The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product network is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.
Patterns are widespread in nature and appear in large plethora of different conformations. From chemistry to biology, passing through physics, beautiful spatially extended motifs are found which spontaneously emerge from an ensemble made of interacting microscopic actors. The spirals in chemical reactions, the colorful patterns on fish skin, and the maculated fur coat of felines are all examples which testify on the intrinsic ability of natural system to self-organize, both in space and in time1,2.
The proto-typical approach to patterns formation in reaction-diffusion processes dates back to Alan Turing’s seminal paper on morphogenesis3. Working within a simplified deterministic model for two species in mutual interactions, Turing proved that a homogeneous fixed point can turn unstable to external perturbations. A symmetry breaking instability can in fact develop which is seeded by diffusion and necessitates of an activator-inhibitor scheme of interaction between factors. When the conditions for the Turing instability are satisfied, the perturbation grows exponentially at short times and the system evolves towards an asymptotic stationary stable attractor, characterized by a spatially inhomogeneous density distribution. Mathematical conditions for the onset of the instability can be obtained via a linear stability analysis, which requires expanding the imposed perturbation on the complete basis formed by the eigenvectors of the Laplacian operators on the chosen domain. Turing instabilities are usually studied on regular lattices or continuous supports. The theory of patterns formation extends however to reaction-diffusion systems defined on a complex graph, as illustrated in the pioneering paper by Othmer and Scriven4, and recently revisited by Nakao and Mikhailov5. In this case the domain of the dispersion relation, from which the instability conditions ultimately descend, is the spectrum of the discrete Laplacian associated to the embedding network. Laplacian eigenvalues determine in fact the spatial characteristics of the emerging patterns, when the system is defined on a heterogeneous complex support. Turing patterns for systems defined on a complex graph materialize in a segregation into activator-rich and activator-poor nodes7. As discussed in6, self-organized patterns can also manifest on multiplex, networks of networks assembled as adjacent layers8,9,10,11,12,13,14,15. Remarkably, patterns on a multiplex can be instigated by a constructive interference between layers, also when the Turing-like instability is prevented to occur on each single layer taken separately. In other cases, inter-layer diffusion can instead act a destructive pressure on the process of pattern formation6.
Building on these premises, we here aim at applying the theory of Turing instability for reaction diffusion systems defined on Cartesian networks. These latter are assembled as the Cartesian product of simpler networks, the fundamental building blocks in the process of hierarchical aggregation. Regular grids, cubes, and their counterparts in higher dimensions are for instance obtained from the Cartesian product of linear chains. Besides the interest from a graph theory point of view16,19, Cartesian product (also referred to as Cartesian networks in the following) have been recently used in the framework of control processes17 and systems synchronization18.
In this paper we shall adapt the linear instability analysis to the relevant setting of the Cartesian networks, and elaborate on the condition for the instability, by expanding the perturbation on a generalized basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, defined on each individual network. For a sake of clarity we will illustrate the theory with reference to the simplified setting where the Cartesian product involves two distinct networks. Clearly, one can straightforwardly extend the theory to Cartesian products made by more than two networks. It is worth emphasizing that a Cartesian network can be equivalently treated as a standard network, specified by a global adjacency matrix and notwithstanding its parcelization in elementary sub-components. Hence, the conditions for the emergence of self-organized patterns in a reaction diffusion system defined on a Cartesian support could be effectively addressed by following the general strategy outlined in5. However, by taking advantage of the peculiar structure of Cartesian networks, one can gain insight into the onset of the instability by tracing it back to the properties of the underlying simplex networks. This level of understanding cannot be achieved when carrying out a direct diagonalization of the global Laplacian matrix associated to the Cartesian product network. To shed light on these aspects, the process of patterns formation on the Cartesian support will be thoroughly discussed in conjunction with the standard analysis which applies to each of the graphs taken independently.
As an interesting application, we will then consider the special case of multiplex networks that can be factorized as Cartesian products of smaller basic networks and prove that the patterns can be created or destroyed by adding more layers to the structure. On a wider perspective, this result applies to reaction-diffusion systems defined on a generic network, which can be factorized as the Cartesian product of smaller networks. The conditions for the instability of the scrutinized system can be hence reformulated in terms of the smaller, hence more tractable, factor networks.
The paper is organized as follows. In the next Section we will present the general theory of Cartesian product networks and formulate the problem of patterns formation for reaction—diffusion systems defined on such networks. For a generic choice of the diffusivities, we shall prove that patterns emerge in the Cartesian product provided they can develop in at least one of the two networks from which the Cartesian support originates. In this respect, Cartesian products are more prone to exhibit Turing instabilities than their corresponding factor networks. In the limiting case when the diffusivities do not depend on the topology of the networks, but just on the species ability to relocate to neighbors sites, Turing patterns can set in if and only if the instability takes place on both factor networks. Our analytical conclusions will be challenged numerically by employing the Mimura—Murray model22 as a representative reaction scheme. We will then turn to investigate the conditions for the emergence of self-organized patterns on degenerate multiplex networks - the same network is repeated on all layers - an important case study which can be handled as an immediate byproduct of our analysis. Finally, in the last Section, we will sum up and conclude.
Given two networks G and H, being respectively characterized by nG and nH nodes, hereby denoted and , and by edges and , one can build16,19 their Cartesian product , that is the network composed by nGnH nodes VG × VH and whose edges are defined by:
Let AG, respectively AH, be the adjacency matrix of the network G, respectively H. Then the adjacency matrix of the Cartesian product network is given by
where is the n × n identity matrix and ⊗ is the Kronecker product. Let us recall that the Kronecker product of two matrices A and B, is the matrix
where aij are the elements of matrix A. Let LG = AG − DG (respectively LH = AH − DH) be the Laplacian matrix of the network G (respectively of H), where DG is the diagonal matrix whose entries are the degrees of the network G (similarly for DH). Then the Laplacian matrix of the Cartesian product network reads:
From the latter equation, it straightforwardly follows that the eigenvalues of are of the form
that is the eigenvalues of the Laplacian matrix associated to the Cartesian product are obtained as the sum of the eigenvalues of the Laplacian operators of each factor networks. Moreover, the eigenvectors of are given by , where and stand for the eigenvectors of respectively LG and LH. In fact:
Let us observe that LG and LH are zero sum symmetric and negative—semidefinite matrices and so it is . Hence, the eigenvalues are all negative, except for the largest one which is identical to zero. We will organize the list eigenvalues so that the first position (α = β = 1) reads always zero, the largest eigenvalue. Hence, .
Reaction-Diffusion systems on Cartesian product networks
Let us now consider a reaction—diffusion system defined on a Cartesian product network . To this end we introduce two species whose continuous densities are labelled u and v. The two species undergo local interaction when they share one of the nGnH nodes of and diffuse among adjacent sites via existing links. Denote with , respectively , the diffusion coefficient of species u, respectively v, on network G. For network H, one can introduce the homologous quantities and . In the following we shall indicate with ugh and vgh the concentrations of respectively u and v at node . As usual, local rules of interaction among species translate in non linear functions of the concentration amount, hereafter f(ugh, vgh) and g(ugh, vgh). The diffusion is in turn modeled by resorting to conventional Laplacian operators. In formulae:
where the diffusion operator (with s = u, v) reads:
Notice that ugh can be written as , and hence . On the other hand, , which implies . Similar considerations hold for , and one can therefore rewrite (5) as:
To progress in the analysis we shall assume that an homogeneous solution of the above equations exists, i.e. , for all g and h such that . In addition, we will require the homogeneous fixed point to be stable, which in turn amounts to impose tr(J) = ∂uf + ∂vg < 0 and det(J) = ∂uf∂vg − ∂vf∂ug > 0, where J stands for the Jacobian matrix evaluated at (to keep the notation simple and because f and g do not depend on the nodes index, we have replaced ugh and vgh by u and v in the former and in their derivatives). Following the standard Turing recipe, we set down to study the conditions that yield an exponential growth of a non-homogeneous perturbation around . We hence define and and linearize system (7) around the equilibrium
where fu, fv, gu and gv are the derivatives of f and g with respect to u and v evaluated at the equilibrium point .
To go one step further we expand δugh and δvgh on the eigenbasis of the Laplacian matrix for and look for solution of system (8) in the form:
By inserting the previous relations into the linearized system (8), one readily finds that the following condition should be met for a non-trivial solution to exist:
where and are defined as:
Let us observe that P is always negative because of the stability assumption (tr(J) < 0) and since and . The exponential instability manifests provided the real part of λαβ gets positive over a bounded portion of the plane . For this reason, we shall solely concentrate on the largest root of equation (9):
whose real part is also called the dispersion relation. Turing instability develops on the Cartesian network provided
within a finite domain in and . In the following we shall set and to make explicit reference to the embedding Cartesian topology.
The above derivation can be formally adapted to the simpler case where the reaction-diffusion system is defined on a standard graph G. In this case the condition for the existence of Turing patterns amounts to imposing , inside a bounded interval of . Importantly, . Similar considerations hold for graph H, which is combined to G to yield the Cartesian network . In practical terms, the dispersion relation which controls the instability on a Cartesian support is a multi-dimensional function (two dimensional, for the case under exam), which reduces to the conventional one dimensional function, when projected on each of the independent subspaces that compose the Cartesian backing. Notice that the above conclusions can be also reached by employing a straightforward two dimensional extension of the network-targeted Fourier transform introduced in20,21 to the current multi-dimensional setting.
Starting from this scenario, it is interesting to elaborate on the mathematical conditions that underly the emergence of collective patterns on a Cartesian support, in relation to the mechanisms which seed the homologous instabilities on the composing graphs, taken separately. Are Cartesian patterns reminiscent of the instability that occur on each layer of the assembly? To answer this question it is entirely devoted the remaining part of the paper.
Different diffusion constants on distinct graphs
Let us start by considering the general case where the diffusion coefficients for each species are assumed to depend on the hosting network, namely and . Imagine that Turing patterns can develop when the inspected reaction-diffusion system is hosted on G. Then, as we shall prove hereafter, the patterns can invade the Cartesian support . Similar conclusions obviously hold when the dual scenario is considered, i.e. when the patterns are allowed to develop on graph H, instead of G.
Since Turing patterns can be found by hypothesis on network G, there exists at least one such that . Consider the eigenvalue (where in the last step we made use of ) and write the following chain of relations:
The same modes which are unstable on G, are also destabilized when the reaction-diffusion system is made to evolve on the Cartesian support . The network can hence exhibit Turing patterns, the perturbation being localized on a set of unstable modes which includes (or coincides with) those active on G. If the spectrum of the Laplacian was continuum, one could always delimit, by continuity of , a finite portion of the parameter plan , adjacent to the degenerate line , for which . However, the spectrum of the Laplacian operator is discrete. One should therefore require a sufficiently small to exist, so that , for non trivial modes of the Cartesian support could be triggered unstable.
To make this concept more explicit, we consider the celebrated Mimura-Murray model22, which we shall assume to specify the reaction terms. More specifically we will set f(u, v) ((a + bu − u2)/c − v)u and g(u, v) = (u−(1 + dv))v, where a, b, c and d are constant parameters. The Mimura-Murray model possesses six equilibria, whose stability depends on the value of the above parameters. We will hereby set a = 35, b = 16, c = 9 and d = 0.4 and focus on the homogeneous stationary solution , where Δ(bd − 2d − c)2 + 4d2 (a + b − 1). It is immediate to realize that det(J) > 0 and tr(J) < 0, hence the selected fixed point is stable. The diffusion coefficients are assigned as discussed in the caption of Fig. 2. In particular, patterns can develop when the Mimura-Murray system is let evolve on graph G. At variance, Turing instability cannot take place on graph H. When the system is instead hosted on the Cartesian support , as obtained by composing together the individual graphs G and H, patterns can materialize, as demonstrated in Fig. 2. The generalized dispersion relation takes indeed positive values over a finite portion of the discrete two dimensional support as it can be appreciated by visual inspection of panel c of Fig. 2.
The diffusion is the same on distinct networks
Consider now the simpler setting where the diffusion coefficients are assumed identical on all graphs composing the Cartesian networks. In formulae we require and . Also the kinetics parameters do not depend on the reaction site. Under this working hypothesis, Turing patterns are allowed on the Cartesian network , if and only if they can also develop on both G and H. To prove our claim, we remark that the assumption of identical diffusivities enables one to simplify the dispersion relation and Eqs. (10) and, in particular, we get:
Hence, the instability takes place on the Cartesian support if Dugv + Dvfu > 0 and . On the other hand, when these latter inequalities are matched, Turing patterns develop on both G and H, provided their discrete Laplacian eigenvalues populate the interval where the dispersion relations for each single graph, and , are positive.
Degenerate multiplex as the Cartesian product of two graphs
Assume G to be an open one dimensional chain, with nearest neighbors connections. This configuration is also termed path in the literature, and differs from a ring or cycle, because it lacks periodic boundary conditions. Then, for any arbitrary choice of network H, the cartesian product is a multiplex with peculiar characteristics. On each layer of the multiplex the same network H is repeated. Inter-layer connections are established only between adjacent layers, as depicted in Fig. 3.
If we are interested in investigating the possibility of a Turing like instability for a generic reaction-diffusion system defined on such a multiplex, one cannot resort to the approach discussed in6. The multiplex is in fact degenerate, meaning that the layers are identical by construction and their associated spectra coincide. This clearly implies dealing with repeated eigenvalues a condition which violates the hypothesis on which the analysis of6 builds. It is however worth emphasizing that the analysis of6 can be extended to the case where the eigenvalues are indeed repeated, at the price of some additional complications in the calculations. Following the above conclusion, we can however expect Turing patterns to materialize on the multiplex support, if the reaction-diffusion system under inspection can undergo a diffusion driven instability when placed on the path network G. As we shall argue in the following, this request translates in a compact condition for the instability to develop on the multiplex support. In fact, the homogenous equilibrium is unstable to external inhomogeneous perturbation, for a reaction diffusion-system evolving on G, provided and . On the other hand, the eigenvalues of the Laplacian operator defined on G can be written in a closed form as:
For a given reaction kinetics, Turing patterns can flourish on G, if and only if there exists at least on eigenvalue for which , namely
are the two positive roots of . When condition (13) is met, and by virtue of the analysis carried out above, the patterns can invade the multiplex support. In Fig. 4 we provide a direct evidence of the phenomenon, employing again the Mimura-Murray reaction model as the reference case study. In our results, no correlation between the asymptotic values of ui (or vi) and the node degree is observed, at variance with the conclusion of5. This is an intriguing difference which deserves to be further investigated, in relation to the system size of the employed networks and their intrinsic topological features.
Another interesting case to consider is when G is the complete graph with nG nodes, namely a network with all-to-all connections but self—loops. Then, for any network H, the Cartesian product is a multiplex, which hosts on every layer a replica of H, each node of a given layer being directly connected to all its specular images on the other layers. The number of nodes of network G determines therefore the number of layers of the Cartesian multiplex. Based on the above, we can readily infer an explicit condition for the existence of Turing instability on the generalized Cartesian support. The only non trivial eigenvalue of the complete graph is −nG (with multiplicity nG − 1) and condition (13) yields q− < −nG < q+. In other words, the number of nodes of G, or equivalently the number of layers in the multiplex, can act as a control parameter to instigate, or alternatively dissolve, the Turing instability. In Fig. 5 we provide a numerical demonstration of the predicted phenomenon. Patterns can be seen on the Cartesian multiplex, for a given choice of H and of the reaction kinetics, only if the number of nodes of the complete graph G falls within a bounded interval. Once again, the asymptotic concentration ui (or, alternatively, vi) does not correlate with the degree of the corresponding node i: all the nodes belonging to the same layer in the left panel of Fig. 5 host the same concentration, irrespectively of their associated degree. At variance, nodes with the same degree across layers can display different concentration amount.
Reaction-diffusion systems on complex networks are gaining attention because of their multifaceted applications to a vast realm of interdisciplinary problems. As follows a symmetry breaking instability, seeded by diffusion, a stable homogeneous fixed point of the examined reaction kinetics can turn unstable to inhomogeneous perturbation. This event is the precursor of a Turing pattern, which is eventually approached by the system in its late time evolution. The conditions that underly the spontaneous emergence of self-organized patterns can be obtained via a standard linear stability analysis, which requires expanding the imposed perturbation on a basis formed by the eigenvectors of the network Laplacian operator. Importantly, the associated eigenvalues play the role of non trivial wavelengths for the embedding network support.
Starting from this setting, we have here analyzed the case of Cartesian networks, namely complex support assembled as the Cartesian product of simpler networks, and elaborated on the conditions for the Turing instability to set in. To this aim, we introduced and exploited a generalized basis for tracking the perturbation in its linear regime of evolution. This is the tensor product of the eigenvectors of the discrete Laplacian operators, defined on each individual network. For simplicity we focused from the beginning on the simplified setting where the Cartesian product involves two distinct networks, but the analysis, as well as the conclusions of our study, apply to a more general setting where several networks can be combined together to give a multidimensional Cartesian Product. The dispersion relation which ultimately determines the onset of the instability is now function of two independent set of discrete wavelengths, the eigenvalues of the Laplacian operators constructed from the two networks that combine in the Cartesian structure. As a consequence, the process of patterns formation for a reaction-diffusion system on Cartesian support can be rationalized via an integrated approach which moves from the analysis of the instability conditions on each of the graphs taken independently. In particular, we could prove that patterns can invade the Cartesian network, if they are supported on one of the graphs that compose its structure. Multiplex networks can be also obtained as a special limiting case and the domain of instability delimited by compact relations. When a generic network is assembled with a complete graph to yield a degenerate multi-dimensional complex lattice, the onset of the instability can be controlled by the number of nodes of the complete sub-structure. Our findings have been corroborated by direct numerical integration of the reaction-diffusion equations, assuming the Mimura-Murray kinetics as a representative model.
How to cite this article: Asllani, M. et al. Turing instabilities on Cartesian product networks. Sci. Rep. 5, 12927; doi: 10.1038/srep12927 (2015).
The work of T.C. presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.