Topological Determinants of Perturbation Spreading in Networks

Xiaozhu Zhang (张潇竹), Dirk Witthaut, and Marc Timme

Phys. Rev. Lett. 125, 218301 – Published 16 November 2020

Abstract

Spreading phenomena essentially underlie the dynamics of various natural and technological networked systems, yet how spatiotemporal propagation patterns emerge from such networks remains largely unknown. Here we propose a novel approach that reveals universal features determining the spreading dynamics in diffusively coupled networks and disentangles them from factors that are system specific. In particular, we first analytically identify a purely topological factor encoding the interaction structure and strength, and second, numerically estimate a master function characterizing the universal scaling of the perturbation arrival times across topologically different networks. The proposed approach thereby provides intuitive insights into complex propagation patterns as well as accurate predictions for the perturbation arrival times. The approach readily generalizes to a wide range of networked systems with diffusive couplings and may contribute to assess the risks of transient influences of ubiquitous perturbations in real-world systems.

Received 20 February 2020
Revised 20 August 2020
Accepted 22 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.218301

Physics Subject Headings (PhySH)

Collective dynamics Dynamics of networks Spreading

Coupled oscillators Power grid networks

Kuramoto model

Nonlinear DynamicsNetworks

Authors & Affiliations

Xiaozhu Zhang (张潇竹)^1,*, Dirk Witthaut ², and Marc Timme ^1,†

¹Institute for Theoretical Physics, Center for Advancing Electronics Dresden (cfaed), and Cluster of Excellence Physics of Life, Technical University of Dresden, 01062 Dresden, Germany
²Institute for Energy and Climate Research—Systems Analysis and Technology Evaluation (IEK-STE), Forschungszentrum Jülich, 52428 Jülich, Germany and Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany

^*xiaozhu.zhang@tu-dresden.de
^†marc.timme@tu-dresden.de

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Issue

Vol. 125, Iss. 21 — 20 November 2020

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Images

Figure 1
Characterizing perturbation spreading in diffusively coupled networks. (a) Topology of a random network created based on Ref. [22]. Each node is color coded with its topological distance to the perturbed node 1. (b) Starting from $t = 0$ , node 1 is perturbed with a sinusoidal signal with frequency 1 Hz. (c) Response time series of three representative nodes 1,2,3 at topological distances $d \in {0, 1, 2}$ . The times $t_{1}^{(1)}, t_{2}^{(1)}, t_{3}^{(1)}$ at which the responses at these nodes cross a threshold $ε_{th} = 1.5 \times 10^{- 3} rad$ are illustrated. The network parameters are set as $K_{i j} = 30 s^{- 2}$ , $α = 1 s^{- 1}$ , $ω_{+} = 4 s^{- 2}$ [squares in (a)], $ω_{-} = - 1 s^{- 2}$ [disks in (a)]. Each node is assigned with a natural frequency of $ω_{+}$ or $ω_{-}$ f[ at random, satisfying $\sum_{i} ω_{i} = 0$ , so that a fixed point exists for the given network topology.
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Figure 2
Universal spreading dynamics across diverse network topologies. (a) Five diverse network topologies: chain (a1), square lattice (a2), cubic lattice (a3), random power grid (a4) [22], and Delaunay triangulation networks (a5). The perturbed node is marked with a black square. (b) The threshold-crossing arrival times ( $ε_{th} = 10^{- 9} rad$ ) from direct simulations are scattered; however (c) they largely collapse across topologies if normalized by the respective topological factors, ${\tilde{t}}_{i}^{(k)} ≔ t_{i}^{(k)} / T_{i}^{(k)}$ . Panels (b)–(f) share the color coding with the network sketches (a1)–(a5). The arrival time differences among network topologies, e.g., the ratio between the arrival times with topological distance $d$ and the ones with the same $d$ in the chain network, are highlighted in the insets of (b) and (c). The dotted line in (c) illustrates the topology-independent factor Eq. (6), explicating the differences from the normalized actual arrival times (solid gray line as a visual aid). (d),(e) In networks of (one-dimensional) Kuramoto oscillators (d), in networks of (three-dimensional) Rössler oscillators (e), and in networks of discrete nonlinear Schrödinger equations (DNLSEs) (f), normalized arrival times all collapse. For (a)–(c), the parameters for the networks ( $ω_{i}$ , $K_{i j}$ , $α$ ) and for the perturbation signal ( $ϵ$ , $Ω$ , $φ$ ) are the same as in Fig. 1, except for (a1)–(a3) $ω_{+} = 1 s^{- 2}$ , $ω_{-} = - 1 s^{- 2}$ are distributed alternatively. For (e) Rössler oscillators ( $a = 0.1$ , $b = 0.1$ , $c = 4$ ) are diffusively coupled in variable $z$ and for (f) diffusive couplings naturally arise in the equation of motion for the complex field $ψ$ (see Supplemental Material Sec. S3 for simulation details [25]).
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Figure 3
Master function of spreading dynamics and prediction performance. (a) A master function (7) estimated from $10^{6}$ arrival time measurements in 100 random network topologies [22] with 100 nodes. The error bars indicate a prediction interval of $3 σ (d)$ . (b) The arrival time predictions for the 5 example networks in Fig. 2 highly correlate with the measurements (Pearson’s correlation coefficient $r > 0.99$ ). (c) In the prediction of $10^{6}$ arrival times in another 100 random networks [22], the hit rate, i.e., the probability that the actual arrival time from simulation falls in the prediction interval, grows rapidly with a broadening prediction interval. (d),(e) The relative prediction error $E ≔ | t_{i, pred}^{(k)} - t_{i}^{(k)} | / t_{i}^{(k)}$ drops for smaller arrival times, e.g., (d) for smaller distances and (e) for lower thresholds. For a lowering threshold, the error decays approximately as a power law [inset of (e)]. (f) The prediction performance is robust against an increasing heterogeneity $a$ in network coupling strength. In panels (d)–(f) the shaded area indicates standard deviations. For panels (a)–(d) and (f) the threshold $ε_{th} = 10^{- 9} rad$ . For panels (a)–(e) the parameters ( $α$ , $ω_{i}$ , $ϵ$ , $Ω$ , $φ$ , $K_{i j}$ ) are set the same as in Fig. 1.
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