# Nonlinear Laplacian Dynamics: Symmetries, Perturbations, and Consensus

Riccardo Bonetto  0000-0001-8075-6147 Hildeberto Jardón Kojakhmetov
0000-0001-8708-7409

###### Abstract

In this paper, we study a class of dynamic networks called Absolute Laplacian
Flows under small perturbations. Absolute Laplacian Flows are a type of
nonlinear generalisation of classical linear Laplacian dynamics. Our main goal
is to describe the behaviour of the system near the consensus space. The
nonlinearity of the studied system gives rise to potentially intricate
structures of equilibria that can intersect the consensus space, creating
singularities. For the unperturbed case, we characterise the sets of
equilibria by exploiting the symmetries under group transformations of the
nonlinear vector field. Under perturbations, Absolute Laplacian Flows behave
as a slow-fast system. Thus, we analyse the slow-fast dynamics near the
singularities on the consensus space. In particular, we prove a theorem that
provides existence conditions for a maximal canard, that coincides with the
consensus subspace, by using the symmetry properties of the network.
Furthermore, we provide a linear approximation of the intersecting branches of
equilibria at the singular points; as a consequence, we show that,
generically, the singularities on the consensus space turn out to be
_transcritical_.

Keywords:  Laplacian dynamics, symmetries, consensus, singular perturbations.

##  1 Introduction

Laplacian dynamical systems are among the most renowned subjects in dynamic
networks science. Indeed, given a graph structure, a linear Laplacian system,
$\dot{\bm{x}}=-L\bm{x}$, where $L$ is the graph Laplacian, describes a
(linear) diffusion process on a network. In fact, the Laplacian matrix plays
the role of discrete counterpart of the Laplacian differential operator,
$\Delta$, present for example in the heat equation, $\dot{u}=\Delta u$.
Unfolding the equations for a linear Laplacian system, we obtain a set of
ordinary differential equations that componentwise read as
$\dot{x}_{i}=\sum_{j}a_{ij}(x_{j}-x_{i})$, where $x_{i}$ represents the state
of the $i$th-agent, or node of the network, and $a_{ij}$ encodes the graph
structure and the ‘strength’ of interaction. A lot of effort has been made to
precisely analyse the behaviour of Laplacian systems, especially for the
linear case [44, 43, 36]. More recently, also nonlinear extensions of the
Laplacian dynamics are being explored [42, 3, 1, 24, 25]. The potential
extensions from a linear Laplacian system to a nonlinear one are various; in
the present paper we aim to look at a class of nonlinear Laplacian systems
called _Absolute Laplacian Flows_ (ALFs) [42, 3], defined by

$\dot{\bm{x}}=-LF(\bm{x})+\epsilon H(\bm{x},\Lambda),$ (1)

where $F(\bm{x})$ is a nonlinear vector field, $0<\epsilon\ll 1$, and
$H(\bm{x},\Lambda)$ is a perturbation; more precise definitions follow in the
next sections.

An ALF can be regarded as a discrete analogue of an extended system’s
diffusion process, namely the _Fokker-Planck Equation_ (FPE) [37]. Let us
recall that for an extended system on the real line, the (one-dimensional) FPE
reads

$\frac{\partial}{\partial t}u(x,t)=\frac{\partial^{2}}{\partial
x^{2}}\left(D(x,t)u(x,t)\right)-\frac{\partial}{\partial
x}\left(\mu(x,t)u(x,t)\right),$ (2)

where $u:\mathbb{R}\times\mathbb{R_{+}}\to\mathbb{R}$, $D(x,t)$ is called the
diffusion coefficient, and $\mu(x,t)$ the drift coefficient. We notice that
the term $F(\bm{x})$ in (1) can be written as $F(\bm{x})=:K(\bm{x})\bm{x}$,
where $K(\bm{x})$ is a diagonal matrix, and the precise structure follows from
the definition of the components of the vector field $F(\bm{x})$, see (4)
below. Then, one can recognise a correspondence between FPE and a perturbed
ALF, i.e.,

$\frac{\partial^{2}}{\partial x^{2}}\left(D(x,t)u(x,t)\right)\mapsto-
LF(\bm{x}),\qquad-\frac{\partial}{\partial
x}\left(\mu(x,t)u(x,t)\right)\mapsto\epsilon H(\bm{x},\Lambda).$ (3)

So, the first term of (1), $-LF(\bm{x})$, can be interpreted as a nonlinear
diffusion process on a network, while the perturbation $\epsilon
H(\bm{x},\Lambda)$ describes a small drift.

The above is just one possible interpretation of a perturbed Absolute
Laplacian Flow. From another perspective, it is possible to recognise a
connection with directed graphs with state-dependent weights. If we consider
the rewriting $F(\bm{x})=:K(\bm{x})\bm{x}$, we can define a state-dependent
Laplacian, $L_{\textup{K}}(\bm{x}):=LK(\bm{x})$. The matrix
$L_{\textup{K}}(\bm{x})$ can be seen as the Laplacian of a directed graph
under the definition of out-degree Laplacian [2, 35]. Then, the diffusion term
in (1) is given by $-L_{\textup{K}}(\bm{x})\bm{x}$, retrieving a form similar
to the linear Laplacian dynamics but for directed graphs with state-dependent
weights. Let us notice that the form of the state-dependent weights is
constrained by the structure of the matrix $K(\bm{x})$, and so in turn by the
definition of $F(\bm{x})$.

Within the context of Laplacian dynamics the diffusion of ‘information’ among
the agents can lead the system to reach a common state, referred to as
_consensus_. As such, Laplacian systems find a remarkable application in
consensus problems [36, 39, 12], which have been extensively studied for their
importance in several branches of science, e.g., synchronisation, opinion
dynamics, coordination of robots, rendezvous problems, among many others.

Recapping, the models we are going to study (1), called ALFs, are relevant in
the study of, for instance, nonlinear diffusion processes on networks, or
state-dependent directed graphs. We consider ALFs subject to a small
perturbation that, as we have argued, acts as a small drift in the undergoing
nonlinear diffusion.

Let us now delineate the main contributions of the paper. Notice that it is
quite natural to consider graphs with positive weights, since, for example,
they can represent the strength of the interaction among the agents of the
system. So, first we prove topological equivalence near consensus between
(unperturbed) ALFs with arbitrary positive weights and with unit weights
(proposition 5). Next, recall that linear Laplacian systems have well-known,
simple, sets of equilibria; but that is not the case for the ALFs we consider.
Thus, we then characterise new equilibria emerging from the invariance of
$F(\bm{x})$ under group transformations (proposition 6). Later, we consider
the perturbation problem (1), which turns out to be a _singular perturbation
problem_ [46]. Due to the more complicated structure of the equilibria, and
the presence of multiple timescales, we study the existence of canard
solutions (theorem 2); and we characterize the local geometry of the consensus
space at the singular points (proposition 17). In particular, such results
show that under certain conditions related to the preservation of symmetry,
_ALFs can only have transcritical singularities on the consensus space_ , and
thus the possible canard solutions are induced by such singularities. Along
this paper, we make a strong connection between dynamic behaviour and symmetry
properties of the networked system (1). In that regard, the cell-network
formalism [19, 21] is a prominent relevant framework. Indeed, in sections 3
and 7, we discuss the connections between ALFs and the cell-network formalism.

The paper is structured as follows: in section 2 we define the (unperturbed)
ALF, and we exploit some general properties. Section 3 is dedicated to
understanding the role of symmetries in ALFs. Later, in section 4 we introduce
a perturbation term leading to (1). As we will see, the perturbations, in
general, give rise to a slow-fast behaviour of the system. A brief summary of
the main ideas of singular perturbation theory useful for the main analysis
can be found in appendix A; while the slow-fast consensus dynamics of ALFs
with complete graph structure is studied in section 5. In order to highlight
our results, in section 6, we show some numerical examples. A brief discussion
and the conclusions are drawn in section 7.

##  2 Definitions and preliminaries

From a mathematical point of view, a network structure can be described by a
graph, $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, which consists of a finite
set of nodes (or vertices) $\mathcal{V}=\\{1,...,n\\}$, and a set of edges
$e_{ij}\in\mathcal{E}$, where each $e_{ij}$ represents a connection between
the node $j$ and $i$ [18]. Furthermore, if we assign a set of weights
$\mathcal{W}$ to the graph $\mathcal{G}$, then we obtain a weighted graph,
$\mathcal{G}_{\textup{w}}=\\{\mathcal{V},\mathcal{E},\mathcal{W}\\}$. We
assume (unless otherwise stated) that the weights, $w_{ij}\in\mathcal{W}$, are
positive reals. Such a choice is natural both for the algebraic properties
that follow from it [4], and for the modelling perspective, since often the
weights represent some measure of distance, or strengths of the coupling [33].
In other contexts negative weights are also considered [41], for example in
models involving inhibition, or antagonism; in the present paper we do not
consider such situations. The edges and the weights are in one-to-one
correspondence, in fact, we assign to each edge $e_{ij}$ a weight $w_{ij}$. If
the network is _directed_ then in general $w_{ij}\neq w_{ji}$, otherwise for
an _undirected_ network we have $w_{ij}=w_{ji}$.

######  Definition 1.

A _simple graph_ is an undirected graph without self-loops or multiple edges.

Unless otherwise stated, we consider simple graphs. For weighted graphs, we
assume a simple unweighted structure, i.e.,
$\mathcal{G}_{\textup{w}}\setminus\mathcal{W}$ is a simple graph. Moreover, we
make no nomenclature distinction between graphs and weighted graphs with unit
weights, because the algebraic structures in such cases are exactly the same.

We now recall a few well-known concepts of graph theory [33, 6, 18]. A _path_
is a sequence of edges that joins a sequence of distinct vertices. A
_connected component_ of a graph is a sub-graph such that every pair of nodes
is connected by a path. A relevant class of graphs is given by the complete
graphs.

######  Definition 2.

A _complete graph_ with $n$ nodes, denoted by $K_{n}$, is a graph such that
for all pair of nodes there is an edge connecting them, i.e., $\forall
i,j\in\mathcal{V}\ \exists\ e_{ij}\in\mathcal{E}$.

Given a graph it is possible to define some algebraic structures on it, we
recall here the ones that are useful for the purpose of the paper.

  * •

The adjacency matrix, $A$, of a graph is the $n\times n$ matrix with
components $A_{ij}=w_{ij}$ if $e_{ij}\in\mathcal{E}$, and $A_{ij}=0$ if
$e_{ij}\notin\mathcal{E}$.

  * •

The degree matrix, $\Delta$, of a graph is the $n\times n$ diagonal matrix
with (diagonal) components $\Delta_{ii}=\sum_{j}w_{ij}$.

  * •

The Laplacian matrix, $L$, of a graph is the $n\times n$ matrix defined by
$L:=\Delta-A$.

For simple graphs, let us recall that the Laplacian matrix, $L$, is symmetric,
degenerate and semi-positive definite; moreover, the following proposition
holds [4].

######  Proposition 1.

The algebraic multiplicity of the $0$ eigenvalue of $L$, $\mu_{a}(0)$, is
equal to the number of connected components of the graph. Furthermore, the
vector $\mathbf{1}=(1,...,1)^{\intercal}$ is an eigenvector of $L$ with
eigenvalue $0$, i.e., $\mathbf{1}\in\textup{ker}(L)$.

We now introduce the class of systems we are going to investigate in the
present paper. Let us assign to each node of a graph, $i\in\mathcal{V}$, a
state $x_{i}\in\mathbb{R}$ and a function

$\displaystyle f_{i}:\mathbb{R}$ $\displaystyle\to\mathbb{R}$ (4)
$\displaystyle x_{i}$ $\displaystyle\mapsto f_{i}(x_{i}),$

which we call _response function_. So, given the set of nodes $\mathcal{V}$,
we obtain a state vector $\bm{x}=(x_{1},...,x_{n})^{\intercal}$, and a
response vector field
$F(\bm{x})=\left(f_{1}(x_{1}),...,f_{n}(x_{n})\right)^{\intercal}$. An
Absolute Laplacian Flow [42, 3] is defined by the equation

$\dot{\bm{x}}=-LF(\bm{x}).$ (5)

Moreover, we say that an ALF is _homogeneous_ if the response functions
satisfy the equalities

$f_{1}=\dots=f_{n}.$ (6)

######  Remark 1.

Due to proposition 1, the Laplacian decomposes in a direct sum
$L=L^{(1)}\oplus\dots\oplus L^{(\mu_{a}(0))}$, and therefore an ALF decomposes
in $\mu_{a}(0)$ indipendent systems. So, in general, we consider connected
graphs.

Note that if we choose the response vector field to be the identity, i.e.,
$F(\bm{x})=\bm{x}$, then (5) becomes the widely studied (linear) Laplacian
dynamics $\dot{\bm{x}}=-L\bm{x}$, [44, 43, 36]. Indeed, the ALF is a direct
nonlinear generalisation of the linear Laplacian dynamics [42, 3, 1, 24]. ALFs
have received considerably less attention than the linear Laplacian dynamics,
although they still have strong relations with consensus problems. For
example, ALFs have been considered as models for nonlinear communications
protocols in [47, 34].

Our objective is to exploit the symmetry properties of ALFs, and characterise
the near consensus behaviour for such systems under small perturbations.

###  2.1 General properties

ALFs have some generic properties deriving from the symmetric and degenerate
structure of the Laplacian.

######  Proposition 2.

ALFs are invariant under the transformation of the response vector field

$F(\bm{x})\mapsto F(\bm{x})+h(\bm{x})\mathbf{1},$ (7)

where $h(\bm{x}):\mathbb{R}^{n}\to\mathbb{R}$ is a $k$-differentiable
function, $k\geq 1$.

###### Proof.

Let $F^{(h)}(\bm{x})=F(\bm{x})+h(\bm{x})\mathbf{1}$. Since
$\mathbf{1}\in\textup{ker}(L)$, we have $LF^{(h)}(\bm{x})=LF(\bm{x})$. ∎

As a consequence of proposition 2 we have that the response vector field is
not unique, in fact there exist an infinite set of vector fields leading to
the same dynamics. This freedom can be used in practice to choose the
“simplest” response field. Such property reminds in some way a gauge freedom
[30], where one can choose among a class of response fields
$F^{(h)}(\bm{x})=F(\bm{x})+h(\bm{x})\mathbf{1}$ without affecting the
evolution equations (5).

Another important property of ALFs is the existence of a constant of motion.

######  Proposition 3.

Given an ALF (5), there exists a constant of motion $k$,

$k=\langle\mathbf{1},\bm{x}\rangle,$ (8)

where $\langle\cdot,\cdot\rangle$ is the usual Euclidean scalar product in
$\mathbb{R}^{n}$.

###### Proof.

The time evolution of $k$ is given by
$\dot{k}=\langle\mathbf{1},\dot{\bm{x}}\rangle$. Then we have
$\langle\mathbf{1},\dot{\bm{x}}\rangle=-\mathbf{1}^{\intercal}LF(\bm{x})=-(L\mathbf{1})^{\intercal}F(\bm{x})=0$.
So, $k$ is a constant (in time) value. ∎

Let us notice that the constant of motion $k$ is related to the arithmetic
mean of the states, $k=n\langle\bm{x}\rangle$, where
$\langle\bm{x}\rangle:=\langle\mathbf{1},\bm{x}\rangle/n$. So, we can
interpret the presence of the constant of motion $k$ as a conservation law for
$\langle\bm{x}\rangle$. More generally, every quantity of the form $ck$, with
$c$ a constant, is a constant of motion for an ALF.

######  Proposition 4.

The set of equilibria of an ALF is given by

$E=\\{\bm{x}\in\mathbb{R}^{n}\ |\ f_{1}(x_{1})=\dots=f_{n}(x_{n})\\}.$ (9)

###### Proof.

In order to have an equilibrium for (5) we need $F(\bm{x})\in\ker(L)$, which
means $f_{1}(x_{1})=\dots=f_{n}(x_{n})$, implying in turn that $\bm{x}$ is an
equilibrium if and only if $\bm{x}\in E$. ∎

######  Corollary 1.

Given a homogeneous ALF the set

$C=\\{\bm{x}\in\mathbb{R}^{n}\ |\ x_{1}=\dots=x_{n}\\}$ (10)

is a subset of equilibria of the system, i.e., $C\subseteq E$.

The set $C$ is the so-called _consensus_ space [25, 36, 39]. Notice that for
heterogeneous ALFs we can expect clustering, for example, if
$f_{1}=f_{2}=\dots=f_{n/2}\neq f_{n/2+1}=\dots=f_{n}$, then two clusters of
equilibria arise. The present work is mainly concerned with homogeneous ALF,
for which we have a global consensus space. Although this assumption might
seem restrictive, we will see that the nonlinearity of the response function
can lead to nontrivial structures of equilibria.

As we already mentioned, one of our goals is to characterise the behaviour
near the consensus space (10). Our characterisation aims to be qualitative, in
the sense of _topological equivalence_ [22, 48].

######  Definition 3.

Two vector fields, $X,Y$, are said to be topologically equivalent if there
exists a homeomorphism which takes orbits of $X$ to orbits of $Y$, preserving
directions but not necessarily time parametrisation.

It is known that for linear Laplacian dynamics, systems associated with
weighted graphs are topologically equivalent to the corresponding unweighted
ones [7]. We show, for the nonlinear case, that weighted homogeneous ALFs are
topologically equivalent to simple homogeneous ALFs near the consensus space.

######  Proposition 5.

Let the response function $f$ be fixed. Then, in a neighbourhood of the
consensus space, a homogeneous ALF with graph structure
$\mathcal{G}_{\textup{w}}$ is topologically equivalent to an ALF with simple
graph structure $\mathcal{G}$.

###### Proof.

Let $L_{\textup{w}}$ be the Laplacian of a weighted graph
$\mathcal{G}_{\textup{w}}$. Then the Jacobian of the associated homogeneous
ALF on the consensus space is given by
$-L_{\textup{w}}\textup{D}F(\bm{x}^{*})$, where $\textup{D}F(\bm{x}^{*})$ is
the Jacobian of $F(\bm{x})$ evaluated at $C$. By setting all the weights to
one on $\mathcal{G}_{\textup{w}}$ we obtain the simple graph $\mathcal{G}$.
So, for the homogeneous ALF with unweighted graph $\mathcal{G}$ the Jacobian
is $-L\textup{D}F(\bm{x}^{*})$, where $L$ is the Laplacian matrix associated
to $\mathcal{G}$.

Let us recall that the Laplacian matrix is a semi-positive definite matrix
[4], which means that its eigenvalues are all greater than or equal to zero,
moreover such property holds for both simple graphs and positive-weighted
graphs. Then, the signs of the eigenvalues of $L_{\textup{w}}$ and $L$ are the
same, compactly we write
$\operatorname{sgn}(\operatorname{spec}(L_{\textup{w}}))=\operatorname{sgn}(\textup{spec}(L))\geq
0$. Now, since for $\bm{x}^{*}\in C$ it holds that
$x^{*}_{1}=\dots=x^{*}_{n}:=x^{*}$, we have
$\textup{D}F(\bm{x}^{*})=\operatorname{Id}\textup{d}_{x}f(x^{*})$, where
$\textup{d}_{x}f$ is a short notation for the derivative of $f$ with respect
to $x$, and $\operatorname{Id}$ is the identity matrix. So, the eigenvalues of
the ALFs weighted and unweighted, at the consensus space $C$, are given by
$\textup{d}_{x}f(x^{*})\operatorname{spec}(L_{\textup{w}})$ and
$\textup{d}_{x}f(x^{*})\textup{spec}(L)$ respectively. Let us first consider
$\textup{d}_{x}f(x^{*})\neq 0$. Since we are assuming connectedness, the
Laplacians have one zero eigenvalue associated with the eigenvector
$\mathbf{1}$ (independently of the weights), see proposition 1. The
eigendirection associated with the zero eigenvalue points along the consensus
space, which is a line of equilibria. So, along the nonhyperbolic
eigendirection $\mathbf{1}$ the two systems are equivalent. Therefore
considering a Lyapunov–Schmidt reduction, along the other directions the
topological equivalence follows form Hartman-Grobman theorem [22]. It is clear
that when $\textup{d}_{x}f(x^{*})=0$ the nonhyperbolic nature is entirely
determined by $\textup{d}_{x}f(x^{*})$, which is a common factor of the two
systems. ∎

Having in mind proposition 5, from now on we assume that homogeneous ALFs have
all weights equal to one, i.e., they are defined by a simple (unweighted)
graph $\mathcal{G}$. In general, we also assume homogeneity of the response
vector field without referring to it, but mentioning only the graph structure.

The results we stated until this point rely mainly on the algebraic properties
of the Laplacian. For simple graphs it is particularly useful to exploit the
symmetry properties induced by the graph, which in turn can provide insightful
information on the ALF.

##  3 Symmetries

Before introducing the role of symmetries in ALFs, it is worth recalling some
basic notions about group theory [38, 17].

######  Definition 4.

Let $\Gamma$ be a set and $\star:\Gamma\to\Gamma$ a binary operation. A group
is a couple $(\Gamma,\star)$ satisfying the following properties

  1. 1.

Associativity:
$(\gamma_{i}\star\gamma_{j})\star\gamma_{k}=\gamma_{i}\star(\gamma_{j}\star\gamma_{k})$,
$\forall\gamma_{i},\gamma_{j},\gamma_{k}\in\Gamma$.

  2. 2.

Identity: $\exists\ e\in\Gamma$ such that $e\star\gamma=\gamma\star e=\gamma$,
$\forall\gamma\in\Gamma$.

  3. 3.

Inverse: $\forall\gamma\in\Gamma$, $\exists\ \gamma^{-1}\in\Gamma$ such that
$\gamma^{-1}\star\gamma=\gamma\star\gamma^{-1}=e$.

The operation $\star$ is called group operation. Note that, closure under the
group operation follows from the definition of the operation itself. Unless
strictly necessary, we refer to a group just by the set, e.g., $\Gamma$,
implicitly considering it equipped with a group operation; we also omit the
group operation symbol, e.g.,
$\gamma_{i}\gamma_{j}=\gamma_{i}\star\gamma_{j}$.

######  Definition 5.

A group is said to be finite if the set $\Gamma$ has a finite number of
elements. The order of a finite group, $\textup{Ord}(\Gamma)$, is the number
of elements in the group.

Given a vector space, a representation, $\psi$, is a homomorphism from the
group to the general linear group of the vector space. Since the states,
$\bm{x}$, of an ALF belong to $\mathbb{R}^{n}$, we are interested on how
$\Gamma$ acts on $\mathbb{R}^{n}$, therefore we consider representations
$\psi:\Gamma\to GL(\mathbb{R}^{n})$. To avoid new unnecessary notation, we
omit to explicitly write the representation. It is implicit that when we write
a group element applied to a vector we are considering the representation on
the appropriate vector space, e.g., $\gamma\bm{x}=\psi(\gamma)\bm{x}$.

######  Definition 6.

Given two groups $\Gamma^{(1)}$, $\Gamma^{(2)}$, the direct product
$\Gamma^{(1)}\times\Gamma^{(2)}$ is a group with set of elements given by the
Cartesian product

$\Gamma^{(1)}\times\Gamma^{(2)}=\left\\{(\gamma^{(1)},\gamma^{(2)})\ |\
\gamma^{(1)}\in\Gamma^{(1)},\gamma^{(2)}\in\Gamma^{(2)}\right\\},$ (11)

and operation defined as follows

$(\gamma^{(1)}_{i},\gamma^{(2)}_{j})(\gamma^{(1)}_{k},\gamma^{(2)}_{l})=(\gamma^{(1)}_{i}\gamma^{(1)}_{k},\gamma^{(2)}_{j}\gamma^{(2)}_{l}),$
(12)

$\forall\gamma_{i,k}^{(1)}\in\Gamma^{(1)}$ and
$\forall\gamma_{j,l}^{(2)}\in\Gamma^{(2)}$.

It follows that the direct product can be extended to an arbitrary number of
groups, $\Gamma^{(1)}\times\Gamma^{(2)}\times\dots\times\Gamma^{(n)}$. Another
important concept is the one of _fixed-point space_ , i.e., the set of points
that are kept fixed by the action of the group.

######  Definition 7.

Let $\Sigma\subseteq\Gamma$ be a subgroup of $\Gamma$. The fixed-point space
(under the action in $\mathbb{R}^{n}$) of the group $\Sigma$ is denoted by
$\operatorname{Fix}(\Sigma)$ and it is given by

$\operatorname{Fix}(\Sigma)=\left\\{\bm{x}\in\mathbb{R}^{n}\ |\
\sigma\bm{x}=\bm{x},\ \forall\sigma\in\Sigma\right\\}.$ (13)

A direct interplay between graph theory and group theory is given by the
graph’s automorphism group. For graphs the automorphism group is a subgroup of
the (finite dimensional) symmetric group $\mathfrak{S}_{n}$ [40], i.e., the
group of permutations of $n$ symbols.

######  Definition 8.

Given a (simple) graph $\mathcal{G}$, the automorphism group,
$\textup{Aut}(\mathcal{G})$, is the group of permutations which preserves the
adjacency structure of the vertices of $\mathcal{G}$.

######  Remark 2.

From definition 8 follows that a graph $\mathcal{G}$ has
$\textup{Aut}(\mathcal{G})=\Gamma$, with $\Gamma\subseteq\mathfrak{S}_{n}$, if
and only if $[A,\sigma]$, $\forall\sigma\in\Gamma$, where $[\cdot,\cdot]$
represent the commutator of matrices, i.e., $[A,\sigma]:=A\sigma-\sigma A$.
Consequently, the same statement holds if we substitute the adjacency matrix
with the Laplacian matrix.

In the following sections, we are going to look at how the group properties of
response functions and graphs affect the equilibria, and the dynamics of ALFs.

###  3.1 Equilibria of homogeneous ALFs

We have proven that the consensus space is always a subset of equilibria for a
homogeneous ALF. We now show that the group transformations leaving the
response function invariant induce new equilibria from the consensus space.

######  Definition 9.

Let $\Gamma$ be a group111Notice that this is not related to
$\textup{Aut}(\mathcal{G})$, we use the same symbol $\Gamma$ for different
groups in different context.. A function $f$ is called $\Gamma$-invariant if

$f\circ\Gamma=f,$ (14)

where $\circ$ denotes the composition of maps.

######  Proposition 6.

Let $f$ be a $\Gamma$-invariant response function. Then, the group
$\Gamma_{n}:=\underbrace{\Gamma\times\cdots\times\Gamma}_{n\textnormal{-times}}$
maps the consensus space $C$ to equilibria, i.e.,

$\Gamma_{n}(C)\subseteq E.$ (15)

###### Proof.

Let us consider $-LF(\Gamma_{n}\bm{x}^{*})$, where $\bm{x}^{*}\in C$. Notice
that $\Gamma\bm{x}^{*}\neq\bm{x}^{*}$ in general, but thanks to homogeneity of
the ALF and invariance of the response function, we have
$-LF(\Gamma_{n}\bm{x}^{*})=-LF(\bm{x}^{*})=0$, where in the last equality we
used the fact that the consensus space is an equilibrium. Then we have that
$\Gamma_{n}(C)$ is also a set of equilibria for the system. ∎

######  Remark 3.

From Proposition 6 it follows that any subgroup of $\Gamma_{n}$ maps $C$ to
equilibria.

The invariance of the response function leads to a richer structure of the
equilibria, similarly to what happens for equivariant bifurcations [20, 9].
Such a richer structure can give rise, for example, to _bipartite consensus_
[49, 3, 31], where the agents converge to two separate clusters of consensus.
A simple exemplification of appearance of bipartite consensus in ALFs is given
by considering the response function $f(x)=x^{2}$. It is clear that the
symmetry group for such a function is $\mathbb{Z}_{2}$, in particular we are
considering the representation $\\{1,-1\\}$. From Proposition 6 we have that
the set of equilibria consists of all the possible combinations $\pm x_{1}=\pm
x_{2}=\dots=\pm x_{n}$, and therefore we have bipartite consensus among the
potential equilibria for the system. Let us notice that the linear set-up for
bipartite consensus requires negative weights on the network structure in
order to obtain such result, while for ALFs bipartite consensus follows form
nonlinearity, and symmetry properties of the response function. It is
interesting to notice that the lines of bipartite consensus, $\pm x_{1}=\pm
x_{2}=\dots=\pm x_{n}$, arising from generic even response functions are
saddles; this property follows by straightforward computations of the
Jacobian.

###  3.2 Equivariant ALFs

The concept of symmetry, in dynamical systems theory, refers to the fact that
there exists a group mapping solutions to solutions. When a vector field is
given, such a concept can be translated in terms of equivariance of the vector
field.

######  Definition 10.

Let $\Gamma$ be a group, and $X$ a vector field. $X$ is called
$\Gamma$-equivariant if

$X\circ\Gamma=\Gamma\circ X,$ (16)

and $\gamma\in\Gamma$ is called a symmetry of the vector field.

######  Remark 4.

It is straightforward to see that if a system is $\Gamma$-equivariant, then
$\Gamma$ maps solutions to solutions.

A solution of a dynamical system is a simple example of an invariant set under
the flow. Invariant sets, in general, are extremely relevant in the study of
ODEs, since they allow to reduce the analysis to potentially simpler
equations. We recall here the general definition of invariant set under the
flow induced by a vector field. Later we state the interplay between
symmetries and invariant sets.

######  Definition 11.

Let $\phi$ be the flow generated by a vector field on $\mathbb{R}^{n}$. A set
$\mathcal{M}\subset\mathbb{R}^{n}$ is said to be invariant (respectively
positively/negatively invariant) under the flow if for every
$x\in\mathcal{M}$, $\phi_{t}(x)\in\mathcal{M}$, $\forall t\in\mathbb{R}$
(respectively $\forall t\in\mathbb{R}_{+}/\mathbb{R}_{-}$).

In order to keep a compact notation, we call a set invariant under the flow a
$\phi$-invariant set. From equivariance of the vector field we can immediately
infer the existence of a $\phi$-invariant set.

######  Proposition 7.

The fixed-point space, $\operatorname{Fix}(\Gamma)$, is a $\phi$-invariant set
for $\Gamma$-equivariant systems [20].

Since we are considering systems with a network structure, the symmetry group
associated to the equivariance properties will be a subgroup of the symmetric
group. Moreover, if an ALF is $\mathfrak{S}_{n}\supseteq\Gamma$-equivariant it
follows that it is also equivariant under any subgroup $\Sigma\subset\Gamma$.
Therefore, also $\operatorname{Fix}(\Sigma)$ is a $\phi$-invariant set, and
$\operatorname{Fix}(\Gamma)\subseteq\operatorname{Fix}(\Sigma)$. From such
considerations, if we are considering a $\Gamma$-equivariant ALF, the analysis
of the dynamics can be reduced to the fixed-point spaces
$\operatorname{Fix}(\Sigma)$, for all $\Sigma$ subgroups of $\Gamma$.

Let us now consider the particular case of homogeneous ALFs with a complete
graph structure, we shortly call them $K_{n}$-ALFs. We notice that
$\textup{Aut}(K_{n})=\mathfrak{S}_{n}$. As a consequence, we show that
$K_{n}$-ALFs are $\mathfrak{S}_{n}$-equivariant. Considering that the phase
space of a $K_{n}$-ALF is $\mathbb{R}^{n}$, when acting on a state of a
$K_{n}$-ALF we use the permutation representation of $\mathfrak{S}_{n}$ in
$\mathbb{R}^{n}$. An element of the permutation representation is called
permutation matrix.

######  Definition 12.

A permutation matrix $\sigma\in\mathfrak{S}_{n}$ is an $n\times n$ matrix with
components

$\sigma_{ij}=\begin{cases}1&\quad\text{if $\sigma$ permutes the $i$-th element
with the $j$-th element },\\\ 0&\quad\text{otherwise}.\\\ \end{cases}$ (17)

######  Proposition 8.

Let $L$ be the Laplacian matrix of a $K_{n}$ graph. Then $[L,\sigma]=0$ for
all $\sigma\in\mathfrak{S}_{n}$, where $[\cdot,\cdot]$ is the commutator.

###### Proof.

The statement follows from the graph completeness and edges
indistinguishability. ∎

Equivalently, thanks to orthogonality of permutations, proposition 8 can be
written as $\sigma^{\intercal}L\sigma=L$; in order words, the Laplacian is
invariant under conjugacy by permutations.

######  Proposition 9.

$K_{n}$-ALFs are $\mathfrak{S}_{n}$-equivariant.

###### Proof.

We need to prove that $LF(\sigma\bm{x})=\sigma LF(\bm{x})$, for all
$\sigma\in\mathfrak{S}_{n}$. Let us start by considering $\sigma LF(\bm{x})$.
Thanks to proposition 8 we have that $\sigma LF(\bm{x})=L\sigma F(\bm{x})$.
Next, the permutation applied to the vector field
$F(\bm{x})=(f(x_{1}),\dots,f(x_{n}))^{\intercal}$ exchanges the order of the
components; since the functions are indistinguishable, we have that $\sigma
F(\bm{x})=F(\sigma\bm{x})$. Therefore, $\sigma LF(\bm{x})=LF(\sigma\bm{x})$,
which proves the statement. ∎

######  Remark 5.

In virtue of homogeneity of the response functions, the action of
$\mathfrak{S}_{n}$ on the (nonlinear) vector field $F(\bm{x})$ is linear.

######  Proposition 10.

The fixed-point space of the symmetric group is the consensus space, i.e.,
$\operatorname{Fix}(\mathfrak{S}_{n})=C$.

###### Proof.

We proceed by contradiction. Suppose that $\bm{x}\notin C$, and
$\bm{x}\in\operatorname{Fix}(\mathfrak{S}_{n})$. From definition 7, we have
$x_{\sigma(i)}=x_{i}$, $i=1,\dots,n$, $\forall\sigma\in\mathfrak{S}_{n}$. From
the fact that $\textup{Ord}(\mathfrak{S}_{n})=n!\geq n$, we obtain
$x_{1}=\dots=x_{n}$, i.e., $\bm{x}\in C$ which is a contradiction. ∎

We established a link between $K_{n}$-ALFs and equivariant systems. Let us
notice that, if we have a dynamical system equivariant under a finite group,
such dynamical system can be seen as a _cell-network_ [19]. A cell-network is
a system of differential equations equivariant under a group(oid), where the
group(oid) is the automorphism group(oid) of a (multi)graph. So, there is a
strict correspondence between cell-networks and the ALF framework. For
example, a $K_{n}$-ALF is a cell-network with $K_{n}$ graph structure. This
alternative point of view, provides an extra tool in the analysis of ALFs.
Indeed, from proposition 7 follows that $\operatorname{Fix}(\mathfrak{S}_{n})$
is $\phi$-invariant, and so, in turn, the consensus space, is an invariant
space for the $K_{n}$-ALFs, or in the language of cell-networks it is a
pattern of synchrony, which is already known from corollary 1. Actually,
corollary 1, not only holds for all homogeneous ALFs, but also tells us that
the consensus space is an equilibrium for the system, not just an invariant
set. However, proposition 10 does not rely on the algebraic structure of the
Laplacian, it is a general result for $\mathfrak{S}_{n}$-equivariant systems.
Let us notice that, the results following from equivariance are, in general,
more robust than the algebraic ones. Therefore, the symmetry perspective could
be very useful when some transformation is applied to the system.

In general, a simple (connected) graph with $n$ nodes can be regarded as a
subgraph of $K_{n}$. As such, a generic simple graph has a subgroup of
$\mathfrak{S}_{n}$ as automorphism group. So, a question that we can ask is:
what are the subgroups of $\mathfrak{S}_{n}$ having $C$ as a fixed-point
space?

######  Proposition 11.

Let $\Gamma$ be a subgroup of $\mathfrak{S}_{n}$, if $\textup{Ord}(\Gamma)\geq
n$ then $\operatorname{Fix}(\Gamma)=C$.

###### Proof.

If $\textup{Ord}(\Gamma)\geq n$, then by considering the equations
$\Gamma\bm{x}=\bm{x}$, we have at least $n$ independent equalities that
constrain all the components of $\bm{x}$ to be equal. Therefore
$\operatorname{Fix}(\Gamma)=C$. ∎

When $\textup{Ord}(\Gamma)<n$ then $\dim\operatorname{Fix}(\Gamma)>1$, so the
fixed-point space is related to clustering, and
$C\subset\operatorname{Fix}(\Gamma)$; we illustrate such a case in the
following second example. Let us first consider the cycle graph $C_{n}$, for
which $\textup{Aut}(C_{n})=\mathbb{Z}_{n}$, where $\mathbb{Z}_{n}$ is the
cyclic group. Since $\textup{Ord}(\mathbb{Z}_{n})=n$, then
$\operatorname{Fix}(\mathbb{Z}_{n})=C$; this can be practically seen by
applying a generator of $\mathbb{Z}_{n}$ to $\bm{x}$, e.g.,

$\displaystyle x_{1}$ $\displaystyle=x_{n}$ (18) $\displaystyle x_{2}$
$\displaystyle=x_{1}$ (19) $\displaystyle\vdots$ (20) $\displaystyle x_{n}$
$\displaystyle=x_{n-1},$ (21)

implying $x_{1}=\dots=x_{n}$. As a second example, we consider the path graph
$P_{n}$ ($n\geq 2$) which has $\mathbb{Z}_{2}$ as group of symmetry. In this
case $\textup{Ord}(\mathbb{Z}_{2})=2$, therefore $P_{n}$ has the consensus as
fixed-point space only if $n=2$. In fact, if we consider the generic case we
can see that the equalities imposed by $\mathbb{Z}_{2}\bm{x}=\bm{x}$ are

$\displaystyle x_{1}$ $\displaystyle=x_{n}$ (22) $\displaystyle x_{2}$
$\displaystyle=x_{n-1}$ (23) $\displaystyle\vdots$ (24) $\displaystyle
x_{n-1}$ $\displaystyle=x_{2}$ (25) $\displaystyle x_{n}$
$\displaystyle=x_{1},$ (26)

which do not lead to consensus in general, but to clustering
$\\{\bm{x}\in\mathbb{R}^{n}\ |\ x_{1}=x_{n},\dots,x_{n}=x_{1}\\}$.

In summary, in this section we outlined a different perspective on the
consensus space. Let us recall that a homogeneous ALF, near the consensus
space, is equivalent to an ALF with simple graph structure (proposition 5).
So, the ALF acquires the property of equivariance under the automorphism group
of the (simple) graph; the symmetry group is a subgroup of the symmetric
group. Therefore, an ALF possesses also a cell-network structure. At this
point, forgetting momentarily the specific properties of ALFs, we can study
the connections between the consensus space and the symmetry properties. It
turns out that the consensus space arises as an invariant space for a class of
subgroups of $\mathfrak{S}_{n}$ (proposition 11).

We conclude this section recalling a theorem highlighting the fundamental role
of the graph structure in the definition of a dynamic network. Although, as we
have seen, graphs and groups are intimately related, if we consider the logic
flow of definitions for dynamic networks the graph structure should be defined
first.

######  Theorem 1 (Frucht [16]).

For any finite group $\Gamma$ there exists a finite graph $\mathcal{G}$ such
that $\textup{Aut}(\mathcal{G})\cong\Gamma$.

Theorem 1, ensures that for any given finite group we can construct a graph
which is invariant under the group. However, uniqueness is not provided, au
contraire, in general there are infinitely many graphs with such property [6].
For this reason, the definition of the graph structure of a dynamic network
precedes the group properties.

##  4 Perturbations

A perturbation is a small ‘change’ of the differential equations governing the
system. The order of magnitude can be quantified by a parameter $\epsilon$,
$0<\epsilon\ll 1$. A perturbation is called _singular_ if the solutions of the
perturbed ODEs have a different qualitative behaviour with respect to the
solutions of the unperturbed system [46]. On the other hand, _regular_
perturbations are the ones that preserve the qualitative structure. For this
reason, singular perturbations are the most interesting if we aim to
characterise qualitatively a system. In appendix A, we briefly review some of
the standard terminology of singular perturbations.

###  4.1 Perturbed ALFs

As already mentioned in the introduction, we now consider perturbations of
ALFs given by:

$\dot{\bm{x}}=-LF(\bm{x})+\epsilon H(\bm{x},\Lambda),$ (27)

where $0<\epsilon\ll 1$,
$H(\bm{x},\Lambda)=(h_{1}(\bm{x},\Lambda),\dots,h_{n}(\bm{x},\Lambda))^{\intercal}$,
and $\Lambda$ is a set of parameters.

Since we are considering homogeneous ALF, the consensus space $C$ is always an
set of equilibrium points for $\epsilon=0$ (corollary 1). Moreover, the
consensus set is a one-dimensional space, then generic perturbations are
singular [46]. The system as written in (27) is in a non-standard form. In
many cases, it is useful to transform a singularly perturbed system into a
standard form, one reason among all is that a large amount of results in
singular perturbation theory are stated for standard forms.

######  Proposition 12.

Given a perturbed ALF in the non-standard form (27), then the coordinate
transformation

$x_{l}\left(k,x_{j\neq l}\right)=k-\sum_{j\neq l}x_{j},$ (28)

where $l\in\left\\{1,\dots,n\right\\}$ is arbitrary, and $k$ is the constant
of motion (8), transforms the system into the standard form

$\displaystyle\dot{x}_{i}$ $\displaystyle=-\sum_{j\neq
l}L_{ij}f(x_{j})-L_{il}f\left(x_{l}\left(k,x_{j\neq l}\right)\right)+\epsilon
g_{i}(x_{j\neq l},k,\Lambda)\quad i\neq l$ (29) $\displaystyle\dot{k}$
$\displaystyle=\epsilon\sum_{j}g_{j}(x_{j\neq l},k,\Lambda),$

where $g_{i}(x_{j\neq
l},k,\Lambda):=h_{i}\left(x_{1},\dots,x_{l}\left(k,x_{j\neq
l}\right),\dots,x_{n},\Lambda\right)$.

###### Proof.

Performing the coordinate change (28) into (27) we obtain the set of equations
for $x_{i}$, $i\neq l$. Then, the equation for $k$ is retrieved evaluating the
scalar product $\langle\mathbf{1},\dot{\bm{x}}\rangle$. ∎

Let us notice that the perturbation transforms the constant of motion into a
slow variable of the system. Since the constant $k$ is related to the
arithmetic mean of the states, $\langle\bm{x}\rangle$, perturbing the system,
in general, means that the states’ density is changing, e.g., the
concentration of the system is increasing/decreasing. In other words, the
perturbation acts as a drift/dissipation in the undergoing diffusive
evolution.

######  Lemma 1.

If $\langle\mathbf{1},H(\bm{x},\Lambda)\rangle=0$, then equation (27) is a
regular perturbation problem.

###### Proof.

If $\langle\mathbf{1},H(\bm{x},\Lambda)\rangle=0$ then $\dot{k}=0$. So, the
standard form (29) is no more a slow-fast system, and therefore the
perturbation is regular. ∎

######  Remark 6.

Notice that lemma 1 holds also for heterogeneous ALFs.

As we already mentioned, we are interested in the singular perturbations
because they are the ones that could lead to a different qualitative behaviour
of the system. In the following section, we look at the near consensus
behaviour under a singular perturbation, and we characterise the dynamics for
a selected class of systems.

##  5 Consensus dynamics

In this section, we further specialise our analysis by studying the consensus
dynamics of (homogeneous) ALFs with a $K_{n}$ graph structure, already
introduced in section 3.2 for their symmetry properties.

###  5.1 The layer problem

Let us consider a perturbed $K_{n}$-ALF in the standard form. The layer
problem, obtained from (29) by setting $\epsilon$ to zero, reads

$\dot{x}_{i}=-\left[\sum_{j\neq
l}L_{ij}f(x_{j})+L_{il}f\left(x_{l}\left(k,x_{j\neq
l}\right)\right)\right]\quad i\neq l.$ (30)

The system described by (30) is $(n-1)$-dimensional with one parameter, $k$.

It is worth noting that although the graph structure in (30) seems to be lost,
some information is still readily available. On the one hand, the term
$\sum_{j\neq l}L_{ij}f(x_{j})$ still corresponds to a $K_{n-1}$-ALF, i.e., on
a complete graph with $n-1$ nodes. On the other hand, the scalar term
$L_{il}f(x_{l})$, where in fact $x_{l}$ is a function of all other nodes and
the constant $k$, can be interpreted as a higher-order interaction [5]
naturally arising from the reduction performed.

Furthermore, and for our specific purposes, there is a more relevant property
that survives the transformation: the symmetry. In fact, starting with a
$\mathfrak{S}_{n}$-equivariant unperturbed system, after the transformation
into the standard form we get a layer problem which is
$\mathfrak{S}_{n-1}$-equivariant. This is a first example of robustness of the
symmetries. Since we are interested in the consensus dynamics, we consider
equation (30) on the invariant space given by
$\operatorname{Fix}(\mathfrak{S}_{n-1})$. Let us notice that, in general, the
invariant space $\operatorname{Fix}(\mathfrak{S}_{n-1})$ is not globally
stable. For example, in a neighbourhood of an attracting branch of the
consensus space it is stable. Instead, in a neighbourhood of a repelling part
of the consensus space it is unstable. This means that the restriction to
$\operatorname{Fix}(\mathfrak{S}_{n-1})$ provides a robust description of the
system only in the basin of attraction of a stable region of
$\operatorname{Fix}(\mathfrak{S}_{n-1})$; in the unstable regions the
restriction works only if the initial conditions lie exactly in the invariant
space and if the perturbation preserves
$\operatorname{Fix}(\mathfrak{S}_{n-1})$. Then, restricting equation (30) on
$\operatorname{Fix}(\mathfrak{S}_{n-1})$ we obtain

$\dot{x}=-\left[f(x)-f(k-(n-1)x)\right],$ (31)

where we used the fact that for a $K_{n}$-graph $L_{ii}=n-1$, and $L_{i\neq
j}=-1$. So, in the invariant space the system reduces to a one dimensional
system (with one parameter). The former consensus space,
$C=\operatorname{Fix}(\mathfrak{S}_{n})$, gives the equilibria of (31). This
is a consequence of the fact that the layer problem is a coordinate
transformation of the unperturbed non-standard form (5), for which we know the
consensus space is a set of equilibria.

######  Proposition 13.

The consensus states of the form

$x^{*}_{k}=\frac{k}{n}$ (32)

are equilibria of (31).

###### Proof.

Inserting $x^{*}_{k}$ into (31) we obtain $\dot{x}=0$, which proves that
$x^{*}_{k}$ is an equilibrium. To prove that it correspond to consensus, we
recall that, since we are in the invariant space
$\operatorname{Fix}(\mathfrak{S}_{n-1})$, then $n-1$ states are already equal
to each other. The only state left to check is the one given by (28), and by
inserting $x_{j\neq l}=k/n$, we obtain $x_{l}=k/n$, which proves the
statement. ∎

######  Remark 7.

Recalling that $k=n\langle\bm{x}\rangle$, we have that the equilibrium is
exactly the mean value of the states of the system, $\langle\bm{x}\rangle$.

Since the equilibria of the layer equation constitute the critical manifold
for the slow-fast system, then the consensus space is a (part of the) critical
manifold of the system. We use the notation
$\bm{x}^{*}_{k}:=(x^{*}_{k},\dots,x^{*}_{k})$ for the points in $C$ having the
form (32).

######  Proposition 14.

The stability of the consensus (32) is determined by the derivative of the
response function. In particular, we can decompose the consensus into three
components,

$\displaystyle C^{a}=\left\\{\bm{x}^{*}_{k}\in C\ |\
\textup{d}_{x}f(x^{*}_{k})>0\right\\},$ (33) $\displaystyle
C^{r}=\left\\{\bm{x}^{*}_{k}\in C\ |\ \textup{d}_{x}f(x^{*}_{k})<0\right\\},$
$\displaystyle C^{s}=\left\\{\bm{x}^{*}_{k}\in C\ |\
\textup{d}_{x}f(x^{*}_{k})=0\right\\},$

where $C^{a}$ is attracting (stable), $C^{r}$ is repelling (unstable), and
$C^{s}$ is singular.

###### Proof.

By straightforward computations we obtain the Jacobian of the system evaluated
at the consensus, $-\textup{d}_{x}f\left(\frac{k}{n}\right)n$, which implies
the proposition. ∎

Thanks to theorem 3 (Fenichel), the regions $C^{a/r}$ are diffeomorphic to
invariant regions of the singularly perturbed system (29). For the
singularities $C^{s}$ we need further analyses. We are going to consider the
case of _transcritical_ singularities which, as we will show, turn out to be
generic due to the geometric properties of the problem.

###  5.2 The slow-fast dynamics around the consensus space

We start by considering some symmetry condition on the perturbation, in order
to preserve the restriction to the invariant space.

######  Lemma 2.

Let $H(\bm{x},\Lambda)$ be a perturbation such that $h_{i}=h_{j}:=h$, $\forall
i,j\neq l$, and $h_{l}=:\tilde{h}$. Then,
$\operatorname{Fix}(\mathfrak{S}_{n-1})$ is an invariant space for the
perturbed system (29).

Assuming a perturbation satisfying lemma 2, we are allowed to continue our
analysis on the invariant space $\operatorname{Fix}(\mathfrak{S}_{n-1})$.
Indeed, under the chosen perturbation the perturbed system (29) becomes
$\mathfrak{S}_{n-1}$-equivariant, preserving the invariant space
$\operatorname{Fix}(\mathfrak{S}_{n-1})$. We remark once again that, in a
neighbourhood of a stable regions of $\operatorname{Fix}(\mathfrak{S}_{n-1})$,
the reduction is robust; while in unstable regions the reduction is preserved
only by fine tuning of initial conditions and perturbation. For the perturbed
system the invariant space reads
$\operatorname{Fix}(\mathfrak{S}_{n-1})=\\{(x,k)\in\mathbb{R}^{2}\ |\
x:=x_{i}=x_{j},\ \forall i,j\neq l\\}$, which we call $(x,k)$-plane. Then, the
perturbed equations (29) evaluated on the $(x,k)$-plane, take the form

$\displaystyle\dot{x}$ $\displaystyle=-\left[f(x)-f(k-(n-1)x)\right]+\epsilon
g(x,k,\Lambda)$ (34) $\displaystyle\dot{k}$
$\displaystyle=\epsilon(n-1)g(x,k,\Lambda)+\epsilon\tilde{g}(x,k,\Lambda),$

where $g(x,k,\Lambda):=h(x,\dots,k-(n-1)x,\dots,x,\Lambda)$, and
$\tilde{g}(x,k,\Lambda):=\tilde{h}(x,\dots,k-(n-1)x,\dots,x,\Lambda)$.

We now state the conditions for (nondegenerate) transcritical singularities.

######  Proposition 15.

Let $\bm{x}^{s}_{k}\in C^{s}$ be a singular consensus point, and assume that
lemma 2 holds. If

$\displaystyle\textup{d}_{x}^{2}f(x^{s}_{k})$ $\displaystyle\neq 0,$ (35)
$\displaystyle n$ $\displaystyle\neq 2,$ (36)
$\displaystyle\langle\mathbf{1},H(\bm{x}^{s}_{k},\Lambda)\rangle$
$\displaystyle\neq 0,$ (37)

then $\bm{x}^{s}_{k}$ is a transcritical singularity of (34)

###### Proof.

We check the conditions for transcritical singularities on planar systems
provided by Krupa and Szmolyan [27]. Given a planar system of the form (67)
then the origin is a transcritical singularity if

$\displaystyle f(0,0,0)$ $\displaystyle=0,$ (38)
$\displaystyle\partial_{x}f(0,0,0)$ $\displaystyle=0,$ (39)
$\displaystyle\partial_{y}f(0,0,0)$ $\displaystyle=0,$ (40)
$\displaystyle\begin{vmatrix}\partial^{2}_{x}f(0,0,0)&\partial^{2}_{xy}f(0,0,0)\\\
\partial^{2}_{yx}f(0,0,0)&\partial^{2}_{y}f(0,0,0)\end{vmatrix}$
$\displaystyle<0,$ (41) $\displaystyle\partial^{2}_{x}f(0,0,0)$
$\displaystyle\neq 0,$ (42) $\displaystyle g_{0}:=g(0,0,0)$ $\displaystyle\neq
0.$ (43)

We now prove that these conditions applied to our system lead to the
conditions of the statement. Conditions (38), (39), (40) are satisfied because
$x^{s}_{k}$ is a singular consensus point. For the non-degeneracy condition
(41), we obtain

$\textup{d}_{x}^{2}f(x^{s}_{k})\neq 0.$ (44)

The transversality condition (42) gives,

$n(n-2)\textup{d}_{x}^{2}f(x^{s}_{k})\neq 0,$ (45)

which implies $n\neq 2$. Finally, the last condition (43) leads to
$\langle\mathbf{1},H(\bm{x}^{s}_{k},\Lambda)\rangle\neq 0$. ∎

As previously mentioned, we consider the case where all the singularities on
the consensus space are transcritical, i.e., the conditions of proposition 15
are met on $C^{s}$. Notice, however, that at least the first two conditions of
proposition 15 are rather mild, we in fact argue about genericity later. The
first says that the singularity is non-degenerate, while the second refers to
networks of at least three nodes. So, in our current setting, the consensus
space consists of a cascade of transcritical singularities changing the
stability from attracting to repelling, and vice-versa. If at $x^{s}_{k}$ the
stability transition is from attracting to repelling, we call $x^{s}_{k}$:
_type-1 transcritical point_. Type-1 transcritical points are characterised by
the sign ratio

$\rho:=\frac{\operatorname{sgn}\left(\textup{d}_{x}^{2}f(x^{s}_{k})\right)}{\operatorname{sgn}\left(\langle\mathbf{1},H(\bm{x}^{s}_{k},\Lambda)\rangle\right)}=-1.$
(46)

If at $x^{s}_{k}$ the stability transition is from repelling to attracting, we
call $x^{s}_{k}$: _type-2 transcritical point_. Type-2 transcritical points
are characterised by the sign ratio $\rho=1$.

At this point, we are going to study the continuation of the consensus space
after a transcritical point. A trajectory reaching a transcritical point could
proceed in three different manners:

  1. 1.

Being attracted to the stable branch of the critical manifold (a curve of
equilibria crossing the consensus),

  2. 2.

Being repelled along the fast flow,

  3. 3.

Continuing along the consensus space (even if it is unstable).

The third characteristic behaviour is called _canard_ , and acts as a
separating case between the other two. Usually, the presence of a canard in a
planar system is associated to a critical value for the parameters of the
system. We are going to show that for ALFs the condition for canards is
geometrical in nature, and can be stated as an algebraic condition on the
perturbation, without specific requirements on the system’s parameters.

######  Proposition 16.

Let $\bm{x}^{s}_{k}\in C^{s}$, if the perturbation satisfies the property

$H(\bm{x}^{s}_{k},\Lambda)\propto\mathbf{1},$ (47)

then, for $\epsilon$ sufficiently small, the system (29) admits canard
solutions.

We call _critical_ , $H^{\textup{crit}}(\bm{x},\Lambda)$, a perturbation
satisfying such a property.

###### Proof.

We use again the results from [27] to check that our statement holds.
Considering a system of the form (67), satisfying conditions (38)-(43), then
there exist a parameter $\lambda$ controlling the behaviour at the
transcritical singularity (which we remind to be at the origin for this
benchmark system). The parameter $\lambda$ is given by

$\lambda=\frac{1}{|g_{0}|\sqrt{\beta^{2}-\gamma\alpha}}(\delta\alpha+g_{0}\beta),$
(48)

where

$\displaystyle\alpha=\frac{1}{2}\partial^{2}_{x}f(0,0,0),$
$\displaystyle\qquad\beta=\frac{1}{2}\partial^{2}_{xy}f(0,0,0),$
$\displaystyle\gamma=\frac{1}{2}\partial^{2}_{y}f(0,0,0),$
$\displaystyle\qquad\delta=\partial_{\epsilon}f(0,0,0).$

So, the parameter $\lambda$ for the system (34), at a transcritical point
$x_{k}^{s}$, reads

$\lambda=-\rho\frac{h(\bm{x}^{s}_{k},\Lambda)+(n-1)\tilde{h}(\bm{x}^{s}_{k},\Lambda)}{\tilde{h}(\bm{x}^{s}_{k},\Lambda)+(n-1)h(\bm{x}^{s}_{k},\Lambda)}.$
(49)

For type-1 transcritical points, we have $\rho=-1$, and the condition for
canards is $\lambda=\lambda^{\textup{crit}}:=1$. So, we obtain the equation

$(n-2)\tilde{h}(\bm{x}^{s}_{k},\Lambda)=(n-2)h(\bm{x}^{s}_{k},\Lambda).$ (50)

Since the case $n=2$ is excluded by proposition 15, we have
$\tilde{h}(\bm{x}^{s}_{k},\Lambda)=h(\bm{x}^{s}_{k},\Lambda)$. Therefore the
critical perturbation $H^{\textup{crit}}(\bm{x},\Lambda)$ at $\bm{x}^{s}_{k}$
reads

$\displaystyle H^{\textup{crit}}(\bm{x}^{s}_{k},\Lambda)$
$\displaystyle=(h(\bm{x}^{s}_{k},\Lambda),\dots,h(\bm{x}^{s}_{k},\Lambda),\dots,h(\bm{x}^{s}_{k},\Lambda))^{\intercal}$
$\displaystyle=h(\bm{x}^{s}_{k},\Lambda)\mathbf{1},$

which agrees with the statement. In order to complete the proof, we show that
for a type-2 transcritical point a critical perturbation admits faux-canards.
Let us recall that faux-canards for planar systems are generic, and appear for
$\lambda<\lambda^{\textup{crit}}$ [27]. By using the fact that for a type-2
transcritical point $\rho=1$, we get $\lambda=-1$. ∎

In figure 1 we sketch the blow-up of the transcritical points under a critical
perturbation [27, 32, 26]. The blow-up picture makes clear the different
behaviour of the type-$1/2$ transcritical points and illustrates how the
continuation of the consensus space occurs.

$\bar{x}^{s,+}_{a}$
---
$\bar{x}^{s,-}_{r}$
---
$\bar{k}^{+}_{r}$
---
$\bar{x}^{s,-}_{a}$
---
$\bar{k}^{-}_{r}$
---
$\bar{x}^{s,+}_{r}$
---
$\bar{k}^{+}_{a}$
---
$\bar{k}^{-}_{a}$
---
$\text{Type}-1$
---
$\text{Type}-2$
---
Figure 1: Sketch of blown-up transcritical points on the consensus space. The
picture displays typical blow-up phase portraits for a type-$1$ transcritical
point (left), and a type-$2$ transcritical point (right); in both cases we are
considering a critical perturbation. For a detailed exposition on the blow-up
phase portraits for transcritical singularities see [27, 32]. The points
$\bar{x}^{s,-/+}_{a/r}$ correspond to the singularity on the consensus space
(solid lines), while the points $\bar{k}^{-/+}_{a/r}$ correspond to the
singularity on the branch of the critical manifold crossing the consensus
space (dashed lines). The labels $-/+$ indicate if we are below, or above, the
transcritical point, and the labels $a/r$ stand for attracting, and repelling,
respectively. The attracting curves of equilibria are shown in green, while
the repelling ones in red. The blue points are entering or exit points
depending on the direction of the curves drawn.

Let us notice that the critical perturbation preserves the full symmetry of
the system on the consensus space, i.e.,
$H^{\textup{crit}}(\bm{x}^{s}_{k},\Lambda)$ is $\mathfrak{S}_{n}$-equivariant.
Indeed, a similar result can be stated in terms of equivariant dynamical
systems theory.

######  Theorem 2.

Let $\mathcal{G}$ be a graph with
$\textup{Aut}(\mathcal{G})=\Gamma\subseteq\mathfrak{S}_{n}$, where
$\textup{Ord}(\Gamma)\geq n$. Let $H_{\Gamma}(\bm{x},\Lambda)$ be a
$\Gamma$-equivariant perturbation of a $\mathcal{G}$-ALF, such that
$H_{\Gamma}(\operatorname{Fix}(\Gamma),\Lambda)\neq 0$. Then the consensus
space is a trajectory of the perturbed system.

###### Proof.

From proposition 11 we have that $\operatorname{Fix}(\Gamma)=C$. So,
projecting equation (27) onto the invariant space $C$ we have

$\dot{\bm{x}}^{*}=\epsilon H_{\Gamma}(\bm{x}^{*},\Lambda),$ (51)

$\bm{x}^{*}\in C$. Since $H_{\Gamma}(\bm{x}^{*},\Lambda)\neq 0$,
$\forall\bm{x}^{*}\in C$, and $\dim(C)=1$, then $C$ is a trajectory for the
system. ∎

In essence, theorem 2 gives some conditions under which a perturbed
equivariant ALF (27) exhibits a maximal canard. Let us also notice that
theorem 2 holds for a class of $\Gamma$-equivariant ALFs where
$\textup{Ord}(\Gamma)\geq n$, and it does not make any particular assumption
on the singularities of the consensus space. This means that in possibly more
complicated cases where the singularities along the consensus space are not
transcritical, the consensus space is still a maximal canard under the
appropriate perturbation.

###  5.3 Polynomial response functions

Until this point, we considered a generic response function $f(x)$. Let us now
consider a more specific class of functions: polynomials. A polynomial
response function can be written as

$f(x)=\sum_{j=0}^{N}a_{j}x^{j},$ (52)

where $a_{j}$ are real constant coefficients, and $N\geq 2$. We know that the
consensus space, for a homogeneous ALF, does not depend on the particular
choice of the response function. On the other hand, the remaining equilibrium
sets, that together with $C$ give rise to the full critical manifold, depend
on the form of the response function. By choosing a polynomial form for $f$,
we are able to study in further detail the nature of the equilibria crossing
the consensus space. We focus our analysis on a neighbourhood of the consensus
space. So, in order to have a local picture, it is sufficient to understand
the shape of the linear approximation of the intersecting branches of
equilibria.

######  Proposition 17.

For a polynomial response function (52), in the $(x,k)$-plane (34), the
branches of the critical manifold intersecting the consensus space locally
have the form

$\kappa(x)=2x^{s}_{k}+(n-2)x.$ (53)

###### Proof.

Let us consider the function $\tilde{\phi}(x,y):=-[f(x)-f(y+x)]$ and the
auxiliary system

$\displaystyle\dot{x}$ $\displaystyle=\tilde{\phi}(x,y)$ (54)
$\displaystyle\dot{y}$ $\displaystyle=0,$

where $y$ acts as a parameter. We are going to evaluate the slope of the
tangents of the curves of zeros of $\tilde{\phi}(x,y)$ intersecting the line
$y=0$. Considering a polynomial response function (52), we expand $f(x+y)$ by
using the binomial theorem,

$\displaystyle f(y+x)$ $\displaystyle=\sum_{j=0}^{N}a_{j}(y+x)^{j}$
$\displaystyle=\sum_{j=0}^{N}a_{j}x^{j}+\sum_{j=0}^{N}\sum_{k=1}^{j}\binom{j}{k}a_{j}y^{k}x^{j-k}.$

So, we can rewrite $\tilde{\phi}(x,y)$ as

$\tilde{\phi}(x,y)=y\sum_{j=0}^{N}\sum_{l=0}^{j-1}\binom{j}{l+1}a_{j}y^{l}x^{j-l-1},$
(55)

where we factored out $y$, and set $l=k-1$. We regularise $\tilde{\phi}(x,y)$
by removing the line $y=0$ from the set of zeros, i.e.
$\tilde{\phi}(x,y)=:y\tilde{\phi}^{\textup{reg}}(x,y)$, where

$\tilde{\phi}^{\textup{reg}}(x,y)=\sum_{j=0}^{N}\sum_{l=0}^{j-1}\binom{j}{l+1}a_{j}y^{l}x^{j-l-1}.$
(56)

Then, the regularized auxiliary system reads

$\displaystyle\dot{x}$ $\displaystyle=\tilde{\phi}^{\textup{reg}}(x,y)$ (57)
$\displaystyle\dot{y}$ $\displaystyle=0.$

Let us compute the Jacobian, $J$, for the regularised auxiliary system,

$J=\begin{pmatrix}\frac{\partial\tilde{\phi}^{\textup{reg}}}{\partial
x}&\frac{\partial\tilde{\phi}^{\textup{reg}}}{\partial y}\\\
0&0\end{pmatrix},$ (58)

where

$\displaystyle\frac{\partial\tilde{\phi}^{\textup{reg}}}{\partial x}$
$\displaystyle=\sum_{j=0}^{N}\sum_{l=0}^{j-1}\binom{j}{l+1}a_{j}(j-l-1)y^{l}x^{j-l-2},$
(59) $\displaystyle\frac{\partial\tilde{\phi}^{\textup{reg}}}{\partial y}$
$\displaystyle=\sum_{j=0}^{N}\sum_{l=0}^{j-1}\binom{j}{l+1}a_{j}ly^{l-1}x^{j-l-1}.$
(60)

Since we are interested on what happens at the intersection with $y=0$, we
evaluate the Jacobian on that line, namely

$\displaystyle J_{0}:=J\big{|}_{y=0}$
$\displaystyle=\sum_{j=0}^{N}j(j-1)a_{j}x^{j-2}\begin{pmatrix}1&\frac{1}{2}\\\
0&0\end{pmatrix}.$ (61)

The eigenvalues of $J_{0}$ are determined by the equation
$\det(J_{0}-\mu\operatorname{Id})=0$, which in our case reads

$-\mu\left[\sum_{j=0}^{N}j(j-1)a_{j}x^{j-2}-\mu\right]=0,$ (62)

and so the eigenvalues are

$\displaystyle\mu_{1}$ $\displaystyle=0,$ (63) $\displaystyle\mu_{2}$
$\displaystyle=\sum_{j=0}^{N}j(j-1)a_{j}x^{j-2}.$ (64)

Let us notice that, in general, for the regularised auxiliary system the
Jacobian $J_{0}$ cannot be interpreted as the linearisation of the system,
because $y=0$ is no more an equilibrium. However, there could be other
equilibria on the line $y=0$ at the intersections with the other curves of
equilibria. At such points the eigenvector with zero eigenvalue gives the
direction of the tangent of the intersecting curve of zeros. The eigenvector
$\tilde{v}^{(\mu_{1})}=(\tilde{v}^{(\mu_{1})}_{x},\tilde{v}^{(\mu_{1})}_{y})^{\intercal}$
must satisfy the equation $J_{0}\tilde{v}^{(\mu_{1})}=0$, that is

$\sum_{j=0}^{N}j(j-1)a_{j}x^{j-2}(\tilde{v}^{(\mu_{1})}_{x}+\frac{1}{2}\tilde{v}^{(\mu_{1})}_{y})=0.$

Therefore $\tilde{v}^{(\mu_{1})}_{x}=-\frac{1}{2}\tilde{v}^{(\mu_{1})}_{y}$,
and so $\tilde{v}^{(\mu_{1})}\propto(1,-2)^{\intercal}$. Notice that
$\tilde{v}^{(\mu_{1})}$ does not depend on $x$, that means wherever the
intersection takes place, then the tangent will have the direction of
$\tilde{v}^{(\mu_{1})}$. At this point, we need to obtain the equivalent
vector for the system of our interest (31). Note that the equation $y=k-nx$
puts in relation the auxiliary system with the layer problem (31). Thanks to
this relation we can construct the transformation connecting the tangent
vectors of the two systems, in particular we have

$\displaystyle\Pi$ $\displaystyle:=\begin{pmatrix}\frac{\partial
x(x,k)}{\partial x}&\frac{\partial x(x,k)}{\partial k}\\\ \frac{\partial
y(x,k)}{\partial x}&\frac{\partial y(x,k)}{\partial
k}\end{pmatrix}=\begin{pmatrix}1&0\\\ -n&1\end{pmatrix},$

and then the inverse transformation reads

$\Pi^{-1}=\begin{pmatrix}1&0\\\ n&1\end{pmatrix}.$ (65)

So, the tangents of the critical manifold at the intersections with the
consensus space, in the original system, have direction given by
$v_{\mu_{1}}=\Pi^{-1}\tilde{v}_{\mu_{1}}\propto(1,n-2)$, which proves that the
slope of the tangent in the $(x,k)$-plane is $n-2$. Therefore, the tangent
lines have an equation of the form $\kappa(x)=\tilde{\kappa}+(n-2)x$. In order
to find $\tilde{\kappa}$, we use the fact that $\kappa(x^{s}_{k})=nx^{s}_{k}$,
obtaining $\tilde{\kappa}=2x^{s}_{k}$; the statement is proven. A sketch of
the result of this proposition is shown in figure 2. ∎

$k$$C$$x$ Figure 2:  Local structure of the critical manifold near the
consensus space in the $(x,k)$-plane.
In the figure, the $(x,k)$-plane is rigidly rotated in order to visualise the
consensus space as a horizontal line. The black dots represent singular
points, the coloured lines crossing the consensus space at the singular points
are other branches of the critical manifold. We draw in red the unstable
sections of the critical manifold, while the stable ones are drawn in green.
We emphasise that, in general, the branches of equilibria that cross the
consensus manifold are not straight lines, but due to proposition 17 they are
all parallel to each other.

Proposition 17 tells us that, up to linear approximation, all other branches
of equilibria of the critical manifold crossing the consensus space are
parallel to each other and are in fact transverse to the consensus space.
Moreover, we notice that proposition 17 is in agreement with the
transversality condition, $n\neq 2$, for transcritical points. Indeed, for
$n=2$ we have that the tangents (53), of the branches of the critical manifold
intersecting the consensus, are parallel to the fast foliation. As a
consequence of proposition 17 we have that pitchfork singularities are not
possible on the consensus space for the class of ALFs under examination, this
statement is also confirmed by checking the conditions for pitchfork
singularities [27]. By a similar reasoning, fold singularities on the
consensus space are also excluded. Finally, Hopf singularities do not appear
either by the fact that the layer system has always real eigenvalues. So, one
can conclude that for $K_{n}$-ALFs transcritical singularities are generic on
the consensus space, in the sense of codimension one bifurcations; while on
other branches of the critical manifold it is possible to have other
singularities, see some examples in section 6.

##  6 Examples and simulations

In this section, we present explicit examples of ALFs, and show numerical
simulations for such concrete systems.

###  6.1 Example 1

For the first example, we choose the response function
$f(x)=(x-1)^{2}(x+1)^{2}$, see figure 3. Let us notice that $f(x)$ is an even
function, and so it is invariant under the symmetry group $\mathbb{Z}_{2}$;
this implies that we expect a nontrivial critical manifold for the ALF. In
fact, if we set the number of nodes $n=3$ we can visualise the critical
manifold in $\mathbb{R}^{3}$, see figure 4.

Figure 3: Response function $f(x)=(x-1)^{2}(x+1)^{2}$. Figure 4: Critical
manifold for n=3 and $f(x)=(x-1)^{2}(x+1)^{2}$

Following the analysis of section 5.2, we consider a symmetry preserving
perturbation and we look at the behaviour of the system in the $(x,k)$-plane.
Considering (34), we fix the perturbation parameter $\epsilon=1/10$, and a
critical perturbation $g(x,k,\Lambda)=\tilde{g}(x,k,\Lambda)=-1$, that is, the
slow-flow is directed downwards in figure 5. We numerically simulate the
system by tracking some characteristic canard solutions, see figure 5. It is
worth recalling now a few issues with numerical simulations of transcritical
points, see more details in [14]. Let us notice that, although the simulations
are performed with a working precision of $50$ digits they still suffer of
numerical problems. In particular we refer to the fact that a canard solution,
under an appropriate perturbation, should spend the same time ‘close’ to the
attracting part as to the repelling part. In this examples, such a time
symmetry should be visible as a spatial symmetry in correspondence of a
transcritical point on the consensus manifold. As one can see in figure 5, the
symmetry described is not very well reproduced by the numerical simulation.
There are several factors that induce such problem, first, we have that the
symmetry is exact only in the limit for $\epsilon\to 0$, so we expect that a
numerical simulation will just approximate such symmetry. On the more
numerical side, we have that the $\epsilon$ parameter is quite large to
clearly see this effect, however lowering $\epsilon$ would imply to further
increase the working precision, leading to problems with the time performance
of the CPU. Since our present work has a theoretical focus, we do not dive
deeper in the numerical analysis of these delicate systems, leaving the
problem of highly accurate simulations for future works. However, there is
another detail to take into account: in figure 5 we have canard solutions that
cross three transcritical points on the consensus space without leaving the
repelling intermediate section (light blue and pink curves); here the question
is: where should we expect the canards to leave the consensus manifold? As we
just discussed, we expect some symmetry, but here we need to be more careful
stating with respect to which transcritical point. We argue that, for these
particular solutions, the intermediate section has a null effect, and
therefore the solution should leave the consensus after the third
transcritical point (counting top to bottom) after a distance (approximately)
equal to the distance travelled close to the attracting section before the
first transcritical point. In order to prove the aforementioned statement, we
use the slow-divergence integral [13, 11, 10]. So, let us consider the slow-
divergence integral along the consensus manifold for the system under
analysis. Since the perturbation is constant the divergence of the vector
field in the $(x,k)$-plane, evaluated on the consensus space, is simply
$-n\textup{d}_{x}f(k/n)$. Therefore the slow-divergence integral becomes

$-n\int\textup{d}_{x}f(k/n)dk.$ (66)

We now notice two facts: first the intermediate section is divided in two
spatially equal sections by a transcritical point. Second, on the first
subsection $\textup{d}_{x}f(k/n)>0$ and on the second subsection
$\textup{d}_{x}f(k/n)<0$. Therefore the slow-divergence integral on the
intermediate section of the consensus space is null, so on the intermediate
section the attracting and repelling effect compensate each other. The
considerations we have done hold for the dynamics near consensus under a
constant critical perturbation. It is worth mentioning that the nonlocal
interaction of transcritical singularities via hysteresis processes could give
rise to an enhanced delay for the canard [15, 45].

(a) Critical manifold in the $(x,k)$-plane

(b) Canards solutions

Figure 5: On the left, figure 5(a), we can see in blue the consensus space,
and in red dashed the other branches of the critical manifold. This critical
manifold is obtained from the critical manifold of figure 4. On the right,
figure 5(b), we computed numerically four canard solutions for a critical
perturbation. Namely, in green we have a canard crossing two transcritical
points (only one on the consensus space), after the first transcritical point
it leaves the consensus space; in light blue and pink we have two canard
solutions that cross three transcritical points without leaving the
intermediate unstable section of the consensus space; in purple we have a
solution that after the first transcritical point leaves the consensus space,
after crossing a second transcritical point (not on consensus) due to the
particular shape of the critical manifold it is attracted again to consensus
and finally crosses a third transcritical point on the consensus. Notice that
all the solutions, when crossing transcritical points on the consensus space,
show the typical canard behaviour. We remark that not all the solutions
displayed are robust, in particular this depends on the stability properties
of the $(x,k)$-plane. We can expect that solutions starting in a neighbourhood
of attracting branches of the consensus are robust, and so the reduction is
representative of the dynamics. In the unstable regions the reduction still
applies if we consider ad-hoc perturbations and initial conditions.

###  6.2 Example 2

Let us consider the larger network, $K_{10}$, in figure 6. We keep the same
response function $f(x)=(x-1)^{2}(x+1)^{2}$ as in example 1, but now the
perturbation is randomly generated, $H=(r_{1},\dots,r_{n})$, where
$r_{1},\dots,r_{n}$ are random reals between $0$ and $1$. For this example, we
consider the equations in the non-standard form (27), and we compute the time-
series for each node state with initial conditions randomly chosen between
$-1$ and $0$, see figure 7. Moreover, we also compute the timeseries for a
weighted counterpart of a $K_{10}$-ALF in order to highlight the equivalence
between the weighted and unweighted ALFs near consensus.

Figure 6: $K_{10}$

(a)

(b)

(c)

(d)

Figure 7: Timeseries for a $K_{10}$-ALF under a random constant and positive
perturbation. Figures 7(a), 7(c) display time in a logarithmic scale, while in
figures 7(b), 7(d) time is linear. Different colours represent different
node’s states. Figures 7(a) and 7(b) represent a simulation for a weighted
$K_{10}$-ALF, where the weights are randomly chosen between $1$ and $5$; on
the other hand, in figures 7(c) and 7(d) one can see the behaviour for a
(unweighted) $K_{10}$-ALF. The plots confirm the topological equivalence
proved in proposition 5. The black dots represent the simulation for the same
system under a critical perturbation, where the expected canard solution takes
place. Let us remark that the initial conditions for the weighted and for the
unweighted case are not the same but they are chosen randomly in the same
interval, between $-1$ and $0$. Such random initial conditions are chosen
nearby an attracting branch of the consensus space, implying the robustness of
the solutions. Indeed, the behaviour of the two simulations is similar, and
well predicted by the reduced system.

The timeseries in figure 7 show a characteristic two-timescale behaviour,
alternating long periods of slow drift and short fast transitions of regime.
In particular after an initial fast convergence to consensus, then the system
slowly reaches the first transcritical point, there we can see that the system
starts a sequence of fast transitions that bring the system out of consensus.
Finally, due to the geometry of the critical manifold the system converges to
consensus again.

###  6.3 Example 3

As a final example, we consider the phenomenon of symmetry breaking. We
briefly recall what we mean by (spontaneous) symmetry breaking. Let us
consider a dynamical system equivariant under a group $\Gamma$. Then, if there
are equilibria that are not fixed under $\Gamma$ we say that the symmetry is
broken. As we have seen in section 3.1, for ALFs the symmetries of the
response function induce, in general, a set of equilibria larger than the
consensus space, which is the fixed-point space for groups of permutations of
order greater than or equal to $n$. So, when we have a response function with
nontrivial symmetry we should expect symmetry breaking. What we just mentioned
works for unperturbed ALFs. When we perturb the system, in general, we destroy
the symmetry of the unperturbed system and the behaviour changes dramatically.
However, if we consider the particular case of a symmetry preserving
perturbation, then the maximal canard becomes the solution fixed under the
symmetry group replacing, in some sense, the role of the consensus space in
the unperturbed system.

Let us consider the response function
$f(x,\lambda)=(x-\lambda)(x+\lambda)^{2}$. We can construct a bifurcation
diagram by plotting the critical manifold on the $(x,k)$-plane for a range of
values of $\lambda$, i.e., $\\{(x,k,\lambda)\in\mathbb{R}^{3}\ |\
-[f(x,\lambda)-f\left(k-(n-1)x,\lambda\right)]=0\\}$, see figure 8(a).

(a)

(b)

Figure 8: Bifurcation diagram for the response function
$f(x,\lambda)=(x-\lambda)(x+\lambda)^{2}$, figure 8(a), and bifurcation
diagram for the response function
$f(x,\lambda)=(x-\lambda)^{2}(x+\lambda)^{2}$, figure 8(b); in both cases we
fixed $n=3$. We show in orange the consensus manifold, in green the branches
of the critical manifold arising by varying $\lambda$ common to both response
functions. Notice that, in figure 8(b), there is an extra branch of the
critical manifold (shown in blue), this is due to the $\mathbb{Z}_{2}$
invariance of the response function, or, in other words, it is due to an extra
degeneracy of the response function.

We notice, by looking at figure 8(a), that for $\lambda=0$ the critical
manifold consists of the consensus space only. Then, for $\lambda>0$, another
manifold appears breaking the symmetry of the system. If we think for a moment
at the response function alone, we could consider another bifurcation diagram:
the one given by $\\{(x,\lambda)\in\mathbb{R}^{2}\ |\ f(x,\lambda)=0\\}$. This
reasoning allows us to compare what the bifurcation diagram of the critical
manifold on the $(x,k)$-plane is, and the bifurcation described by an
hypothetical one dimensional system, $\dot{x}=f(x,\lambda)$, which we call
_response bifurcation_. For the function
$f(x,\lambda)=(x-\lambda)(x+\lambda)^{2}$ the response bifurcation diagram is
the one of a degenerate transcritical singularity. On the other hand, if we
consider the response function $f(x,\lambda)=(x-\lambda)^{2}(x+\lambda)^{2}$,
it is also representative of a degenerate transcritical singularity; in
particular for $\lambda=1$ it becomes the response function considered in the
first and second examples. It is interesting to notice that, if we compare the
response bifurcation diagrams for the function
$f(x,\lambda)=(x-\lambda)^{2}(x+\lambda)^{2}$ and
$f(x,\lambda)=(x-\lambda)(x+\lambda)^{2}$ the branches of equilibria have the
same structure of degenerate transcritical bifurcations. However, if we
consider the bifurcation diagrams of the critical manifold on the
$(x,k)$-plane induced by those functions, we notice that the degeneracy
induces different critical manifolds, see figure 8. Thus, with this example,
we have argued that the network structure plays a highly non-trivial part when
qualitatively studying networked dynamics and their bifurcations.

##  7 Conclusions and discussion

This paper has been dedicated to the analysis of a class of nonlinear
Laplacian dynamical systems, ALFs, and their perturbations. The analysis
starts by looking at (unperturbed) ALFs on graphs with positive weights. We
emphasise the importance of the topological equivalence proved in proposition
5. Indeed, a weighted graph, in general, loses the symmetry properties that
its own unweighted counterpart has. The fact that a positive-weighted graph
induces dynamics of the ALF that are topological equivalent to the one induced
by the unweighted counterpart, allows us to consider unweighted simple graphs.
In turn, an ALF with simple graph structure inherits the symmetry properties
of the underling graph, enabling us to use the tools of equivariant dynamical
systems theory. In such a context, the cell-network formalism was developed
with the purpose of exploiting network symmetries in dynamical systems,
actually going even beyond the group symmetries. In the language of cell-
networks, the linear invariant spaces arising from the symmetry properties of
the underlying network are referred to as pattern of synchrony. So, in fact,
the result of theorem 2 is a particular case of a robust pattern of synchrony;
indeed, the existence of a maximal canard is obtained by using the symmetry
properties of the network. We note that theorem 2 brings more stringent
conditions for the consensus than being just an invariant space. Specifically,
when the system is unperturbed, given the specific form of the ALF’s equation,
we know that the the consensus space is a set of equilibria. When a
perturbation is switched on, we require the perturbation not only to preserve
the symmetry, but also to be non-zero along the consensus space. So, asking
the consensus space to be a robust pattern of synchrony is a necessary, but
not a sufficient, condition to have a maximal canard. It is also worth
remarking that, theorem 2 does not rely on specific tools of slow-fast
systems, which are the classical techniques used to determine the existence of
canards.

Several new directions arise from the results of our paper. For example, one
could consider response functions ($F(\bm{x})$) depending also on
“neighbouring” nodes; in such a case the network structure appears also inside
the response vector field, and not only through the Laplacian matrix.
Moreover, one may be interested in studying the case of heterogeneous response
functions, i.e., different nodes might contribute differently to the mutual
interaction. Regarding the graph structure, one may take into account negative
weights, which even for the linear case give rise to much richer dynamics than
the positive-weights case; directed graphs could be considered as well.
Another possibility is to study Laplacian systems where the states have
dimension greater than one, for example, one might consider Laplacian systems
where the states evolve in the real plane, or even in a two-dimensional
manifold. The dynamics of planar systems entail interesting phenomena, such as
periodic orbits, that could potentially give rise to new phenomena unseen in
the one-dimensional framework. For similar reasons, states of dimension three
might be relevant as well, since it is well-known that it is for three-
dimensional systems that chaos, at the level of the nodes, may appear. In all
aforementioned cases, a natural question is to investigate, and if possible
classify, the possible singularities that arise and the influence of the
network on them (for example in this paper we showed that complete ALFs can
only have transcritical singularities along the consensus manifold). Moreover,
one would then be interested in the behaviour under small perturbations, which
as seen here, is also closely related to the network structure.

##  Appendix A Singular perturbations

In general, the symbols used in this section are not related to those in the
main text.

A singularly perturbed system of ODEs, in the standard form, is also called
_slow-fast system_ and it has the form

$\displaystyle\dot{x}$ $\displaystyle=f(x,y,\epsilon)$ (67)
$\displaystyle\dot{y}$ $\displaystyle=\epsilon g(x,y,\epsilon),$

where $0<\epsilon\ll 1$, $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, and the
time-parametrization $t$ is called _fast-time_. Equation (67) is said to be in
the standard form because the time separation between the fast-variables, $x$,
and the slow-variables, $y$, is explicit. In the non-standard form a
singularly perturbed system has the form

$\dot{z}=F(z,\epsilon),$ (68)

where $z\in\mathbb{R}^{n+m}$, and the vector field $F(z,0)$ has at least one
equilibrium of dimension greater than zero [46].

Defining the _slow-time_ , $\tau:=\epsilon t$, equation (67) can be rewritten
as

$\displaystyle\epsilon x^{\prime}$ $\displaystyle=f(x,y,\epsilon)$ (69)
$\displaystyle y^{\prime}$ $\displaystyle=g(x,y,\epsilon).$

Setting $\epsilon$ to zero in equation (69), we obtain the _reduced system_

$\displaystyle 0$ $\displaystyle=f(x,y,0)$ (70) $\displaystyle y^{\prime}$
$\displaystyle=g(x,y,0),$

which is a set of constrained differential equations. Instead, setting
$\epsilon$ to zero in equation (67) we obtain the _layer problem_

$\displaystyle\dot{x}$ $\displaystyle=f(x,y,0)$ (71) $\displaystyle\dot{y}$
$\displaystyle=0,$

where the slow-variables, $y$, play the role of parameters.

######  Definition 13.

The critical manifold is defined as the set

$\mathcal{C}_{0}=\\{(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}\ |\
f(x,y,0)=0\\}.$ (72)

######  Remark 8.

Although the commonly accepted name, the critical manifold it is not
necessarily a manifold.

The critical manifold is the constraint space for the reduced system (70), or
equivalently, it is the set of equilibria of the layer problem (71).

######  Definition 14.

A point $x^{*}\in\mathcal{C}_{0}$ is called hyperbolic if the Jacobian matrix
at $x^{*}$ has no eigenvalues with zero real part.

######  Lemma 3.

A subset $\mathcal{M}_{0}\subseteq\mathcal{C}_{0}$ such that
$x^{*}\in\mathcal{M}_{0}$ is hyperbolic, $\forall x^{*}\in\mathcal{M}_{0}$, is
a normally hyperbolic invariant manifold (NHIM) [23].

######  Theorem 3 (Fenichel).

For $\epsilon$ sufficiently small, compact subsets of NHIMs of the layer
system (71) are diffeomorphic to compact subsets of NHIMs of the singularly
perturbed system (67) (or, equivalently (69)).

######  Remark 9.

Theorem 3 is restricted to the results relevant for the present paper. An
exhaustive version can be found in [28].

Another way to state theorem 3 is that normally hyperbolic critical manifolds
persist as slow manifolds, and inherit all stability properties of the
critical manifold. For nonhyperbolic points, $x^{*}\in\mathcal{C}_{0}$, also
known as _singularities_ , Fenichel’s theorem does not provide information
about the perturbed system in a neighbourhood of $x^{*}$. The process of
analysis of the system in a neighbourhood a singularity is called
desingularization. We briefly introduce the _blow-up_ (desingularization) for
_nilpotent_ singularities [26].

######  Definition 15.

A point $x^{*}\in\mathcal{C}_{0}$ is a nilpotent singularity if all the
eigenvalues of the Jacobian, at $x^{*}$, are zero.

When a singular point has a mixture of zero eigenvalues, and non-zero (real
part) eigenvalues it is possible to reduce the problem to a nilpotent
singularity via a _center manifold_ reduction [8].

######  Definition 16.

Let $x^{*}$ be a nilpotent singularity. The spherical blow-up is a (local)
coordinate transformation

$\displaystyle\beta:\mathbb{R}^{n+m+1}$
$\displaystyle\to\mathbb{S}^{n+m}_{x^{*}}\times I$ (73)
$\displaystyle(x,y,\epsilon)$
$\displaystyle\mapsto(\bar{x},\bar{y},\bar{\epsilon},r),$

where $\mathbb{S}^{n+m}_{x^{*}}$ is the $n+m$-dimensional sphere (embedded in
$\mathbb{R}^{n+m+1}$) centred at $x^{*}$,
$(\bar{x},\bar{y},\bar{\epsilon})\in\mathbb{S}^{n+m}_{x^{*}}$, and $r\in
I\subseteq\mathbb{R}_{\geq 0}$, $0\in I$.

Let us notice that, in a neighbourhood of the singularity the transformation
(73) is a diffeomorphism. Instead at the singular point the blow-up transforms
the point $x^{*}$ into a $n+m$-dimensional sphere. The core idea of the blow-
up is to analyse the induced flow on the sphere and deduct the behaviour of
the singularity. However, the blow-up transformation by itself does not
desingularize the system. After the blow-up it is necessary to perform a
conformal transformation (time reparametrization) in order to regularise the
system, i.e., retrieve hyperbolicity. We described the spherical blow-up, but
several other blow-up transformations are possible, e.g., quasihomogeneous,
directional, or even to other manifolds, not necessarily spheres [29]. The
choice of the appropriate blow-up should be done by considering the
regularisation, and feasibility of the analysis. Since this work is not
concerned with a direct application of the blow-up, but rather with the
implementation of results involving the blow-up technique, we presented the
spherical set-up which provides the most clear geometrical visualisation.

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