Spontaneous Formation of Dynamical Groups in an Adaptive Networked System

1. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Li Menghui, Guan Shuguang, Lai Choy-Heng Temasek Laboratories National University of Singapore 17/10/2010

2. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Outline • Motivation of the study • The model • Analytical and simulating results • Conclusions and discussion

3. Spontaneous Formation of Dynamical Groups in an Adaptive Networked 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).

4. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Motivation 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]. [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).

5. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Motivation 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] . [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).

6. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System 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.

7. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Model: 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.

8. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Model: 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.

9. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Analytical 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.

10. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Analytical Results II: Many-oscillator system • The phase difference is as follows

11. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Description 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.

12. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Simulation Results I: <w>=1 • Characterization of the formation of the dynamical groups and the modular structure of the network.

13. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Simulation Results II: <w>=1 • Characterization of the dynamical and topological properties of the network after extremely long time evolution.

14. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Simulation 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.

15. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Simulation 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.

16. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Conclusion • 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.

17. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Discussion • 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.

18. Spontaneous Formation of Dynamical Groups in an Adaptive Networked System Thank you very much