This section investigates the interesting, special case of periodically varying C(s)
which satisfies Assumption 6.2.
Motivation for Periodic Variations: Consider the example on issue-varying topology in Section 6.1.1 of a government cabinet that meets to discuss the issues of defence, economic growth, social security programs and foreign policy. Since these issues are vital to the smooth running of the country, one expects the issues to be discussed
regularly and repeatedly. Regular meetings on the same set of issues for decision ma- king/governance/management of a country or company then points to periodically varyingC(s), i.e., social networks with periodic topology.
The system Eq. (6.4), with periodically switching C(s), can be described by a switching signal σ(s)of the form σ(0) = P, and for s ≥ 1, σ(Pq+p) = p,4 where
P < ∞ is the period length, p ∈ P = {1, 2, . . . ,P} andq ∈ Z≥0 is any nonnegative
4Any given s ∈ S can be uniquely expressed by a given fixed positive integer P, a nonnegative
integer. Note that in general,Ci 6=Cjfori,j∈ P andi6= j. Theorem 6.1 immediately allows one to conclude that the system Eq. (6.4) with periodic switching converges exponentially fast to the system’s unique limiting trajectoryx∗(s). This section’s key contribution is to obtain additional, insightful information on the limiting trajectory, including an observation that the limiting trajectory itself is periodic.
For simplicity, analysis begins with the assumption that P = {1, 2}, i.e., there are two different C matrices, and the switching is of period 2. In the sequel, it will be shown that analysis for P = {1, 2, . . . ,P}, with arbitrarily large but finiteP, is a simple recursive extension on the analysis forP ={1, 2}. For theP=2, one obtains
x(s+1) =
(
F1(x(s)) ifsis odd,
F2(x(s)) ifsis even.
(6.10)
The periodic system is now transformed into a time-invariant system. Define a new statez∈R2n (different to the differential transformδzemployed in Sections 5.2 and 6.3 for nonlinear contraction analysis) as
z(2q) = z1(2q) z2(2q) = x(2q) x(2q+1) . (6.11)
The evolution ofz(2q)is studied forq∈ {0, 1, 2, . . .}. Note that z(2(q+1)) = z1(2(q+1)) z2(2(q+1)) = x(2(q+1)) x(2(q+1) +1) . (6.12)
In view of the fact thatx(2(q+1)) = F1(x(2q+1))andx(2(q+1) +1) = F2(x(2q+
2))for anyq∈ {0, 1, 2, . . .}, it follows that z(2(q+1)) = F1(x(2q+1)) F2(x(2q+2)) . (6.13)
Similarly, notice that x(2q+1) = F2(x(2q)) and x(2q+2) = F1(x(2q+1)) for any
q∈ {0, 1, 2, . . .}. From this, forq∈ {0, 1, 2, . . .}, one obtains
z(2(q+1)) = F1 F2(z1(2q)) F2 F1(z2(2q)) = G1(z1(2q)) G2(z2(2q)) , (6.14)
with the time-invariant composition maps G1 = F1◦F2 andG2 = F2◦F1. Thus, one
can express the periodic system Eq. (6.10) as the nonlinear time-invariant system z(2q+2) =G¯(z(2q)), G¯ = [G1>,G2>]> (6.15)
mapFpgiven in Eq. (6.5) for p=1, 2, obeying
x∗(s) =
(
z1∗ if s is odd,
z2∗ if s is even, (6.16)
where z∗1 ∈ int(∆n) and z∗2 ∈ int(∆n) are the unique fixed points of, respectively, G1
and G2, defined above Eq. (6.15). For all initial conditions 0 ≤ xi(0) < 1,∀i ∈ I and
∃j∈ I :xj(0)>0,lims→∞[x(s)−x∗(s)] =0nexponentially fast.
Proof. As mentioned above, one can immediately apply Theorem 6.1 to show that
lims→∞[x(s)−x∗(s)] = 0n. This proof focuses on using the time-invariant transfor- mation to show thatx∗(s)has the properties described in the theorem statement.
Part 1: In this part, the map Gi,i=1, 2 is proved to have at least one fixed point. First, note that it was proved in Theorem 6.1 that the system Eq. (6.4), with initial conditions 0≤ xi(0)<1,∀iand for at least onej,xj(0)>0, will havex(s)∈int(∆n) for alls >0, which implies thatx∗(s)∈ int(∆n). For p=1, 2, Property 6.1 established that Fp is continuous on∆n. The composition of continuous functions is continuous, which impliesG1 = F1◦F2:∆n 7→∆nandG2 =F2◦F1 :∆n 7→∆n are continuous.
The proof of Theorem 6.1 also showed that for all p,Fp ∈A¯ where ¯A={x ∈∆n : 1−r¯ ≥xi ≥0,∀i∈ I }and ¯ris some small strictly positive constant. For the system in Eq. (6.10) with p = 1, 2, it follows that F1(A¯) ⊂ A ⇒¯ F2(F1(A¯)) ⊂ A¯, which
implies that G1(A¯) ⊂ A¯. Similarly, G2(A¯) ⊂ A¯. Brouwer’s Fixed Point Theorem
then implies that there exists at least one fixed pointz∗1 ∈ A¯ such that z∗1 = G1(z∗1)
(respectively at least one fixed point z2∗ ∈ A¯ such that z∗2 = G2(z∗2)) because G1
(respectivelyG2) is a continuous function on the compact, convex setA.
Part 2: In this part, it is proved that the unique limiting trajectory of Eq. (6.10) obeys Eq. (6.16) and thatz∗1 andz2∗are the unique fixed point ofG1 andG2, respecti-
vely. Letz∗1 be a fixed point ofG1. Observe thatz∗1 =F2(F1(z1∗)). Definez∗2 =F1(z1∗).
One has that z∗1 = F2(z∗2). Observe that F1(z∗1) = F1(F2(z2∗)), which implies that
z∗2 = F1(F2(z∗2)) = G2(z∗2). In other words,z∗2 is a fixed point of G2 (but at this stage, and
similarly forz∗1, its uniqueness has not yet been proved).
Next, it will be established that the unique limiting trajectory is the periodic sequence in Eq. (6.16). Recall that Theorem 6.1 yields the conclusion thatall trajecto- riesof Eq. (6.10) converge exponentially fast to a unique limiting trajectory x∗(s) ∈
int(∆n). It follows, from Eq. (6.15) and the definition of z(2q), that for all s ≥ 0, Eq. (6.16)is a trajectoryof the system Eq. (6.10); the critical point here is that Eq. (6.16)
holds for all s. Combining these arguments, it is clear that Eq. (6.16) is precisely the unique limiting trajectory.
Last, it will be shown that z∗1 and z∗2 are the unique fixed point of G1 and G2,
respectively. To this end, suppose that, to the contrary, at least one of z∗1 and z∗2 is not unique. Without loss of generality, suppose in particular thatz106=z1∗is any other
fixed point of G1. Then,z02=F1(z01)is a fixed point ofG2, and
x(s) =
(
z01 ifs is odd,
z02 ifs is even, (6.17)
is a trajectory of Eq. (6.10) that holds for all s ≥ 0, and isdifferent from the trajectory Eq. (6.16) becausez01 6=z∗1. On the other hand, Theorem 6.1 implies that all trajectories of Eq. (6.10) converge exponentially fast to a unique limiting trajectory, which yields a contradiction. Thus, z∗1 andz∗2 are the unique fixed point of G1 and G2, respecti-
vely, and the system in Eq. (6.10) converges exponentially fast to the unique limiting trajectory Eq. (6.16), which completes the proof5.
The generalisation is now provided for periodically switching topology C(s) =
Cσ(s), where σ(s) is of the form σ(0) = P, and for s ≥ 1, σ(Pq+p) = p. Here,
2 ≤ P < ∞, p ∈ P = {1, 2, . . . ,P} and q ∈ Z≥0. The periodic DeGroot–Friedkin
model is described by x(s+1) = ( FP(x(s)) fors=0 Fp(x(s =Pq+p)) for all s≥1. (6.18) A transformation of Eq. (6.18) to a time-invariant system follows a procedure similar to the one detailed forP =2 (in the signal processing and control literature, this technique is sometimes referred to as “lifting”). A new state variablez∈ RPn is defined as z(Pq) = z1(Pq) z2(Pq) .. . zP(Pq) = x(Pq) x(Pq+1) .. . x(Pq+P−1) , (6.19)
and the evolution ofy(Pq)is studied forq∈ {0, 1, . . .}. It follows that zp(P(q+1)) =x(P(q+1) +p−1), ∀p∈ P.
Following the logic in theP=2 case, but with the precise steps omitted, one obtains
z(P(q+1)) = FP−1(FP−2(. . .(FP(z1(Pq))))) FP(FP−1(. . .(F1(z2(Pq))))) .. . FP−2(FP−1(. . .(FP(zP(Pq))))) =G¯(z(Pq)), (6.20)
5During the thesis examination process, an examiner has identified an alternative proof, which we
summarise here for the interested reader. The existence ofx∗(s)was established in Theorem 6.1. Since
x(2s+2) is also a trajectory, kx∗(2s+2)−x∗(2s)kconverges to 0 exponentially fast. From this, one has that for anys≥0 andk∈N,kx∗(s+2k)−x(2s)kconverges to 0 exponentially fast; that isx∗(2s)
is a Cauchy sequence in ¯A, which must converge to a point, denoted asz∗1. From Eq. (6.14), and
by continuity,z∗1 must be a fixed point ofF1(F2(.)). A similar approach shows that x∗(2s+1) must
where ¯G(z) = [G1(z1)>,G2(z2)>, . . . ,GP(zP)>]>. This leads to the following gene- ralisation of Theorem 6.4.
Theorem 6.5. There exists a unique periodic sequence x∗(s)for the system Eq. (6.18), with mapFpgiven in Eq. (6.5) for p=1, 2, . . . ,P, which for any nonnegative integer q, obeys
x∗(Pq+p−1) =z∗p, for all p∈ {1, 2, . . . ,P} (6.21)
where z∗p ∈ int(∆n) is the unique fixed point of Gp defined in Eq. (6.20). For all initial
conditions satisfying 0 ≤ xi(0) < 1,∀i ∈ I and for at least one j ∈ I, xj(0) > 0, lims→∞[x(s)−x∗(s)] =0nexponentially fast.
Proof. The proof is obtained by recursively applying the same techniques used in the
proof of Theorem 6.4. The detailed mechanical calculations are omitted.
This section is concluded by noting that Theorem 6.3, Lemma 6.1 and Corol- lary 6.1 are all applicable to the periodic system Eq. (6.18) because Eq. (6.18) is just a special case of the general switching system Eq. (6.4).
6.5
Simulations
In this section, simulations are provided to illustrate the key conclusions of this chap- ter. To begin, a simulation for general switching topology is given, and then the spe- cial case of periodically-varying topology is illustrated. The set of topologies is given as C = {C1, . . . ,C5}, i.e., P = {1, 2, . . . , 5}. The switching signal σ(s) is generated such that for any given s, there is equal probability that σ(s) = p,∀p ∈ P. The precise numerical forms ofCp are stated in Section 6.7.
Figure 6.1 shows the evolution of individual social power over a sequence of issues for the system as described in the above paragraph, initialised from an ar- bitrarily chosen initial condition vector6, bx(0) = [0.95, 0.95, 0.95, 0, 0, 0]
>. For each individual, with ¯γi = maxp∈Pγp,i, it is computed that ¯γ1 = 0.474, ¯γ2 = 0.237, ¯γ3 =
0.244, ¯γ4 = 0.244, ¯γ5 = 0.244, ¯γ6 = 0.239. Note that ∑iγ¯i 6= 1 in general due to the definition of ¯γi. Corollary 6.1 yields that x∗(s)≤ [0.9, 0.311, 0.323, 0.323, 0.323, 0.314], and this can be seen in Fig. 6.1. Since only ¯γ1>1/3, it is observed that after the first
10 or so issues, only x1∗(s) > 0.5. That is, in the limit, only individual 1 may have more than half the social power at some issue. Note thatx∗4(s)>0 for alls, although for several issues,x∗4(s)is close to 0.
Figure 6.2 shows the system with a different arbitrarily selected vector of initial conditions ex(0) = [0.1, 0.325, 0, 0.8, 0.45, 0.7]
> 6=
b
x(0). Notice that for the initial con- dition vector xb(0)individuals 1, 2, 3 have large perceived social power xbi(0) = 0.95,
while individuals 4, 5, 6 havebxi(0) =0. In contrast, for the other initial condition vec-
torxe(0), the perceived social powerxei(0)is large fori=4, 6 and small fori=1, 2, 3.
Through sequential discussion and reflected self-appraisal, the initial conditions are
6The initial condition vector b
5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 S e lf -W e ig h t, x i (s ) Issue, s Individual 1 Individual 2 Individual 3 Individual 4 Individual 5 Individual 6
Figure 6.1: Evolution of individuals’ social powers x(s) for initial condition vector xb(0). Viewed in conjunction with Fig. 6.2, it is clear that each individuali’s social power trajectory
xi(s)converges tox∗i(s)by abouts=10.
exponentially forgotten and both Fig. 6.1 and 6.2 show convergence to the same uni- que limiting trajectory x∗(s) by abouts = 10. This is highlighted in Fig. 6.3, which displays the individual social powers of selected individuals 1, 3 and 6. The solid and dotted lines correspond to initial condition vectorsxe(0)andbx(0), respectively.
Simulations for periodically-varying topology are now presented. The period is selected to be P = 4, with C3 = C4. That is, the switching signal is σ(s) = [3, 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, . . .]. The matrices used areC1,C2,C3given in Section 6.7. The
same two initial condition vectors bx(0) and xe(0) from the previous simulation are
used. The simulation result forbx(0)is shown in Fig. 6.4. Fig. 6.5 shows a comparison
of the social power trajectory for select individuals 1,3 and 6, with the two different vectors of initial conditions xe(0) and bx(0). It can clearly be seen that the initial
conditions are forgotten exponentially fast, and the unique limiting trajectory x∗(s)
is a periodic sequence.
Remark 6.4. It is worth noting that the unique limiting trajectory depends on thesequence of switching topologies, i.e. the switching signal σ(s). For example, given the same Cp
in the periodically-varying topology simulation, but with a different switching signal (i.e. ordering ofC(s)), the unique limiting trajectory will also be different.
6.6
Conclusions
This chapter has extended the DeGroot–Friedkin model through the incorporation of issue-varying, dynamic relative interpersonal interactions. The nonlinear contraction analysis framework of Chapter 5 has been used to study the system with dynamic
5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 S e lf -W e ig h t, x i (s ) Issue, s Individual 1 Individual 2 Individual 3 Individual 4 Individual 5 Individual 6
Figure 6.2: Evolution of individuals’ social powers x(s) for initial condition vector xe(0). Viewed in conjunction with Fig. 6.1, it is clear that each individuali’s social power trajectory
xi(s)converges tox∗i(s)by abouts=10.
Figure6.3:Evolution of selected individuals’ social powersxi(s), fori=1, 3, 6. The trajectory
xi(s), fori=1, 3, 6, beginning from the two different initial conditions, converges to the same
5 10 15 20 25 Issue, s 0 0.2 0.4 0.6 0.8 1 Individual 1 Individual 2 Individual 3 Individual 4 Individual 5 Individual 6
Figure6.4:Evolution of individuals’ social powersx(s)for initial condition vectorbx(0), with periodically-varying topology. Clearly,x(s)becomes a periodic sequence by abouts=10.
5 10 15 Issue, s 0 0.2 0.4 0.6 0.8 1
Figure6.5: Evolution of selected individuals’ social powers xi(s): a comparison of different
initial condition vectors xb(0) and ex(0), with periodically-varying topology. The trajectory
xi(s), for i = 1, 3, 6, beginning from the two different initial conditions, converges to the
topology, leading to the establishment of a general exponential convergence result. This has been interpreted from the context of a social network discussing a sequence of topics, where the DeGroot opinion process is coupled with the socio-psychological mechanism of reflected self-appraisal. It is concluded that this results in every indivi- dual forgetting his/her initial (perceived) social power exponentially fast. In the limit of the topic sequence, the dynamics of each individual’s social power depends only on the dynamically-varying topology associated with the topic sequence. An upper bound on an individual’s limiting social power trajectory and the convergence rate for a class of topologies has also been obtained. Last, the special case of periodically switching topologies has been investigated in further detail.