# The asymptotic behavior of the Altafini model on signed graphons

Raoul Prisant, Federica Garin, Paolo Frasca

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Raoul Prisant, Federica Garin, Paolo Frasca. The asymptotic behavior of the Altafini model on signed graphons. ECC 2025 - 23rd European Control Conference, Jun 2025, Thessaloniki, Greece. pp.1-6.
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# The asymptotic behavior of the Altafini model on signed graphons

Raoul Prisant, Federica Garin, Paolo Frasca

**Abstract**—The Altafini model, also known as opposing dynamics, is an opinion dynamics model on signed graphs. In this paper, we study an extension of this model to signed graphons, mathematical objects that approximate large undirected graphs. After defining natural extensions of the notions of connectivity and structural balance to signed graphons, we analyze the asymptotic behavior of the dynamics, through the use of semigroup theory and the study of the properties of the opposing Laplacian operator. Our results are consistent with known facts about the classical Altafini model on signed graphs: on a connected signed graphon with degree essentially bounded away from zero, the dynamics converges to a bipartite consensus if the signed graphon is structurally balanced, and converges to zero if it is not.

## I. INTRODUCTION

In recent years, a topic of interest for the control community has been that of social dynamics, which describe the evolution of opinions of individuals interacting over a network. The classical models postulate that the opinion of an individual is attracted by the average opinion of his/her neighbours, which leads too easily to consensus over the entire network—not a very realistic scenario for the evolution of opinions in societies. For this reason, a number of more sophisticated models have been proposed, to try and describe a wider variety of behaviors, such as disagreement and polarization of opinions [1], [2]. One possible way to achieve such richness of behaviors is to allow for negative interactions between agents, as was done in [3], [4].

Another potential limitation of classical opinion dynamics models is their ability to effectively describe large-scale dynamics. Indeed, often times social interactions happen on a society scale, e.g. through online social networks, with millions of agents interacting, thus making the problem impossible to effectively study through the use of graphs. A possible solution to scale network dynamics up to larger networks is to approximate large graphs with graphons, extension of graphs that have a continuum of nodes, and transform the dynamical system of $n$ ordinary differential equations into a single integro-differential equation. This approach is explored by a growing literature [5]–[11], which includes several opinion dynamics models [12], [13].

In recent work [14], we combined these broad research questions by considering two graphon dynamics with negative interactions, which we referred to as the opposing dynamics (or Altafini model [3]) and the repelling dynamics. In [14] we showed how the graphon dynamics well approximates the dynamics on graphs over finite horizons of time.

This work has been partly supported by the French National Research Agency through grant COCOON ANR-22-CE48-0011.

All authors are with Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mails: raoul.prisant@gipsa-lab.fr; federica.garin@inria.fr; paolo.frasca@gipsa-lab.fr).

This fact leaves open the question of the behavior of the system for $t \to \infty$. For the finite graph case, the asymptotic behavior is well-known [4], but the latter has not been studied on graphons yet.

In this paper, we focus on the study of the asymptotic behavior of the Altafini model extended to signed graphons. We show that, in analogy with the classical Altafini model on graphs, the dynamics on a connected signed graphon (with degree essentially bounded away from zero) converges to a bipartite consensus if the signed graphon is structurally balanced, and converges to zero otherwise. Our proofs are in part inspired by their finite-dimensional counterparts, although with significant variations to cope with the infinite-dimensional setting: these variations include the detailed study of a relevant operator on the state space, which we refer to as the opposing Laplacian operator.

**Outline:** The paper is structured as follows: in Section II we recall some preliminary definitions and facts from semigroup theory, which are necessary for the analysis; in Section III we formulate the problem, by defining signed graphons and introducing the dynamics; in Section IV we state and prove some properties of the opposing Laplacian $\mathcal{L}_W^{\text{opp}}$ that defines our dynamics; finally, in Sections VI and VII, we show the asymptotic behavior of the dynamics depending on whether or not the signed graphon is structurally balanced, and then we conclude the paper with some simulations in Section VIII and conclusions in Section IX.

## II. PRELIMINARIES ON INFINITE-DIMENSIONAL SYSTEMS

In this section, we recall some definitions from semigroup theory, which will be the framework used to study our problem. Throughout this section, let $X$ be a Hilbert space. In the rest of the paper (from Section III), we will consider the particular choice $X = L^2(I)$ with $I = (0,1)$, and denote by $\|\cdot\|_2$ the corresponding norm.

**Definition 1 (Strongly continuous semigroup):** A strongly continuous semigroup is an operator $(T(t))_{t \ge 0} : X \to X$ such that

1) for all $t \ge 0$, $T(t)$ is a bounded linear operator on $X$,

2) $T(0) = \text{Id}$,

3) for all $t_1, t_2 \ge 0$, $T(t_1+t_2) = T(t_1)T(t_2)$,

4) $\|T(t)u_0 - u_0\|_X \to 0$ for $t \to 0^+$ and for all $u_0 \in X$. $\square$

Semigroups are associated with solutions of dynamical systems

$$ \begin{cases} \dot{u}(t) = \mathcal{A}u \\ u(0) = u_0, \end{cases} $$

where the operator $\mathcal{A}$ is said to generate the semigroup. The proper definition is as follows.
**Definition 2 (Generator of a semigroup):** The operator $\mathcal{A}: D(\mathcal{A}) \subseteq X \to X$ defined as $\mathcal{A}u = \lim_{t \to 0^+} \frac{1}{t}(T(t)u - u)$ is called the generator of the strongly continuous semigroup $(T(t))_{t \ge 0}$. $\square$

The idea is to show that an operator generates a strongly continuous semigroup, so that we can study the properties of the solutions of the dynamical system through the analysis of the operator $\mathcal{A}$. In our case, we will show that our operator generates a contraction semigroup, a particular case of strongly continuous semigroup.

**Definition 3 (Contraction semigroup):** A strongly continuous semigroup $(T(t))_{t \ge 0}$ is called a contraction semigroup if $\|T(t)\| \le 1$ for all $t \ge 0$, where $\|T(t)\| := \sup\{\|T(t)f\|_X \text{ s.t. } f \in X, \|f\|_X \le 1\}$ is the operator norm. $\square$

Another particular class of strongly continuous semigroups is that of uniformly continuous semigroups:

**Definition 4 (Uniformly continuous semigroup):** A strongly continuous semigroup $(T(t))_{t \ge 0}$ is called uniformly continuous if the map $t \mapsto T(t)$ is continuous with respect to the operator norm. $\square$

A useful property of these semigroups is that their asymptotic behavior can be studied through the spectrum of the generator $\mathcal{A}$.

### III. ALTAFINI MODEL ON GRAPHONS

Dynamics on networks can be hard to study when the number of nodes is large. One possible solution is to approximate the large graph, which we will consider undirected, by a graphon. Graphons mirror the role of adjacency matrices, describing the interactions between a continuum of nodes $I = [0, 1]$, instead of a finite discrete set like in the graph case. A graphon is indeed a symmetric measurable function $W: I \times I \to [0, 1]$. Even though classical graphons are defined taking values in $[0, 1]$, in this paper we extend them to take values in $[-1, 1]$, as the dynamics we will study allows for negative interactions.

**Definition 5 (Signed graphon):** A signed graphon is a symmetric measurable function $W: I \times I \to [-1, 1]$, where $I = [0, 1]$. $\square$

For finite graphs, a well-known dynamics featuring negative interactions is the so-called Altafini model or opposing dynamics. It is an extension of the classical Abelson<fnref n="1" /> model that allows for negative interactions between agents.

We thus define the opposing dynamics associated to the signed graphon $W(x, y)$ as

$$ \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t}(x,t) = \int_I W(x,y)u(y,t) \, dy - \int_I |W(x,y)|u(x,t) \, dy \\ \\ u(x,0) = u_0(x), \end{array} \right. \qquad (1) $$

where $u: I \times \mathbb{R}^+ \to \mathbb{R}$ represents the opinion, evolving with respect to the time $t$, of the continuum of nodes $I = [0, 1]$, labeled by $x$. The function $u_0: I \to \mathbb{R}$ represents the initial distribution of opinions and is assumed to be $L^2$.

When $W(x, y) \ge 0$, the dynamics (1) coincides with the averaging dynamics on graphons [13],

$$ \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t}(x,t) = \int_I W(x,y)(u(y,t) - u(x,t)) \, dy \\ \\ u(x,0) = u_0(x), \end{array} \right. \qquad (2) $$

which is the natural extension of the Abelson model to graphons. In this dynamics each agent is attracted to the opinions of the agents it has positive interaction with. In the general Altafini model, $W(x, y)$ can be negative, in that case the agent is attracted to the opposite of the opinion of the disliked agent.

The aim of the paper is to study the asymptotic behavior of (1), for this purpose we will need the following definitions.

**Definition 6 (Connected graphon):** A graphon $W: I \times I \to [0, 1]$ is connected if for all measurable $A \subseteq [0, 1]$ with measure $\mu(A) \in (0, 1)$,

$$ \int_A \int_{A^c} W(x,y) \, dx \, dy \neq 0, $$

where $A^c = I \setminus A$ is the complement of A. $\square$

The definition of connected graphon is analogous to the fact that a graph is connected if and only if you cannot separate the vertex set $V$ into two non-empty subsets $A$ and $V \setminus A$ such that there is no edge between $A$ and $V \setminus A$.

**Definition 7 (Degree):** The degree of a graphon $W: I \times I \to [0, 1]$ is the function $d_W(x): I \to \mathbb{R}$ defined as

$$ d_W(x) = \int_I W(x,y) \, dy. $$

We say that the degree of $W$ is essentially bounded away from $0$ if $\inf_{x \in I} d_W(x) > 0$ $\square$.

Notice that we have defined connectedness and degree only for graphons with values in $[0, 1]$, as we will only need to consider them for $|W(x, y)|$.

A key definition in the study of the Altafini dynamics on signed graphs is that of structural balance, that here we extend to signed graphons.

**Definition 8 (Structurally balanced graphon):** A signed graphon $W: I \times I \to [-1, 1]$ is called structurally balanced if there exists a measurable set $A \subseteq [0, 1]$ such that

* $W(x, y) \ge 0$ almost everywhere on $(A \times A) \cup (A^c \times A^c)$,

* $W(x, y) \le 0$ almost everywhere on $(A \times A^c) \cup (A^c \times A)$.

**Remark 1 (Well-definition of structural balance):** If $|W|$ is connected, then the set $A$ is unique, modulo interchanging it with its complement and perturbing it by sets of null measure. Namely, if there are two measurable sets $A, B \subseteq I$ such that

* $W \ge 0$ a.e. on $(A \times A) \cup (A^c \times A^c)$ and $(B \times B) \cup (B^c \times B^c)$,

* $W \le 0$ a.e. on $(A \times A^c) \cup (A^c \times A)$ and $(B \times B^c) \cup (B^c \times B)$, then $\mu(A \setminus B) = \mu(B \setminus A) = 0$ or $\mu(A \setminus B^c) = \mu(B^c \setminus A) = 0$.

To show this, let $C = (A \cap B) \cup (A^c \cap B^c)$, and notice that $C^c = (A \cap B^c) \cup (A^c \cap B)$. Then notice that on $C \times C^c$ we have both $W \ge 0$ a.e. and $W \le 0$ a.e., which further imply that $|W| = 0$ a.e. on $C \times C^c$. Since $|W|$ is connected, this in turn implies that either $\mu(C) = 0$ or $\mu(C^c) = 0$. Notice that an equivalent definition of $C$ is $C = (A \setminus B^c) \cup (B^c \setminus A)$ and, similarly, we can write $C^c = (A \setminus B) \cup (B \setminus A)$. One of them must have null measure, yielding the result. $\square$

<footnote n="1">¹The Abelson model is the continuous-time counterpart of the better-known French-DeGroot model [1].</footnote>
IV. THE OPPOSING LAPLACIAN OPERATOR

It is useful to write dynamics (1) in operator form by using
the following definition.

Definition 9 (Opposing Laplacian): We call opposing
Laplacian the operator $\mathcal{L}_W^{\text{opp}}: L^2(I) \to L^2(I)$ defined as

$$
\mathcal{L}_W^{\text{opp}} u(x,t) = \int_I |W(x,y)| u(x,t) \, dy - \int_I W(x,y) u(y,t) \, dy. \quad \square
$$

This definition allows us to write equation (1) as

$$
\frac{\partial u}{\partial t}(x,t) = -\mathcal{L}_W^{\text{opp}} u(x,t).
$$

We can now study the properties of $\mathcal{L}_W^{\text{opp}}$ in order to
understand the behavior of the solutions of (1).

First of all, we observe that $\mathcal{L}_W^{\text{opp}}$ is a linear and bounded
operator. Hence it is closed. Moreover, its domain is the
whole space $L^2(I)$. Furthermore, the operator $\mathcal{L}_W^{\text{opp}}$ can be
written as

$$
\mathcal{L}_W^{\text{opp}} = M_{d|W|} - T_W, \quad (3)
$$

where:

* for any function $d(x) \in L^\infty(I)$, $M_d: L^2(I) \to L^2(I)$ is the multiplication operator defined by $(M_d u)(x) = d(x)u(x)$,

• $d_{|W|}(x)$ is the degree function of $|W(x,y)|$, and

* $T_W: L^2(I) \to L^2(I)$ is the integral operator defined by $(T_W u)(x) = \int_0^1 W(x,y) u(y) dy$.

In the particular case where $W(x,y) \ge 0$ almost everywhere, $\mathcal{L}_W^{\text{opp}}$ is equal to the graphon Laplacian $\mathcal{L}_W = M_{d_W} - T_W$, which allows to write the classic averaging dynamics (2) as

$$
\frac{\partial u}{\partial t}(x,t) = -\mathcal{L}_W u(x,t).
$$

In the finite case, knowing the spectrum of the matrix
defining the system is enough to understand the asymptotic
behavior. In the infinite-dimensional case, we want to use
the same approach, but we need to combine several results
of semigroup theory to justify it.

We start by showing that $\mathcal{L}_W^{\text{opp}}$ is self-adjoint, which accomplishes two goals. First, it helps us characterize the spectrum of $\mathcal{L}_W^{\text{opp}}$, and second, it is one of the elements needed to prove that $-\mathcal{L}_W^{\text{opp}}$ generates a contraction semigroup.

**Proposition 1 (Self-adjointness):** The operators $M_{d|W|}$, $T_W$ and $\mathcal{L}_W^{\text{opp}}$ are self-adjoint.

*Proof:* Considering expression (3), we have that $\mathcal{L}_W^{\text{opp}}$ is symmetric if both $M_{d|W|}$ and $T_W$ are symmetric. Operator $M_{d|W|}$ is a multiplication operator, so it is symmetric, as $d|W|$ is real-valued. Operator $T_W$ is symmetric as well:

$$
\begin{align*}
\langle T_W u, v \rangle &= \int_I T_W u(x) v(x) \, dx = \int_I v(x) \int_I W(x, y) u(y) \, dy \, dx \\
&= \int_I \int_I W(y, x) u(x) v(y) \, dx \, dy,
\end{align*}
$$

where in the last equality we just switched the names of
integration variables x and y. Now thanks to the symmetry
of W(x,y) and Fubini's theorem we can write

$$
\begin{align*}
\int_I \int_I W(y,x) u(x) v(y) \, dx \, dy &= \int_I \int_I W(x,y) u(x) v(y) \, dy \, dx \\
&= \int_I u(x) T_W v(x) \, dx = \langle u, T_W v \rangle.
\end{align*}
$$

Furthermore, the domain of $M_{d|W|}$, $T_W$ and $\mathcal{L}_W^{\text{opp}}$ is the
entire space $L^2(I)$ and hence $M_{d|W|}$, $T_W$ and $-\mathcal{L}_W^{\text{opp}}$ are self-
adjoint. ■

**Remark 2 (Spectrum of L<sup>opp</sup><sub>W</sub>):** Since L<sup>opp</sup><sub>W</sub> is a self-adjoint operator on a separable Hilbert space, its spectrum is the union of the discrete spectrum, made of at most countably many isolated eigenvalues with finite multiplicity, and its complement, which is called the essential spectrum.

The multiplication operator $M_{d|W|}$ has an essential spectrum equal to the essential range of $d|W|$ (see e.g. [13, Prop. 2.6]). Moreover, $T_W$ is a compact operator, and hence the essential spectrum of $\mathcal{L}_W^{\text{opp}}$ is equal to the essential spectrum of $M_{d|W|}$, since a compact perturbation does not modify the essential spectrum. In particular, the minimum of the essential spectrum of $\mathcal{L}_W^{\text{opp}}$ is equal to the essential infimum of $d|W|$.

Before moving on to show that $-\mathcal{L}_W^{\text{opp}}$ generates a semi-group, we prove the following useful expression for the quadratic form $\langle u, \mathcal{L}_W^{\text{opp}} u \rangle$.

**Proposition 2 (Quadratic form):** Given a signed graphon W and its decomposition in positive and negative parts W⁺ and W⁻, for any u ∈ L²(I) the following holds:

$$
\begin{equation}
\begin{split}
\langle u, \mathcal{L}_W^{\text{opp}} u \rangle &= \frac{1}{2} \int_I \int_I W^+(x,y) (u(x) - u(y))^2 \, dy dx \\
&\quad + \frac{1}{2} \int_I \int_I W^-(x,y) (u(x) + u(y))^2 \, dy dx.
\end{split}
\tag{4}
\end{equation}
$$

Proof:

$$
\begin{align*}
\langle u, \mathcal{L}_W^{\text{opp}} u \rangle &= \int_I u(x) \mathcal{L}_W^{\text{opp}} u(x) \, dx \\
&= \int_I \int_I (|W(x,y)| u(x)^2 - W(x,y) u(x) u(y)) \, dy \, dx \\
&= \int_I \int_I (W^+(x,y) u(x)^2 - W^+(x,y) u(x) u(y)) \, dy \, dx \\
&\quad - \int_{I_I} \int_{I_I} (-W^-(x,y) u(x)^2 - W^-(x,y) u(x) u(y)) \, dy \, dx.
\end{align*}
$$

We now consider the first term. Similarly to proof of Prop. 1, we can use the symmetry of $W(x,y)$, the change of variables $x=y,y=x$ and Fubini's Theorem to write

$$
\int_I \int_I W^+(x,y)(u(x)^2 - u(x)u(y)) \, dy \, dx \quad (5)
$$

$$
= \int_I \int_I W^+(y,x)(u(y)^2 - u(y)u(x)) \, dx \, dy
$$

$$
= \int_I \int_I W^+(x,y)(u(y)^2 - u(x)u(y)) \, dy \, dx. \quad (6)
$$

Since (5) and (6) are equal, (5) is also equal to half the sum
of the two, so that

$$
\begin{align*}
\int_I \int_I W^+(x,y)(u(x)^2 - u(x)u(y)) \, dy \, dx \\
= \frac{1}{2} \int_I \int_I W^+(x,y)(u(x) - u(y))^2 \, dy \, dx.
\end{align*}
$$

Similar manipulations on the second term yield the claim.

Now, we quickly show that $-\mathcal{L}_W^{\text{opp}}$ is dissipative, which is the last element needed to show that $-\mathcal{L}_W^{\text{opp}}$ generates a contraction semigroup.
**Proposition 3 (Dissipativity):** The operator $-\mathcal{L}_W^{\text{opp}}$ is dissipative, i.e., for all $u \in L^2(I)$, $\langle -\mathcal{L}_W^{\text{opp}}u, u \rangle \le 0$.

*Proof:* By Prop. 1, $\langle -\mathcal{L}_W^{\text{opp}}u, u \rangle = -\langle u, \mathcal{L}_W^{\text{opp}}u \rangle$ where $\langle u, \mathcal{L}_W^{\text{opp}}u \rangle$ is non-negative by Prop. 2. ■

**Proposition 4 (Contraction):** The operator $-\mathcal{L}_W^{\text{opp}}$ generates a contraction semigroup.

*Proof:* Theorem [15, Thm 6.1.8] states that a linear, closed and densely defined operator (as $\mathcal{L}_W^{\text{opp}}$ is) generates a contraction semigroup if the operator and its adjoint are both dissipative. Using Proposition 1 and Proposition 3 we have that $-\mathcal{L}_W^{\text{opp}} = (-\mathcal{L}_W^{\text{opp}})^*$ is dissipative, which yields the statement. ■

What we actually need to study the asymptotic behavior is that $-\mathcal{L}_W^{\text{opp}}$ generates a uniformly continuous semigroup. In fact, for such a semigroup, as we will use in Section VII, exponential stability is equivalent to having negative real part for every element in the spectrum. We can easily show that the contraction semigroup generated by $-\mathcal{L}_W^{\text{opp}}$ is also uniformly continuous.

**Proposition 5:** The operator $-\mathcal{L}_W^{\text{opp}}$ generates a uniformly continuous semigroup.

*Proof:* Proposition 4 states that $-\mathcal{L}_W^{\text{opp}}$ generates a contraction semigroup, a particular case of strongly continuous semigroup. From [16, Cor II.1.5] we have that a strongly continuous semigroup is also a uniformly continuous semigroup if its generator is bounded, which $-\mathcal{L}_W^{\text{opp}}$ is. ■

## V. SOLUTIONS AND EQUILIBRIA

After obtaining several properties of $-\mathcal{L}_W^{\text{opp}}$ in the previous section, we now give two properties of the dynamical system (1): existence and unicity of the solution, and characterization of the equilibria.

**Proposition 6 (Solutions):** System (1) with $u_0 \in L^2(I)$ has a unique classical solution of the form $u(x,t) = e^{-t\mathcal{L}_W^{\text{opp}}}u_0(x)$, where $e^{-t\mathcal{L}_W^{\text{opp}}} := \sum_{k=0}^{\infty} \frac{1}{k!}(-t\mathcal{L}_W^{\text{opp}})^k$.

*Proof:* From [16], Comments I.3.9 (iii), if $\mathcal{A}: L^2(I) \to L^2(I)$ generates a strongly continuous semigroup $T(t)$, then, for all $u_0 \in L^2(I)$, the Cauchy problem of the form

$$ \begin{cases} \frac{\partial u}{\partial t}(x,t) = \mathcal{A}u(x,t) \\ u(x,0) = u_0(x) \end{cases} $$

has a unique classical solution of the form $u(x,t) = T(t)u_0(x)$. From [16, Thm I.3.7], every uniformly continuous semigroup is of the form $e^{t\mathcal{A}}$ where $\mathcal{A}$ is the generator. In our case, by Proposition 5, $-\mathcal{L}_W^{\text{opp}}$ generates a uniformly continuous semigroup, from which the statement follows. ■

**Proposition 7 (Equilibria):** Let $u \in L^2(I)$ be an equilibrium of (1), i.e., $\mathcal{L}_W^{\text{opp}}u = 0$. Assume that $|W|$ is connected. Then, the modulus of $u$ is essentially constant, i.e., $|u(x)| = K$ a.e. for some $K \in \mathbb{R}$. Moreover,

* if $W$ is not structurally balanced, then $u(x) = 0$ a.e.;

* if $W(x,y) \ge 0$ a.e., then $u$ is essentially constant, i.e. $u(x) = K$ a.e. for some $K \in \mathbb{R}$.

*Proof:* If $\mathcal{L}_W^{\text{opp}}u(x) = 0$, then also $\langle u, \mathcal{L}_W^{\text{opp}}u \rangle = 0$. We manipulate the expression (4) in the following way: we define the set $A = \{x \in I \text{ s.t. } u(x) \ge 0\} = u^{-1}([0, \infty))$, which is measurable since $u$ is measurable. Considering the first

term in (4), notice that for $(x,y) \in (A \times A) \cup (A^c \times A^c)$, since $u(x)u(y) \ge 0$, we have

$$ \begin{aligned} & \int_{(A \times A) \cup (A^c \times A^c)} W^+(x,y)(u(x) - u(y))^2 \,dx\,dy \\ & = \int_{(A \times A) \cup (A^c \times A^c)} W^+(x,y)(|u(x)| - |u(y)|)^2 \,dx\,dy \end{aligned} $$

whereas for $(x,y) \in (A \times A^c) \cup (A^c \times A)$, since $u(x)u(y) \le 0$,

$$ \begin{aligned} & \int_{(A \times A^c) \cup (A^c \times A)} W^+(x,y)(u(x) - u(y))^2 \,dx\,dy \\ & = \int_{(A \times A^c) \cup (A^c \times A)} W^+(x,y)(|u(x)| - |u(y)|)^2 \,dx\,dy \\ & \quad + 4 \int_{(A \times A^c) \cup (A^c \times A)} W^+(x,y) |u(x)u(y)| \,dx\,dy. \end{aligned} $$

Similar computations can be done on the second term of (4) to obtain

$$ \begin{aligned} & 0 = 2 \langle u, \mathcal{L}_W^{\text{opp}}u \rangle \\ & = \int_{I \times I} W^+(x,y)(u(x) - u(y))^2 \,dx\,dy \\ & \quad + \int_{I \times I} W^-(x,y)(u(x) + u(y))^2 \,dx\,dy \\ & = \int_{I \times I} (W^+(x,y) + W^-(x,y))(|u(x)| - |u(y)|)^2 \,dx\,dy \\ & \quad + 4 \int_{(A \times A^c) \cup (A^c \times A)} W^+(x,y) |u(x)u(y)| \,dx\,dy \\ & \quad + 4 \int_{(A \times A) \cup (A^c \times A^c)} W^-(x,y) |u(x)u(y)| \,dx\,dy. \end{aligned} \quad (7) $$

Notice that the three integrals in the right-hand side are all non-negative, and hence their sum is zero if and only if each of them is zero.

Recalling that $W^+ + W^- = |W|$, the first of the three integrals is equal to

$$ \int_{I \times I} |W(x,y)| (|u(x)| - |u(y)|)^2 \,dx\,dy = 0. \quad (8) $$

From this, we can show that $|u|$ is essentially constant, as follows. Define $f(x) = |u(x)|$, and for any $t \in \mathbb{R}$, define $S_t := f^{-1}((t, \infty))$, which is measurable, since $f$ is measurable. Since the integrand in (8) is non-negative, (8) implies that the integrand is zero almost everywhere on $I \times I$, and hence in particular, $(f(x) - f(y))^2|W(x,y)| = 0$ almost everywhere on $S_t \times S_t^c$. Notice that $(f(x) - f(y))^2 > 0$ for all $(x,y) \in S_t \times S_t^c$ and hence $|W(x,y)| = 0$ a.e. on $S_t \times S_t^c$. The assumption that $|W|$ is connected then implies that either $\mu(S_t) = 0$ or $\mu(S_t) = 1$. Now we have a measurable function $f$ such that for all $t \in \mathbb{R}$ the set $S_t$ either has measure zero or has measure one. By [17, Lemma 3.5], this implies that $f$ is essentially constant, i.e., $|u(x)| = K$ a.e., for some $K \in \mathbb{R}$.

We will now show that this constant $K$ can be non-zero only when $W$ is structurally balanced. Recall that all three integrals in the right-hand side of (8) are zero. The first being zero gives $|u(x)| = K$ a.e., which, replaced in the two other terms gives

$$ 4K^2 \int_{(A \times A^c) \cup (A^c \times A)} W^+(x,y) \,dx\,dy = 0 \quad (9) $$
and $4K^2 \int_{(A \times A) \cup (A^c \times A^c)} W^{-}(x, y) \, dx dy = 0$. If $K \neq 0$, this implies $W^{+}(x, y) = 0$ (i.e. $W \leq 0$) a.e. on $(A \times A^c) \cup (A^c \times A)$ and $W^{-}(x, y) = 0$ (i.e. $W \geq 0$) a.e. on $(A \times A) \cup (A^c \times A^c)$, which means that the graphon is structurally balanced.

Finally, we consider the case where $W \geq 0$ a.e.. In this case, (9) is equivalent to $4K^2 \int_{(A \times A^c) \cup (A^c \times A)} W(x, y) \, dx dy = 0$. Also, in this case $|W|$ connected means $W$ connected, and hence this implies that either $A$ or $A^c$ has zero measure, and hence $u(x)$ has the same sign for almost all $x$. ■

## VI. STRUCTURALLY BALANCED: BIPARTITE CONSENSUS

Analogously with the finite case, the asymptotic behavior of (1) changes depending on whether $W(x,y)$ is structurally balanced or not. In this section, we show that if it is, then under some hypotheses it converges to a bipartite consensus.

**Theorem 1 (Convergence to bipartite consensus):** Let $W(x,y)$ be a signed graphon, structurally balanced with respect to the partition $I = A \cup A^c$, and such that $|W(x,y)|$ is connected and $d_{|W|}$ is essentially bounded away from zero. Let $u$ be the solution of (1), with $u_0 \in L^2(I)$.

Then, $\|u(\cdot, t) - u^*\|_2 \to 0$ exponentially for $t \to \infty$, where

$$u^*(x) = \begin{cases} K, & \text{for almost all } x \in A, \\ -K, & \text{for almost all } x \in A^c, \end{cases}$$

$$\text{with } K = \int_A u_0(x) \, dx - \int_{A^c} u_0(x) \, dx.$$

*Proof:* Consider the gauge transformation

$$v(x,t) = \begin{cases} u(x,t) & \text{if } x \in A, \\ -u(x,t) & \text{if } x \in A^c. \end{cases}$$

Then if $x \in A$, by definition of structural balance $W(x,y) \geq 0$ if $y \in A$ and $W(x,y) \leq 0$ if $y \in A^c$. By decomposing the integral over $I$ into two integrals over $A$ and $A^c$, and substituting $W(x,y) = \pm|W(x,y)|$ according to the sign of $W(x,y)$, we can write

$$\begin{aligned} \frac{\partial v}{\partial t}(x,t) &= \frac{\partial u}{\partial t}(x,t) = \int_I W(x,y)u(y) \, dy - \int_I |W(x,y)|u(x) \, dy \\ &= \int_A |W(x,y)|v(y) \, dy - \int_A |W(x,y)|v(x) \, dy \\ &\quad + \int_{A^c} |W(x,y)|v(y) \, dy - \int_{A^c} |W(x,y)|v(x) \, dy \\ &= \int_I |W(x,y)| (u(y) - u(x)) \, dy. \end{aligned}$$

Similarly, if $x \in A^c$

$$\begin{aligned} \frac{\partial v}{\partial t}(x,t) &= -\frac{\partial u}{\partial t}(x,t) = -(\int_I W(x,y)u(y) \, dy - \int_I |W(x,y)|u(x) \, dy) \\ &= -(\int_A |W(x,y)|v(y) \, dy + \int_A |W(x,y)|v(x) \, dy \\ &\quad - \int_{A^c} |W(x,y)|v(y) \, dy + \int_{A^c} |W(x,y)|v(x) \, dy) \\ &= \int_I |W(x,y)| (u(y) - u(x)) \, dy. \end{aligned}$$

We have found that $v(x,t)$ obeys the standard consensus dynamics (2) according to the [0,1]-valued graphon $|W(x,y)|$.

Convergence to consensus of (2) has been studied in [13]. Define

$$\lambda_2(\mathcal{L}_{|W|}) = \inf_{w \in L^2(I), w \perp 1_I} \frac{\langle \mathcal{L}_{|W|} w, w \rangle}{\|w\|_2^2},$$

where $1_I$ denotes the function $1_I(x) = 1$ for all $x \in I$. By [13, Theorem 4.11], if $\lambda_2(\mathcal{L}_{|W|}) > 0$, then $v(x,t)$ converges to average consensus: $\|v(\cdot, t) - \bar{v}\|_2 \to 0$ exponentially, where $\bar{v}(x) = (\int_I v_0(\xi) d\xi) 1_I(x)$.

We can now conclude the proof that $v(x,t)$ converges exponentially to average consensus, by showing that the assumptions $|W|$ connected and $d_{|W|}$ essentially bounded away from zero ensure that $\lambda_2(\mathcal{L}_{|W|}) > 0$. From Prop. 7 applied to the non-negative graphon $|W|$, since $|W|$ is connected, 0 is an eigenvalue of $\mathcal{L}_{|W|}$ with multiplicity 1 and eigenvector equal to $1_I$. By Remark 2, since $d_{|W|}$ is essentially bounded away from zero, the infimum of the essential spectrum is positive. Then, the min-max theorem for self-adjoint operators ensures that $\lambda_2(\mathcal{L}_{|W|}) > 0$.

Finally, by reverting the change of coordinates, we obtain the statement for $u(x,t)$. ■

**Remark 3:** Theorem 1 states that, under the suitable connectivity assumptions, if the signed graphon $W$ is structurally balanced, then the solutions of (1) converge to a bipartite consensus, i.e., $A$ and $A^c$ act as communities, each of which individually reaches a consensus but disagrees with the other community. As a particular case, if $W(x,y) \geq 0$ a.e., then we recover the classical consensus. In fact, in this case the graphon is structurally balanced with only one community (i.e., either $A$ or $A^c$ has null measure), which reaches consensus.

## VII. NON-STRUCTURALLY BALANCED: CONVERGENCE TO ZERO

Similarly to the graph case, if the graphon is not structurally balanced, then under mild assumptions all solutions converge to zero. In this section, we prove this fact by combining the properties stated in Section IV.

**Theorem 2:** Let $W(x,y)$ be a signed graphon, non structurally balanced, and such that $|W(x,y)|$ is connected and $d_{|W|}$ is essentially bounded away from zero. Let $u$ be the solution of (1). Then, for all $u_0 \in L^2(I)$, $\|u(\cdot, t)\|_2 \to 0$ exponentially for $t \to \infty$.

*Proof:* We have shown that $-\mathcal{L}_W^{\text{opp}}$ generates a uniformly continuous semigroup; as such, the semigroup is uniformly exponentially stable (i.e., $\|T(t)\| \leq Me^{-\epsilon t}$ for some constants $M \geq 1, \epsilon > 0$) if and only if $\text{Re} \, \lambda < 0$ for all $\lambda$ in the spectrum of $-\mathcal{L}_W^{\text{opp}}$, see [16, Thm I.3.14 and Prop I.3.12].

Since $-\mathcal{L}_W^{\text{opp}}$ is self-adjoint, its spectrum is real and, moreover, from Proposition 3 $-\mathcal{L}_W^{\text{opp}}$ is dissipative, so its eigenvalues are all non-positive. Furthermore, 0 is not an eigenvalue, as Proposition 7 implies that for a non-structurally balanced graphon, the equation $-\mathcal{L}_W^{\text{opp}}u = 0$ is solved only by $u = 0$.

At the same time, Remark 2 implies that the essential spectrum of $-\mathcal{L}_W^{\text{opp}}$ is the essential range of $-d_{|W|}$, which is non-positive and assumed to be essentially bounded away from zero, hence strictly negative. Thus, $T(t)$ is uniformly
<figure><img src="image_2.png" /><figcaption>Fig. 1. Plots of the simulations described in Section VIII, showing the evolution in time (first column) and in the state space (second column) for the structurally balanced case (first row) and non-structurally balanced case (second row).</figcaption></figure>

exponentially stable. Recalling that solutions of (1) are of the form $u(x,t) = T(t)u_0(x)$ we thus have

$$ \|u(\cdot, t)\|_2 = \|T(t)u_0\|_2 \le \|T(t)\| \|u_0\|_2 \le Me^{-\epsilon t} \|u_0\|_2, $$

i.e., the system is exponentially stable.

<figure><img src="image_3.png" /></figure>

## VIII. SIMULATIONS

The asymptotic behavior of (1) described in Theorems 1 and 2 is shown in Figure 1. The plots show the evolution in time of (1) with $W(x,y) = \frac{1}{3}(x+y+1)\sin(f(x))$, where $f(x) = (x - \frac{1}{3})(y - \frac{1}{3})$ for the structurally balanced case, and $f(x) = \sin(10xy)$ for the non structurally balanced case. In both simulations, initial conditions are $u_0(x) = \sin(2\pi x) + \frac{1}{2}$. We can observe the convergence to zero in the non structurally balanced case, and the emergence of bipartite consensus over the partition $A = [0, \frac{1}{3}]$ and $A^c = (\frac{1}{3}, 1]$ when W is structurally balanced.

## IX. CONCLUSION

In this paper we have studied the asymptotic behavior of the Altafini model extended to signed graphons. We have extended the definition of structural balance to graphons and shown that the model's behavior in infinite dimension is consistent with the known behavior for the graph case: convergence to bipartite consensus if the graphon is structurally balanced, and convergence to zero if it is not.

In future work, we will consider how the asymptotic behavior of the graphon dynamics translates to dynamics on finite graphs sampled from the graphon. Related questions about the finite-time behavior of these dynamics on sampled graphs have been partly addressed in [14].

## REFERENCES

[1] A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysis of dynamic social networks. Part I,” *Annual Reviews in Control*, vol. 43, pp. 65–79, 2017.

[2] —, “A tutorial on modeling and analysis of dynamic social networks. Part II,” *Annual Reviews in Control*, vol. 45, pp. 166–190, 2018.

[3] C. Altafini, “Consensus problems on networks with antagonistic interactions,” *IEEE Transactions on Automatic Control*, vol. 58, no. 4, pp. 935–946, 2012.

[4] G. Shi, C. Altafini, and J. S. Baras, “Dynamics over signed networks,” *SIAM Review*, vol. 61, no. 2, pp. 229–257, 2019.

[5] G. S. Medvedev, “The nonlinear heat equation on dense graphs and graph limits,” *SIAM Journal on Mathematical Analysis*, vol. 46, no. 4, pp. 2743–2766, 2014.

[6] M. Avella-Medina, F. Parise, M. T. Schaub, and S. Segarra, “Centrality measures for graphons: Accounting for uncertainty in networks,” *IEEE Transactions on Network Science and Engineering*, vol. 7, no. 1, pp. 520–537, 2020.

[7] S. Gao and P. E. Caines, “Graphon control of large-scale networks of linear systems,” *IEEE Transactions on Automatic Control*, vol. 65, no. 10, pp. 4090–4105, 2019.

[8] R. Vizuete, P. Frasca, and F. Garin, “Graphon-based sensitivity analysis of SIS epidemics,” *IEEE Control Systems Letters*, vol. 4, no. 3, pp. 542–547, 2020.

[9] N. Ayi and N. Pouradier Duteil, “Mean-field and graph limits for collective dynamics models with time-varying weights,” *Journal of Differential Equations*, vol. 299, pp. 65–110, 2021.

[10] S. Erol, F. Parise, and A. Teytelboym, “Contagion in graphons,” *Journal of Economic Theory*, vol. 211, p. 105673, 2023.

[11] J.-F. Delmas, P. Frasca, F. Garin, V. C. Tran, A. Velleret, and P.-A. Zitt, “Individual based SIS models on (not so) dense large random networks,” *ALEA, Lat. Am. J. Probab. Math. Stat.*, vol. 21, p. 1375–1405, 2024.

[12] G. Aletti and G. Naldi, “Opinion dynamics on graphon: The piecewise constant case,” *Applied Mathematics Letters*, vol. 133, p. 108227, 2022.

[13] B. Bonnet, N. Pouradier Duteil, and M. Sigalotti, “Consensus formation in first-order graphon models with time-varying topologies,” *Mathematical Models and Methods in Applied Sciences*, vol. 32, no. 11, pp. 2121–2188, 2022.

[14] R. Prisant, F. Garin, and P. Frasca, “Opinion dynamics on signed graphs and graphons: Beyond the piece-wise constant case (Extended version),” 2024. [Online]. Available: https://arxiv.org/abs/2404.08372

[15] B. Jacob and H. J. Zwart, *Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces*. Birkhäuser Basel, 2012.

[16] K.-J. Engel and R. Nagel, *One-Parameter Semigroups for Linear Evolution Equations*. Springer New York, NY, 2000.

[17] A. Khetan and M. Mj, “Cheeger inequalities for graph limits,” *Annales de l'Institut Fourier*, vol. 74, no. 1, pp. 257–305, 2024.