ? ? ?? to be identified, con-
sider the minor matrix???which is obtained by removing the first
four row–column pairs of??. The maximum eigenvalue of???satis-
fies ????????? ? ???????? ? ??????? ? ??? ? ?????? ? ??? ?
????????? ? ???? ? ??????.
Choose the initial values as ?? ? ?, ?? ? ???, ????? ? ?, ????? ?
?????????,?????? ? ????????,????? ? ??????,? ????? ? ???????
for ? ? ? ? ???, and ? ?????? ? ? with ? ? ? ? ??? ? ? ? ? in the
numerical simulation. Fig. 2(a) shows the evolution of inner-coupling
functions ??????? and ???????. From Fig. 2(a), it is easy to see that
??????? and ??????? are linearly independent, for example, choosing
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1682IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 10, OCTOBER 2009
Fig. 2. (a) Evolution of inner-coupling functions ? ??? ? ?? ? ????. It is obvious that ? ??? ? and ? ??? ? are linearly independent. (b) Tracing errors?? and
? ? ?? ? ? ? ???? versus time ?. It is seen that all the state variables can be traced.
Fig. 3. (a) Estimations of the unknown coupling set ??
time ?. It is shown that the estimations approach to the corresponding weight values perfectly.
??????
? versus time ?. (b) Estimations of the unknown coupling set ??
?? ????
? versus
two different times ??? ? and ??? ?, where ? is a large constant, it
is seen that
???????????
???????????
???????????
???????????
?? ??
The synchronous errors??? and? ???? ? ? ? ???? are plotted in
Fig. 2(b), which shows that all the state variables have been traced.
The topology estimations are illustrated in Fig. 3. It is found that the
weights have been estimated precisely.
If attime
??
? ????
????????????????????????????????? switches from ????????????
???????????????
to
???????????????????????????,
proposed topology identification approach also performs well,
which is exhibited in Figs. 4 and 5. Besides the successful topology
identification, all the state variables have been traced.
theunknowncouplingset
the
V. CONCLUSION
In this brief, a criterion has been presented for identifying the uncer-
tain topology of a neurobiological network by using an adaptive feed-
back controlling method. Unlike similar approaches which monitor all
the states of all the nodes to reconstruct network topology, we have
presented a different mechanism. By receiving the membrane poten-
tials of only a fraction of the neurons, an estimated model is designed
to identify the unknown weight couplings in the original neural net-
work. Simulated examples are shown to illustrate the effectiveness of
the proposed approach. In addition to the application in neurobiology,
this technology is expected to be implemented on many other fields in
whichthedynamicsofeachagentscanbemonitoredandreceived,such
as remote control and diagnostics, disease transmission, management,
and administration of Internet cafe, and so on.
APPENDIX I
Lemma 1 (Schur complement [32], [33]): The following linear ma-
trix inequality (LMI):
????
?????
????
????
? ?
where ?????? ???? and ?????? ????, is equivalent to one of the
following conditions:
a) ???? ? ? and ???? ? ??????????????? ? ?;
b) ???? ? ? and ???? ? ???????????????? ?.
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ZHOU et al.: IDENTIFYING THE TOPOLOGY OF A COUPLED FITZHUGH–NAGUMO NEUROBIOLOGICAL NETWORK VIA A PINNING MECHANISM1683
Fig. 4. Evolution of inner-coupling functions and variable tracing of the neural network whose topology switches at ? ? ????. (a) Evolution of inner-coupling
functions ? ??? ? ?? ? ????. (b) Tracing errors?? and? ? ?? ? ? ? ???? versus time ?. It is seen that all the state variables can be traced.
Fig.5. Topologyidentificationfortheneuralnetworkwhosetopologyswitchesat? ? ????.(a)Theestimationsoftheunknowncouplingset??
vs. time ?. (b) The estimations of the unknown coupling set ??
?? ??
values perfectly.
???? ??
?
??
? vs. time ?. It is shown that the estimations approach to the corresponding weight
APPENDIX II
The Proof of Lemma 2: On the one hand, if ? ? ? ? ?, one has
??? ? clearly.
On the other hand, we will prove that if ??? ?, then ? ? ? ? ?.
It is obvious that
? ? ????
? ? ? ?
??
??
??
?
where ??? represents the ? ? ? identity matrix, and ?? is the cor-
responding matrix with compatible dimension. Since ? ? ???? ?
????????
? ? ? when ? ? ? is a sufficiently large constant, we have
? ? ????
??
??
??
?
? ?
if??? ?accordingtoLemma1.Thatis,??? ?leadsto??? ? ?.
Thus, the proof is completed.
APPENDIX III
The Barabási–Albert model (widely known as the BA model) [40]
introduced in 1998 explains the power-law degree distribution of net-
works by considering two main ingredients: growth and preferential
attachment. The algorithm used in the BA model is as follows.
• Growth: Starting with ?? fully connected nodes, at every time
step, a new node is introduced and connected to ??? ??? ex-
isting nodes in the network.
• Preferential attachment: Assume that the probability ? that a new
node is connected to node ? depends on the degree ?? of node ?,
such that ? ? ????
Numerical simulations and analytic results indicate that this algorithm
evolves a scale-free network.
????.
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