Functional Control of Network Dynamics Using Designed Laplacian Spectra

Open Access

Functional Control of Network Dynamics Using Designed Laplacian Spectra

Aden Forrow, Francis G. Woodhouse, and Jörn Dunkel

Phys. Rev. X 8, 041043 – Published 7 December 2018

Abstract

Complex real-world phenomena across a wide range of scales, from aviation and Internet traffic to signal propagation in electronic and gene regulatory circuits, can be efficiently described through dynamic network models. In many such systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow. Spectral graph theory has traditionally prioritized analyzing unweighted networks with specified adjacency properties. Here, we introduce a complementary framework, providing a mathematically rigorous weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of various archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces generalized chimera states in Kuramoto-type oscillator networks, tunes or suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. Our approach can be generalized to design continuous band gaps through periodic extensions of finite networks.

Received 5 January 2018
Revised 17 September 2018

DOI:https://doi.org/10.1103/PhysRevX.8.041043

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Chimera states Collective behavior in networks Network structure

Deterministic networks Weighted networks

Phase oscillator models

NetworksNonlinear Dynamics

Authors & Affiliations

Aden Forrow^1,*, Francis G. Woodhouse^2,†, and Jörn Dunkel^1,‡

¹Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
²Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*aforrow@mit.edu
^†Present address: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom.
^‡dunkel@mit.edu

Popular Summary

Many seemingly disparate real-world phenomena, from Internet traffic to the interaction of genes within our body, can be described as dynamic networks: interlinked hubs exchanging information along links according to known rules. Often the key behavior is captured entirely by the interconnection structure of the network. Like a wine glass resonating at a particular spectrum of notes, these network structures also have their own spectrum of resonances dictating how the system behaves. But if we know what behavior we want—and what spectrum we need—how do we design the right network? Our work shows how to create a network with any desired spectrum and demonstrates the power of this control in areas like pattern formation and quantum mechanics.

Specifically, we focus on controlling network dynamics by tailoring the Laplacian, a critical matrix associated with any network that determines the spread of forces and flows. To this end, we derive a formula that shows how to build a network with any desired Laplacian spectrum, using only positive feedback between nodes. Such control enables deeper theoretical studies of the effect of the Laplacian spectrum in specific physical models. Building on this framework, we show how suitably designed networks can be used to achieve desired behaviors in a variety of physical systems, from mixed states of simultaneous synchrony and asynchrony in biophysical oscillators to quantum localization and continuous band gaps.

Our examples illustrate ways to create new exotic behaviors in different experimental systems, including computer-coupled chemical oscillators, cold atomic systems with precisely specified optical potentials, or etched superconducting waveguide resonators.

Key Image

Article Text

Click to Expand

Supplemental Material

Click to Expand

References

Click to Expand

Issue

Vol. 8, Iss. 4 — October - December 2018

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Designing networks from spectra. (a) Schematic of DBG network construction. Given a spectrum of eigenvalues distributed in two (or more) groups, we build a graph with non-negative edge weights that realizes this spectrum exactly (1). Sparsification of this complete DBG network with the Spielman-Srivastava [33] algorithm (2) yields a new network with wider eigenvalue distributions and a smaller gap (3). (b) Example graphs used in applications below: Starting from a DBG graph on 200 vertices with 100 eigenvalues set to independent and identically distributed (IID) $N (5, 0.25)$ and 99 set to IID $N (20, 0.25)$ (left), sparsification with $ε = 0.5$ creates a new graph (top) with the number of edges reduced from 19900 to 3758. As a control, we also compare to a gapless random graph (bottom) with 362 edges and the same weighted vertex degrees as the original DBG graph (Appendix pp3). (c) The eigenvalues for the graphs in (b). The mode on the complete DBG network with the $k$ th largest nonzero eigenvalue is supported on the first $k + 1$ vertices, counted counterclockwise from the top red vertex, and highly localized on vertex $k + 1$ , which is colored to match in (b). Gray lines indicate the borders of the unstable region for the Swift-Hohenberg model with the parameters used in Fig. 3. (d) Sparsified networks retain a significant gap even for relatively large $ε$ . Each point shows the mean number of edges and gap size at fixed $ε$ between 1 (left) and 0.01 (right), starting from a graph on 200 vertices designed to have $100 \times$ eigenvalue 5 and $99 \times$ eigenvalue 20. The solid curve shows the worst-case gap estimate, reduction by a factor of $1 - \frac{5}{3} ε$ . Sample size is 1000 for $ε \geq 0.1$ and 300 for $ε < 0.1$ . Error bars are $\pm 1$ standard deviation; horizontal error bars are smaller than the marker size.
Reuse & Permissions
Figure 2
DBG networks lead to staggered synchronization and chimeras. (a)–(f) In the Kuramoto model with $α = 0$ , the complete (first row) and sparsified (second row) graphs synchronize much faster than the random graph (third row). For the complete graph the gap affects the rate of synchronization, with highly connected vertices synchronizing faster (a), while on the sparsified graph the gap is only visible in the mode basis (e). (g)–(i) Weak chimera states appear when $α = 1$ . Both the complete (g) and sparsified (h) graphs have two dominant groups of phase-locked oscillators, with the complete graph more fully synchronized. Dynamics on the random graph (i) are much less coherent. Solid black lines indicate the predicted approximate frequency difference for a network with two distinct eigenvalues, 5 and 20. (j)–(l) Order parameter $r = | \sum_{j} e^{i θ_{j}} |$ for the simulations in (g)–(i) for the strongly connected vertices (red), weakly connected vertices (teal), and all vertices (gray). See Video 1 in Supplemental Material [56] for animation.
Reuse & Permissions
Figure 3
Generic suppression of pattern formation with a designed discrete band gap. (a) Pattern formation in the Swift-Hohenberg system is completely suppressed by constructing a gap around the range of unstable eigenvalues [Fig. 1]. (b) On a sparsified graph that has a few eigenvalues just within the unstable region, some modes settle at small nonzero values. (c) On the random graph many more eigenvalues are well within the unstable region and the corresponding modes settle at larger amplitudes. The inset graphs show the final steady state on each graph. All simulations used identical initial conditions $u_{i} \sim N (0, 1)$ and parameters $α = 90$ , $D_{1} = - 20$ , $D_{2} = 1$ . See Video 2 in Supplemental Material [56] for animation.
Reuse & Permissions
Figure 4
Controlling pattern formation with a designed discrete band gap (Video 3 in Supplemental Material [56]). (a) Instead of placing a gap in the spectrum around the unstable pattern-forming range, as in Fig. 3, we deliberately place particular eigenvalues in the middle of that range corresponding to eigenvectors localized on a desired pattern. (b) From random initial conditions, the system settles into a state where only the chosen modes have non-negligible amplitudes. (c)–(e) Time series of pattern evolution on a designed network, with vertices colored according to the stability of the mode localized there as in (a). The size of the vertices indicates $| u |$ . (c) The encoded pattern is not obvious from either the designed network or the random initial conditions. (d) By time $t = 0.07$ the stable modes have nearly all vanished. (e) The steady state reveals the eigenmode-designed pattern. Because the modes are highly localized, selecting a set of modes to activate is approximately equivalent to selecting a set of vertices to activate. Thus, we can encode an arbitrary pattern as the steady state. Depending on initial conditions, the system may settle into other stable states with slight variations in the vertex activations; the picture typically shows up at least as clearly as it does here and often is more uniform. The parameters $α = 90$ , $D_{1} = - 20$ , and $D_{2} = 1$ were identical to those in Fig. 3; the tuning parameters to control pattern formation are only the network edge weights. See Video 3 in Supplemental Material [56] for animation and other initial conditions.
Reuse & Permissions
Figure 5
Localization on a DBG quantum network (Video 4 in Supplemental Material [56]). (a)–(c), When the wave function in the Gross-Pitaevskii model of Eq. (4) is initialized at a weakly connected vertex with low kinetic energy, localization or delocalization (indicated by high or low potential energy, respectively) is controlled by the interplay between the graph spectrum and the rate of potential energy loss $g$ . The random graph (purple) always delocalizes, due to its dense spectrum. However, while the sparsified graph (yellow) can delocalize for low $g$ (a) and high $g$ (c), again due to available eigenmodes, intermediate $g$ (b) places the range of allowed modes inside the spectral gap, preventing delocalization. The complete graph (blue) always inhibits spreading due to the extreme localization of its eigenvectors.
Reuse & Permissions
Figure 6
Designed spectra on a discrete network are preserved when extended periodically in one dimension. (a) We extend a finite network to an infinite one by rewiring a subset of the edges to cross between adjacent copies of the original network. Here, we take the network with the spectrum in (b) and rewire the edge between vertices $j$ and $k$ if $| k - j | > n / 2$ . This rewires roughly one quarter of the edges. (b) One unit cell in (a) would have a discrete spectrum with $λ_{j} = 21 - j$ . (c) Most of the eigenvalue bands do not change significantly with $q$ , so the density of states consists of 21 sharp peaks with low- or zero-density regions between. (d) The same construction as in (a) can be repeated for any spectrum; this is the result for a gapped network. (e) One unit cell in (d) would have a gapped spectrum, with 10 eigenvalues equal to 20 and 10 equal to 5, in addition to the always-present zero eigenvalue. (f) Again, most of the eigenvalue bands are roughly constant, even though the eigenvectors do depend strongly on $q$ . The gap in the middle of the spectrum is nearly perfectly preserved; a small gap remains between the bottom two bands. Note the log scale on both density of states plots.
Reuse & Permissions

Physical Review X