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arXiv:1512.07150v1 [math.OC] 22 Dec 2015
Exponential Convergence of the Discrete-Time Altafini Model∗
Ji Liu Xudong Chen Tamer Ba¸sar Mohamed Ali Belabbas
December 23, 2015
Abstract
This paper considers the discrete-time version of Altafini’s model for opinion dynamics in
which the interaction among a group of agents is described by a time-varying signed digraph.
Prompted by an idea from [1], exponential convergence of the system is studied using a graphical
approach. Necessary and sufficient conditions for exponential convergence with respect to each
possible type of limit states are provided. Specifically, under the assumption of repeatedly
jointly strong connectivity, it is shown that (1) a certain type of two-clustering will be reached
exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs
is repeatedly jointly structurally balanced corresponding to that type of two-clustering; (2)
the system will converge to zero exponentially fast for all initial conditions if, and only if, the
sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on
the convergence rate is also provided.
1 Introduction
Over the past decade, there has been considerable interest in developing algorithms intended to
cause a group of multiple agents to reach a consensus in a distributed manner [2–15]. Consensus
processes have a long history in social sciences and are closely related to opinion dynamics [16].
Probably the most well-known opinion dynamics is the DeGroot model which is a linear discrete-
time consensus process [17]. Recently, quite a few models have been proposed for opinion dynamics,
including the Friedkin-Johnsen model [18–20], the Hegselmann-Krause model [21–23], the Deffuant-
Weisbuch model [24], and the DeGroot-Friedkin model [25–28]. A particularly interesting opinion
dynamics model, which was first proposed by Altafini [29,30] and can be viewed as a more general
linear consensus model, has received increasing attention lately [1, 31–38].
The Altafini model deals with a network of n > 1 agents and the constraint that each agent
is able to receive information only from its “neighbors”. Unlike the existing models for opinion
dynamics or consensus, the neighbor relationships among the agents are described by a time-
dependent, signed, digraph (or directed graph) in which vertices correspond to agents, arcs (or
directed edges) indicate the directions of information flow, and, in particular, the signs represent
the social relationships between neighboring agents in that a positive sign means friendship (or
cooperation) and a negative sign indicates antagonism (or competition). Each agent ihas control
over a time-dependent state variable xi(t) taking values in R, which denotes its opinion on some
issue. Each agent updates its opinion based on its own current opinion, the current opinions of
its current neighbors, and its relationships (friendship or antagonism) with respect to its current
∗This research was supported in part by AFOSR MURI Grant FA 9550-10-1-0573. J. Liu, X. Chen, T. Ba¸sar, and
M.-A. Belabbas are with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign ({jiliu,
xdchen, basar1, belabbas}@illinois.edu).
1
neighbors. Specifically, for those neighbors with friendship, the agent will trust their opinions; for
those neighbors with antagonism, the agent will not trust their opinions and, instead, the agent
will take the opposite of their opinions in updating, which is the key difference between the Altafini
model and other opinion dynamics models.
The continuous-time Altafini model has been studied in [31–33, 38], and papers [1, 34–37, 39]
have studied the discrete-time counterpart. This paper will focus on the latter and present a more
comprehensive treatment of the work in [39]. Specifically, the paper provides proofs for the main
theorems, addresses the convergence rate issue, and discusses a time-invariant case with a less
restrictive connectivity condition, which were not included in [39].
The most general result in the literature regarding the discrete-time version of Altafini’s model
is that for any “repeated jointly strongly connected” sequence of signed digraphs, the absolute
values of all the agents’ opinions will asymptotically reach a consensus, which is called “modulus
consensus”, having standard consensus and “bipartite consensus” as special cases [37]. The result is
independent of the structure of signs in the digraphs which can be described by the term structural
balance (or structural unbalance) from social sciences [40] in that different types of structural bal-
ance correspond to different clusterings of opinions in the network (Section 3.1). Notwithstanding
this, the following questions remain. What are necessary and sufficient conditions on the sequence of
signed digraphs that will lead to a specific clustering? When will the convergence be exponentially
fast and how can the rate of convergence be characterized? What will happen if the assumption
of strong connectivity is relaxed? This paper aims to answer these questions and will appeal to a
graphical approach prompted by an idea introduced in [1], as further discussed below.
In the recent work by Hendrickx [1], a very nice lifting approach was proposed to establish the
equivalence between the Altafini model and an expanded DeGroot (or consensus) model with a
special structure; with this equivalence relationship, the convergence results of the Altafini model
were extended under the so-called “type-symmetry” assumption (i.e., the interaction and signs
between each pair of neighboring agents are symmetric). In the discussion section of [1], it was
mentioned that this lifting approach might “prove harder to treat systems where the presence of
interaction is symmetric but their signs are not”. This is precisely what we consider in this paper
and with more generality, we make use of the lifting approach and consider the most general case
where both interaction and signs between each pair of neighboring agents can be asymmetric. The
same approach was used in [36] which provides sufficient conditions for asymptotic zero consensus
and bipartite consensus for a certain class of time-varying “rooted graphs” without the “type-
symmetry” assumption.
The main contributions of this paper are first, development of a graphical approach to the anal-
ysis of the discrete-time version of Altafini’s model; second, derivation of necessary and sufficient
conditions for exponential convergence with respect to different limit states, under a strong connec-
tivity assumption; third, providing an upper bound on the convergence rate; and last, description
of the system behavior in the case when the neighbor graph is rooted, but not strongly connected.
The remainder of the paper is organized as follows. Some notations and preliminaries are
introduced in Section 1.1. In Section 2, the discrete-time Altafini model is introduced and the
existing modulus consensus result is reviewed. In Section 3.1, the concepts of structural balance
and clustering are introduced. The main results of the paper are presented in Section 3.2, whose
analysis and proofs are given in Section 4. The convergence rate issue is addressed in Section 3.3.
In Section 5, the system behavior, when the strong connectivity assumption is relaxed, is discussed
for the time-invariant case. The paper ends with some concluding remarks in Section 6.
2
1.1 Preliminaries
For any positive integer n, we use [n] to denote the index set {1,2,...,n}. We view vectors as
column vectors and write x′to denote the transpose of a vector x. For a vector x, we use xito
denote the ith entry of x. For any real number y, we use |y|to denote its absolute value. For any
matrix M∈Rn×n, we use mij to denote its ijth entry and write |M|to denote the matrix in Rn×n
whose ijth entry is |mij |. A nonnegative n×nmatrix is called a stochastic matrix if its row sums
are all equal to 1.
For a digraph G, we use (i, j) to denote a directed edge from vertex ito vertex j. We say that
Ghas an undirected edge between vertex iand vertex j, denoted by [i, j], if either (i, j) or (j, i)
is a directed edge in G. A directed walk of Gis a sequence of vertices v0, v1,...,vmin Gsuch
that (vi−1, vi) is a directed edge in Gfor all i∈[m]. If the vertices v0and vmare the same, then
the directed walk is called closed. If all the vertices are distinct, then the directed walk is called a
directed path. If all the vertices are distinct except that vertices v0and vmare the same, then the
directed walk is called a directed cycle.
We write Gsa to denote the set of all digraphs with nvertices, which have self-arcs at all vertices.
The graph of an n×nmatrix Mwith nonnegative entries is an nvertex directed graph γ(M) defined
so that (i, j) is an arc from ito jin the graph only when the jith entry of Mis nonzero. Such a
graph will be in Gsa if and only if all diagonal entries of Mare positive. For purposes of analysis,
we write ¯
Gsa to denote the set of all digraphs with 2nvertices which have self-arcs at all vertices.
A digraph Gis strongly connected if there is a directed path between each pair of its distinct
vertices. A digraph Gis rooted if it contains a directed spanning tree. A digraph Gis weakly
connected if there is an undirected path between each pair of its distinct vertices. Note that every
strongly connected graph is rooted and every rooted graph is weakly connected. The converse
statements, however, are false.
2 The Discrete-Time Altafini Model
Consider a network of n > 1 agents labeled 1 to n.1Each agent ihas control over a real-valued scalar
xi(t) which the agent is able to update from time to time. Each agent ican receive information only
from certain other agents called agent i’s neighbors. Neighbor relationships among the nagents are
described by a signed digraph G(t), called neighbor graph, on nvertices with an arc from vertex i
to vertex jwhenever agent iis a neighbor of agent jat time t. For simplicity, we always take each
agent as a neighbor of itself. Thus, each G(t) has self-arcs at all nvertices. Each arc is associated
with a sign, either positive “+” or negative “–”, which indicates that agent iregards agent jas a
cooperative neighbor if arc (j, i) is associated with a “+” sign, or a competitive neighbor if (j, i) is
associated with a “–” sign. It is natural to assume that each self-arc is associated with a “+” sign
since an agent cannot compete with itself.
We are interested in the following discrete-time iterative update rule. At each time tand for
each i∈[n], agent iupdates its opinion by setting
xi(t+ 1) = X
j∈Ni(t)
aij (t)xj(t) (1)
where Ni(t) denotes the set of neighbors of agent iat time t, and each aij (t) is a real-valued weight
whose sign is consistent with the sign of the arc (j, i). The weights aij (t) are assumed to satisfy
1The purpose of labeling of the agents is only for convenience. We do not require a global labeling of the agents
in the network. We only assume that each agent can identify and differentiate between its neighbors.
3
the following assumption.
Assumption 1 For each i∈[n], there hold aii(t)>0and
n
X
j=1
|aij (t)|= 1
for all time t. There exists a positive number β > 0such that |aij (t)| ≥ βwhen |aij (t)|>0for all
i, j ∈[n]and t.
The update rule (1) is an analog of the continuous-time update rule in [30]. The nupdate
equations in (1) can be combined into one linear recursion equation
x(t+ 1) = A(t)x(t) (2)
in which x(t) is a vector in Rnwhose ith entry is xi(t) and A(t) is an n×nmatrix whose ijth
entry is aij (t). From Assumption 1, it can be seen that the infinite norm of each A(t) equals
1 and |A(t)|is a stochastic matrix with positive diagonal entries. In the case when all the arcs
have positive signs, the system becomes the standard linear consensus process which will reach a
consensus exponentially fast if and only if G(t) is “repeatedly jointly rooted” [6, 15]. Thus, system
(2) can be regarded as a generalized model of standard linear consensus.
For each matrix A(t) which satisfies Assumption 1, we define the graph of A(t) to be a signed
digraph so that (i, j) is an arc in the graph whenever aji(t) is nonzero and the sign of (i, j) is the
same as the sign of aji (t). It is straightforward to verify that the graph of A(t) is the same as the
neighbor graph G(t). We will use this fact without any special mention in the sequel.
2.1 Modulus Consensus
We say that system (2) reaches a modulus consensus if the absolute values of all nagents |xi(t)|,
i∈[n], converge to the same value as time t→ ∞. If, in addition, the limiting value does not
equal zero and the agents’ limiting values have opposite signs, system (2) is said to reach a bipartite
consensus. It should be clear that consensus (including zero consensus and nonzero consensus) and
bipartite consensus are two special cases of modulus consensus.
To state the existing modulus consensus result, we need the following concepts.
We say that a finite sequence of digraphs G1,G2,...,Gmwith the same vertex set is jointly
strongly connected if the union2of the digraphs in this sequence is strongly connected. We say that
an infinite sequence of digraphs G1,G2,...with the same vertex set is repeatedly jointly strongly con-
nected if there exist positive integers pand qfor which each finite sequence Gq+kp ,Gq+kp+1,...,Gq+(k+1)p−1,
k≥0, is jointly strongly connected. It is worth emphasizing that the above connectivity con-
cepts are also applicable to signed digraphs, without taking signs into account. Repeatedly jointly
strongly connected graphs are sometimes called “B-connected” graphs in the consensus litera-
ture [41].
The following result states that system (2) asymptotically reaches modulus consensus for all
initial conditions under an appropriate connectivity assumption.
Proposition 1 (Theorem 2.1 in [37]) Suppose that all nagents adhere to the update rule (1) and
Assumption 1 holds. Suppose that the sequence of neighbor graphs G(1),G(2),... is repeatedly
jointly strongly connected. Then, system (2) asymptotically reaches a modulus consensus.
2The union of a finite sequence of unsigned digraphs with the same vertex set is an unsigned digraph with the
same vertex set and the arc set which is the union of the arc sets of all digraphs in the sequence.
4
This proposition was proved in [37], although the form of the connectivity condition is slightly
different. We provide an alternative, more transparent proof in the appendix.
3 Main Results
As mentioned earlier, modulus consensus includes as special cases bipartite consensus, zero consen-
sus, and nonzero consensus. In the sequel, we will explore necessary and sufficient conditions for
each type of limit states. It should be clear that the conditions will naturally depend on the sign
structure of the sequence of neighbor graphs. We will appeal to the concept of “structural balance”
from social sciences [40]. The concept of structurally balanced networks was first introduced to
consensus problems in [30]. For recent developments on this topic, see [42–44].
3.1 Structural Balance
A signed digraph Gis called structurally balanced if the vertices of Gcan be partitioned into
two sets such that each arc connecting two agents in the same set has a positive sign and each
arc connecting two agents in different sets has a negative sign. Otherwise, the graph Gis called
structurally unbalanced.
For our purposes, we extend the same definition of structural balance (and structural unbalance)
to “signed multidigraphs”, where a signed multidigraph is a signed digraph in which for some ordered
pairs of two distinct vertices iand j, there exist two directed edges from vertex ito vertex j, with
positive and negative signs respectively.3Our reason for doing so will be clear shortly.
In this paper, we regard signed multidigraphs as a subset of signed digraphs. We call those
signed digraphs which are not signed multidigraphs, signed simple digraphs. In other words, a signed
simple digraph does not have multiple directed edges. Thus, the sets of signed simple digraphs and
signed multidigraphs are a partition of the set of signed digraphs.
There is an equivalent condition for checking structural balance (and structural unbalance) for
signed simple digraphs in the literature [30, 40], as follows. Let Gbe a signed simple digraph. For
each directed (or undirected) cycle Cin G, we say that Cis negative if it contains an odd number
of negative signs, and positive otherwise. It has been shown that Gis structurally balanced if
and only if it does not have negative undirected cycles [30, 40]. Consequently, Gis structurally
unbalanced if and only if it has at least one negative undirected cycle. Using arguments similar to
those as in [30, 40], the above equivalent conditions also hold for signed multidigraphs, and thus
hold for all signed digraphs.
A simple example of a structurally unbalanced digraph is the signed digraph in which there exists
one pair of distinct agents iand jsuch that the arcs (i, j) and (j, i) have different signs. Another
example is the signed multidigraph in which there exist two arcs from vertex ito another vertex
jand they have different signs. The two graphs in the above examples both have an undirected
cycle, consisting of the vertex sequence i, j, i, which is negative.
In the sequel, we will differentiate between different types of structurally balanced signed di-
graphs by introducing the concept of clustering.
Let Ibe a set of vectors in Rnsuch that for each b∈ I, there hold b1= 1 and biequals either
1 or −1 for all i∈[n] and i6= 1. The set Iis a finite set and 1∈ I where 1denotes the vector in
Rnwhose entries all equal 1. Each element bin Iuniquely defines a clustering of all the agents in
3In graph theory, a “multidigraph” is usually allowed to have multiple (can be more than two) directed edges
from a vertex ito another vertex j. For our purposes, we restrict our attention on those digraphs which have at most
two directed edges from ito j. Thus, the set of such n-vertex (signed) digraphs is a finite set.
5
the network by the signs of the entries of b. Specifically, we use V+
bto denote the set of indices in
[n] such that bi= 1 for all i∈ V +
band V−
bto denote the set of indices in [n] such that bi=−1 for
all i∈ V−
b. It can be seen that V+
band V−
bare disjoint and V−
b∪ V+
b= [n]. Since b1= 1, it follows
that V+
bis nonempty. In the case when V−
bis nonempty, the vector bdefines a unique biclustering
among the agents in the network. In the special case in which V−
bis an empty set (i.e., b=1), all
the agents belong to the same cluster.
It is worth noting that each possible nonzero modulus consensus (i.e., all the absolute values of
the opinions of the agents in the network reach a consensus at some nonzero value) can be uniquely
represented by an element b∈ I in that the agents, including agent 1, whose labels in V+
bhave
the same sign, and the agents whose labels in V−
bhave the same sign. In particular, the vector
1represents the standard consensus, and every other element in Irepresents a unique bipartite
consensus. Thus, all possible nonzero limit states of modulus consensus can be partitioned into
different types, each corresponding to a unique vector in I.
Each element bin I, as well as the unique associated clustering, also corresponds to a class of
signed digraphs with nvertices. To be more precise, each element b∈ I such that b6=1corresponds
to a class of structurally balanced digraphs, in which the arcs connecting two vertices in V+
b(or V−
b)
have positive signs and the arcs connecting one vertex in V+
band one vertex in V−
bhave negative
signs, and the element 1corresponds to the class of signed digraphs whose signs are all positive.
We call each of the above classes of signed digraphs a structurally balanced class and denote it by
Cb,b∈ I. The remaining signed digraphs with nvertices are all structurally unbalanced and we call
this class of graphs the structurally unbalanced class, denoted by Cu. In general, a signed digraph
Gmay belong to different classes. But in the case when Gis weakly connected, it belongs to a
unique class.
3.2 Exponential Convergence
For each type of nonzero modulus consensus and zero (modulus) consensus, we will provide nec-
essary and sufficient conditions under which the modulus consensus can be reached exponentially
fast. To state our main results, we need the following concepts.
The union of two signed digraphs Gpand Gqwith the same vertex set is the signed digraph with
the same vertex set, and the signed arc set being the union of the signed arcs of the two digraphs.
It is clear that the union can be a signed multidigraph. Note that union is an associative binary
operation; because of this, the definition extends unambiguously to any finite sequence of signed
digraphs including signed multidigraphs. Since we are interested in signed digraphs with positive
self-arcs at all vertices, the signed digraph generated by the union operation will also have positive
self-arcs at all vertices.
We say that a finite sequence of signed digraphs G1,G2,...,Gmwith the same vertex set is
jointly structurally balanced with respect to a clustering b∈ I (or jointly structurally unbalanced)
if the union of the graphs in this sequence is structurally balanced with respect to the clustering b
(or structurally unbalanced). We say that an infinite sequence of signed digraphs G1,G2,... with
the same vertex set is repeatedly jointly structurally balanced with respect to a clustering b∈ I
(or repeatedly jointly structurally unbalanced) if there exist positive integers pand qfor which each
finite sequence Gq+kp,Gq+kp+1,...,Gq+(k+1)p−1,k≥0, is jointly structurally balanced with respect
to the clustering b(or jointly structurally unbalanced).
Note that if a finite sequence of signed digraphs with the same vertex set is jointly structurally
balanced with respect to a clustering b∈ I, then each graph in the sequence is structurally bal-
anced with respect to the clustering b. Thus, in any sequence of signed digraphs which is jointly
6
structurally balanced with respect to the clustering b, all those signed digraphs which are not in
the structurally balanced class Cbcan appear in the sequence for only a finite number of times.
It is also worth noting that if a finite sequence of signed digraphs Σ is jointly structurally
unbalanced, then any finite sequence of signed digraphs which contains Σ as a subsequence must
be jointly structurally unbalanced.
Remark 1 If an infinite sequence of signed digraphs G1,G2,... with the same vertex set is both
repeatedly jointly strongly connected and repeatedly jointly structurally balanced with respect to a
clustering b∈ I (or repeatedly jointly structurally unbalanced), then it can be seen that there
must exist positive integers pand qfor which each finite sequence Gq+k p,Gq+kp+1,...,Gq+(k+1)p−1,
k≥0, is jointly strongly connected and jointly structurally balanced with respect to the clustering b
(or jointly structurally unbalanced). We will use this fact without any special mention in the sequel.
The main results of this paper are then as follows. We begin with the cases of nonzero modulus
consensus.
Theorem 1 Suppose that all nagents adhere to the update rule (1) and Assumption 1 holds.
Suppose that the sequence of neighbor graphs G(1),G(2),... is repeatedly jointly strongly connected.
Then, for each b∈ I, system (2) reaches the corresponding nonzero modulus consensus exponentially
fast for almost all initial conditions if, and only if, the graph sequence G(1),G(2),... is repeatedly
jointly structurally balanced with respect to the clustering b.4
The sufficiency of the repeatedly jointly structurally balanced condition is more or less well
known, although the form of the condition may vary slightly from publication to publication; see
for example [1, 34, 36]. The pro of of necessity w ill b e given in Section 4.2.
Remark 2 Suppose that the sequence of neighbor graphs G(1),G(2),... is repeatedly jointly strongly
connected and structurally balanced with respect to a clustering b∈ I. Without loss of generality,
assume that each G(t),t≥1, is structurally balanced with respect to b. Let Bbe the n×n
diagonal matrix whose ith diagonal entry equals bifor all i∈[n]. Note that B2=Iand for
each A(t), the matrix BA(t)Bis a stochastic matrix. Since the sequence of the (signed) graphs of
A(1), A(2),... is repeatedly jointly strongly connected, so is the sequence of the (unsigned) graphs of
BA(1)B , BA(2)B,.... With these facts and the well-known result of standard discrete-time linear
consensus processes [6–8,11], it is straightforward to verify that the matrix product A(t)···A(2)A(1)
converges to a rank one matrix of the form bc′exponentially fast, where cis a nonzero vector, and
thus x(t)converges to bc′x(1). It follows that system (2) will reach the corresponding nonzero
modulus consensus if c′x(1) does not equal zero. Since the set of those vectors xwhich satisfy the
equality c′x= 0 is a thin set, the nonzero modulus consensus will be reached exponentially fast for
almost all initial conditions.
Remark 3 From the definition of repeatedly jointly structural balance, Theorem 1 implies that if
the system reaches a nonzero modulus consensus corresponding to a clustering b, then all those
signed digraphs which do not belong to the structurally balanced class Cbcan appear in the sequence
of neighbor graphs for only a finite number of times.
The next theorem addresses the case of zero (modulus) consensus.
4We are indebted to Lili Wang (Department of Electrical Engineering, Yale University) for pointing out a flaw in
the original version of this statement and suggesting how to fix it.
7
Theorem 2 Suppose that all nagents adhere to the update rule (1) and Assumption 1 holds.
Suppose that the sequence of neighbor graphs G(1),G(2),... is repeatedly jointly strongly connected.
Then, system (2) converges to zero exponentially fast for all initial conditions if, and only if, the
graph sequence G(1),G(2),... is repeatedly jointly structurally unbalanced.
The proof of this theorem will be given in Section 4.2.
It is worth emphasizing that a repeatedly jointly structurally unbalanced sequence of signed
digraphs does not require each graph in the sequence to be structurally unbalanced. Actually, in
extreme cases, all graphs in a repeatedly jointly structurally unbalanced sequence can be struc-
turally balanced, but with respect to different clusterings. Let us consider the following example.
Suppose that for each odd time step 2k−1, k≥1,
A(2k−1) =
0.5 0 0.5
−0.5 0.5 0
0−0.5 0.5
and for each even time step 2k,k≥1,
A(2k) =
0.5 0 −0.5
0.5 0.5 0
0−0.5 0.5
It can be seen that the graphs of A(2k−1) and A(2k) are both structurally balanced, but the
union of the two graphs is structurally unbalanced (and strongly connected). Thus, according to
Theorem 2, system (2) with A(2k−1) and A(2k) will converge to zero exponentially fast for all
initial conditions.
Remark 4 It is well known that for discrete-time linear consensus processes, repeatedly jointly
strong connectivity guarantees exponentially fast consensus [6–8, 11]. But this is not the case for
modulus consensus. Examples can be generated to show that modulus consensus may be reached
only asymptotically, but not exponentially fast.
3.3 Convergence Rate
Theorems 1 and 2 provide necessary and sufficient conditions for system (2) to converge exponen-
tially fast to different types of modulus consensus. Of particular interest is the rate at which a
modulus consensus is reached. To characterize the convergence rate, we need the following concept
and results.
Let {S(t)}be a sequence of stochastic matrices. A sequence of stochastic vectors {π(t)}is an
absolute probability sequence for {S(t)}if
π′(t) = π′(t+ 1)S(t), t ≥1
It has been shown by Blackwell [45] that every sequence of stochastic matrices has an absolute
probability sequence. More can be said.
Lemma 1 (Theorem 4.8 in [46]) Let {S(t)}be a sequence of n×nstochastic matrices that satisfy
the following conditions:
(a) (Aperiodicity) The diagonal entries of each S(t)are positive, i.e., sii(t)>0for all tand
i∈[n].
8
(b) (Uniform Positivity) There is a scalar β > 0such that sij (t)≥βwhenever sij (t)>0.
(c) (Irreducibility) The sequence of graphs {γ(S(t))}is repeatedly jointly strongly connected.
Let {π(t)}be an absolute probability sequence for {S(t)}. Then, there is a positive scalar δ > 0
such that πi(t)≥δfor all i∈[n]and t.
A class of stochastic matrices which have this property was introduced in [46] (as class P∗).
The convergence rate of discrete-time linear consensus with a repeatedly jointly strongly con-
nected sequence of neighbor graphs has been recently characterized with an explicit dependence on
the graph structure including the longest shortest directed path [47, 48]. In the sequel, we will apply
the convergence rate result for the discrete-time Altafini model. We begin with nonzero modulus
consensus.
Proposition 2 Suppose that all nagents adhere to the update rule (1) and that Assumption 1 holds.
Suppose that each neighbor graph G(t)is strongly connected and structurally balanced with respect
to a clustering b∈ I. Then, system (2) reaches the corresponding nonzero modulus consensus as
fast as ρtconverges to zero, where
ρ= 1 −δβ2
4p∗
in which δ > 0is the uniform lower bound on the entries of the absolute probability sequence for
the sequence of stochastic matrices {|A(t)|},β > 0is given in Assumption 1, and p∗= maxtp∗(t)
where p∗(t)is the longest shortest directed path of spanning trees contained in G(t).
The proof of this proposition is straightforward using arguments similar to those in [48] and is
therefore omitted.
We next consider zero (modulus) consensus.
Proposition 3 Suppose that all nagents adhere to the update rule (1) and that Assumption 1
holds. Suppose that each neighbor graph G(t)is strongly connected and structurally unbalanced.
Then, system (2) converges to zero as fast as ¯ρtconverges to zero, where
¯ρ= 1 −¯
δβ2
4¯p∗
in which ¯
δ > 0is the uniform lower bound on the entries of the absolute probability sequence for the
sequence of stochastic matrices {¯
A(t)}(matrix ¯
A(t)is defined in (3)), β > 0is given in Assumption
1, and ¯p∗= maxt¯p∗(t)where ¯p∗(t)is the longest shortest directed path of spanning trees contained
in the graph of ¯
A(t).
This proposition is a consequence of Proposition 5 in Section 4.1.
The results of Propositions 2 and 3 can be easily extended to the cases when the sequence of
neighbor graphs is repeatedly jointly strongly connected and structurally balanced (or unbalanced).
Remark 5 We will see in the next section that each n×nmatrix A(t)satisfying Assumption 1
uniquely determines a 2n×2nstochastic matrix ¯
A(t)defined in (3). Thus, ¯
δand ¯p∗in Proposition
3 are the counterparts of δand p∗in Proposition 2. Using the arguments in the proof of Proposition
5, it can be shown that ¯p∗≤2p∗+c∗in which c∗= maxtc∗(t)where c∗(t)is the longest directed
cycle contained in G(t). Characterization of the relationship between δand ¯
δis a subject of future
research.
9
4 Analysis
In this section, we present a graphical approach to analyze the discrete-time Altafini model. As
mentioned earlier, the approach we adopt is inspired by a novel idea from [1] which lifts the system
to an expanded system, as follows.
Define a time-dependent 2n-dimensional vector z(t) such that for each time t,
z(t) = x(t)
−x(t)
Then, for all i∈[2n],
zi(t+ 1) =
2n
X
j=1
¯aij (t)zj(t)
in which
¯aij (t) = ¯ai+n,j+n(t) = max{0, aij (t)}
¯ai+n,j (t) = ¯ai,j +n(t) = max{0,−aij (t)}
It has been shown in [1] that the above expanded system is equivalent to the discrete-time Altafini
model.
It is straightforward to verify that the expanded system is a discrete-time linear consensus
process in which the states are coupled. Thus, it can be written in the form of a state equation
z(t+ 1) = ¯
A(t)z(t) (3)
where each ¯
A(t) = [¯aij (t)] is a 2n×2nstochastic matrix. With this fact, the graph of ¯
A(t) is an
unsigned digraph with 2nvertices.
The graph of ¯
A(t) has the following properties, whose proofs are straightforward and are there-
fore omitted.
Lemma 2 For all i, j ∈[n], if aij (t)>0, then the graph of ¯
A(t)has an arc from vertex jto vertex
iand an arc from vertex j+nto vertex i+n; if aij (t)<0, then the graph of ¯
A(t)has an arc from
vertex jto vertex i+nand an arc from vertex j+nto vertex i. In particular, the graph of ¯
A(t)
has self-arcs at all 2nvertices.
Lemma 3 Suppose that the graph of A(t)has a directed path from vertex ito vertex jwith i, j ∈[n].
Then, the graph of ¯
A(t)has a directed path from vertex ito vertex jor j+n. In particular, if the
directed path from ito jin the graph of A(t)is positive, then the graph of ¯
A(t)has a directed path
from ito j; if the directed path from ito jin the graph of A(t)is negative, then the graph of ¯
A(t)
has a directed path from ito j+n.
Lemma 4 If the graph of ¯
A(t)has a directed path from vertex ito vertex jwith i, j ∈[n], then
it has a directed path from vertex i+nto vertex j+n, and vice versa. If the graph of ¯
A(t)has a
directed path from vertex ito vertex j+nwith i, j ∈[n], then it has a directed path from vertex
i+nto vertex j, and vice versa.
10
4.1 Graphical Results
In this section, we will establish two key relations between the signed digraph of A(t) and the
expanded unsigned digraph of ¯
A(t), which will play important roles in the proofs of the main
results.
Proposition 4 Suppose that the graph of A(t)is strongly connected and structurally balanced with
respect to a clustering b∈ I. Then, the graph of ¯
A(t)consists of two disjoint strongly connected
components of the same size, n. In particular, the first component consists of vertices i,i∈ V+
b,
and j+n,j∈ V−
b, and the other one consists of vertices i,i∈ V−
b, and j+n,j∈ V+
b.
This proposition has been proved in [36] (see Lemma 1 in [36]), as well as the conference version
of this paper [39] (see Proposition 1 in [39]).
Proposition 5 Suppose that the graph of A(t)is strongly connected and structurally unbalanced.
Then, the graph of ¯
A(t)is strongly connected.
This result was claimed in [36] (see Lemma 2 in [36]), but with an incomplete5proof. In the
sequel, we will provide a more complete proof. The proof relies on the following results.
Lemma 5 Suppose that a signed digraph Gis strongly connected and structurally unbalanced.
Then, there exists a negative directed closed walk in G.
Proof: Suppose that, to the contrary, there does not exist a negative directed walk in G. Since
Gis structurally unbalanced, there must exist a negative undirected cycle Cin G. Label the vertices
of Cas 0,1,...,m−1, with [0,1],[1,2],...,[m−1,0] the associated undirected edges. In the proof,
we adopt the convention that if an integer iis not in the range 0,1,...,m −1 but referring to a
vertex of C, then it refers to the vertex (imod m).
By assumption, the subgraph Ccannot be a directed cycle. Thus, there exists at least a vertex
iof Csuch that
1. either iis a source, i.e., both (i, i −1) and (i, i + 1) are arcs of G,
2. or iis a sink, i.e., both (i−1, i) and (i+ 1, i) are arcs of G.
Let Sbe the collection of all such vertices. Then, it should be clear that the cardinality of Sis
even. Let i1, i2,...,i2kbe the elements of S, with
i1< i2<···< i2k
Without loss of generality, we assume that i1is a source. Then, i3, i5,...,i2k−1are all sources while
i2, i4,...,i2kare all sinks. For each j∈[k], let
p+
j:= (i2j−1, i2j−1+ 1) ···(i2j−1, i2j)
be a directed path of Gon Cfrom the source i2j−1to the sink i2j. Similarly, let
p−
j:= (i2j−1, i2j−1−1) ···(i2j−2+ 1, i2j−2)
5In the proof of Lemma 2 in [36], it is stated that “since Gis structurally unbalanced, there is a negative (directed)
cycle in G[40].” But what [40] actually proved is that if Gis structurally unbalanced, there is a negative undirected
cycle (called semicycle in [40]) in G.
11
be a directed path of Gon Cfrom the source i2j−1to the sink i2j−2.
Now fix a j∈[k] and choose a directed path q+
jof Gfrom i2j−2to i2j−1. Such a directed
path exists since Gis strongly connected. Let n(p−
j) and n(q+
j) be the numbers of negative signs
contained in directed paths p−
jand q+
j, respectively. Then, n(p−
j)≡n(q+
j) mod 2 since otherwise,
by concatenating the two directed paths p−
jand q+
j, we have a negative directed closed walk. Note
that this argument applies for all j. Now consider a directed closed walk by concatenating directed
paths
w:= q+
1p+
1q+
3p+
3···q+
2k−1p+
2k−1
Similarly, we let n(p+
j) be the number of negative signs contained in p+
j. Then, by the previous
arguments, we have
k
X
j=1 n(p−
2j−1) + n(p+
2j−1)
≡
k
X
j=1 n(q+
2j−1) + n(p+
2j−1)mod 2
On the other hand, the left hand side of the expression is the total number of negative signs in C
which is an odd number. Thus, the right hand side of the expression is also an odd number. In
other words, there exists a negative directed closed walk.
Following Lemma 5, we have the next result.
Corollary 1 Suppose that a signed digraph Gis strongly connected and structurally unbalanced.
Then, there exists a negative directed cycle in G.
Proof: Let wbe the negative directed closed walk in G. Such a directed closed walk exists by
Lemma 5. Choose a vertex in was a start point and we label it as vertex i1. Express was
w= (i1, i2)(i2, i3)···(in, i1)
If wis itself a directed cycle, then the statement is true. Suppose not, then we choose the least
integer number jsuch that
ij=ij+k
for some k. In other words, ijis the first vertex, other than i1, in w(with respect to the start
point) which appears at least twice in w. For this vertex ij, we may choose the integer kto be the
least positive number that ij=ij+kholds. It should be clear that
(ij, ij+1)···(ij+k−1, ij+k)
is a directed cycle. If this directed cycle is negative, then the statement is true. Suppose not, we
then remove this positive directed cycle out of the directed closed walk. Then, what remains is still
a directed closed walk, denoted by
w′:= (i1, i2)···(ij−1, ij)(ij, ij+k+1)···(in, i1)
Moreover, this directed closed walk is also negative. For convenience, we call such an operation
acycle reduction of a directed closed walk. If w′is a directed cycle, then the statement is true.
12
Suppose not, then we can apply the operation of cycle reduction on w′. Thus, we get a sequence
of negative directed closed walks as
w→w(1) →w(2) → · · ·
This sequence stops
1. either at certain step, the removed directed cycle is negative.
2. or there is an integer lsuch that w(l)is itself a negative directed cycle.
Then, in either of the two cases above, we have found a negative directed cycle. This completes
the proof.
We are now in a position to prove Proposition 5.
Proof of Proposition 5: By Corollary 1, there exists a directed negative cycle in the graph
of A(t). For any pair of i, j ∈[n], since the graph of A(t) is strongly connected, there must exist a
directed path from vertex ito a vertex kon the directed cycle, and a directed path from vertex k
to vertex j. Thus, there is a directed path from ito j, through k, in the graph of A(t). By Lemma
3, there exists a directed path in the graph of ¯
A(t) from ito jor j+n, depending on the sign of
the directed path from ito jin the graph of A(t). Now consider the directed walk from ito jin
the graph of A(t) consisting of the above directed path and a complete round of the directed cycle.
Since the directed cycle is negative, the directed walk has a different sign from the above directed
path. Thus, there exist two directed paths in the graph of ¯
A(t) from ito jand to j+n. By Lemma
4, there exist directed paths in the graph of ¯
A(t) from i+nto jand j+n. This completes the
proof.
The results of Propositions 4 and 5 can be extended to the cases when a finite sequence of
neighbor graphs is jointly strongly connected and structurally balanced (or unbalanced), as follows.
Corollary 2 Suppose that a finite sequence of the graphs of A(p), A(p+ 1),...,A(q),q≥p, is
jointly strongly connected and structurally balanced with respect to a clustering b∈ I. Then, the
union of the graphs of ¯
A(p),¯
A(p+1),..., ¯
A(q)consists of two disjoint strongly connected components
of the same size, n. In particular, the first component consists of vertices i,i∈ V+
b, and j+n,
j∈ V−
b, and the other one consists of vertices i,i∈ V−
b, and j+n,j∈ V+
b.
Corollary 3 Suppose that a finite sequence of the graphs of A(p), A(p+ 1),...,A(q),q≥p, is
jointly strongly connected and structurally unbalanced. Then, the union of the graphs of ¯
A(p),¯
A(p+
1),..., ¯
A(q)is strongly connected.
The proofs of Corollaries 2 and 3 are fairly straightforward using the arguments similar to those
used in the proofs for Propositions 4 and 5, and are therefore omitted.
4.2 Proofs of Main Results
In this section, we provide proofs for the main results stated in Section 3.2.
Proof of Theorem 1 (Necessity): Note that if all those signed digraphs that do not belong to
the structurally balanced class Cbappear in the sequence of neighbor graphs only a finite number of
times, then the sequence of neighbor graphs is repeatedly jointly structurally balanced with respect
to the clustering b, and thus, from Remark 2, the system will reach the corresponding nonzero
modulus consensus exponentially fast for almost all initial conditions. Thus, to prove the necessity,
13
it is enough to show that all those signed digraphs that do not belong to Cbwill only appear a finite
number of times.
First consider the signed digraphs in the structurally unbalanced class Cu. Since the sequence
of neighbor graphs G(1),G(2),... is repeatedly jointly strongly connected, by definition, there exist
two positive integers pand qsuch that each finite sequence G(q+kp),G(q+kp + 1),...,G(q+ (k+
1)p−1), k≥0, is jointly strongly connected. For each k≥0, let
Hk=G(q+kp)∪G(q+kp + 1) ∪... ∪G(q+ (k+ 1)p−1)
Then, each signed digraph Hkis strongly connected. Note that if a signed digraph Gis struc-
turally unbalanced, then any finite sequence of signed digraphs which contains Gmust be jointly
structurally unbalanced. Thus, if the graphs in Cuappear infinitely many times, then the graphs
in the sequence Hkwill be structurally unbalanced for infinitely many times. By Corollary 3, for
each k≥0, the union of the graphs of ¯
A(q+kp),¯
A(q+kp + 1),..., ¯
A(q+ (k+ 1)p−1) is strongly
connected if Hkis structurally unbalanced. Thus, the expanded system is a standard discrete-time
linear consensus process whose graph is (jointly) strongly connected infinitely many times. By
Theorem 2 in [12], the expanded system will asymptotically reach a consensus. From the definition
of z(t), it follows that x(t) must converge to zero for all initial conditions. Thus, the graphs in Cu
can only appear in the sequence of neighbor graphs a finite number of times.
Next consider the signed digraphs in the structurally balanced classes. If only one class of
structurally balanced graphs appear infinitely many times and this class is not Cb, then from
Remark 2, for almost all initial conditions, the system will reach a nonzero modulus consensus not
corresponding to the clustering b. Now suppose that more than one class of structurally balanced
graphs appear infinitely many times. By Theorem 2 in [12], each type of modulus consensus
corresponding to those structurally balanced classes will be reached. Since different structurally
balanced classes correspond to different types of clusterings, by the coupled structure of the entries
of z(t), the state vector x(t) must converge to zero for all initial conditions. Therefore, all the
structurally balanced graphs that do not belong to Cbcan only appear in the sequence of neighbor
graphs a finite number times. This completes the proof.
Proof of Theorem 2: We first prove the sufficiency. Suppose that the sequence of neighbor
graphs G(1),G(2),... is repeatedly jointly strongly connected and structurally unbalanced. Then,
there exist two positive integers pand qsuch that each finite sequence G(q+kp),G(q+kp +
1),...,G(q+ (k+ 1)p−1), k≥0, is jointly strongly connected and structurally unbalanced.
By Corollary 3, the union of the graphs of ¯
A(q+kp),¯
A(q+kp + 1),..., ¯
A(q+ (k+ 1)p−1) is
strongly connected for each k≥0. Thus, the expanded system is a standard discrete-time linear
consensus process whose sequence of graphs is repeatedly jointly strongly connected. It is well
known that in this case, the expanded vector z(t) will reach a consensus exponentially fast for all
initial conditions [6–8, 11]. By the definition of z(t), it follows that x(t) must converge to zero
exponentially fast for all initial conditions.
Now we prove the necessity. Since uniform asymptotic stability and exponential stability are
equivalent for linear systems, it is enough to show that uniform asymptotic stability of system (2)
implies that the sequence of neighbor graphs is repeatedly jointly structurally unbalanced. Suppose
therefore that system (2) is uniformly asymptotically stable.
To establish the claim, suppose that, to the contrary, the sequence of neighbor graphs G(1),G(2),...
is not repeatedly jointly structurally unbalanced. Then, for every pair of positive integers land
m, there is an integer k0> m such that the graph sequence G(k0),G(k0+ 1),...,G(k0+l−1) is
jointly structurally balanced.
14
Let Φ(k, j) be the state transition matrix of A(k). Since x(k+ 1) = A(k)x(k) is uniformly
asymptotically stable, for each real number e > 0, there exist integers ke>0 and Ke>0 such that
kΦ(k+Ke, k)k< e for all k > ke. Set e= 1. It follows from the preceding arguments that there
must exist an integer k0> kesuch that the graph sequence G(k0),G(k0+ 1),...,G(k0+Ke−1) is
jointly structurally balanced.
Suppose that the union of the graphs of G(k0),G(k0+ 1),...,G(k0+Ke−1) is structurally
balanced with respect to b∈ I . Let Bbe the n×ndiagonal matrix whose ith diagonal entry equals
bifor all i∈[n]. Then, from Remark 2,
Φ(k0+Ke, k0) = A(k0+Ke−1) ···A(k0+ 1)A(k0)
=B(BA(k0+Ke−1)B)···
(BA(k0+ 1)B) (BA(k0)B)B
in which BA(i)Bis a stochastic matrix for all i∈ {k0, k0+ 1,...,k0+Ke−1}. It follows that
Φ(k0+Ke, k0) has an eigenvalue at 1 and thus kΦ(k0+Ke, k0)k= 1, which is a contradiction.
Therefore, the sequence of graphs G(1),G(2),... must be repeatedly jointly structurally unbalanced.
5 Discussions
For a fixed signed digraph G, it has been shown in [34] that
1. If Gis structurally balanced and rooted (or strongly connected), then all |xi(t)|converge to
the same value exponentially fast.
2. If Gis structurally unbalanced and strongly connected, then all xi(t) converge to 0 exponen-
tially fast.
In this section, we discuss the cases when Gis neither structurally balanced nor strongly connected.
Since the least restrictive condition for standard consensus in discrete-time linear consensus
processes is rooted connectivity [6, 15], we consider the case in which Gis structurally unbalanced
and rooted, but not necessarily strongly connected. To state our results, we need the following
concept.
We say that vertices iand jin a digraph Gare mutually reachable if each is reachable from the
other along a directed path in G. Mutual reachability is an equivalence relationship on the set of
vertices of Gwhich partitions the vertex set into the disjoint union of a finite number of equivalence
classes.
Lemma 6 Each rooted graph has a unique mutually reachable class of roots.
The proof of this lemma is fairly straightforward and is therefore omitted.
Lemma 6 implies that if a digraph has more than one root, they are mutually reachable from
each other. Given a rooted digraph G. Let Rdenote the set of roots of Gand GRfor the subgraph
induced by R. From Lemma 6, GRis either a single-vertex graph or a strongly connected graph.
Proposition 6 Suppose that a signed digraph Gis structurally unbalanced, rooted, but not strongly
connected. If GRis structurally unbalanced, then all the eigenvalues of Aare less than 1in magni-
tude. If GRis structurally balanced, then Ahas an eigenvalue at 1and all the remaining eigenvalues
are less than 1in magnitude.
15
To prove Proposition 6, we need the following result.
Lemma 7 (Theorem 8.1.18 in [49]) For any M∈Rn×n,ρ(M)≤ρ(|M|), where ρ(M)denotes the
spectral radius of matrix M.
Proof of Proposition 6: In the case when Gis strongly connected, every vertex of Gis a
root. Thus, the unique mutually reachable class of roots of Gcontains all the vertices of G. Since
Gis not strongly connected, the set of roots of Gmust be a proper subset of the vertex set of G.
Let R={i1, i2,...,im},m < n, be the set of roots. Let πbe any permutation map for which
π(ij) = j,j∈[m], and let Pbe the corresponding permutation matrix. Then,
P AP ′=B0
C D
where Bcorresponds to the set of roots R. Note that |P AP ′|is a stochastic matrix with positive
diagonal entries. Then, |B|is also stochastic with positive diagonal entries. From Lemma 6, the
graph of B,GR, is strongly connected. It has been proved in [34] that if GRis structurally balanced,
Bhas an eigenvalue at 1 and all remaining ones are strictly less than 1 in magnitude, and if GRis
structurally unbalanced, all the eigenvalues of Bare strictly less than 1 in magnitude.
Note that the eigenvalues of Aconsist of the eigenvalues of Band D. Thus, to prove the lemma,
it is sufficient to show that all the eigenvalues of Dare strictly less than 1 in magnitude. Suppose
the contrary, namely that Dhas an eigenvalue λwhose magnitude is 1. Since all the diagonal
entries of Aare positive, by the Gersgorin circle theorem, Amay have eigenvalues at 1 and all
other possible eigenvalues must be strictly less than 1 in magnitude, so does P AP ′. Thus, λ= 1.
It is straightforward to verify that
P|A|P′=|B|0
|C| |D|
By Lemma 7, |D|must have an eigenvalue at 1. Note that |B|is a stochastic matrix, it also
has an eigenvalue at 1. Thus, P|A|P′has at least two eigenvalues at 1, so does |A|. But this
contradicts the fact that |A|has a unique eigenvalue at 1 since |A|is a stochastic matrix with
positive diagonal entries whose graph is rooted. Therefore, all the eigenvalues of Dare strictly less
than 1 in magnitude and the proof is complete.
More general cases in which the sequence of neighbor graphs is “repeatedly jointly rooted” is a
subject of future research.
6 Conclusion
In this paper, the discrete-time version of Altafini’s opinion dynamics model has been studied
through a graphical approach. Necessary and sufficient conditions for exponential convergence of
the system with respect to different limit states have been established under the assumption of
repeatedly jointly strong connectivity. The rate of convergence has been provided and a time-
invariant case without the assumption of strong connectivity has been discussed. The time-varying
case without the strong connectivity assumption, which was partially studied in [36], is a direction
for future research.
16
7 Acknowledgment
The authors wish to thank Mahmoud El Chamie (University of Texas at Austin), Julien M. Hen-
drickx (Universit´e Catholique de Louvain), A. Stephen Morse (Yale University), Angelia Nedi´c
(University of Illinois at Urbana-Champaign), Lili Wang (Yale University), and Zhi Xu (Mas-
sachusetts Institute of Technology) for useful discussions which have contributed to this work.
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8 Appendix
Proof of Proposition 1: Since the sequence of neighbor graphs G(1),G(2),... is repeatedly
jointly strongly connected, by definition, there exist two positive integers pand qsuch that each
finite sequence G(q+kp),G(q+kp + 1),...,G(q+(k+ 1)p−1), k≥0, is jointly strongly connected.
For each k≥0, let
Hk=G(q+kp)∪G(q+kp + 1) ∪... ∪G(q+ (k+ 1)p−1)
Then, each signed digraph Hkis strongly connected. By Corollaries 2 and 3, the union of the
graphs of ¯
A(q+kp),¯
A(q+kp + 1),..., ¯
A(q+ (k+ 1)p−1) is either strongly connected or consists
of two strongly connected components.
Note that the set of all Hkis a finite set, since the number of all possible signed digraphs on n
vertices is finite. This finite set can be partitioned into the sets of structurally balanced graphs Cb,
b∈ I, and the set of structurally unbalanced graphs Cu. It should be clear that there is at least
one of these sets whose graphs appear in the Hksequence infinitely many times.
Suppose first that structurally unbalanced graphs appear in the Hksequence infinitely many
times. By Corollary 3, for each k≥0, the union of the graphs of ¯
A(q+kp),¯
A(q+kp + 1),..., ¯
A(q+
(k+ 1)p−1) is strongly connected if Hkis structurally unbalanced. Thus, the expanded system
is a standard discrete-time linear consensus process whose graph is (jointly) strongly connected
infinitely many times. By Theorem 2 in [12], the expanded system will asymptotically reach a
consensus. From the definition of z(t), it follows that x(t) must converge to zero for all initial
conditions.
Now suppose that structurally unbalanced graphs appear in the Hksequence only a finite
number of times. In this case, there exist at least one of the sets Cb,b∈ I , whose graphs appear
infinitely many times. If only one such set exists, say Cb0, then from Remark 2, system (2) will
reach the bipartite consensus with respect to b0exponentially fast. If at least two such sets exist,
say Cb1and Cb2, then by Theorem 2 in [12], both types of bipartite consensus, with respect to b1
and b2, will be reached asymptotically. Since different structurally balanced classes correspond to
different types of clusterings, by the coupled structure of the entries of z(t), the state vector x(t)
must converge to zero.
Combining the above cases, we reach the conclusion that system (2) will always asymptotically
reach a modulus consensus.
20
