平成24年度 電気関係学会四国支部連合大会

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Synchronization for Social Network withLocal Bridge using Coupled Rulkov Maps
Tomoya Shima† , Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio†(† Tokushima University), (‡ Zurich University of Applied Sciences)

1. Introduction
In this study, we focus on “local bridge” in social network. Example of network including the local bridge isshown in Fig. 1.

Figure 1: Example of network including the local bridge.
In Fig. 1, the shortest route from 1 to 25 is 1-25. Oneof second shortest route from 1 to 25 is 1-4-5-8-9-11-1314-15-16-20-22-23-25. The 1-25 route is predominantly
shorter than other routes. In this case, the 1-25 route iscalled the local bridge. In the social network, the localbridge aﬀects the entire network in terms of propagation of information. In order to analyze complex phenomena, we use coupled maps. Generally, the coupledmaps is used as general models for the complex dynamics, e.g. biological systems. Coupled oscillatory systems
can also produce interesting phase patterns, “clustering”. Moreover, a discrete map for spiking-burstingneural behavior was proposed by Rulkov [1]. Rulkov
maps can be useful for understanding the dynamicalmechanism of oscillators in “large scale network”. In
this study, we use Rulkov maps applying for the coupled maps because the social network has features ofclustering and large scale network.

2. Social network of Rulkov maps

Here, “full synchronization” is that all maps are synchronized at the in-phase, and “clustering” is that maps
is synchronized at the in-phase locally. The simulation
results are shown in Fig. 2. The horizontal axis is iteration time n and the vertical axis is space i. Thecomplex propagation wave can be observed when g is0.03 and ∆αmax is 0.7 (see. Fig. 2). From Fig. 2 (a),we assume that the maps tend to converge full synchronization by repeating update. However, in the case ofFig. 2 (b), the maps switch between full synchronization and clustering. Also, when the maps switch fromfull synchronization to clustering, phase deviation occurs from the local bridges. In the case of Figs. 2 (a)and (b), the phase deviation occurs from local bridge
14-15 and 22-23. From these result, in the social network with local bridge, the social network of Rulkov
maps does not converges full synchronization. Because
the maps switch between full synchronization and clustering by the local bridge.

(a) g = 0.03, ∆α = 0.7, n = 2000 − 5000.

We consider the social network (see. Fig. 1) of coupled maps. The number of coupled map is set to 25. We
describe equations based on a chain of coupled maps.
g ∑xi,n+1 = f (xi,n , xi,n−1 , yi,n ) +(xj,n − xi,n ),Nj∈Cjyi,n+1 = yi,n − µ(xi,n + 1) + µσi + µg ∑(xj,n − xi,n ),Nj∈Cji = 1, ..., N,(1)where x and y are the fast and slow dynamical variables, g is the coupling strength, N is the number ofcells in the social network, µ = 0.001, σ = 0.15, Cj 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 andf (xn , yn ) =(2)xn−1 ≤ 0−1, xn ≥ α + yn or xn−1 > 0.
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 timing of bursting wave between two maps is right.

Powered by TCPDF (www.tcpdf.org)(b) g = 0.03, ∆α = 0.7, n = 297000 − 300000.

Figure 2: Propagation wave for the social network.

3. Conclusions
In this study, we have studied inﬂuence of the local
bridge in the social network via synchronization phenomena of Rulkov maps applying for the coupled maps.
From simulation results, in the social network with local bridge, the maps switch between synchronizationand clustering by the local bridge.
References[1] S. Mark, “The Strength of Weak Ties,” Amerian Journal ofSociology, vol 78, pp 1360-1380, 1973.