Functional control of network dynamics using designed Laplacian spectra

Spectral band gaps control the behavior of physical systems in areas as diverse as topological insulators [1, 2], phononic crystals [3], superconductors [4], acoustic metamaterials [5], and active matter [6]. In addition to ubiquitous physical network models [7, 8, 9, 10] ranging from aviation [11] to electronics [12], there is also considerable interest in virtual or computational networks [13] with fewer physical constraints, such as those recently used to create spiral wave chimeras in coupled chemical oscillators [14]. Often, dynamics in such systems depend on the graph Laplacian [15, 16] and in particular on its spectrum of eigenvalues. Traditionally studied in periodic lattice graph models [3, 5, 6, 17] and more recently also in hyperuniform systems [18], the targeted design of spectra of any desired shape remains a major challenge in modern materials science [5, 19]. Recent breakthroughs in 3D printing [20, 21, 22, 23] and lithography [24] make it possible now to produce and explore network structures that go beyond the traditionally considered periodic lattice geometries.

Building on such experimental and theoretical progress, we present here a mathematically rigorous solution to the longstanding question of how any desired spectrum can be realized exactly on a suitably designed positively-weighted network. Specifically, our construction of networks with specified eigenvalues allows us to place arbitrary gaps in the spectrum of the network Laplacian $L=D-A$, where $D$ and $A$ are the weighted degree and adjacency matrices respectively. These gaps, finite analogs to band gaps in continuous systems, enable precise control over the dynamics in a wide range of graph-based physical systems. While in a strict sense band gaps can only exist in an extended system with continuous energy bands, to follow the analogy we will name an eigenvalue-free region in our finite networks that is comparable to the range of eigenvalues a discrete band gap (DBG). That said, our construction can also be used to create continuous band gaps (Section IV). Designing a suitably weighted network topology in this way presents an alternative to control procedures based on adjusting model parameters or initial conditions on a given network [25]. The spectral approach towards functional control of network dynamics proposed here can, for example, be directly implemented with recently developed computer-coupled oscillator setups [14].

After summarizing the main mathematical result, we demonstrate its broad applicability explicitly for classical and quantum systems, by showing how suitably placed DBGs can induce chimera states [26] in oscillator networks, inhibit structural growth in generic higher-order pattern formation models, and facilitate state localization in quantum networks. In parallel, we illustrate how our construction can be combined with sparsification algorithms [27, 28] to yield simplified networks preserving DBGs. This approach complements the more traditional procedure of constructing graph ensembles with predefined statistical adjacency characteristics [29, 30, 31, 32]. Finally, we discuss periodic extensions of finite networks as a systematic procedure for designing continuous band gaps.

The problem of recovering a network from its eigenvalues has been studied extensively, both from an algorithmic [33, 34, 35] and mathematical [36, 37] perspective. However, with a few limited exceptions [37], most prior work has focused only on unweighted networks [38], where there are a finite number of graphs on $n$ vertices and thus only a finite number of possible spectra. We here construct an exact solution for the weighted case.

Our main result is that, given a set $\left\{{\lambda }_i\right\}$ of desired eigenvalues ordered so ${\lambda }_1⩾\dots ⩾{\lambda }_{n-1}⩾{\lambda }_n=0$, there is a weighted graph $G$ on $n$ vertices with non-negative edge weights whose Laplacian $L$ has spectrum ${\lambda }_1,\dots ,{\lambda }_{n-1},0$. The Laplacian, which determines the graph, can be reconstructed from its eigenvalues and eigenvectors with the eigenvalue decomposition; we therefore need to find a set of eigenvectors that together with $\left\{{\lambda }_i\right\}$ give a graph Laplacian. In fact, the same set of eigenvectors $v^{\left(k\right)},k=1,\dots ,n-1,$ given by

suffices for any spectrum. These eigenvectors are strongly localized: the inverse participation ratio (4-norm) ${\Vert v^{\left(k\right)}\Vert }_4^4=1-2k^{-1}+O\left(k^{-2}\right)$ indicates near-perfect localization ${\Vert v^{\left(k\right)}\Vert }_4^4\to 1$ for almost all $k$, itself a desirable phenomenon [39, 16]. As the $v^{\left(k\right)}$ are mutually orthonormal and orthogonal to the vector of all ones, the matrix $L={\sum }_{k=1}^{n-1}{\lambda }_k{v}^{\left(k\right)}{v}^{\left(k\right)\top }$ has the desired spectrum with $\frac{1}{\sqrt{n}}1$ as the final eigenvector for $k=n$ with eigenvalue zero. By explicitly computing the sum over $k$ for $i<j$, we find

that is, that the off-diagonal elements of $L$ are all nonpositive (Appendix B); $L$ therefore corresponds to a graph with nonnegative weight $-L_{ij}$ between vertices $i$ and $j$. If all of the eigenvalues are nonzero, all of the off-diagonal elements of $L$ will be nonzero and the resulting graph will be complete.

Some spectra can only be realized on complete graphs. A graph $G$ with approximately constant spectrum must be complete: if $L$ has nonzero eigenvalues ${\lambda }_k=\lambda +{ϵ}_k$ for $k<n$ and ${\lambda }_n=0$, then

The off-diagonal elements of $\lambda I-L$, which equal the original edge weights of $G$, are therefore $\frac{\lambda }{n}+O\left(ϵ\right)$. For small ${ϵ}_k$, every edge has nonzero weight. More generally, our construction also shows that the spectrum of any non-complete weighted graph with no isolated vertices cannot uniquely specify that graph, in line with older results on, for example, the spectra of trees [36].

This construction allows us to create networks with precisely specified gaps. For instance, choosing ${\lambda }_1={\lambda }_2=\dots ={\lambda }_{n/2-1}$ and ${\lambda }_{n/2}={\lambda }_{\left(n/2\right)+1}=\dots ={\lambda }_{n-1}$ leads to a graph with edge weights $-L_{ij}={\lambda }_{n-1}/n$ if $i>n/2$ or $j>n/2$ and $-L_{ij}=(2{\lambda }_1-{\lambda }_{n-1})/n$ otherwise (Appendix A); that is, there are two groups of vertices, one strongly connected within itself and one weakly connected to everything. Adding a small amount of noise to each eigenvalue then lifts the eigenvalue degeneracy while preserving the connectivity structure and retaining a gap (Fig. 1b,c).

Since complete graphs can be difficult to realize physically, we explore the effect of the sparsification-by-resistances algorithm developed by Spielman and Srivastava [27]. Given an accuracy parameter $ϵ$, this sparsification creates a network with $O\left(n\log \right(n\left)/{ϵ}^2\right)$ edges whose eigenvalues match the eigenvalues of the original network to within a multiplicative factor $1\pm ϵ$ with high probability. Sparsification by resistances aims to preserve the entire spectrum, not just a gap; future sparsification algorithms directly constructed to preserve a gap could therefore improve on its efficiency. In other applications, the networks of interest are virtual ones [14] and sparsification may not be necessary.

We now demonstrate the practical potential of DBGs with three generic network models. In each case, we compare the dynamics on a complete DBG network (Fig. 1b, left) both to a sparsified approximate DBG network (Fig. 1b, top) and to a random connected network (Fig. 1b, bottom) constructed to have the same weighted vertex degrees as the DBG network (Appendix C). The gap is approximately preserved in the sparsified network and vanishes entirely in the random graph (Fig. 1c,d). Matching the degrees in the random graph to the DBG network ensures that any differences in dynamics are due to the gap and not differences in coarse features like the average connectivity. Often, the behavior of optimized networks is sensitive to small perturbations [40]; here, behaviors preserved in the sparsified graph are robust to significant changes.

Simulations were performed using a third or fourth order Adams-Bashforth linear multistep method with a time step $\Delta t={10}^{-4}$. All simulations were written in C++ using Armadillo [41].

We can construct infinite periodic networks in a standard way from any base network $G$ by tiling periodically and rewiring edges (Fig. 5a,d). Starting from the original vertex set $\left\{j\right\}$ for $1⩽j⩽n$ and edge weights $-L_{jk}$, we make an infinite string $G^{\infty }$ of copies of $G$ with vertices indexed by $j$, the label in $G$, and $c\in Z$, the unit cell. This will give a new, infinite Laplacian $L^{\infty }$. For the edges that will not be rewired, we set $L_{jc,kc}^{\infty }=L_{jk}$ for all $c$. Doing this for all edges would leave the copies of $G$ disconnected. To connect them, we choose a subset of edges $\left\{\right(j,k\left)\right\}$ and rewire them to cross between unit cells; for example, if $(j,k)$ is an edge to be rewired to have $k$ in a unit cell to the left of $j$ we can set $L_{jc,k(c-1)}^{\infty }=L_{jk}$ for all $c$ and symmetrically set $L_{k(c-1),jc}^{\infty }=L_{kj}$. The remainder of the entries of $L^{\infty }$ are set to zero.

Since $L^{\infty }$ is periodic, Bloch’s Theorem allows us to write the eigenvectors as

where $q$ is a wavenumber in the first Brilloun zone $-\pi <q<\pi$. The $\tilde{U}$ then satisfy

which reduces to a new eigenvalue equation for a matrix of size $n$:

where the matrix elements of $\tilde{L}\left(q\right)$ are the same as those of $L$ for edges within a single unit cell and differ by a factor $e^{iq(c-d)}$ for edges that cross between unit cells $c$ and $d$.

Using these transformations, which are standard in the study of lattice systems [17], we can find the continuous spectra of periodic tilings of our designed networks. Even without any optimization of which edges to rewire, the spectral characteristics persist in the infinite system. If we rewire all edges with $|j-k|>n/2$, for example, a spectrum of equally-spaced eigenvalues leads to a density of states with corresponding equally-spaced large spikes (Fig. 5a-c), while a discrete band gap is almost entirely preserved (Fig. 5d-f). In both cases, only the bottom few bands vary significantly with $q$. Note that, because we moved edges incident to the first vertex, all of the eigenvectors do change and are not localized for nonzero $q$.

Controlling dynamics on a network typically requires detailed understanding of its spectral properties. Here we have reversed the conventional approach by starting from a desired spectrum and providing a mathematically rigorous construction of a matching network. This enabled us to induce chimera states, suppress pattern formation, and control wavefunction localization [53] using suitably designed gapped spectra. Our method, which starts from global properties, complements traditional approaches using small-scale local rules to build and analyze networks [29, 54, 55, 56]. In the future, the above results may also prove useful as a standard of comparison for other networks. Contrasting the dynamics on an important class of networks with the dynamics on networks designed to have identical spectra can help identify the important features of that class. Moreover, as dynamics are often related to matrices other than the Laplacian [57], it will be interesting to investigate control of their spectra for weighted networks as well. Although our construction works optimally with fully-connected graphs, one can expect that improved sparsification algorithms together with recent progress in 3D printing and lithography [24, 18] may soon lead to physically-realizable networks with arbitrary gaps; since any graph can be embedded in 3D [58], the framework introduced here lays a conceptual foundation for the targeted design of complex non-periodic metamaterials with desired spectral properties. Currently, the approach can be applied directly to interfacing biochemical oscillators with computational networks [14]. Such hybrid networks can also be naturally extended to periodic systems, where the spectral properties are preserved well without any further optimization.

Acknowledgements. The authors would like to thank Jon Kelner and Philippe Rigollet for helpful discussions. This work was supported by Trinity College, Cambridge (F.G.W.), an Edmund F. Kelly Research Award (J.D.), and a Complex Systems Scholar Award of the James S. McDonnell Foundation (J.D.).