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Opinion dynamics using Altafini’s model with a time-varying directed graph
Anton V. Proskurnikov and Ming Cao
Abstract— Distributed control policies (or protocols) for
multi-agent consensus have been extensively studied in recent
years, motivated by numerous applications in engineering and
science. Most of these algorithms assume the agents to be mutu-
ally cooperative and “trustful” and correspondingly attractive
couplings between the agents bring the values of the agents’
states closer. Opinion dynamics of real social groups, however,
require beyond conventional models of multi-agent consensus
due to ubiquitous competition and distrust between some pairs
of agents, which are usually characterized by the repulsive cou-
pling. Antagonistic interactions prevent the averaging tendency
of the opinions, which cooperative consensus protocols promote,
and may lead to their polarization and clustering. A simple yet
insightful model of opinion dynamics with both attractive and
repulsive couplings was proposed recently by C. Altafini, who
examined first-order consensus algorithms over static signed
interaction graphs, where arcs of positive weights connect
cooperating agents, and of negative weights correspond to
antagonistic pairs. This protocol establishes modulus consensus,
where the opinions become the same in modulus but may differ
in sign. In the present paper, we extend the modulus consensus
model to the case where the network topology is arbitrary
time-varying, directed, signed graph. We show that under mild
condition of uniform strong connectivity of the network, the
protocol establishes agreement of opinions in moduli, whose
signs may be opposite, so that the agents’ opinions either
reach consensus or polarize. This result is further extended
to nonlinear consensus protocols. We show also that, unlike
cooperative consensus algorithms, uniform strong connectivity
cannot be relaxed to uniform quasi-strong connectivity (UQSC).
I. INTRO DUC TIO N
The striking phenomenon of global consensus in a multi-
agent network, which is caused by only local interactions
between the agents, has attracted long-standing interest from
the research community. The interest is motivated by nu-
merous natural phenomena and engineering designs, that
are based, implicitly or explicitly, on reaching synchronism
between the agents (components of a complex system).
Examples include, but are not limited to, intelligence of
large-scale biological populations and multi-agent robotics.
We refer the reader to [14], [21], [22] for excellent surveys of
recent research on distributed consensus protocols and their
applications, as well as historical milestones.
Originated from iterative procedures of decision-making
[6], early consensus algorithms were based on the principle
The work was supported by the European Research Council (ERCStG-
307207), and RFBR, grants 12-01-00808, 13-08-01014 and 14-08-01015.
A.V. Proskurnikov is with the Institute of Technology and Management,
University of Groningen, 9747AG Groningen, the Netherlands and also with
St. Petersburg State University and Institute for Problems of Mechanical
Engineering RAS, St.Petersburg, Russia; avp1982@gmail.com
M. Cao is with the Institute of Technology and Management, University
of Groningen, 9747AG Groningen, The Netherlands; m.cao@rug.nl
of contraction, or averaging: every agent’s state constantly
evolves to the relative interior of the convex hull spanned by
its own and neighbors’ states. The convex hull spanned by all
the agents states, driven by such a protocol, is shrinking over
time. Based on the Lyapunov-like properties of this convex
hull and its diameter [3], [11], [16] and relevant results on
convergence of infinite products of stochastic matrices [5],
[14], [21], convergence properties of averaging consensus
protocols were examined intensively for multi-agent systems
with special attention on the effect of time-variant interaction
topology. Necessary and sufficient conditions for consensus
over undirected [4], [12], [16] and cut-balanced graphs [10]
boil down to infinite joint connectivity of the network. For
general directed graph the condition of uniform quasi-strong
connectivity (UQSC) [11], in spite of being only sufficient,
is considered to be “the weakest assumption on the graph
connectivity such that consensus is guaranteed for arbitrary
initial conditions” [17]. This common belief has recently
been confirmed by a fundamental result from [24], stating
that the UQSC is necessary and sufficient for consensus
robustness against bounded disturbances. It is also neces-
sary and sufficient for consensus in a stronger “uniform”
sense [11]. Many high-order protocols extend their first-order
counterparts [21], [22] or are squarely based on them [23].
Despite many natural and engineered teams of agents
are known to achieve common goals due to cooperation,
real-world networks often involve repulsive couplings which
represent competition or antagonism between some pairs
of agents. In multi-agent swarms, repulsive couplings may
prevent collisions between close agents [26]. In real social
groups, such relations between the individuals taking the
form of competition or hostility, are also ubiquitous and
seriously influence the dynamics of opinions, leading in
general to polarized or clustering behavior. Unlike coop-
erative consensus algorithms, the protocols with both at-
tractive and repulsive couplings still demand more indepth
mathematically rigorous analysis. A simple yet instructive
model of opinion dynamics where the agents may be both
“friendly” or “hostile” were examined by C. Altafini [1],
[2]. Altafini considered a first-order consensus protocol over
astrongly connected signed interaction graph, where the
positive weight of an arc implies cooperation between the
two agents and the negative one corresponds to their antago-
nism. This protocol establishes “modulus consensus”, where
the opinions agree in modulus but may differ in signs. If
the consensus value of modulus is non-zero, the modulus
consensus is referred to as the bipartite consensus, where the
agents divide into two groups with polarized opinions (one
of these groups, however, may be empty, so that all opinions
agree). The latter situation is possible only if the network is
structurally balanced, that is, the community splits into two
hostile camps (e.g. adherence of two political parties), the
relations inside each faction being cooperative.
The result from [1], [2] is mainly concerned with networks
with static topologies and is based on techniques of gauge
transformations. In the very recent paper [20] we extended
these results to the case of time-varying undirected graphs
and obtained complete necessary and sufficient conditions
for modulus consensus, including both consensus and polar-
ization (bipartite consensus). Similarly to classical consen-
sus protocols, joint integral connectivity of the network is
sufficient for modulus consensus and is “almost necessary”,
except for the degenerate situation where the opinions con-
verge to zero independently of initial values. In the present
paper, we consider Altafini’s model on opinion dynamics
over general directed time-varying graphs. Though one could
expect the UQSC to be sufficient for the modulus consensus,
under signed graph it does not imply consensus even for
static graphs. We show, however, that if the topology is
uniformly strongly connected, then modulus consensus is
achieved. We also examine nonlinear consensus protocols
analogous to those in [1], [2] under switching graphs.
The paper is organized as follows. Section III gives tech-
nical preliminaries and the setup of the problem in question.
Section IV presents the main results, and their application is
discussed in Section V. Appendix gives a proof of a technical
lemma.
II. PRELIMINARIES
Throughout the paper m:n, where m≤nare integers,
stands for the sequence {m, m + 1, . . . , n}. The sign of
a number x∈Ris denoted by sgn x ∈ {−1,0,1}.
The abbreviation “a.a.” stands for “almost all” (except for
the set of zero measure). Given a matrix L= (Lij), let
|L|∞:= maxij|Lij |, e.g. for column vector x∈RN,
one has |x|∞= maxi|xi|. It is easy to show that |L|∞=
sup
|x|∞≤1
|Lx|∞, where xis appropriately dimensioned.
A signed (weighted) graph is a triple G= (V, E , A),
where V={v1, . . . , vN}stands for the set of nodes,E⊂
V×Vis a set of arcs and A∈RN×Nis a signed adjacency
matrix such that ajk ̸= 0 if and only if (vk, vj)∈E. We
always assume the graph has no self-loops: ajj = 0. We say
the graph is bidirectional if ajk ̸= 0 ⇔akj ̸= 0. Throughout
the paper, we confine ourselves to digon sign-symmetric
graphs [2] which means that opposite arcs (if exist) cannot
have different signs, e.g. ajkakj ≥0. Digon sign-symmetric
graph is structurally balanced [2], [7] if the set of nodes is
divided into two sets V=V1∪V2,V1∩V2=∅, such that
ajk ≥0if (j, k)∈Viand aj k ≤0for j∈V1, k ∈V2.
Aroute connecting nodes vand v′is a sequence of
nodes vi0:= v, vi1, . . . , vin−1, vin:= v′(n≥1) such
that (vik−1, vik)∈Efor k∈1 : n. A route where
vi0=vinis referred to as a cycle. The cycle is positive if
ai0i1ai1i2. . . ain−1in>0and negative otherwise. The digon-
symmetric strongly connected graph is structurally balanced
if and only if all its oriented cycles are positive [2], [7]. A
node is called root if it can be connected with a route to any
other node of the graph. A graph is quasi-strongly connected
(QSC) if it has at least one root and strongly connected if
any two of its nodes may be connected by a route.
Any matrix A∈RN×Nmay be assigned to a signed graph
G[A] = (1 : N, E [A], A)where E[A] := {(j, k) : akj ̸= 0}.
Following [2], its Laplacian matrix L=L[A]is defined by
L= (Ljk )N
j,k=1, Lj k := −ajk, j ̸=k
N
m=1 |ajm |, j =k. (1)
In the case where ajk ≥0∀j, k and |aj k|=ajk , the matrix
(1) is the conventional Laplacian of a weighted graph [18].
We say the graph G[A]is strongly (respectively, quasi-
strongly) ε-connected if G[Aε]is strongly (respectively,
quasi-strongly) connected where Aε= (aε
ij );aε
ij =aij
when |aij | ≥ εand aε
ij = 0 otherwise. Time-variant graph
G[A(t)], where Ais locally summable, is called uniformly
strongly connected (respectively, uniformly quasi-strongly
connected, UQSC), if there exist ε > 0, T > 0such that
Gt+T
tA(s)dsis strongly (respectively, quasi-strongly)
ε-connected for any t≥0.
III. PROB LEM SE TUP
Consider a group of N≥2agents indexed 1through N,
the opinion of the i-th agent is denoted by xi∈Rand we
define x:= (x1, . . . , xN)T∈RN. The agents update their
opinions in accordance with a distributed protocol as follows:
˙x(t) = −L[A(t)]x(t), t ≥0,(2)
which may be written componentwise as
˙xj(t) =
N
k=1
|ajk (t)|(xk(t)sgn ajk (t)−xj(t)) ∀j. (3)
Here the matrix A(t) = (ajk (t)) with ajj (t)≡0is
locally bounded and describes the interaction topology of
the network. At time t≥0, the opinion of the j-th agent
is influenced by agents for which ajk (t)̸= 0 (“neighbors”).
Unlike usual consensus protocols [18] this influence may be
both cooperative (when ajk >0) or competitive (when ajk <
0). The coupling term |ajk |(xksgn ajk −xj)in (3) drives
infinitesimally the opinion of the j-th agent, respectively,
either towards the opinion of the k-th one or against it.
In the paper [2] the protocol (2) was thoroughly examined
in the case of a constant signed interaction graph (A(t)≡A),
assumed to be digon sign-symmetric and strongly connected.
Unlike purely cooperative consensus algorithms (ajk(t)≥
0), (2) does not guarantee shrinking of the convex hull of
opinions. It was shown that (−L[A]) is a Hurwitz matrix
and thus opinions converge to 0independent of initial values,
unless the graph is structurally balanced. This property
implies that a community is divided into two hostile camps
(such as votaries of two political parties), where each agent
cooperates with its camp-mates, competing with “opponents”
from the other camp. The case of unsigned weighted graph is
a special case of a structurally balanced sign graph where one
of the antagonistic camps is empty. For structurally balanced
graphs, two situations of the network behavior are possible
depending on the graph and initial condition: the opinions
may reach consensus or agree in modulus, differing in sign
(bipartite consensus [2]). Summarizing, the protocol (2) with
A(t) = const and G[A]strongly connected provides the
modulus consensus [13]:
Definition 1: The protocol (2) establishes modulus con-
sensus, if for any x(0) a number x∗≥0exists such that
lim
t→+∞|xi(t)|=x∗.(4)
It is easy to show that under modulus consensus the opinion
of each agent has a finite limit lim
t→+∞xi(t). The following
lemma, proved in Appendix, describes the structure of these
ultimate opinions.
Lemma 1: Suppose that protocol (2) establishes modu-
lus consensus. Then there exist vectors v, ρ ∈RNwith
ρ1, . . . , ρN=±1such that for any solution of (2) one has
lim
t→+∞x(t) = ρvTx(0) ⇔lim
t→+∞xj(t) = ρj
N
k=1
vkxk(0).
Lemma 1 shows that time-varying protocol (2) may establish
modulus consensus of the following types:
1) trivial consensus: v= 0 =⇒lim
t→+∞xj(t) = 0∀x(0);
2) nontrivial consensus: v̸= 0,ρ1=. . . =ρN, so
opinions agree on some value dependent on x(0);
3) bipartite consensus: v̸= 0,ρihave different signs, so
opinions polarize for any initial data with vTx(0) ̸= 0.
The goal of the present paper is to disclose conditions
which guarantee modulus consensus for general time-varying
graphs. In the paper [2] this problem was considered only for
the very special case where the graph is not only constantly
strongly connected but also weight-balanced ( although not
explicitly stated, this follows from the proof relying on [18,
Theorem 9]) with constant signs of the arcs. Below we relax
these restrictions, requiring only the graph to be uniformly
strongly connected. Dealing with real-world social networks,
the time-invariance of such relationships between individ-
uals as friendship and hostility is evidently a non-realistic
assumption. What is more important, the opinion dynamics
in social networks are usually considered to be nonlinear
[8], [9], however, such models are often reducible to the
linear case by introducing time-variant gains, depending on
the solution. Considering general time-varying graphs allows
us to examine both linear and nonlinear consensus protocols
from [1], [2] in the common framework.
It should be noticed that the problem of determining to
which behavior of 1)-3) in Lemma 1 a given graph corre-
sponds seems to be non-trivial and is a subject of ongoing
research. In the important case of bidirectional interactions
it was solved in our recent paper [20], where necessary and
sufficient criteria for each type of the modulus consensus
were obtained. Another situation where this classification
may be given and vectors ρ, v be explicitly found is the
static topology case: A(t)≡A. The relevant result from [2]
deals with strongly connected graphs only, which restriction
will be discarded in Section IV-A.
IV. MAIN RES ULTS
The section is organized as follows. We start with the
case of static graphs (Subsection IV-A), where necessary and
sufficient conditions are offered for non-trivial (Lemma 2)
and trivial (Lemma 3) modulus consensus. Subsection IV-
B deals with the switching topology case that is the main
concern of the present paper. Lemma 4 shows that the
maximal of opinion moduli is non-increasing independently
of the graph properties and hence has a limit at infinity.
Under additional restriction of uniform strong connectivity
the protocol establishes modulus consensus; this is the main
result of the paper (Theorem 1). The proofs of the mentioned
results are omitted due to space limitations, available upon
request and to appear in [19]. In the final Subsection IV-C we
demonstrate that, unlike the cooperative consensus protocols,
the conventional UQSC condition is neither necessary nor
sufficient for modulus consensus over general signed graph.
A. Modulus consensus over static signed graphs
The first lemma gives a necessary and sufficient condition
for behaviors 2) and 3) under static topology.
Lemma 2: Let the topology be static: A(t)≡A. The
protocol (2) establishes non-trivial modulus consensus (with
v̸= 0) if and only if G[A]is structurally balanced and
quasi-strongly connected. If this is the case, the matrix L[A]
has zero eigenvalue of algebraic and geometric multiplicity
1. The vectors ρand vare respectively right and left
eigenvectors at zero, that is vTL[A] = L[A]ρ= 0, and
vTρ= 1. For the correspondent subdivision of agents into
hostile camps V1∪V2= 1 : N, one may choose ρand vso
that ρi= 1 for i∈V1and ρi=−1when i∈V2. Nontrivial
consensus is established only if ajk ≥0for any j, k and
thus either V1or V2is empty; otherwise opinions polarize.
If the graph G[A]is not structurally balanced, only trivial
kind of modulus consensus may be reached. An obvious
obstacle to this behavior is existence of a subcommunity,
which is independent of the remaining agents and is able
to reach non-trivial modulus consensus. Given a graph G=
(V, E , A), any graph G′= (V′, E′, A′), where V′⊆V,
E′=E∩(V′×V′)and A′is the corresponding submatrix
of A, is said to be a subgraph of G. We say a subgraph G′
is isolated if ajk = 0 whenever j∈V′and k̸∈ V′, that
is, agents from V′are not aware of the opinions of their
teammates from V\V′. In accordance with Lemma 2, if
the graph G=G[A]has a non-empty isolated structurally
balanced subgraph, its nodes agree on some non-zero (for
a.a. initial conditions) opinion which entails that the whole
community is not able to reach trivial consensus. As shown
by the following lemma, the inverse proposition is also valid.
Lemma 3: The protocol (2) establishes trivial consensus,
i.e. −L[A]is a Hurwitz matrix, if and only if the graph G[A]
has no non-empty structurally balanced isolated subgraphs
(in particular, G[A]is not structurally balanced itself).
B. Modulus consensus under switching signed graphs
We start with the following useful lemma, which does
not rely on any connectivity assumptions and shows, in
particular, that solutions to (2) are always bounded.
Lemma 4: For any solution of system (2), the function
|x(t)|∞= maxi|xi(t)|is monotonically non-increasing:
|x(t)|∞≤ |x(t0)|∞whenever t≥t0≥0. Equivalently, the
Cauchy evolutionary matrix Φ(t;t0)of system (2) satisfies
the inequality |Φ(t;t0)|∞≤1for t≥t0.
Lemma 4 shows, in particular, that the maximal modulus
always has a limit lim
t→+∞|x(t)|∞. However, to guarantee that
all other moduli converge to the same limit, one requires
additional connectivity assumption.
Assumption 1: The functions ajk(t)are essentially
bounded: there exists M > 0such that |ajk (t)| ≤ Mfor a.a.
t≥0. The graph G[A(·)] is uniformly strongly connected.
We are now in a position to present our main theorem
which gives a sufficient condition for modulus consensus.
Theorem 1: Under Assumption 1 the protocol (2) estab-
lishes modulus consensus.
As a corollary, we obtain the result from [2] for static
graphs and consensus criterion for cooperative protocols.
Corollary 1: Suppose that A(t)≡Aand G[A]is strongly
connected. Then protocol (3) establishes modulus consensus.
The proof is immediate since a static strongly connected
graph is also uniformly strongly connected.
Corollary 2: Let Assumption 1 hold and ajk (t)≥0. Then
protocol (2) establishes non-trivial consensus.
Proof: By virtue of Theorem 1, modulus consensus
is established. Accordingly to Lemma 1 only three types of
such a consensus are possible, which are trivial consensus,
polarization and consensus. The common feature of the first
two types is that for a.a. x(0) there exists i∈1 : Nsuch that
lim
t→+∞xi(t)≤0. It is well known [11], [15] that the convex
hull of the agents’ states ∆(t) = [minixi(t),maxixi(t)] is
non-expanding over time and in particular, if xi(0) ≥1, then
xi(t)≥1for any t≥0. This obviously excludes the options
of trivial consensus and bipartite consensus.
C. UQSC Is Not Sufficient for Modulus Consensus
In this subsection we are going to point out a crucial
differences between purely cooperative protocols (2) (ajk ≥
0) and those with antagonistic interactions. Namely, the
uniform quasi-strong connectivity (UQSC) property, com-
monly adopted as sufficient and “almost necessary” condition
for consensus with cooperative interactions, appears to be
insufficient for modulus consensus over signed graph. As was
discussed in Section II, UQSC is also not necessary.
It is widely known that the result from Corollary 2 is in
fact valid under weaker assumption of uniform quasi-strong
connectivity [11], [15], [25], whereas in a general situation
the USQ property cannot be relaxed to UQSC even for the
case of static topology A(t)≡A, as follows from Lemma 3
and is illustrated by the following simple example.
Example 1. Consider a team of N= 3 agents with states
x1(t), x2(t), x3(t). Assume that a12 =a21 =−1,a31 =
a32 = 1 (see Fig. 1), and hence the equations are
˙x1= (−x2−x1),˙x2= (−x1−x2),˙x3= (x1+x2−x3).
Fig. 1. Static quasi-strongly connected graph: no modulus consensus
The system has the set of equilibria x1=a, x2=−a, x3=
0,a∈Rand hence does not reach modulus consensus.
Example 1 deals with structurally unbalanced static graph
(agents 1and 2are constantly antagonistic, so the structural
balance requires the agent 3to cooperate with only one of
them, competing or interacting not with the other, whereas
in reality it cooperates with both of agents 1 and 2). In the
same time, nodes 1and 2and two connecting them arcs
constitute a structurally balanced isolated subgraph. Were the
graph static and structurally balanced, QSC would establish
modulus consensus due to Lemma 2. Moreover, this result
remains valid for dynamically changing structurally balanced
graph, provided that the “hostile camps” remain unchanged.
Lemma 5: Suppose that V=1:N=V1∪V2, where
ajk (t)≥0for any t≥0if j, k ∈V1or j, k ∈V2; otherwise,
ajk (t)≤0for any t≥0. If the graph G[A(·)] is UQSC,
the protocol establishes non-trivial consensus (if V1=∅or
V2=∅) or bipartite consensus (when V1, V2̸=∅).
Proof: Introducing a diagonal matrix D=
diag(d1, . . . , dN)by di= 1 for i∈V1and di=−1for
i∈V2, one can easily check that the gauge transformation
[2] x7→ z:= Dx transforms the system (2) into
˙z(t) = −L[|A(t)|]z(t),|A|= (|ajk |).(5)
Obviously G[A]is UQSC if and only if G[|A|]is UQSC.
Therefore, protocol (5) establishes consensus, corresponding
to either non-trivial consensus (if V1or V2is empty) or
bipartite consensus of network (2) since zi=±xi.
However, if the relations of friendship and hostility be-
tween the agents also evolve over time, the UQSC property
is not sufficient for modulus consensus. Moreover, in the
following example we construct a protocol (2) with periodic
piecewise-constant matrix A(t), such that the graph G[A(t)]
is UQSC and structurally balanced for any t≥0, but
nevertheless the protocol fails to establish consensus.
Example 2. Consider the more general system
˙x1(t) = (−x2(t)−x1(t)),˙x2(t) = (−x1(t)−x2(t)),
˙x3(t) = a31(t)(x1(t)−x3(t)) + a32 (t)(x2(t)−x3(t)).
(6)
The functions a31, a32 are constructed as follows. Consider
first system (6) with a31(t)≡1, a32 (t)≡0and the solution
to (6) launched at the initial state x1(0) = 1, x2(0) =
−1, x3(0) = −1/2. It is evident that x1(t) = 1 = −x2(t)for
any t≥0and x3(t)↑1as t→+∞. Therefore, there exists
the first time instant T0>0such that x3(T0) = 1/2. Notice
that in the symmetric situation where a31(t)≡0, a32 (t)≡1
and x(t)is a solution to (6) starting at x1(0) = 1, x2(0) =
−1, x3(0) = 1/2, one has x3(t)↓ −1and T0is the first
instant where x3(T0) = −1/2. Taking
a31(t) = 1−a32 (t) = 1, t ∈[0; T0)∪[2T0; 3T0)∪. . .
0, t ∈[T0; 2T0)∪[3T0; 4T0)∪. . . ,
one finally gets a 2T0-periodic A(t)which, evidently, corre-
sponds to uniformly quasi-strongly connected graph G[A(·)].
Moreover, this graph is also quasi-strongly connected and
structurally balanced at any time. Even so the solution to
(6) starting at x1(0) = 1, x2(0) = −1, x3(0) = −1/2does
not achieve modulus consensus. It can easily shown that
x1(t) = −x2(t)=1for any t≥0. Since a31(t)=1
and a32(t)=0when t<T0, one has x3(T0)=1/2by
definition of T0. On the next interval t∈[T0; 2T0)one has
a31(t) = 0 and a32(t) = 1 and hence x3(2T0) = −1/2,
so the solution x(t)is periodic and x3(t)∈[−1/2; 1/2]
whereas |x1(t)|=|x2(t)|= 1.
V. APPLICATIONS: NONLINEAR PROTOCOLS
In this section we consider some applications of The-
orem 1 to nonlinear consensus protocols, similar to those
studied in [1], [2].
A. Additive Laplacian protocols
Our first example concerns with nonlinear consensus
algorithms that are referred in [2] as the “additive Laplacian
feedback schemes”. The first of them is
˙xi(t) =
N
j=1
|aij (t)|(hij (xj(t)sgn aij (t))−hij (xi(t))),(7)
and the second protocol has the form
˙xi(t) =
N
j=1
|aij (t)|hij (xj(t)sgn aij (t)−xi(t))) ∀i. (8)
We adopt the following assumption about the nonlineari-
ties.
Assumption 2: For any i, j = 1, . . . , N the map hij ∈
C1(R)is strictly increasing (and hence h′
ij >0) with
hij (0) = 0.
Defining the functions Hij [y, z]as follows: Hij [y, z] :=
(hij (y)−hij (z))/(y−z)for y̸=zand Hij [z, z] := h′
ij (z).
It is easily noticed that Hij is a continuous function and
hij (y)−hij (z) = Hij [y, z](y−z)∀y, z. Under Assumption 2
Theorem 1 appears to be applicable to the protocols (7),(8),
as shown by the following lemma.
Lemma 6: Let x(t)be a solution to system (7), which is
defined for t≥0. Define the matrix A(t) = (aij (t)) by
aij (t) := aij (t)Hij [xj(t)sgn aij (t), xi(t)]. Then
˙x(t) = −L[A(t)]x(t).(9)
If the matrix-valued function A(·)satisfies Assumption 1, the
same is true for A(·). The same claims hold for the protocol
(8), taking aij (t) := aij (t)Hij [xj(t)sgn aij (t)−xi(t),0].
Proof: We consider system (7), and the protocol (8)
may be studied in the same way. Equation (9) is im-
mediate from the definitions of aij and Hij . As follows
from Lemma 4, the solutions of (9) remain bounded since
|x(t)|∞≤ |x(0)|∞. Since Hij >0are continuous functions,
there exist M > m > 0such that m≤Hij [y, z]≤M
whenever |y|,|z| ≤ |x(0)|∞, and these inequalities hold,
in particular, for y:= xj(t)sgn aij (t)and z:= xi(t).
Therefore, m|ajk | ≤ |ajk| ≤ M|aj k|, and hence if the
matrix A(·)is bounded and the associated graph G[A(·)]
is uniformly strongly connected, the same claims hold for
A(t).
Application of Theorem 1 to (9) yields the following
result.
Theorem 2: Under Assumption 2, the solutions to systems
(7),(8) exist, are unique and infinitely prolongable for any
initial condition. If Assumption 1 holds, the protocols (7),(8)
establish modulus consensus.
Proof: Since the right-hand sides of (7),(8) are smooth
in x, the solutions exist locally and are unique. According
to Lemma 4 and 6, the solutions remain bounded and
thus infinitely prolongable. Under Assumption 1, modulus
consensus follows from Theorem 1, applied to (9).
Comparing the result of Theorem 2 with that of [2,
Theorem 3,4], one notices that our assumption about the
nonlinearities hij differs from [2], where they are not as-
sumed to be smooth, but only monotonic with some integral
constraint. However, unlike [2, Theorem 3,4], functions hij
may be heterogeneous and not necessarily odd; the graph
may be time-varying, satisfying Assumption 1.
B. Nonlinear Laplacian Flow
In this subsection we examine the nonlinear consensus
protocol similar to that addressed in [2, Section IV-B]:
˙xi(t) =
N
j=1
|Fij (t, x)|(xj(t)sgn Fij (t, x)−xi(t)),(10)
here i∈1 : Nand Fij : [0; ∞)×RN→Rare Caratheodory
maps, i.e. Fij (t, ·)are continuous for a.a. tand Fij (·, x)are
measurable for any x. We assume also that for any compact
set K⊂[0; ∞)×RNone has
sup{Fij (t, x) : (t, x)∈K}<∞ ∀i, j. (11)
Theorem 3: For any initial condition x(0) a solution of
(10) exists for t≥0. If the matrix-valued function A(t) :=
Fij (t, x(t)) satisfies Assumption 1, the protocol (10) estab-
lishes modulus consensus.
Proof: Using Lemma 4, one proves that the solution
is bounded and hence its derivative also remains bounded
due to (11), so any solution is infinitely prolongable. The
remaining claim follows now from Theorem 1.
To guarantee the boundedness of the matrix A(t), one
in practice has to strengthen the condition (11), assuming
that for any compact C⊂RNone has sup{|Fij (t, x)|:
t≥0, x ∈C}<∞. Although in general it is hard to
verify the uniform strong connectivity of G[A(·)], where
A(t) = Fij (t, x(t)) depends on the concrete solution, in
special cases such a property may also be proved. For
instance, it is implied by the global strong ε-connectivity [2,
Section IV-B]: the graph G(ˆ
Fij (t, x)) is strongly ε-connected
for any t, x. The result of Theorem 3 extends the result from
[2, Section IV-B] in several ways. First of all, it deals with
time-variant gains Fij(t, x)and does not require them to
have a constant sign. In particular, system (10) does not
necessarily generate order-preserving flow [1]. Moreover, we
do not assume that the graph G[A(t)] is weight-balanced
which can hardly be provided for nonlinear functions Fij.
At last, we replace global ε-connectivity where ε > 0with
uniform strong connectivity.
VI. CONCLUSION
In the present paper, we extend a model of opinion dy-
namics in social networks with both attractive and repulsive
interactions between the agents, which was proposed in
recent papers by C. Altafini, who considered the conventional
first-order consensus protocols over signed graphs. Altafini
showed, in particular, the possibility of opinion polarization
if the interaction graph is structurally balanced. In general,
the protocol establishes modulus consensus, where the agents
agree in modulus but differ in signs. In the present paper,
we have examined dynamics of Altafini’s protocols with
switching directed topologies and offer sufficient conditions
for reaching modulus consensus that boil down to uniform
strong connectivity of the network. We are currently working
with sociologists to test the theoretical results presented in
this paper using data from human social groups.
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APP END IX
PROO F OF LEM MA 1
Consider the protocol (2), establishing modulus consensus.
Let Φ(t)stands for the Cauchy evolutionary operator so
that x(t) = Φ(t)x(0). Note that since functions xi(t)are
continuous, the existence of the limits lim
t→+∞|xi(t)|=x∗
implies that the limits lim
t→+∞xi(t)also exist (and equal to
±x∗). Therefore Φ(t)−→
t→∞ Φ∗:= [ϕ1, . . . , ϕN]as t→ ∞,
where the columns ϕjhave entries with equal modules.
The same applies to any linear combination N
j=1 αjϕj. If
Φ∗= 0, the statement of Lemma 1 is evident, taking v= 0.
Assume that one of ϕj, say, ϕ1is nonzero, thus ϕ1=v1ρ
where v1̸= 0 and ρis a vector with entries ±1. Notice that
for any real numbers α, β ̸= 0 we have |α−β| ̸=|α+β|.
Therefore, if ϕj̸= 0 for some j̸= 1, all entries of
ϕj−ϕ1have the same module if and only if ϕj=vjρ,
vj̸= 0. If ϕj= 0, we put by definition vj= 0. Therefore,
ϕj=vjρfor any jand lim
t→∞ xj(t)=Φ∗=ρvTx(0), where
v:= (v1, . . . , vN)T.
