Motion-induced synchronization in metapopulations of mobile agents

Jesús Gómez-Gardeñes, Vincenzo Nicosia, Roberta Sinatra, and Vito Latora

Phys. Rev. E 87, 032814 – Published 28 March 2013

Abstract

We study the influence of motion on the emergence of synchronization in a metapopulation of random walkers moving on a heterogeneous network and subject to Kuramoto interactions at the network nodes. We discover a mechanism of transition to macroscopic dynamical order induced by the walkers’ motion. Furthermore, we observe two different microscopic paths to synchronization: depending on the rule of the motion, either low-degree nodes or the hubs drive the whole system towards synchronization. We provide analytical arguments to understand these results.

Received 9 June 2012

DOI:https://doi.org/10.1103/PhysRevE.87.032814

Authors & Affiliations

Jesús Gómez-Gardeñes^1,2,*, Vincenzo Nicosia^3,4,*, Roberta Sinatra⁵, and Vito Latora^3,6

¹Departamento de Física de la Materia Condensada, University of Zaragoza, Zaragoza 50009, Spain
²Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50018, Spain
³School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
⁴Computer Laboratory, University of Cambridge, Cambridge CB3 0FD, United Kingdom
⁵Center for Complex Network Research and Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
⁶Dipartimento di Fisica e Astronomia, Università di Catania and INFN, 95123 Catania, Italy

^*J.G.G. and V.N. contributed equally to this work.

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Vol. 87, Iss. 3 — March 2013

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Images

Figure 1
Phase diagram $r (α, λ)$ of the metapopulation model on uncorrelated SF networks with (a) $γ = 2.7$ , (b) $γ = 3.0$ , and (c) on the U.S. air-transportation network. The three networks have $N = 500$ nodes. The dashed curves in panels (a) and (b) are the corresponding lower-bound analytical predictions for the onset of synchronization in uncorrelated graphs obtained from Eq. (9) for $α > - 1$ , and from Eq. (10) for $α < - 1$ . Panel (d) shows $r (α)$ for $λ = 0.08$ , corresponding to the horizontal line in panel (b).Reuse & Permissions
Figure 2
(a), (b) The average local order parameter $r_{k}$ of nodes of degree $k$ as a function of $k$ , for various values of $λ$ , and for two fixed values of the bias, respectively, (a) $α = - 2.0$ and (b) $α = - 0.25$ , corresponding to the two vertical lines in Fig. 1b. (c), (d) $r_{k}$ for $λ = 0.08$ and various values of $α$ [respectively, $α < - 1$ in panel (c) and $α > - 1$ in panel (d)] corresponding to the horizontal line in Fig. 1b.Reuse & Permissions
Figure 3
Theoretical predictions for the critical coupling strength $λ_{c} (α)$ as a function of $k_{max}$ for scale-free random graphs with exponent $γ = 2.0$ (top panel) and $γ = 3.0$ (bottom panel). When $k_{max}$ increases, global synchronization is attained for larger values of $λ$ . Also an increase of $γ$ , which corresponds to more homogeneous degree distributions, produces an increase of the critical coupling strength.Reuse & Permissions
Figure 4
Phase diagram reporting the local and global order parameters $r_{loc}$ (upper panel) and $r$ (lower panel) as a function of $α$ and $λ$ for the metapopulation model in the limit $Δ \to \infty$ ( $W = 5000$ agents on a scale–free network with $N = 500$ nodes and $P (k) \sim k^{- 3}$ ). When there is no motion, the system is globally incoherent, even if the local synchronization at the nodes can be enhanced at will by increasing the value of the interaction strength $λ$ .Reuse & Permissions
Figure 5
Cross-section plots of the two phase diagrams in Fig. 4. We report $r_{loc}$ and $r$ as a function of the coupling strength $λ$ for three different values of $α$ . The absence of motion hinders global synchronization ( $r ≃ 0$ , orange dashed lines), even if, for an appropriately large value of $λ$ , all the nodes of the network can reach complete local synchronization ( $r_{loc} = 1$ , solid lines).Reuse & Permissions
Figure 6
Global order parameter $r$ as a function of $Δ$ for $λ = 0.1$ and two values of the motion bias, respectively, $α = - 2.0$ (red open circles) and $α = 0.5$ (blue filled circles). Global synchronization is attained when agents move frequently ( $Δ \to 0$ ), while the system remains in an incoherent state if the motion is too slow ( $Δ \to \infty$ ).Reuse & Permissions

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