State-dependent effective interactions in oscillator networks through coupling functions with dead zones.

The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have 'dead zones', that is the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories, the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Introduction

Many systems in applied sciences can be seen as systems of coupled units that mutually influence each other, such as interacting neurons of an animal's nervous system. Moreover, the system's functionality often depends on emergent properties of this dynamical system. The dynamical behaviour of a network composed of coupled systems depends on a number of factors. One might wish to separate these factors into three broad categories: firstly, the dynamics of component systems (the unit dynamics) in isolation; secondly, the structure of the graph of couplings between the systems (the network structure viewed as a directed graph where the nodes are systems and the edges are connections); and, thirdly, the nature of the interactions (we refer, in general, to this as coupling functions [1]).

This approach can be limiting in several ways. First, it is well known that coupling isolated systems with simple unit dynamics may result in relatively simple dynamics (such as synchronization); equally, it can result in the emergence of qualitatively different dynamics [2]. While a graph structure—as a graph of connections—is an efficient way to encode linearly weighted coupling between dynamical units, it may not capture higher-order, multi-way coupling. Such interactions are typically not ‘pairwise’ if there are nonlinear interactions between three or more nodes; for example, if the input from unit 2 to unit 1 is modulated by unit 3. This leads to more general non-pairwise coupling that has recently been investigated in coupled oscillators (for example [3-5]) and in a broader contexts such as ecological networks [6]. Second, even if one assumes linearly weighted pairwise couplings, the coupling function itself is often assumed to be fairly simple in some sense: in the case of weakly coupled oscillatory units, the interaction is typically never assumed to vanish on any interval of phase differences.

Here, we analyse networks with more complex coupling functions that allow dynamical units to (effectively) dynamically decouple and recouple as the system evolves. More specifically, the dynamical systems we investigate have state-dependent interactions that arise through dead zones in the coupling function; this concept has been previously identified (for example [7]). Intuitively speaking, in the dead zones of a coupling function there is no interaction. We call the complements live zones and formalize these concepts below. Although a wide range of adaptive and time-dependent networks have been studied in the past, this sort of coupling between network structure and dynamics has been overlooked in many contexts, probably because the coupling functions considered may be thought to be pathological (the functions have non-trivial variation at some regions and trivial variation at others and so cannot be analytic). State-dependent dynamics induced by coupling with dead zones relate dynamical models in systems biology, through piecewise linear dynamics characterized by thresholds (for example [8-13]) or continuous modelling (e.g. [14,15]). However, the investigation in these specific settings have focused almost exclusively on the asymptotic behaviour (whether synchronized, periodic [16-19] or chaotic [20,21]) of the underlying high-dimensional coupled system rather than on the dynamics of the effective interactions between individual subsystems of a network. This is also the case in [22,23], where the authors have related the existence of circuits in the graph of interactions to the existence of multistable or stable limit cycles in phase space.

In this paper, we elucidate the interplay between dynamics and effective interactions in networks of coupled phase oscillators. We first define the notion of a dead zone for such systems, which leads to the definition of an effective coupling graph at a particular state. The network dynamics are determined by the effective interactions: the dynamics at a particular point are determined by the (state-dependent) effective interactions which may change over time. Hence, the dynamics are determined by both the structural properties (what effective interactions are possible) and the dynamical properties (whether and how do the effective interactions change over time) of the system. Our contribution is threefold. First, we consider the structural question what effective interaction graphs are possible for a given network structure by careful design of coupling functions and examination of the dead zones. Second, we give a result on how the effective interactions do (or do not) change as time evolves. Third, we give instructive examples how dead zones shape the network dynamics; in particular, for a fully connected network and a coupling function for which the dynamics are fully understood, a single dead zone can induce non-trivial periodic dynamics.

While we restrict ourselves to dead zones in coupled phase oscillators, we note that these concepts are likely to be applicable to more general network dynamical systems in a wider range of contexts. Indeed, even in networks that appear structurally simple (for example, networks that are all-to-all coupled from a structural point of view) the existence of dead zones can induce new dynamics. A decomposition of phase space into regions of identical effective interactions yields a natural coarse-graining of the system: it can be understood in terms of the transitions between effective interactions similar to the state transition diagrams in [9]. Such a dynamical decomposition can provide a framework for network dynamical systems with coupling that has ‘approximate’ dead zones—regions where the coupling is small but non-zero. Hence, we anticipate that notable examples and applications may arise in systems biology as discussed above, where many studies use this type of active/inactive interactions, neuroscience, where for example state- and time-dependent interactions may arise, for example, through mechanisms such as spike-time-dependent plasticity [24] or refractory periods, or continuous opinion models [25,26] where agents only interact if their opinion is sufficiently close.

Dead zones for phase oscillators

The particular class of system that we study in this paper has particularly simple unit dynamics (coupled phase oscillators) and pairwise coupling so that a network of interactions and pairwise coupling functions are appropriate. These models arise naturally in a range of applications where there are coupled limit cycle oscillators and the coupling is weak compared to the limit cycle stability [27,28]. More precisely, we assume that the phase of oscillator evolves according to where ω is the fixed intrinsic frequency of all oscillators, encode the coupling topology between oscillators (we assume no self-coupling, A = 0), and the (non-constant) coupling function determines how the oscillators influence each other. A graph is associated with the adjacency matrix (A)—the structural coupling graph that encodes whether oscillator k can receive input from oscillator j. We constrain ourselves here to phase oscillator networks where the oscillators have the same intrinsic frequency and the coupling function g is the same between all pairs of oscillators.

The coupling topology relates to dynamical properties of network dynamical systems such as (1.1). For commonly studied coupling functions, properties of the structural coupling graph such as its spectrum [29,30] determine, for example, synchronization properties (complete or partial) of the network (1.1): this is used in the master stability function approach of Pecora & Caroll [31,32] and the work of Wu & Chua [33], who revealed the role played by the spectral gap and the spectral radius of . Various conditions for complete synchronization of networks with arbitrary graph structures have been found using spectral properties [34-38]. Consequently, this helps understand the effects of structural perturbations on the synchronizability of networks, including (1.1) and networks where the unit dynamics are more complex (see [39-43]).

Note that it is not necessarily sufficient to consider the structural coupling graph to determine dynamical properties: this is particularly the case if the coupling function g has dead zones, i.e. if it is zero over some interval of phase differences. In the presence of dead zones, we will define an effective coupling graph of (1.1) as a subgraph of , which encodes the effective interactions between oscillators at a particular point in phase space. By definition, the effective coupling graph is state-dependent and may change dynamically with time. As the system evolves, the network may even decouple into several components under the influence of dead zones. These networks with time- and state-dependent links can also be viewed in the framework of ‘asynchronous networks’ [44,45].

Even though we assume that the uncoupled units are very simple and the functional form of interactions is the same, the possible dynamics of (1.1) may be very complex [28]. We, therefore, mostly restrict equation (1.1) to the case where the coupling is all-to-all (and thus fully symmetric), that is A = 1 for all j≠k, and the phase evolves according to for k = 1, …, N. In spite of the high degree of symmetry of the system, the system shows a very rich variety of behaviour that includes synchronization [46], clustering [47], heteroclinic dynamics [48] and chaos [49]; see also [47,50,51] for a discussion of the dynamics and bifurcations of (1.2) and see [52] for a recent review.

Looking at the structural coupling graph, the network associated with (1.2) is rather simple, since the network is fully connected. However, this also means that there is a rich set of 2 subgraphs corresponding to setting (off-diagonal) entries of to zero. For a coupling function with dead zones, this means there is a very rich set of effective coupling graphs that can occur.

Our goal here is to explore some connections between properties of such dead zones, the effective coupling and the typical dynamics for such networks with these coupling functions for networks of the form (1.1) or (1.2). As an example, figure 1 shows some possible effective couplings that can be achieved by (1.2) with N = 5 and choices of coupling function with dead zones. Section 2 presents a setting in which these dead zones can be defined. It also presents conditions in proposition 2.12 for local skew product structure that appears in the dynamics due to the dead zones. We then address the following questions:

Figure 1.

(a) Coupling for the graph 5 corresponding to the fully connected network (1.2) with N = 5. Panels (b–f ) show five examples of the 25×4 = 1 048 576 possible embedded subgraphs of (a), i.e. having the same number of nodes as (a): by proposition 3.4, we can show that all of these and more can be realized as effective coupling graphs for a coupling function g with dead zones. Panels (b,d) show graphs with more than one component: (b) is the ‘empty’ graph with no edges, (c) is a cycle of length 5 and (d,f ) have non-trivial structure. A typical trajectory of the system for such a system will visit several different effective coupling graphs under time-evolution. While (e,f ) shows similar structure proposition 4.3 shows that only (e) can be realized in a dynamically stable manner as it contains a spanning diverging tree.

Given any subgraph of the structural coupling graph, is there a coupling function such that this subgraph is realized as the effective coupling graph for some point in the phase space?

What is the relation between the coupling function, the set of possible subgraphs that can be realized, and the points where these realizations happen?

How do the dynamics and effective couplings influence each other?

Section 3 discusses some results related to these questions concerning effective coupling graphs: proposition 3.4 answers Q0 positively in that it shows that any directed graph can be realized as an effective coupling for a suitable coupling function. A refinement of this proposition is given by proposition 3.8 by showing it is possible to do it at a given length of live zone (small enough with respect to the number of nodes). We also consider specific subcases of Q1 by showing how symmetries of the points in the torus determine partially the effective coupling graphs. Section 4 moves on to Q2 and consider the interaction between effective coupling and dynamics: proposition 4.3 (and corollary 4.4) show that any graph that admits a spanning diverging tree can not only be realized at some point in phase space but also this can be made dynamically stable. More generally, it seems that the interaction of dynamics and dead zones may be quite complex and so we explore some examples. Section 5 looks in detail at the realized effective coupling graphs for N = 2 and N = 3 with one dead zone. Finally, §6 explores uses, generalizations and applications of these ideas.

Graph theoretic preliminaries

We briefly introduce some notions (and notation) that are used in this paper (see, e.g., [53] for more background on graph theory). Recall that a directed graph is a pair  = (V, E) with a finite set of vertices V and directed edges E⊂V2 between vertices. Depending on the context, we write V (), E() to denote the vertices and edges of . A pair (j, k)∈E is an edge from vertex j to vertex k. Since the graphs we consider here relate to network dynamics, we use terms vertex/node and edge/link interchangeably. We will assume that the graphs do not contain self-loops, i.e. (k, k)∉E for any k. A vertex k∈V is said to have an incoming edge if there exists another vertex j such that (j, k)∈E. Any graph can be identified with an adjacency matrix A with coefficients A = 1 if (k, j)∈E() and A = 0 otherwise. We say the graph is undirected if A is symmetric, i.e. A is equal to its transpose, or equivalently, (j, k)∈E if and only if (k, j)∈E. This means that an undirected graph has an even number of directed edges. Finally, a graph  = (V ′, E′) is a subgraph of , and we write ⊂, if V ′⊂V and E′⊂E. In this paper, it will be convenient to consider a subgraph (V ′, E′)⊂ as an embedded subgraph (V, E′) by including all vertices V of . As an example, figure 1a illustrates the fully connected graph on five vertices and figure 1b–f some embedded subgraphs.

Write . If  = (V, E) and ⊂ is an embedded subgraph, then the associated adjacency matrices A, A are N × N matrices. The fully connected graph is the graph on V with (j, k)∈E() for all j≠k; it has N(N − 1) edges (figure 1a). Similarly, let ∅ = (V, ∅) denote the empty graph with no edges; cf. figure 1b. Note that and ∅ are undirected (figure 1). For (p1, p2, …, p)∈V, let denote the directed path with vertices p1, …, p, i.e. the subgraph of with edge set and similarly, let be the directed cycle with vertices p1, …, p and edges ; cf. figure 1c. The undirected path and undirected cycle are obtained by adding the reverse edges to , , respectively. Finally, let be the fully connected subgraph on the set of nodes {p1, …, p}. When convenient, we will identify the graphs , and (and their undirected versions) with their corresponding embedded subgraphs with vertices V.

A directed graph  = (V, E) is strongly connected if given any two vertices in V there exists a directed path of edges in E between these two nodes. A directed graph is said to be weakly connected if it is not strongly connected and its underlying undirected graph, obtained by ignoring the orientations of the edges, is strongly connected. A spanning diverging tree of is a weakly connected subgraph of such that one node (the root node) has no incoming edges and all other nodes have one incoming edge (for instance, the graph in figure 1e contains such a tree). Lastly, for any graph  = (V, E), an independent set S is a set of nodes included in V , for which any two nodes of S are never connected by an edge in E. We say that is a k-partite graph if V admits a partition in k distinct independent sets.

Symmetries, dynamics and graphs

Let be a graph with V () = V and let be the symmetric group of all permutations of . The automorphisms of , denoted by form a subgroup of under composition. Define the set of embedded subgraphs and write . Note that the group Γ() naturally acts on : For and γ∈Γ() the image γ is the graph with vertices V and edges for γ∈Γ(). For this action, the isotropy group of the graph ⊂ is Note that the isotropy group does not uniquely identify the subgraph, for example one can reverse the edges and get the same isotropy; however, it is a useful characterization of the graph.

The group acts on by permuting components. Let . For we define the fixed point space . For a given , the isotropy subgroup of θ is . The symmetries have a number of dynamical consequences for the oscillator network (1.1) for above being the structural coupling graph : note that (1.1) is equivariant with respect to the action of where acts via permutation of the oscillators and acts via phase shifts The fixed point space of any isotropy subgroup of is dynamically invariant [50]. It is often useful to consider behaviour of (1.1) in terms of the group orbits of . Equivalently, the quotient by corresponds to considering the dynamics in phase difference coordinates, and relative equilibria (equilibria for the quotient system) typically correspond to periodic orbits for the original system.

For the structural coupling graph  = , we obtain the all-to-all coupled oscillator network (1.2), which is equivariant. In this case, the dynamics on the full phase space are completely determined by the dynamics on the canonical invariant region (CIR) [50,51] The full synchrony and splay phase configurations are relative equilibria of the dynamics. There is a residual action of on the CIR and Θsplay is the fixed point of this action [50].

From dead zones to effective coupling graphs

In this section, we define dead zones for a coupling function and introduce the resulting effective coupling graph and its properties. We will restrict to a suitable class of coupling functions that have dead zones but are otherwise smooth and in general position; clearly, this could be easily generalized, for example to coupling functions with only finite differentiability.

Definition 2.1.

Suppose that is a smooth 2π-periodic function.

A coupling function g is locally constant at if there is an open set U with such that g(U)≡c. Define LC(g) to be the set of locally constant points of g.

A coupling function g is locally null at if it is locally constant with c = 0. Let DZ(g)⊂LC(g) denote the set of locally null points of g.

A coupling function g has simple dead zones if DZ(g) has finitely many connected components and LC(g) = DZ(g), i.e. if there is a finite set of locally constant regions, and all are locally null.

Definition 2.2.

Let g be a coupling function with simple dead zones. Any connected component of DZ(g) is a dead zone of g. Connected components of the complements are interaction or live zones.

Here, we will only consider the case of simple dead zones: in the rest of the paper, we will implicitly consider only coupling functions with simple dead zones. The class of coupling function with simple dead zones excludes (smooth approximation of) piecewise constant coupling functions. These may have non-trivial dynamics that are solely given by the different frequencies in the region where the coupling function is locally constant. Such non-trivial dynamics are clearly of interest in some applications, but is beyond the scope of this paper.

Definition 2.3.

We say that g is dead zone symmetric if −DZ(g) = DZ(g) modulo 2π, i.e. if whenever is in a dead zone, then −ϕ also belongs to a dead zone.

As an illustration, figure 7 in §5 provides examples of coupling functions (which are dead zone symmetric or not) with one dead zone.

Figure 7.

The coupling functions (5.2) provide examples of coupling functions with one dead zone; here ε = 5 × 10−3 and α = 1.3. The shaded area indicates the dead zone of the coupling function. (a) We have a dead zone symmetric coupling function with DZ(g) = (5π/6, 7π/6); cf. figure 5b. (b) We have DZ(g) = (π/3, 3π/2); cf. figure 6a. (c) We have DZ(g) = ( − (π/3), 11π/12); cf. figure 6b. (d) We have DZ(g) = (0.5, 1.5).

Effective coupling graphs and their symmetries

Suppose that g is a coupling function for (1.1) with structural coupling graph given by the adjacency matrix (A) and let . We say a node k is g-effectively influenced by node j at θ for (1.1) if A = 1 and θ − θ∉DZ(g).

Definition 2.4.

The effective coupling graph of (1.1) with coupling function g at is the graph on N vertices with edges

Conversely, an edge if A = 0 (the edge is not contained in ) or θ − θ∈DZ(g) (the phase difference is in a dead zone).

Clearly , and this will be a proper subgraph (that is, it differs from by at least one edge) for some if g has at least one dead zone. For the system (1.1) with coupling function g and given ⊂, define

Definition 2.5.

If Θg, () is not empty, then is realized as an effective coupling graph for (1.1) with coupling function g. Moreover, a graph can be realized as an effective coupling graph for (1.1) if there exists a coupling function g for which Θg, () is not empty.

For the special case  = , that is the oscillator network (1.2), we simply write for the effective coupling graph with edges Similarly, we write for the regions of phase space with a particular effective coupling graph. Note that the sets Θg(), , partition the CIR .

For particular structural coupling graphs of (1.1), there is a large number of symmetries, i.e. the automorphism group Γ() may be large [31]; it is maximal for  = . At the same time, Γ() acts on the underlying phase space. We now show how the symmetry of a point relates to the symmetries of the effective coupling graph at θ.

Lemma 2.6.

Consider the system (1.1) with structural coupling graph and any coupling function g. For any , we have for all γ∈Γ().

Proof.

Note that [γθ] = θ, where [  ·  ] refers to the kth component, and so [γθ] − [γθ]∈DZ(g) if and only if θ − θ∈DZ(g). These are the edges of . ▪

Corollary 2.7.

Consider the system (1.1) with structural coupling graph and any coupling function g. For any , we have

To see this, note that if γ∈Σ then γθ = θ and so which implies that . ▪

Note that the reverse containment of corollary 2.7 does not necessarily hold, for example if there are no dead zones (i.e. if DZ(g) is empty) then clearly Σ = Γ() for all θ.

Remark 2.8.

Note that while is a fundamental region for the dynamics of the all-to-all coupled network (1.2), the effective coupling graphs can differ between symmetric copies of .

Local (skew-)product structure and asynchronous networks

In this section, we show that the effective coupling graph at a point captures essential dynamical information. In particular, we have that, locally around a generic point, the vector field factorizes into factors that correspond to (weakly) connected components of the effective coupling graph.

For let denote the projection of onto the coordinates in v. We write and θ = (θ, …, θ) are the coordinates in . Suppose that v1, v2⊂V partition V, that is, v1∩v2 = ∅, v1∪v2 = V. Write r = |v| for the length of the vector v and we identify with elements θ = (θ, θ) through the natural isomorphism that reorders coordinates appropriately.

Definition 2.9.

Consider a general ODE on the N-torus, with , some smooth function, and v = (v1, v2) a partition of V.

The system has a local skew-product structure v1 → v2 at if there is an open neighbourhood U of θ and functions F(1), F(2) such that F(θ) = (F(1)(θ), F(2)(θ, θ)) for all θ∈U.

The system has a local product structure with respect to v at if there is an open neighbourhood U of θ and functions F(1), F(2) such that F(θ) = (F(1)(θ), F(2)(θ)) for all θ∈U.

The second statement is equivalent to F having a local skew product structure v1 → v2 and v2 → v1 at θ. For let int(A) denote the interior of A.

Lemma 2.10.

Consider the dynamics (1.1) with a coupling function g. Generically, θ∈ int(Θg, ()) for some .

We have that is a union of finitely many algebraic sets. ▪

Note that this does not imply that all Θg, () have non-empty interior. Consider for example the fully connected network (1.2) with  =  and a coupling function g such that 0∈∂DZ(g). Then Θg() has an isolated point.

Lemma 2.11.

Consider the dynamics of (1.1) written in the form (2.3) for a coupling function g. Suppose that θ∈int(Θg, ()). Then (j, k)∉E() if and only if there exists a neighbourhood of θ such that (∂F)(θ) = 0 for all θ∈U.

Write θ = (θ1, …, θ)∈int(Θg, ()). First, suppose that (j, k)∉E(). Since θ∈int(Θg, ()) there exists a neighbourhood U⊂int(Θg, ()) of θ. Now (∂F)(θ) = Ag′(θ − θ) = 0 for all θ = (θ1, …, θ)∈U since either A = 0 or θ − θ∈DZ(g). Conversely, suppose that there exists an U such that ∂F = 0 on U. But 0 = ∂F = Ag′(θ − θ) on U which implies A = 0 or g′(θ − θ) = 0 (on U). In either case, we have (j, k)∉E() by definition. ▪

Let  = (V, E) be a graph. A partition {v1, v2} of V is a graph cut. Write for the edges from vertices in v to vertices in v. The cut-set of {v1, v2} is E()∪E(). The graph cut is directed if E()≠∅, E() = ∅. The partition is disconnected if E() = E() = ∅. The following result relates properties of the effective coupling graph at a given point θ with the local properties of the dynamical system (1.1).

Proposition 2.12.

Consider a generic point .

There is a directed graph cut v1 → v2 for the effective coupling graph iff the system (1.1) has a local skew-product structure v1 → v2 at θ.

The partition {v1, v2} of the effective coupling graph is disconnected iff the system (1.1) has a local product structure v1 → v2 at θ.

Write . By lemma 2.10, we may assume θ∈int(Θg, ()). Suppose that {v1, v2} is a partition. The assertion now follows from applying lemma 2.11 for any edge in E() and E(), respectively. ▪

Recall that two nodes v, w in a directed graph are weakly connected if there is a path of edges (irrespective of their direction) between them. A weakly connected component is a maximal weakly connected subgraph. The following is a direct consequence of proposition 2.12.

Corollary 2.13.

Consider a generic point and let . If v1, …, vℓ is the partition of the vertices corresponding to the weakly connected components of then there is a neighbourhood U of θ such that (1.1) can be written in the form (2.3) with on .

Remark 2.14.

A network of coupled oscillators naturally defines a non-trivial asynchronous network as defined in [44]. Events occur when the effective coupling graph changes along a trajectory, which defines the ‘event map’ . Now the coupled oscillator network (1.1) written as (2.3) is determined by the state-dependent ‘network vector field’ . Moreover, the network structure is ‘additive’ in the sense that the dynamics of each oscillator is determined by a sum of the contributions from other oscillators and the condition whether only depends on (θ, θ). In the language of [44,45], this means that the asynchronous network is ‘functionally decomposable’ and ‘structurally decomposable’. Finally, note that proposition 2.12 implies that, generically, we have a local product structure, a condition for the spatio-temporal decomposition of the factorization of dynamics theorem [45].

Realizing effective coupling graphs

In this section, we aim to relate dead zones and effective coupling graphs, noting that the effective coupling graph depends both on coupling function g and choice of θ. Recall that Q0 and Q1 in §1a concern the set of effective coupling graphs that can be realized and the goal of this section is to tackle such structural problems. Recall that if Θg, ()≠∅ then we say that the effective coupling graph is realized for g. Proposition 3.4 answers Q0 in the positive: for typical choice of θ (with trivial isotropy), all effective coupling graphs can be realized. We consider two special cases of Q1:

Given a point , a structural coupling graph and a graph , is there a coupling function g for (1.1) (resp. (1.2)) such that ? (resp. )?

Is there a coupling function that realizes all possible effective coupling graphs? Specifically, for (1.2), is there a coupling function g such that[1] ?

Proposition 3.1 gives some constraints on answers of Q1a. We do not have a complete answer to Q1a, while corollary 3.5 answers Q1b positively. We also consider what possible effective coupling graphs will be realized for a coupling function g: this is important if we wish to understand the dynamics of (1.2) with a fixed coupling function. To some extent this is simply a computational question: for any θ = (θ1, …, θ) one has to determine which phase differences θ − θ lie in a dead zone. We give some general results in this direction in §3b. We believe that typical coupling functions will not be able to realize more than a small subset of effective coupling graphs. Henceforth, we mainly restrict to discussion of the fully connected network (1.2) although several of the results easily generalize to (1.1).

Restrictions on the effective coupling graph imposed by θ

Here, we tackle the questions Q1a, Q1b above by putting the emphasis on the point θ. Specifically, given , what do the properties of θ impose on the effective coupling graphs of (1.2)? The isotropy of has some important consequences on the possible effective coupling graphs realized at θ:

Proposition 3.1.

Consider the all-to-all coupled oscillator network (1.2) with coupling function g. Let be fixed.

If θ has isotropy Σ then must have at least the same isotropy.

For full synchrony Θsync = (a, …, a) we have .

Suppose there exists 0 < a < 2π/N such that θ − θ = a for any . Then one of the following cases occurs:

The directed path is a subgraph of but 1,2,…, is not.

The directed path 1,2,…, is a subgraph of but is not.

The undirected path is a subgraph of .

is a n-partite graph (with n = [N/2] if N is even or n = [N/2] + 1 if not).

(i) Is an application of corollary 2.7, notably θ with non-trivial isotropy will limit the possible networks to those that have at least the same isotropy.

(ii) The claim follows from (i): the effective coupling graph must have full symmetry and hence be either or ∅.

(iii) We have one of the following cases as (we illustrate case (4) in figure 2): (1) a∈LZ(g), 2π − a∈DZ(g), (2) a∈ DZ(g), 2π − a∈LZ(g), (3) a, 2π − a∈ LZ(g), (4) a, 2π − a∈DZ(g). Since all the differences θ − θ are equal to a (and thus all θ − θ equal to 2π − a), then in case (1) we have that is a subgraph of but not 1,2,…,. Similarly for the cases (2) and (3). In the case (4), the vertices of can be partitioned into n = [N/2] independent sets A if N is even (or into n = [N/2] + 1 such sets if N is odd): namely the successive sets , , etc. This means that is a n-partite graph. ▪

Figure 2.

Illustration of case (4) in proposition 3.1(iii) for N = 6 oscillators: the coupling function g shown in (a) has two live zones centred at 2a and 3a, the remainder consists of two dead zones. The diagram in (b) shows the phases θ at one instant in time such that θ − θ = a(j − i) for all j > i. The effective coupling graph for the coupling function in (a) is indicated by black arrows between the phases corresponding to individual nodes. This coupling graph is tripartite as indicated by the node colouring. (Online version in colour.)

While proposition 3.1(iii) limits to the splay configuration Θsplay in a special case, the next proposition gives a characterization of effective coupling graphs that are realized for Θsplay.

Proposition 3.2.

Consider the coupled oscillator network (1.2) with coupling function g.

If 2π/N∈LZ(g) then the directed cycle 1,2,…, is a subgraph of .

Let 1 < n < N and suppose that 2nπ/N∈LZ(g). Take indices modulo N.

If N = nm then for any the directed cycle is a subgraph of .

If n does not divide N, then the directed cycle 1,1+ is a subgraph of .

(i) As in the proof of proposition 3.1(iii), we consider successive phase differences for Θsplay = (θ1, …, θ). We have θ − θ = θ1 − θ = 2π/N. Since 2π/N∈ LZ(g) by assumption, we have which proves that .

(ii) A similar argument proves the second assertion. Let 1 < n < N and since θ − θ∈LZ(g) we have . Now suppose that N = nm + q with 0 ≤ q < m. If q = 0 we have r + pn = r mod  N for p = m < N which proves case (a). If q≠0 then r + pn = r mod  N only if which corresponds to case (b). ▪

Recall that Θsplay is the only fixed point of the residual action of on the CIR [50]. The graph is invariant under the action of the symmetry by proposition 3.1(i). In cases (i) and (iib) of proposition 3.2, contains one cycle involving all vertices, that is mapped to itself. By contrast, in case (iia) of proposition 3.2, there are m disjoint cycles that are permuted by the symmetry. The existence of cycles has also some immediate consequences for the connectedness of .

Corollary 3.3.

Consider system (1.2) with coupling function g and suppose that the number of oscillators N is prime. Then either or contains a directed cycle involving all N vertices, and so is strongly connected.

If 2nπ/N∈DZ(g) then . Now suppose that there is an 1 ≤ n < N such that 2nπ/N∈LZ(g). Since N is prime, either proposition 3.2(i) and (iib) applies. In either case, contains a directed cycle involving all N vertices. ▪

The next result gives a positive answer to Q0, providing that we avoid ‘non-generic’ choices of θ.

Proposition 3.4.

For a generic choice of , and for any subgraph , there exists a coupling function g such that .

Generically all the difference terms θ − θ (when j, k are ranging in ) are distinct: therefore, we can specify live zones that contain points θ − θ if and only if the edge (j, k) is contained in . ▪

Notice that the proof gives an upper bound on the number of dead and live zones needed to realize a given as an effective coupling graph (by choosing a coupling function and a point θ), namely the bound given by the number of edges of . This bound is far from being optimal, notably for very ‘regular’ graphs: for instance for any , the graph itself can be realized as a where g has only one live zone and no dead zone. Corollary 3.5 extends the same method of proof to show that one can, in principle, realize all subgraphs using one and the same g.

Corollary 3.5.

There exists a coupling function g such that for any subgraph there exists such that .

Enumerate all graphs in and choose a set such that all phase differences θ − θ are distinct. Now take a coupling function g such that for any n we have θ − θ∈LZ(g) if and only if (j, k) is in E(). ▪

A similar proviso holds here: such a constructed g will typically have a very large number of dead zones.

Coupling functions for an interaction graph

Given a coupling function g, which properties of g imply certain effective coupling graphs realized by g? On the other hand, given , a structural coupling graph and , how can one construct a coupling function g such that ? Among the different parameters characterizing the coupling function g, the number of dead zones plays a major role in these questions, since it determines the shapes of the resulting effective coupling graphs. We thus make the following definition.

Definition 3.6.

Let . We denote by the set of coupling functions having n dead zones.[2]

Proposition 3.7.

Consider system (1.2) with coupling function g.

The coupling function g is dead zone symmetric if and only if all effective coupling graphs for g are undirected.

Assume that is dead zone symmetric with LZ(g) = [ − a, a]. If a < 2π/N, then for any 1 ≤ k ≤ N and any sequence k, …, k + p in we have that ∅, and the embeddings of and can be realized as effective coupling graphs for g. If a = 2π/N, then and the embeddings of graphs and can be realized as effective coupling graphs for g.

Assume that is dead zone symmetric with LZ(g) = [π − a, π + a] and a ≤ 2π/N. Then ∅ and can be realized as effective coupling graphs for g.

(i) This follows directly from the definition of a dead zone symmetric function.

(ii) Suppose that a < 2π/N. Taking such that all the successive differences θ − θ are in (a, 2π/N), we have that : indeed any phase difference θ − θ, with j > k, will belong to the interval (a, 2π(N − 1)/N) and therefore will be in DZ(g). Similarly, taking θ such that all the successive differences θ − θ are strictly smaller than a/(N − 1) we have . Now consider 1 ≤ k, p ≤ N and a sequence k, …, k + p in . Taking θ such that where ϵ is a sufficiently small positive real number, we have that . Similarly taking a point such that with ϵ small enough, we have in this case that . If a = 2π/N, then the same reasoning applies.

(iii) This follows along similar lines as (ii). ▪

Observe that, for the particular points θ = Θsync and θ = Θsplay the coupling functions considered in propositions 3.1(i) and 3.2 can be taken in and , respectively. The proof of proposition 3.4 constructs functions using many dead zones that might be very small. There are various questions one can pose about optimality. For example, given , a structural coupling graph and a graph , what is the minimum n such that there is a such that ? The proof of proposition 3.4 gives an upper bound to this question in the case of System (1.2), namely n ≤ 2# (where #E() is the number of edges of ), but does not give any information on the length of the live zones involved, which can possibly be arbitrarily small, and for which one may need to control the size. Proposition 3.8 below gives a lower bound, as a function of the number of nodes N (namely π/2), on the length of live zones δ > 0 for which it is possible to realize any as an effective coupling graph (and this thanks to a coupling function of which live zones have length δ).

Proposition 3.8.

Let 0 < a < π/2 and let 0 < δ < a. Then, for any with and and for any , there exists an integer n ≤ #E() and a coupling function with live zones of length at least δ such that .

Suppose that θ satisfies the conditions required and is a subgraph of . The idea is to construct a coupling function for which each live zone is precisely associated with only one edge of (figure 3).

Figure 3.

An example of directed graph (b) within 5 with seven edges realized as an effective coupling graph with a coupling function g in (a) constructed via the method described in the proof of proposition 3.8: in the bottom of the graph of g, we show the successive phase differences θ − θ for which the edge (i + 1, i) is in E() (the ones for which (i + 1, i)∉E() are not shown), and at which we impose the existence of a live zone of g. Above the graph of g, we show the values of the phase differences that are determined by the values of the successive phase differences.

To do this, we start with the coupling function identically equal to 0. Now, if (2, 1)∈E() then we put a live zone of length δ centred at θ2 − θ1. If not, then we set g to be locally null at θ2 − θ1. Next, we consider θ3 − θ2≥θ2 − θ1 + a: If (3, 2)∈E() we put a live zone of length δ centred at θ3 − θ2. This second live zone does not intersect the first one since θ3 − θ2 > θ2 − θ1 + δ. If (3, 2)∉E() we let g be locally null at θ3 − θ2. Then we have θ3 − θ1≥θ3 − θ2 + a which satisfies θ3 − θ1 > θ3 − θ2 + δ; we can thus put a live zone of length δ centred at θ3 − θ1 if (3, 1)∈E().

Repeating the process, we construct g by imposing the existence of a live zone of length δ centred at any of the θ − θ (with 1 ≤ k ≤ i ≤ N) such that (i, k) is in E(). Since all these terms are smaller than π, all the opposite values are determined and separated as well by a distance larger than a. It is, therefore, possible to add live zones at the θ − θ (with 1 ≤ k ≤ i ≤ N) for which the edge (k, i) is in E(). ▪

Dynamics of effective coupling graphs

The previous section considered the structural problem of understanding the effective coupling graph at some point in phase space. Now let θ(t) = φ(θ) be the solution of the phase oscillator network (1.1) with initial condition θ. Clearly, defines an evolution on the set of effective coupling graphs. In this section, we briefly consider possible dynamics of these effective coupling graphs.

Suppose that is an effective coupling graph realized for (1.1) with coupling function g for some θ.

Definition 4.1.

The graph can be stably realized if there is an asymptotically stable invariant open set B such that B⊂Θg, (). Moreover, if B = Θg, (), then we say that the graph is completely stably realized.

In other words, for a stably realized effective coupling graph , there is an open set of θ such that for large enough t. If is completely stably realized then this holds for all θ∈Θg, ().

By constructing a coupling function with a stable (relative) equilibrium, we now strengthen proposition 3.4 to show that for any ‘sufficiently connected’ there exists a g such that the effective coupling graph can be stably realized. To prove this result, recall the following spectral graph property [30,54].

Proposition 4.2 ([54, Corollary 1]).

Let be a graph admitting a spanning diverging tree. Consider the Laplacian matrix L with coefficients where A denotes the adjacency matrix of the graph . Then the multiplicity of the eigenvalue 0 in the spectrum of L is one.

We use this to prove the following result.

Proposition 4.3.

For any admitting a spanning diverging tree, there is a coupling function g such that the oscillator network (1.2) has a locally asymptotically stable relative equilibrium (Ωt + θ1, …, Ωt + θ) satisfying . In other words, there exists a coupling function g that stably realizes .

Proof.

Let θ be a generic point such that all the terms θ − θ, j≠k, are distinct. The proof now proceeds in two steps; see also [47]. First, we construct a coupling function satisfying specific conditions (specified below) and show that θ is a relative equilibrium of (1.2). Second, we show that for g, the relative equilibrium θ is locally asymptotically stable. In the following, we write and let Ωt · 1 + θ denote the point (Ωt + θ1, …, Ωt + θ).

Step 1. We now construct a coupling function g that satisfies the conditions By assumption all θ − θ, j≠k, are distinct and hence we can specify g and g′ in sufficiently small neighbourhoods of these points. To satisfy (i) for given , choose g such that θ − θ∈DZ(g) if and only if (j, k)∉E(). For phase differences θ − θ∈LZ(g), it is easy to see that we can specify g, g′ such that conditions (ii) and (iii) are satisfied. This yields a coupling function g that satisfies all three conditions.

Note that by condition (iii), we have Ω such that for all of . This implies that Ωt · 1 + θ is a relative equilibrium of (1.2).

Step 2. Second, we linearize (1.2) at the relative equilibrium θ and check that the conditions (i),(ii),(iii) on g also ensure asymptotic stability of θ. For χ = θ − (Ωt · 1 + θ), we have This yields the linearized equation at θ. Therefore, by condition (ii), the linearized equation at Ωt · 1 + θ read Now set T = A g′(θ − θ) for j≠k and to write (4.2) in matrix form The stability conditions are given by the spectrum of the matrix T. In fact −T is a Laplacian matrix with spectrum where the eigenvalue 0 corresponds to the direction 1 = (1, …, 1) along the group orbit of the phase-shift symmetry. This means that stability of Ωt + θ is only determined by the eigenvalues λ2, …, λ.

Note that none of the eigenvalues λ2, …, λ are equal to zero, i.e. zero is a simple eigenvalue of T. Indeed, let denote the graph of which adjacency matrix is defined by for j≠k. By condition (ii) we have if and only if A = 0. This means that and therefore has a spanning diverging tree: by proposition 4.2 the eigenvalue 0 of the Laplacian matrix of (i.e. of the matrix −T) is simple, which means that 0 is a simple eigenvalue of T. We also remark that by condition (ii) and by the Gershgorin circle theorem, all the eigenvalues of belong to the discs centred in T < 0 and of radius −T. We can thus conclude that the eigenvalues λ2, …, λ have all a negative real part and so Ωt · 1 + θ is an asymptotically stable relative equilibrium. ▪

In fact proposition 4.3 can be generalized to (1.1) in a similar way.

Corollary 4.4.

Assume that admits a spanning diverging tree. Then there is a coupling function g such that (1.1) has an asymptotically stable relative equilibrium (Ωt + θ1, …, Ωt + θ) satisfying . In other words, there exists a coupling function g that stably realizes for (1.1).

The proof works in an exactly similar way as in proposition 4.3. Under the same conditions (i), (ii), (iii) above, the linearized equation at Ωt · 1 + θ reads and the coefficients of the corresponding Laplacian matrix T are and Denoting again by the graph with adjacency matrix for j≠k, we have that if and only if A = 0 by the assumptions made for . Thus, admits a spanning diverging tree. As in the proof of proposition 4.3, this gives again the asymptotic stability of the relative equilibrium (Ωt + θ1, …, Ωt + θ). □

We finish with a brief discussion of a sufficient condition for an effective coupling graph to be completely stably realized. Suppose that θ∈int(Θg, ()) and write the coupled oscillator network (1.1) as . Suppose that the boundary ∂Θg, () is a locally N − 1-dimensional semialgebraic set; this is typically the case as discussed in §2b. Let (θ) denote the piecewise defined normal vector pointing into the interior. One can now obtain sufficient conditions for being invariant by imposing that 〈(θ), F(θ)〉 > 0 for almost all θ∈∂Θg, ().

Effective coupling graphs for networks of two and three oscillators

For coupling functions with an arbitrary number of dead zones, all effective coupling graphs can be realized as outlined above. But what is the global picture of the dynamics for the minimal case of a coupling function with a single dead zone? In this section, we concentrate on this question by exploring small all-to-all coupled phase oscillator networks (1.2) with a coupling function . First, we briefly consider a network of N = 2 oscillators where there are only four possible effective coupling graphs. Then we paint the picture for N = 3 oscillators; there are 64 possible effective coupling graphs and we explore the dynamics numerically.

Networks of two oscillators

One can easily demonstrate that a single dead zone and a single live zone is sufficient to realize all effective coupling graphs for (1.2) with N = 2 oscillators. More precisely, choose any with LZ(g) = [ − a, 2a] for a < π/2, where all inequalities are understood in the interval [ − π, π]. Then This shows that there is a single coupling function that realizes all four subgraphs of 2. Note that if g is dead zone symmetric then only the undirected graph 2 and ∅2 can be realized (cf. proposition 3.7(iii)).

Networks of three oscillators

We now consider all-to-all coupled networks of N = 3 oscillators. Since 3 has six edges, there are 26 = 64 different possible effective coupling graphs. By assigning a colour to each edge, we create a scheme that assigns a unique colour to each graph and permutations of the three nodes correspond to permutations of the colour channels. This assignment is outlined in figure 4 together with examples of graphs coloured in their respective colour. Moreover, the possible effective coupling graphs can be numbered according to the edges that are present. More specifically, let and write A for the associated 3 × 3 adjacency matrix. Define the graph number which uniquely encodes the realized effective coupling graph as an integer. In particular, we have ν(∅3) = 0 and ν(3) = 63; more examples are given in figure 4.

Figure 4.

The colour scheme to identify the graphs in . (a) The shades of cyan, magenta and yellow identified with each directed edge of 3. If multiple edges are present, the colours are added. Examples of graphs in their associated colours, as well as the corresponding graph numbers ν(), as defined in (5.1), are shown in (b). The subgraphs where all edges to/from a given node are present (and no others) are associated with the colours red, green and blue. The symmetry that permutes the three nodes acts on the colour scheme by permuting the colour channels. Hence, graphs which are invariant under this symmetry operation have a colour that is invariant under permutation of the colour channels; this includes white for ∅3, black for 3, and shades of grey for the directed cycles 1,2,3, 3,2,1. (Online version in colour.)

For a given coupling function , the sets Θg(), , partition the CIR . To visualize which region of phase space is associated with a given effective coupling graph, we colour the part of accordingly. If g is dead zone symmetric, the dead zone can be parametrized by a single parameter a∈(0, π) that represents the beginning (or end) of the dead zone. Now suppose that 0∈LZ(g) so —the case 0∈DZ(g) is analogous by ‘inverting’ edges (or colours). There are two qualitatively different cases that are shown in figure 5: if 0 < a < 2π/3 then and if 2π/3 < a < π then . Many more cases are possible for a general function ; rather than paint a complete picture, we illustrate some cases in figure 6.

Figure 5.

The sets Θg() for different partition the canonical invariant region for the fully symmetric system of N = 3 oscillators. The CIR is sketched in (a): its boundary is given by the sets θ1 − θ2 = 0, θ2 − θ3 = 0 and θ3 − θ1 = 0 (black lines) which intersect in Θsync (black dot, •). The splay phase Θsplay is the centroid (hollow dot, °) and is the fixed point of the residual symmetry which rotates the CIR (indicated by grey lines). Dashed lines indicate phase configurations where one phase difference is equal to π. For a dead zone symmetric coupling function only the undirected subgraphs of 3 can be realized; these correspond to the ones shown in figure 4 excluding the cycles. (b) The partition of the CIR for DZ(g) = (π/3, 5π/3). (c) The partition for a dead zone symmetric coupling function with DZ(g) = (5π/6, 7π/6). (Online version in colour.)

Figure 6.

Many effective coupling graphs are possible for N = 3 oscillators and a general coupling function with one dead zone. As in figure 5, the sets Θg() are plotted in the colour corresponding to the effective coupling graph in the colour scheme of figure 4. We have DZ(g) = (π/3, 3π/2) in (a), DZ(g) = ( − (π/3), 11π/12) in (b) and DZ(g) = (π/3, 11π/12) in (c).

The previous considerations were purely in terms of the structure of the effective coupling graph. We now look at examples of the system's dynamics and explore how the effective coupling graph changes along trajectories. To this end, we examine the dynamics of (1.2) with N = 3 and the coupling function for constants a∈[0, 2π), b∈[0, π), ε > 0 and α∈[0, 2π). This coupling function is a modulated Kuramoto–Sakaguchi coupling with phase-shift parameter α. We call the approximate dead zone of the coupling function (5.2) since in the limit the coupling function (5.2) has a single dead zone centred at a of half-width b; here the inequality is to be understood modulo 2π. In the following, we fix ε = 5 × 10−3 and α = 1.3.

We explore the dynamics for four examples of coupling functions (5.2) with approximate dead zones shown in figure 7. Recall that without dead zones (and for any N), the dynamics for Kuramoto–Sakaguchi coupling is well known: depending on the parameter α, either full synchrony Θsync or an anti-phase configuration is stable; see also [55]. Now for each of the Kuramoto–Sakaguchi coupling functions with a dead zone, figure 8 shows a partition of phase space by effective coupling graph (using the same colour scheme as in figures 5 and 6) together with a phase portrait for trajectories of (1.2) integrated forwards from a grid of initial conditions. Note that the dynamics in Θg(∅3)—coloured in white—are trivial and we find, for example, periodic trajectories that visit Θg() for multiple as time evolves. Such dynamics are impossible for Kuramoto–Sakaguchi coupling without dead zones.

Figure 8.

The phase space for (1.2) with N = 3 oscillators and coupling function g with one dead zone as in (5.2) and parameters as in figure 7a–d respectively. As in figures 5 and 6, black lines indicate the sets θ1 − θ2 = 0, θ2 − θ3 = 0 and θ3 − θ1 = 0 which intersect in Θsync (black dot, •) that bound and its symmetric image. The splay phases are indicated by hollow dots (°) and dashed lines indicate phase configurations where one phase difference is equal to π. As above, the colouring indicates the effective coupling graph overlaid by trajectories started on a regular grid, shown in white—a very wide range of effective coupling graphs are realised; see figure 9. For (b) and (c), there are white regions of trivial dynamics where no trajectories are present: these correspond to the effective coupling graph ∅3. Finally, note that for (a) and (c) there are trajectories that visit Θg() for multiple as time evolves. (Online version in colour.)

Figure 9.

The possible effective coupling graphs realized using N = 3 and (5.2) for parameters as in figure 7 and some θ. Black indicates Θg()≠∅ for with a given graph number, and white indicates Θg() = ∅. Since (a) is a dead zone symmetric coupling function, only undirected subgraphs are realized. By contrast, taking all effective coupling graphs that are realized by coupling functions (b) and (c) together, one obtains all possible subgraphs of 3.

Finally, we consider the entirety of the effective interaction graphs that are realized by a given coupling function. Figure 9 shows the set of realized effective coupling graphs corresponding to the coupling functions in figure 7 sorted by their graph numbers. We are unable to find a single coupling function g with one live zone that can realize all possible effective coupling graphs—however, we note that a combination of two coupling functions (for example, (b) and (c)) suffice to cover all cases.

Conclusion

In this paper, we have demonstrated that the effective coupling graph of a dynamical network is subtly related to the network structure, the system state and the presence of dead zones in the interaction. Working with coupled phase oscillator networks (1.1), we give constructions of coupling functions g that achieve any desired subnetwork, possibly using the same g (corollary 3.5), even in the special and highly symmetric case of all-to-all coupling (1.2).

In terms of structural questions, we obtain a number of conditions on g and θ that guarantee the presence of certain coupling structures in . There are several natural questions that relate to the number, location and lengths of the dead zones to the set of realizable effective coupling graphs. For example, the coupling functions in figure 7b,c together can realize all possible (embedded) subgraphs of 3. Two specific questions in this direction for (1.2) are

What is the minimum number n of dead zones such that there is a that realizes all ?

What is the minimum m for which one can find m coupling functions, each with at most n dead zones, such that any is realized by at least one of these functions?[3]

In terms of the dynamics, probably the most interesting problems relate to how the dynamics of the coupled system interacts with the effective coupling as changes along a trajectory starting at θ. This is briefly explored in §4 and in the examples in §5, but we do not have a complete picture as yet. For example, can one determine which effective coupling graphs can be stably realized, and which can be only transiently realized? What does the passage through effective coupling graphs tell us about the underlying dynamics? How does the partition of phase space into basins of attraction map on to the partition of phase space by Θg()?

Here, we only considered phase oscillator networks with coupling functions that have simple dead zones, i.e. there are a finite set of non-trivial intervals on which the coupling function vanishes. This could be developed in three directions: first, one may want to examine coupling with an infinite set of dead zones (though this is likely to be not of much relevance to applications). Second, one could look at the case where the coupling function is locally constant on several intervals where it takes distinct values. Third, one would like to get explicit results for coupling functions with approximate dead zones, i.e. intervals on which the coupling functions are small but non-zero. In this direction, it would be desirable to prove explicit results concerning how well (and over what time scale) networks with dead zones approximate networks where interaction between nodes is small (but non-zero) in parts of phase space.

Finally, we have restricted ourselves here to discussion of these questions for coupled phase oscillators where all interactions are governed by a single periodic phase interaction/coupling function g. On the one hand, it would be desirable to link the coupling functions considered here to nonlinear oscillator networks through a phase reduction. On the other hand, it would be interesting to explore how these results can be generalized, for example, to more general dynamical systems with pairwise coupling of the form for , where g(x, x) is null in some open subset of . Finally, dead zones could also be present in multi-way interactions [1,4], i.e. interactions where the coupling to x depends simultaneously on the relative position of several of the x with j≠k, and not only on one of them.