Investigation of Synchronization for Social Networkwith Local Bridge via 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
phone:+81-88-656-7470, fax: +81-88-656-7471
Email:{s-tomoya, uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Applied Simulation, Zurich University of Applied Sciences,
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerlandphone:+41-058-934-56-84
Email:thomas.ott@zhaw.ch

Abstract— It is important to understand dynamics of socialnetwork. We focus on “local bridge” in the social network.
In this study, we investigate influence of the local bridge inthe social network via synchronization phenomena of Rulkov
maps applying for coupled maps. From simulation results, in thesocial network with local bridge, the propagation wave switches
between full synchronization and clustering by the local bridge.

I. Introduction
Understanding dynamics of social network is importantbecause “social network analysis” has been used not only for
analyzing modern society but also for physics, biology andinformation 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 and so on.
In this study, we focus on “local bridge” which is found by
M. Granovetter [5] and is one of phenomena in social network.
We need to explain keyword “bridge” before explaining thelocal bridge. The bridge is an path in network which provides
the only route between two points. Example of the bridge isshown 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 path happens only rarely inlarge network actually. However, the bridging function may
be provided locally. Its phenomenon shows 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-

978-1-4577-1729-1/12/$26.00 ©2012 IEEE.

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-bursting neural
behavior was proposed by Rulkov [6]. Rulkov maps canbe useful for understanding the dynamical mechanism of
oscillators in “the large scale network”. In this study, weuse Rulkov maps applying for the coupled maps. Because
the social network has features of cluster and the large scalenetwork, and we consider that the propagation of information499

on the social network is realized by the Rulkov maps applying
for the coupled maps.
In this study, we show synchronization of the social networkof Rulkov maps if a coupling strength g and a control
parameter α are changed. Thereby, we observe the influenceof the local bridge on the social network.

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X1,n1II. Two coupled Rulkov Maps

0In this section, we observe fundamental synchronization
for two coupled Rulkov maps if the coupling strength g andthe control parameter α are changed in order to compare
synchronization for the social network of Rulkov maps whichis large scale network of two coupled Rulkov maps. The
equation of the Rulkov maps is shown as follows.

-1-2-34

32xi,n+1 = f (xi,n , yi,n + βi,n ),yi,n+1 = yi,n − μ(xi,n + 1) + μσi + μσi,n ,X2,n1(1)0⎧⎪xn ≤ 0α/(1 − xn ) + y,⎪⎪⎨
<α+ y and xn−1 ≤ 0α+y,0<xf (xn , y) = ⎪n⎪⎪⎩ −1,x ≥ α + y or x > 0,n-1(2)-2n−1-30where x and y are the fast and slow dynamical variables, αis a control parameter of the map. The coupling between twocells is provided by the current flowing from one cell to theother. This coupling is modeled by
βi,n = g ji βe (x j,n − xi,n ),σi,n = g ji σe (x j,n − xi,n ),Fig. 3.

100

200

300

400500n (time)

600

700

800

900

1000

In phase synchronization (α1 = 5.0, α2 = 5.5, g = 0.03).

4(3)32X1,n10-1-2-34321X2,nwhere g denotes the coupling strength. In the numericalsimulations the values of the coeﬃcients are set to be equal:βe = 1.0 and σe = 1.0. The other parameters has the followingvalues: μ = 0.001, and σ = 0.15. The coupling between themaps is symmetrical, i.e., g ji = gi j = g.
First, we investigate the boundary of synchronization andasynchronization for two coupled Rulkov maps if the difference between the control parameter α of one cell andthe other increases, or the coupling strength g is changed.
By this investigation, Here, α1 of one cell is fixed at 5.0,
α2 of the other cell is α1 + Δα. Two bursting waves aresynchronized at the in-phase as shown in Fig. 3. And Fig. 4
is shown asynchronization. The boundary of synchronization
and asynchronization for two coupled Rulkov maps is shownin Fig 5. From this result, if g or Δα is getting larger, twocoupled Rulkov maps tends to synchronize. Also, if g is larger
than 0.05, two coupled Rulkov maps has highly synchronous.

0-1-2-3III. Social network of Rulkov maps

0

We consider the case of the social network. In this study,
this social network is shown in Fig. 2. In order to simulate on
the social network, we write an equation based on a chain ofFig. 4.

100

200

300

400

500

600

700

800

900

1000n (time)

500

Asynchronization (α1 = 5.0, α2 = 7.9, g = 0.03).

Fig. 5.

Boundary of synchronization and asynchronization.

coupled maps [7].

g (x j,n − xi,n ),N j∈Cjg (x j,n − xi,n ),yi,n+1 = yi,n − μ(xi,n + 1) + μσi + μN j∈Cxi,n+1 = f (xi,n , xi,n−1 , yi,n ) +ji = 1, ..., N,(4)where x and y are the fast and slow dynamical variables, g is
the coupling strength, N is the number of cells in the socialnetwork, μ = 0.001, σ = 0.15, C j is set of nodes which areconnected to node i. The function f () has the following form:⎧⎪xn ≤ 0α/(1 − xn ) + yn ,⎪⎪⎨,0<x<α+yn and xn−1 ≤ 0α+yf (xn , yn ) = ⎪nn⎪⎪⎩ −1,x ≥ α + y or x > 0,nn(5)n−1Similarly, we investigate the boundary of synchronization andasynchronization for the social network of Rulkov maps if thediﬀerence between the control parameter α of each cell in thesocial network increases, or the coupling strength g is changed.
Here, we determine α0 which is standard value and is 5.0. αof each cell is α0 + Δα. Δα is set for randomly distributedin the interval [0 : Δα]. The boundary of synchronization
and asynchronization for the social network of Rulkov maps
is shown in Fig. 6. From this result, the social network ofRulkov maps tends to lowly have synchronous. Also, if g islarger than 0.07, the social network of Rulkov maps has highly
synchronous like two coupled Rulkov maps.

Next, we show synchronization of the social network ofRulkov maps which produce spiking-bursting behavior. In
this study, we define two maps as synchronization if timingof bursting wave between two maps is right. Here, “fullsynchronization” is that all maps are synchronized at the inphase, and “clustering” is that maps is synchronized at thein-phase locally. The simulation results are shown Fig. 7. The
horizontal axis is iteration time n and the vertical axis is space
i. The complex propagation wave can be observed when g is
0.05 and Δαmax is 1.2 (see. Figs. 6 (a) and (b)). From Fig. 6 (a),we assume that the maps tend to converge full synchronization
by repeating update. However, in the case of Fig. 6 (b),the maps switch between full synchronization and clustering.
Also, when the maps switch from full synchronization toclustering, phase deviation occurs from the local bridges. In
the case of Figs. 6 (a) and (b), the phase deviation occurs
from local bridge 8-9 and 14-15. For Figs. 7 (c) and (d), weset g = 0.04 and Δαmax = 1.0. For Figs. 7 (e) and (f), we setg = 0.03 and Δαmax = 0.7. Similarly, for Figs. 7 (c) and (d), thephase deviation occurs from local bridge 8-9, for Figs. 7 (e)and (f), the phase deviation occurs from local bridge 14-15
and 22-23. From these result, in the social network with localbridge, the social network of Rulkov maps does not converges
full synchronization. Because the maps switch between fullsynchronization and clustering by the local bridge.
IV. Conclusion
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,
in the social bridge with local bridge, the propagation wave
switches between synchronization and asynchronization by thelocal bridge. In the future works, we investigate synchronization phenomena of another social network which has larger
scale.
AcknowledgmentThis work was partly supported by JSPS Grant-in-Aid forYoung Scientists 23700269.
References[1] S. Milgram, “The Small World Problem,” Psychology Today, vol. 2, pp.
60-67, 1967.
[2] D. Watts and S. Strogatz, “Collective Dynamics of small-world’ networks,” Nature, vol. 393, pp. 440-442, 1998.
[3] A.L. Barabsi and R. Albert, “Emergence of scaling in random networks,”
Science, vol. 286, pp. 509-512, 1999.
[4] D. Watts, “Small worlds : the dynamics of networks between order andrandomness,” Princeton University Press, 2003.
[5] S. Mark, “The Strength of Weak Ties”, Amerian Journal of Sociology,
vol 78, pp 1360-1380, 1973.
[6] N.F. Rulkov, “Modeling of Spiking-Bursting Neural Behavior using Twodimensional Map,” Physical Rev., E, vol. 65, 041922, 2002.

Fig. 6.

Boundary of synchronization and asynchronization.

[7] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattice,” Prog.
Theor. Phys, vol. 75, pp. 1033-1044, 1985.

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(a) g = 0.05, Δα = 1.2, n = 2000 − 5000.

(b) g = 0.05, Δα = 1.2, n = 297000 − 300000.

(c) g = 0.04, Δα = 1.0, n = 2000 − 5000.

(d) g = 0.04, Δα = 1.0, n = 297000 − 300000.

(e) g = 0.03, Δα = 0.7, n = 2000 − 5000.

(f) g = 0.03, Δα = 0.7, n = 297000 − 300000.

Fig. 7.

Propagation wave for the social network.

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