IEEE Workshop on Nonlinear Circuit Networks
December 14-15, 2012

Investigation of Clustering and Local Bridges ina Complex Network by Coupled Rulkov Maps
Tomoya Shima†, Yoko Uwate†, Thomas Ott‡ and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Email:{s-tomoya, uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Applied Simulation, Zurich University of Applied Sciences,
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email:thomas.ott@zhaw.ch

Abstract—It is important to understand dynamics of complex
networks. We focus on network with “local bridge” which is oneof the complex networks. In this study, we investigate influence ofthe local bridge for network using synchronization phenomena of
coupled Rulkov maps. From simulation results, when the maps
switch from full synchronization to clustering, phase deviation
occurs from the local bridges.

I. Introduction
We focus on social network which is one of complex
network. Understanding dynamics of the social network isimportant because “social network analysis” has been used
not only for analyzing modern society but also for physics,
biology and information science by many researchers [1]-[4].
The social network is structure which shows social relation,e.g. aerial line, infection with a virus, World Wide Web andso on.
In this study, we focus on “local bridge” which is foundby M. Granovetter [5]. We need to explain keyword “bridge”
before explaining the local bridge. The bridge is an edge which
provides the only route between two nodes. Here, edge is line
connecting two nodes. Example of the bridge is shown in
Fig. 1.

Fig. 1.

Example of the bridge.

Here, the bridge between A and B provides the only routewhich can flow information or influence from any contact of
A to any contact of B. The function of the bridge is called“bridging function”. The bridge happens only rarely in large
network actually. However, in the case of large scale network,
the bridging function may be provided locally. Example of thelocal bridge is shown in Fig. 2.
In Fig. 2, the shortest route from 1 to 25 is 1-25. Oneof second shortest route from 1 to 25 is 1-4-5-8-9-11-13-- 97 -Fig. 2.

Example of the local bridge.

14-15-16-20-22-23-25. The 1-25 route is not strictly bridge
because other routes construct. However, the 1-25 route is
predominantly shorter than other routes. In this case, thebridging function is provided locally by the 1-25 route andthis bridge is called the local bridge. Similarly, it is foundthat 8-9, 14-15, 15-16 and 22-23 route are local bridge. In thesocial network, the local bridge aﬀects the entire network interms of propagation of information. Because the local bridge
creates more and shorter routes.
In order to analyze complex phenomena of the social
network with local bridge, we use coupled maps [7]. Generally,
the coupled maps is used as general models for the complex
dynamics, e.g. biological systems, economic activities andneural network. Coupled oscillatory systems can also produce
interesting phase patterns, “clustering” and complex phasepatterns. Moreover, a discrete map for spiking and spikingbursting neural behavior was proposed by Rulkov [6]. Rulkov
maps can be useful for understanding the dynamical mechanism of oscillators in “the large scale network”. In this study,
we use Rulkov maps applying for the coupled maps. Because
the social network has features of cluster and the large scale
network, and we consider that the propagation of informationon the social network is realized by the Rulkov maps applying
for the coupled maps.
In this study, we show synchronization of the social network

of Rulkov maps when a coupling strength g and a control
parameter α are changed. Thereby, we observe the influenceof the local bridge on the social network.
II. Social Network of Coupled Rulkov Maps
We consider the case of the social network with local bridge.
In this study, this social network is shown in Fig. 2. The
equation of the Rulkov maps is described as follows.
g ∑xi,n+1 = f (xi,n , xi,n−1 , yi,n ) +(x j,n − xi,n ),N j∈Cjg ∑yi,n+1 = yi,n − µ(xi,n + 1) + µσi + µ(x j,n − xi,n ),N j∈C(a) g = 0.03, ∆α = 0.

ji = 1, ..., N,(1)where x and y are the fast and slow dynamical variables, g is
the coupling strength, N is the number of nodes in the socialnetwork, µ = 0.001. C j is set of nodes which are connected
to node i. The function f () has the following form:
α/(1 − xn ) + yn ,xn ≤ 0
α + yn ,0 < xn < α + yn and xn−1 ≤ 0f (xn , yn ) =  −1,x ≥ α + y or x > 0,nn(2)n−1(b)g = 0.001, ∆α = 0.5.

Here, we determine α0 which is standard value. The control
parameter α of each cell is α0 + ∆α. ∆α is set for randomly
distributed in the interval [0 : ∆α].

Fig. 4. Space-time diagram for spiking behavior. (a) Full synchronization.
(b) Asynchronization.

A. Synchronization Phenomena of Social Network

asynchronization at steady state if the coupling strength g is
low and the control parameter α vary with each node.
However, from Fig. 5 (a), the maps switch from fullsynchronization and clustering at steady state. When the maps
switch from full synchronization to clustering, phase deviation
occurs from the local bridge 22-23. From Fig. 5 (b), the maps
show full synchronization and clustering even if the parameters
are changed. Moreover, the maps show more cluster than Fig. 5(a). Phase deviation occurs from the local bridges even if themaps show many clusters. Thereby, the local bridge has astrong influence on network.

First, when the coupling strength g and the control parameter α are changed, we show synchronization phenomena of
the social network of Rulkov maps which produce spiking
behavior. These behaviors are shown in Fig. 3.
10.5

xn0-0.5

III. Conclusion-1

In this study, we have studied influence of the local bridge inthe social network via synchronization phenomena of Rulkov
maps applying for the coupled maps. From simulation results,
when the maps switch from full synchronization to clustering,phase deviation occurs from the local bridges.

-1.50Fig. 3.

200400600

800

1000n(time)

1200

1400

1600

1800

2000

Typical wave forms of Rulkov maps of Spiking behavior.

In this simulation, we define two synchronization phenomena for spiking behavior. First, “full synchronization” is that
all maps are synchronized at the in-phase. Second, “clustering”
is that maps is synchronized at the in-phase locally. The
simulation results are shown in Fig. 4. The horizontal axis
is iteration time n. The vertical axis is space i. The color ofFig. 4 is the dynamical variable x. The simulation results areshown in Fig. 4. Here, we determine that the standard value
α0 is 4.0 and σ is 0.01. From Fig. 4 (a), the maps show
full synchronization at steady state if the coupling strength gis high. On the other hand, from Fig. 4 (b), the maps show

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(b) g = 0.009, ∆α = 0.03.
Fig. 5.

Space-time diagram by changing parameters for spiking behavior.

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