spontaneous formation of dynamical groups in an adaptive networked system li menghui, guan shuguang,...

Spontaneous Formation of Dynamical Groups in an Spontaneous Formation of Dynamical Groups in an Adaptive Networked SystemAdaptive Networked System

Li Menghui, Guan Shuguang, Lai Choy-HengLi Menghui, Guan Shuguang, Lai Choy-Heng

Temasek Laboratories National University of SingaporeTemasek Laboratories National University of Singapore

17/10/201017/10/2010

Temasek Laboratories

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Spontaneous Formation of Dynamical Groups in an Adaptive Networked System

OutlineOutline

Motivation of the study

The model

Analytical and simulating results

Conclusions and discussion


Motivation I: Properties of empirical system Motivation I: Properties of empirical system

Modularity is common in social and biological systems[1]. In many social systems, with the evolution of the network topology, the system may form different groups corresponding to different attributes.

In email networks[2], the links are formed between individuals with similar attributes( e.g., status, gender, age, departmental affiliation, and number of years in the community).

In modular network, the intra links are stronger than the inter links[3]. In general, the distributions of degree, vertex strength and link weights follow power law.

[1] M. Girvan, M.E.J. Newman, Proc. Natl. Acad. Sci. USA 99 7821-7826 (2002)[2] G. Kossinets, D. J. Watts, Science 311 88 (2006).[3] G. Palla, A.-L. Barabasi, T.Vicsek, Nature 446 664-667 (2007)A. E. Krause, K. A. Frank, D. M. Mason, R. E. Ulanowicz, W. W. Taylor, Nature 426 282 (2003).



Motivation II: Background of complex networksMotivation II: Background of complex networks

In the past decade, there are extensive works exploring networked complex systems, mainly focusing on to the topological structure of the networks [4] and the dynamics on the networks [5].

In various realistic systems, the network topology and dynamics are strongly dependent on each other. Thus any formed network structures and dynamical patterns are actually the results of the coevolution of network dynamics and structure[6].

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[4] Albert R, Barabasi A-L 2002 Rev. Mod. Phys. 74 47; Newman M E J 2003 SIAM Rev. 45 167;Boccaletti S, Latora V., Moreno Y, Chavez M, Hwang D-U 2006 Phys. Rep. 424 175

[5] Dorogovtsev S N, Goltsev A V, Mendes J F F 2008 Rev. Mod. Phys. 80 1275; Arenas A, D´ıaz-Guilera A, Kurths J, Moreno Y, Zhou C 2008 Phys. Rep. 469 93

[6] T. Gross and B. Blasius, J. R. Soc. Interface 5, 259-271 (2008).


Motivation III: Adaptive systemsMotivation III: Adaptive systems

The conversation time is determined by the personality of mobile agents in the mobile networks[7].

The change of the synaptic coupling strengths between neurons depends on the relative timing of the presynaptic and postsynaptic spikes[8]. This can be regarded as adaptive networks.

Recently attentions have been paid to the adaptive coevolutionary networks, including the adaptively rewiring links [9], and the adaptively altering link weights[10] .

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[7] Onnela J-P, et al, Proc. Nat. Acad. Sci. USA 104 7332-7336 (2007).[8] G.-Q. Bi, M.-M. Poo, J. Neurosci. 18 10464-10472 (1998); Y. Dan, M.-M. POO, Physiol. Rev. 86 1033-1048 (2006).[9] J. Sun, M. W. Deem, Phys. Rev. Lett. 99 228107 (2007).[10] T. Aoki, T. Aoyagi, Phys. Rev. Lett. 102 034101 (2009).



Motivation IV: Question Motivation IV: Question

How the dynamical groups are generated during the coevolution of network structure and dynamics has not been investigated from the point of view of complex networks.

The dynamical group is defined as a sub-network, where the nodes share similar dynamical states.

Motivated by this idea, in the present work, we set up a toy model consisting of phase oscillators.

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Model: Collective dynamicsModel: Collective dynamics

(1)

According to the equation (1), the oscillators are spontaneously divided into two groups. Within the same groups, the oscillators have similar dynamical states.

(2)

The link weight evolves according to the phase difference between oscillators based on the equation (2). If the phase difference is small, the link weight will be enhanced. Otherwise, the link weight will be weakened.

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Model: Order parameterModel: Order parameter

The order parameter represents whether the

global synchronization emerges or not.

The order parameter measures the fraction

of all links synchronized in networks.

The order parameters R and F can be jointly used to characterize whether the local synchronization within sub-networks takes place or not.

For example, when R ≈0 and F ≫ R, it indicates that the local synchronization within sub-networks emerges rather than the global one, i.e., the dynamical groups have formed in the system.

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Analytical Results I: Two-oscillator systemAnalytical Results I: Two-oscillator system

If link weight is fixed, the dynamics can be rewritten as follows.

We can obtain the phase difference as follows

If , these two states correspond to the in-phase synchronization and the anti-phase synchronization of the two oscillators, respectively. If , they only tend to in-phase or anti- phase synchronization.

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Analytical Results II: Many-oscillator systemAnalytical Results II: Many-oscillator system

The phase difference is as follows

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0 5 10 15

-3

-2

-1

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

i(t)

Time(a)

Simulation Analytical

mn

/(sm+s

n)(b)



Description of conditionDescription of condition

Type of networks: Random networks, N=100, <k>=20, <w>=1

We do not consider the rewinding of the network connections, and only focus on how the connection strengths co-evolve with the dynamics.

At every time steps, we normalize the connection strength, i.e., <w>= 1.

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Simulation Results I: <w>=1Simulation Results I: <w>=1

Characterization of the formation of the dynamical groups and the modular structure of the network.

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Simulation Results II: <w>=1Simulation Results II: <w>=1

Characterization of the dynamical and topological properties of the network after extremely long time evolution.

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Simulation Results III: <w>=1Simulation Results III: <w>=1

Characterization of the properties of the observable networks consisting of the active connections, whose strength is larger than a certain threshold.

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Simulation Results IVSimulation Results IV

Characterization of the dynamical and topological properties of the network after extremely long time evolution, where the total connection strength is not limited.

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ConclusionConclusion

We have investigated a coevolutionary networked model. In this model, the node dynamics are described by phase oscillators, and the connections among oscillators are coupled with the dynamical states.

With the formation of the dynamical pattern, the network also converts to the final modular network with power-law distribution of connection strength.

If the total connection strength is limited as a constant, the two dynamical groups will almost decouple when the inter connection is too weak.

If the total connection strength has not limit, the two dynamical groups will finally merge into one and all oscillators achieve in-phase synchronization.

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DiscussionDiscussion

We only investigate the particular case with two groups, i.e., h = 2. In fact, the above analysis can be conveniently generalized to a general case with h groups. The results are similar to those with two groups.

The size and topology are fixed. The growth model of adaptive networks can be investigated in further work.

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Thank you very muchThank you very much

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spontaneous formation of dynamical groups in an adaptive networked system li menghui, guan shuguang,...

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