How Crucial Is Small World Connectivity for Dynamics?

Gade, Prashant M.; Sinha, Sudeshna

doi:10.1142/s0218127406016458

Cited by 10 publications

(3 citation statements)

References 26 publications

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“…We also note Fig. 2 in our previous paper [12]. There we have plotted the power spectra of the collective field for small world lattices at different values of p. We note that high frequency (i.e.…”

Section: Discussionsupporting

confidence: 61%

Dynamic transitions in small world networks: Approach to equilibrium limit

Gade

Sinha

2005

Phys. Rev. E

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We study the transition to phase synchronization in a model for the spread of infection defined in a small world network. It was shown [Phys. Rev. Lett. 86, 2909 (2001)] that the transition occurs at a finite degree of disorder p, unlike equilibrium models where systems behave as random networks even at infinitesimal p in the infinite-size limit. We examine this system under variation of a parameter determining the driving rate and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to p=0 in the very slow driving limit, just as in the equilibrium case.

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Section: Discussionsupporting

confidence: 61%

Dynamic transitions in small world networks: Approach to equilibrium limit

Gade

Sinha

2005

Phys. Rev. E

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“…Therefore it is expected that systems on a small-world network shows mean-field type behavior. Several equilibrium systems on a small-world network display behavior in the universality class as mean-field models for infinitesimal small values of p. However in non-equilibrium case, there are several examples in which mean-field behavior is not reached even for p = 1 [12,32,33]. On border II, we observe mean-field type behavior for large rewiring probability.…”

Section: Phase Diagram and Dynamic Behaviormentioning

confidence: 82%

Stretched exponential dynamics of coupled logistic maps on a small-world network

Mahajan¹,

Gade²

2018

J. Stat. Mech.

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We investigate the dynamic phase transition from partially or fully arrested state to spatiotemporal chaos in coupled logistic maps on a small-world network. Persistence of local variables in coarse grained sense acts as an excellent order parameter to study this transition. We investigate the phase diagram by varying coupling strength and small-world rewiring probability p of nonlocal connections. The persistent region is a compact region bounded by two critical lines where bandmerging crisis occurs. On one critical line, the persistent sites shows a nonexponential (stretched exponential) decay for all p while for another one, it shows crossover from nonexponential to exponential behavior as p → 1. With an effectively antiferromagnetic coupling, coupling to two neighbors on either side leads to exchange frustration. Apart from exchange frustration, non-bipartite topology and nonlocal couplings in a small-world networks could be reason for anomalous relaxation. The distribution of trap times in asymptotic regime has a long tail as well. The dependence of temporal evolution of persistence on initial conditions is studied and a scaling form for persistence after waiting time is proposed. We present a simple possible model for this behavior.

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“…This is thought to be of relevance for 'feature binding' (Masuda and Aihara 2004), efficiency in local and global information exchange (Latora and Marchiori 2001;, and for facilitating global synchrony of coupled phase oscillators Barahona and Pecora 2001). In general, this collective behavior presupposes the non-locality of at least some connections (Gade and Sinha (2006): a requirement which, of course, small-world networks meet. However, small-word networks (graphs) comprise several classes depending on their vertex connectivity; one of them is identical with 'scale-free networks' (Amaral et al 2000;Newman 2000).…”

Section: On Brain Network Connectivitymentioning

confidence: 99%

Consciousness related neural events viewed as brain state space transitions

Werner

2008

Cogn Neurodyn

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This theoretical and speculative essay addresses a categorical distinction between neural events of sensory-motor cognition and those presumably associated with consciousness. It proposes to view this distinction in the framework of the branch of Statistical Physics currently referred to as Modern Critical Theory (Stanley, Introduction to phase transitions and critical phenomena, 1987; Marro and Dickman, Nonequilibrium phase transitions in lattice, 1999). Based on established landmarks of brain dynamics, network configurations and their role for conveying oscillatory activity of certain frequencies bands, the question is examined: what kind of state space transitions can systems with these properties undergo, and could the relation between neural processes of sensory-motor cognition and those of events in consciousness be of the same category as is characterized by state transitions in nonequilibrium physical systems? Approaches for empirical validation of this view by suitably designed brain imaging studies, and for computational simulations of the proposed principle are discussed.

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How Crucial Is Small World Connectivity for Dynamics?

Cited by 10 publications

References 26 publications

Dynamic transitions in small world networks: Approach to equilibrium limit

Dynamic transitions in small world networks: Approach to equilibrium limit

Stretched exponential dynamics of coupled logistic maps on a small-world network

Consciousness related neural events viewed as brain state space transitions

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