LIU JIE and LU JUNAN

1
Long Range Connection Based Semi-random
Small-world Network

• LIU JIE and LU JUN-AN
• School of Mathematics and Statistics
• Wuhan,China

2
Outline
Main Structure I. Introducing a new
long-range connection based small-world model
based on NWs modified model II.
Comparing the synchronizability of the two
related small-world model Firstly, recall of
existing models is briefly given Then, the
semi-random strategy for creating a SW is given
with some comparison At last, synchronizability
of our model and NW small-world is compared with
each other based on two related Criterions mostly
used by researchers.
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Small-world network is highly clustered network
with small distance among the nodes. There are
many real-world networks that present this kind
of connection, such as the WWW, Transportation
system, Social or Biological network, achieve
both a strong
local clustering ( node have
many mutual neighbors ) and a small
average shortest path length
( maximum distance between any two
nodes ). These networks now have been verified
and characterized as small-world (SW) networks
1-21.
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'How crucial is the long-range links in such
networks produced by the semi-random SW
strategy', This
question is indeed worth reasoning. Recently,
Adilson E. Motter et al., have investigated the
range-based attack on links in scale-free
networks, they found that, the small world
property of scale free networks is mainly due to
short range links3. Further more, Takashi
Nishikawa et al., in the same research group,
numerically and analytically studied the
synchronizability of heterogeneous networks20.
5
In the context of network design, the semi-random
SW strategy (typically described as modeling
related real networks by the addition of
randomness to regular structures) now is shown to
be an efficient way of producing synchronically
networks when compared with some standard
deterministic graphs networks and even to fully
random and constructive schemes. A great deal
of research interest in the theory and
applications of small-world networks has arisen
since the pioneering work of D Watts and H
Strogatz. Here, we omit the recall of classical
SW network models, since most of us are familiar
with them. ?
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In this paper, we will try to
investigate some related questions by
constructing a modified version of small world
network through only randomly adding 'long range
links' (in some sense of 'space-distance') to a
regular lattice. We will try to compare the
modified model with the most used small-world
networks in some researching works introduced by
Newman and Watts, recently.
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• SOME most used existing SW network models
• SW rewired small-world network
• NW modified small-world model
• (Another presentation version is by Wang X
  F and Chen G., differing by fraction of existing
  edges or all possible edges, See Wu Ch W(2000))
• III. Deterministic small-world networks
• IV. Semi-random small-world network reserving
  degrees.

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Main two steps of Creating A SW Based On Adding
Long-range Links

• Initialize
• Start with a nearest-neighbor coupled ring
  lattice with
• N nodes, in which each node i is connected
  to its K neighboring nodes

where K is an even integer. The nodes are
numbered sequentially from 1 to N (For
simplicity, we suppose that N is a
multiple of 4), Thus, the lattice
distance12-13 between two nodes
numbered i and j can be calculate by
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(ii) Randomize with re-choosing shortcuts
Randomly adding links between a pair of nodes in
the network with probability p, during the whole
process, duplicated links are excluded. Defined
Long-range links as links between nodes whose
lattice distance dij is larger than or equal to
D(d), where D() is a function of d, and d
denotes the diameter of the original regular
lattice. Re-choosing the links added in the
above lattice network, reserving only new links
those lattice distance longer than setting value
D(d). Thus, we obtain a network only with
randomly added long-range links. Obviously, D(d)
can be set on the intervalK/2 1 N/2 , and if
D(d) set at K 1, it generates the same network
as that generated by NW model.
10
Appendix ONLY from the basic procedures of NW,
Wang-Chen, D_SW and M_SW Different constructing
prices can be compared with each other in that
way
Especially
11
The original regular lattice network and
the randomized network models constructed in our
simulations are illustrated respectively as
follows ( for the straight intuitive
purpose, choosing N 24K
4 p 001D (d) N4 See
Following Figures ).
12
Figure 1 A regular lattice started from N
24, K 4 for illustration. Figure 2
Deterministic SW network, where Hub 6
13
Figure 3 NW small-world network, p 005435.
Figure 4 M-SW network D(d) N4, for
comparison, links 13-17,
19-22 in Fig.3 are un-chosen here.
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Local rewired Small-world network, which we
have proposed in another paper.
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Some basic characteristics of NW SW model
16
Characteristic index of long-range links based
small-world networks
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Fig.6 log-linear scale figure of Fig.5
18
Probability distribution of the connectivity
for degreek, where p0.05, NW_ ? M-NW_ ?
N/3 ? N/4 ?_N/5. The distribution curves
be transported from right to left.
19
Probability distribution of the connectivity
for degreek, where p0.01, NW_ ? M-NW_ ?
N/3 ? N/4 ?_N/5. The distribution curves
be transported from right to left.
20
Remark 1 We have do large amount of numerical
experiments about the change of C and L about the
modified model besides Fig. 5-6.
Remark 2 For any given value of p, Cluster
Coefficients C(N p) clearly increase with the
increasing of D(d) but the average path lengths
L increase very slightly with the increasing of
D(d). For the similar average path length L,
the cluster coefficients C in the modified model
is much larger than that in the NW models. That
is to say, by using the modified procedure, one
can obtain better clustered networks with similar
average path length. (eg.,
, where C is the changed fraction of cluster
coefficients, \Delta Nadd is changed fraction of
added edges.)
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In the second part of this paper, We discuss
the Synchronizability of the two types of
semi-random networks after adding the SAME number
of new shortcuts into a regular lattice, which is
differ from prices comparison mentioned as
before. NW model v.s Modified model TWO RELATED
VIEWPOINTS.
Similarly, we recall some related works firstly
on this topic. Previous work I. A node is
periodic or chaotic, but the coupling topology is
regular. II. The coupling topology is SW or NW
Schemes. III. The coupling topology is NW Scheme
plus Time-varying.
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Some information about complex system (Chaotic
ones) synchronization
Crickets Synchronize their chirps
A Synchronized Clap
We are very interested in this topic
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Some interesting books on synchronization

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I. A typical Dynamical Network Model
Where cgt0 coupling strength,
\Gamadiagd_1,d_n. A is the coupling
matrix satisfies the diffusive coupling
conditions. If there is a connection between node
i and node j (j?i), then aijaji1 otherwise,
aijaji0 A is a symmetric and irreducible
matrix. The eigenvalues of A can be arranged as
?10gt ?2gtgt?n (n2,3,,N) , Specially
we firstly concern such a dynamical network on
certain schemes
N identical linearly and diffusively coupled
nodes, with each node being a n-dimensional
dynamical system
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Cited Paper X.F.Wang and G. Chen Int. J.
Bifurcation Chaos (2002)
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Some Information about existing results
The synchronizability is said to be strong if the
networks can be synchronized with a small
coupling strength c. In X F Wang, and G Chens
2002 IJBC paper, they had found that, for given
systems, on typical coupling scheme, both
theoretically and numerically investigations show
that, a sufficiently large coupling strength c
will lead to synchronization. They stated, in
some situation, the networks synchronizability
can be characterized by the second-largest
eigenvalue of coupling matrix. (A small value of
?2 corresponds to a large value of ?2 which
implies that network can synchronize with a small
coupling strength c.)
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Globally coupled network No matter how small
the coupling strength is, a global coupled
network will synchronize if its size is
sufficiently large. Locally coupled network No
matter how large the coupling strength is, a
locally coupled network will not synchronize if
its size N is sufficiently large.
By Comparing The Synchronizability of Two
Extremely Type Networks,
We want to know, which type of Non-local
connections added in can lead to better
synchronizability indeed?
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Our new model seems to give some hints for this
interesting question
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Address X.F.Wang and G. Chen (2002) results
again Synchronizability can be greatly enhanced
by just adding a tiny fraction of Shortcuts.
This reveals Similar semi-random small-worldfy
strategies show an advantage for improving the
synchronizability. In another paper Wu Ch W.,
Phys. Lett. A (2002), he studied this question
from the viewpoint of graph theory. When N \to
\infty, perturbations which only add local
coupling will not change \lambda_2, On the other
hand, there exist perturbations which modify an
arbitrarily small percentage of matrix elements,
each of which is changed by arbitrarily small
amount ant yet can make \lambda_2 arbitrarily
large.
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Using the \lambda_2 criterion proposed by Wang
and Chen , adding the same number of links into
pure NW small-world model and our model,
respectively, Simulations are shown as follows
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Amplification of formers
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• It can be seen from our results
• For any given value of N, \lambda_2 decreases
  with the increasing of added edges (See Fig.9)
• (ii) Adding the same number of new edges, the
  value of \lambda_2 increases with the increasing
  of D(d) (See Fig.9)
• (iii) It is strange that, from Figure 11-13, we
  found that the contribution for synchronization
  of networks (1) caused by intentionally adding n
  long-space-range edges is almost no difference
  when randomly adding n edges to the original
  regular lattice, when p \in 0, 0.01.
• If p \in 0.01, 0.1, we found that
  additional long-space-edges have not special
  effects for improving the synchronization for the
  dynamical networks (1).
• It is even worse than randomly adding the
  same number of edges to the original lattice from
  the view point of consider-ing constructing
  prices.

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This fact hints us that, in practice,
according to both physical and synchronizing
mechanism reasons, it is not a economic way by
constructing too many long-space-edges to
obtain better synchronization of networks ( 1 ),
Although it will cause more clustered small-world
structure as mentioned before.
In a recent research of Barahona and Pecora ,
they stated the small- world property does not
guarantee in general that a network will be
synchronizable. We will discuss this paper LATER
!. The results further confirmed the fact that,
the random coupling in some sense the best
candidate for such non-local coupling for
synchronization in the semi-random strategy,
which is Coincide with Wu Ch Ws research.
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Considering the Synchronizability Question
From another Criterion Proposed By

Barahona, M. Pecora, L.M 2002
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Firstly, recall a most cited paper on
synchronization in SW networks and their general
model by Barahona, M. Pecora, L.M.,
Synchronization in small-world Systems, Phys.
Rev. Lett., 89, 054101, 2002.
Their conclusion is Even sufficiently
strong coupling do not guarantee
synchronization. Some networks are
just not synchronized no matter how the
global coupling is tuned.
Maximum LE \lambda_max v.s. Generic coupling
strength \alpha
They introduce the master stability function
to analysis local linear synchronizability
of a dynamical networks. the criterion is
described as a eigenratio \lambda_max/
\lambda_2lt \alpha_2/\alpha_1 \beta
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The same FIG. as former pagein Linear-Linear
Scale
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Change of index \lambda_max/lambda_2 of the
coupling scheme mentioned in the paper (GD-A,
i.e., AAT) where Random N/3 Diamond
N/4lt N/5gt.
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Two in One
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The results further confirmed our results!
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J. Lü et al. derive a sufficient and necessary
condition of chaos synchronization for the
time-varying network. See ILv, J., Yu, X.
Chen, G., Chaos synchronization of general
complex dynamical networks, Physica A, 334,
281-302, 2004 IILv, J., Yu, X., Chen, G.,
Cheng, D. Characterizing the Synchronizability
of Small-World Dynamical Networks, IEEE
Transactions on CSI 51.787-796, 2004 Belykh et
al. proposed a new general method to determine
global stability of system synchronization, which
combines the Lyapunov method with graph
connectivity theory. He use this method analyzed
a blinking small world model successful.
See III V. Belykh, I. Belykh, and M. Hasler
"Connection graph stability method for
synchronized coupled chaotic systems." Physica
D, 195, 1-2, 159-187, 2004. IVI. Belykh, V.
Belykh, M. Hasler "Blinking model and
synchronization in small-world networks with a
time-varying coupling", Physica D, 195, 1-2,
188-206, 2004. VV. Belykh, I. Belykh, M.
Hasler, K. Nevidin "Cluster synchronization in
three-dimensional lattices of diffusively coupled
oscillators", International Journal of
Bifurcation and Chaos, 13, 4, 755-779, 2003.
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Motter, A.E., Zhou, C. Kurths, J., Enhancing
complex-network synchronization,
cond-mat/0406207. Nishikawa, T., Motter, A. E.,
Lai, Y.-C., Hoppensteadt F. C., Hetero-geneity
in Oscillator Networks Are Smaller Worlds Easier
to Synchronize? Phys. Rev. Lett. 91, 014101,
2003.
Li, C.G. Chen, G., Phase synchronization in
small-world networks of chaotic oscillators,
Phys. A, 341, 73-79, 2004 Li, X. Chen, G.,
Synchronization and desynchronization of complex
dynamical networks An engineering viewpoint,
IEEE Trans on CS-I 50, 1381-1390, 2003.
Wu, C.W., Perturbation of coupling matrices and
its effect on the synchronizability in arrays of
coupled chaotic systems, Phys. Lett. A, 319,
495-503, 2003.
42
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Thank You !
By Liu J., and Lu J. A Wuhan University School
of Mathematics Statistics Hubei Wuhan
430072 P.R. China Email liujie_at_jerry.cn