How zealots affect the energy cost for controlling complex social networks
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How zealots affect the energy cost for controlling complex social networks
Hong Chen1, 2 and Ee Hou Yong1, a)
1)
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,
Nanyang Technological University, Singapore, 637371, Singapore
2)
Business Analytics Centre, National University of Singapore, Singapore 119613,
Singapore
The controllability of complex networks may be applicable for understanding how to control a complex social
network, where members share their opinions and influence one another. Previous works in this area have
focused on controllability, energy cost, or optimization under the assumption that all nodes are compliant,
passing on information neutrally without any preferences. However, the assumption on nodal neutrality should
be reassessed, given that in networked social systems, some people may hold fast to their personal beliefs. By
introducing stubborn agents, or zealots, who hold steadfast to their beliefs and seek to influence others, the
arXiv:2107.11744v2 [physics.soc-ph] 5 Jan 2022
energy cost needed to control such a network with neutral and non-neutral nodes are calculated and compared
against those where there were no zealots. It was found that the presence of zealots alters the energy cost
at a quadratic rate with respect to their own fixed beliefs. However, whether or not the zealots’ presence
increases or decreases the energy cost is affected by the interplay between different parameters such as the
zealots’ beliefs, number of drivers, final control time regimes, network effects, network dynamics, number and
configurations of neutral nodes influenced by the zealots. For example, when a network dynamics is linear but
does not have conformity behavior, it could be possible for a contrarian zealot to assist in reducing control
energy. With conformity behavior, a contrarian zealot always negatively affects network control by increasing
energy cost. The results of this paper suggest caution when modelling real networked social systems with the
controllability of networked linear dynamics, since the system dynamical behavior is sensitive to parameter
change.
There has been a lot of interest in studying the i in a communication network3 , the transcription fac-
controllability of complex networks because many tor concentration in a gene regulatory network4 , or the
complex systems may be modelled as dynami- opinion of an agent in a consensus network5–8 and so
cal systems, thus understanding how to control on. It has its roots in control theory9,10 , and the dy-
a high dimensional networked dynamical system namics of the networked system is assumed to be linear
has the potential to lead to technological break- time-invariant (LTI), which is also suitable for modelling
throughs. Typically, in these studies, the type of opinion networks5–8 . For modelling complex systems
networked system is not specified, and the sys- with nonlinear dynamics, such as epidemic spreading in
tem is generically assumed to be linear dynam- networks11 , LTI dynamics is adequate for capturing the
ical. This may pose some problems when one linearized dynamics of the nonlinear system around its
is interested in specifically modelling social dy- equilibrium points2 . Within the literature of network
namics in networks, where the network interac- control, an important consideration is the energy cost,
tions may have higher complexity. For example, which measures the amount of energy that each of the
some people may be stubborn (zealots) and refuse control signal needs to consume to drive the state vector
to accept new ideas, yet they continue to influ- of the network12 . Therefore, if the energy cost required
ence the rest within the social network. This pa- for performing certain tasks is too high, the system can-
per addresses this niche by modelling zealots into not in practice be controlled.
the framework of network control and investigates
In statistical physics of social dynamics13 (or socio-
how they would affect the energy cost.
physics), zealots, agents with unwavering opinions, have
been researched in various social dynamics models14–22 .
For example, if considering a network of opinions on oper-
I. INTRODUCTION ating systems (such as Microsoft Windows, Apple MAC
O/S, or Linux), then a zealot is someone who is fiercely
loyal to a particular opinion, who refuses to accept any
The controllability of complex networks1,2 refers to the other views, while advocating theirs to others19 . The-
modelling of complex dynamical systems with state vec- oretically, zealots represent interesting modifications to
tor evolving in time and driven by external control sig- existing models to examine altered system behavior. Em-
nals toward the desired node states. Depending on the pirically, partisanship23 and confirmation-bias24 in so-
system being considered at hand, the state vector rep- cial networks have been reported, suggesting that zealots
resents the amount of traffic which passes through node could also be a realistic feature of networked social sys-
tems, since not all members are truly neutral. While
zealots have become well-understood in the setting of
a) Electronic mail: eehou@ntu.edu.sg socio-physics, it is still unclear how they affect the be-2
havior of the controllability of complex networks and its a zealot-influenced node (colored blue). While the driver
associated energy cost. For example, in a campaign to node is still able to control all normal agents toward con-
steer the opinions of individuals in a complex social net- sensus c, less/more effort may be needed, depending on c
work, would the presence of zealots assist or sabotage the and z. The states evolution in time are displayed in Fig.
campaigning effort? 1(b), where they stabilize in long time in the absence of
In this paper, zealots are introduced to the framework control, and in Figs. 1(e) and (h), where the network is
of network controllability, focusing on networked control- driven in the absence and presence of zealots respectively.
lability in the context of socio-physics. Conceptually, this Correspondingly, the normal agents’ state space trajec-
research is similar to an earlier work25 , which consid- tories are shown in Figs. 1(c), (f), and (i). In Fig. 1(i),
ers two types of drivers, one effecting local influences, the 3rd state dimension would have shown the zealot’s
and the other the canonical driver nodes, which steers fixed node state, which constraints x1 (t) and x2 (t) to the
the state vector globally. However, Ref.25 focuses on fixed x3 = z plane. For this example, z = −5, contrary
the context of infrastructure networks and presents nu- to the control goal of driving toward c = 2, and it can be
merical results when the competing driver nodes induce seen in Fig. 1(i) that the state space trajectory elongates
exponentially increasing local influences to simulate in- as compared to Fig. 1(f).
frastructural damages, and examines the amount of en- There are a few rules that this model should follow:
ergy needed to neutralize these attacks. On the other
hand, the present research focuses on competing driver • The mutual exchange of information between nor-
nodes that induce constant local influences, simulating mal agents is modelled by undirected links between
zealots’ unwavering opinions, and studies the amount them
of energy needed to control the social network in com- • Zealots receive no directed links from any other
petition or cooperation with the zealots. Furthermore, nodes or control signals as they hold steadfast to
detailed analytical and numerical results are presented, their beliefs
where the number of canonical driver nodes and control
time regimes are varied. In addition, beyond the canon- • Zealots advocate their beliefs to other normal
ical continuous-time linear dynamics1 , the analyses ex- agents through directed links
tend toward discrete-time linear dynamics with confor-
mity behavior26 , which may be of particular interest to • External control signal u(t) steers the opinions of
socio-physics, since it models social networks where each normal agents with directed links, and a single con-
agent conforms to their nearest neighbors. Taken to- trol signal can only attach to one normal agent.
gether, this paper presents a nuanced characterization of
Although self-dynamics links are not shown in the Fig.
how zealots affect the energy cost when trying to control
1 models, they should be present for continuous-time
a complex social network, where it shows that the in-
network dynamics. Self-dynamics stabilize the dynamics
terplay between different parameters such as the zealots’
of the system27 , which is crucial for modelling complex
beliefs, number of drivers, final control time regimes, net-
dynamical systems realistically28 , for example, opinion
work effects, network dynamics, number and configura-
dynamics29 . While there can be multiple zealots present
tions of nodes influenced by the zealots, can lead to dif-
in an arbitrary network, to simplify the scope of the re-
ferent energy cost behaviors.
search, only one group of zealots holding the same opin-
ion z is considered, and so it is mathematically the same
to model only one zealot node affecting multiple nor-
II. CONTINUOUS-TIME LINEAR DYNAMICS MODEL mal agents. Throughout the rest of the paper, “zealot”,
“zealot node”, or “zealots” are used interchangeably. For
To study zealots affecting the energy cost in control- notational conciseness, the total system size is denoted
ling a complex social network, an example is given in to be n, where n = N + 1, of which N are normal agents,
Fig. 1. Think of the node states, xi (t), as opinions on and the n-th node is always the zealot node.
a particular topic, where a positive xi (t) models sup- Generalizing to an arbitrary network with n nodes, the
port of a particular idea, and a negative xi (t) denotes continuous-time model with zealots influencing normal
opposition. In Fig. 1(a), there are N = 2 number of nor- agents within the network can be modelled using target
mal agents (nodes 1 and 2), neutral nodes without any control30 :
preferred opinion, who are open to adopting new ideas,
and communicate with one another to exchange informa- ẋ(t) = Ax(t) + Bu(t),
(1)
tion. In Fig. 1(d), a control signal u1 (t) attaches to node y(t) = Cx(t),
1, making it the driver node1 (colored red); by directly
changing the state of node 1 with u1 (t), the state of node where x(t) = [x1 (t), x2 (t), ..., xN (t), z]T ∈ Rn×1 is the
2 becomes affected and all normal agents are controlled time-varying state vector, with the first N elements de-
toward consensus c. In Fig. 1(g), a stubborn agent, or noting the node states of the N normal agents, and
zealot node (node z), with fixed opinion z, is introduced the n-th element denoting the zealot node’s fixed be-
into the system; the zealot influences node 2, making it lief, z. A is the full n × n network structure such that3
(a) (b) (c)
1 2
(d) (e) (f)
1 2
u1(t)
(g) (h) (i)
1 2
u1(t) z
FIG. 1: (a) Network in the absence of control signal and zealot. (b) Node states evolution of the network in the absence of
external influences. (c) State space trajectory of x1 (t) and x2 (t) in the absence of external influences. (d) Network with
control signal u1 (t) attached to node 1. (e) Node states evolution of the network being driven toward consensus c = 2. (f)
State space trajectory of nodes being driven toward consensus. (g) Network with control signal u1 (t) attached to node 1 and
zealot node influencing node 2. (h) Node states evolution of the network being driven toward consensus, while node 2 is under
the influence of the zealot node with fixed opinion z = −5. (i) State space trajectory of nodes being driven toward consensus
while under the influence of the zealot.
aij is non-zero if there is a directed link from node j to lates which node states are being steered by u(t), and
node i (aij = 0 otherwise), and comprises the symmet- cij = 1 if node j is the i-th node (where i, j = 1, 2, ..., N )
ric reduced N -dimensional principal submatrix à (re- to be target controlled, and cij = 0 otherwise. While
move n-th row/column), where non-zero undirected link Eqn. (1) borrows the language of target control30 , note
ãij = ãji represents normal agent nodes i and j that that for this research, full controllability of all N normal
are connected with each other and exchange ideas, with agents are considered, thus identity IN is the reduced
the final row ain = 1 if zealot node influences node i N -dimensional principal submatrix of C, such that only
(ain = 0 otherwise, and ani is always zero because the the first N nodes of the state vector is being driven (fi-
zealot node cannot be influenced by any other nodes). nal column cin = 0 for i = 1, 2, ..., N since the zealot
B ∈ Rn×M is the control input matrix, where M is node has fixed opinion, z, controlling the n-th node is
the number of control signals (such that 1 ≤ M ≤ N ), not permitted).
and bij = 1 if control signal j attaches to node i (for The energy cost is defined to be9
i = 1, 2, ..., N , where i = n is not permitted as the zealot Z tf
node has unwavering opinion, and j = 1, 2, ..., M ). Nodes
J= uT (t)u(t)dt, (2)
which have a control signal attached to them are called t0
driver nodes. u(t) = [u1 (t), u2 (t), ..., uM (t)]T ∈ RM ×1
is the input vector of external control signals. y(t) = where t0 is the initial time, tf is the final control time
[y1 (t), y2 (t), ..., yN (t)]T ∈ RN ×1 is the output state vec- (the amount of time allocated to the control signals to
tor. C ∈ RN ×n is the target control matrix which re- steer the state vector), and when minimized leads to the4
energy-optimal target control signal30 where M̃ = Q̃◦F̃ is the simplified controllability Gramian
of the reduced matrix, and
T
u∗ (t) = BT eA (tf −t)
CT (CWCT )−1 (yf −CeA(tf −t0 ) x0 ),
[e(λi +λj )tf − 1]
(3) M̃ij =Q̃ij F̃ij = [P̃T B̃B̃T P̃]ij , (6)
R tf A(t −t) T AT (tf −t) λi + λj
where W = t0 e f
BB e dt is the controlla-
bility Gramian, yf = [c, c, ..., c]T ∈ RN ×1 is the final out- with Q̃ij = [P̃T B̃B̃T P̃]ij , and F̃ij = [e(λi +λj )tf −1]
. The
λi +λj
put state vector where c is the consensus opinion that the
inverse matrices follow similarly as
system is being steered toward, and x0 = [0, 0, ..., 0, z]T ∈
Rn×1 is the initial state vector of the system, where it is (CWCT )−1 = W̃−1 = (P̃M̃P̃T )−1 = P̃M̃−1 P̃T . (7)
assumed that all node states begin initially with neutral
opinions at zero. When B matrix is chosen appropri- Substituting the eigen-decompositions A = PDP−1 ,
ately such that the system is controllable, (CWCT ) is A = VDV−1 , Ã = P̃D̃P̃T , and Eqn. (7) into Eqn. (4),
T
invertible30 . Substituting Eqn. (3) into Eqn. (2), and the energy cost becomes
setting t0 = 0, the energy cost when using u∗ (t) to steer N X
N
the complex system is E =c2
X
[P̃M̃−1 P̃T ]ij
T i=1 j=1
E =yfT (CWCT )−1 yf − 2xT0 eA tf
CT (CWCT )−1 yf N X
N
X
+
T
xT0 eA tf CT (CWCT )−1 CeAtf x0 − 2cz [VeDtf V−1 ]ni [P̃M̃−1 P̃T ]ij
(4) i=1 j=1
(for derivation of Eqn. (4) and subsequent derivations, N X
X N
see Supplementary Information). + z2 [VeDtf V−1 ]ni [P̃M̃−1 P̃T ]ij [PeDtf P−1 ]jn .
The reduced network connection matrix, Ã, which con- i=1 j=1
sists of only normal agents’ mutual interactions (undi- (8)
rected links), is symmetric and can be diagonalized as By inspection, Eqn. (8) is a quadratic function in terms of
à = P̃D̃P̃T , where P̃ is the N × N orthonormal z, given that c is fixed. Solving the turning point ∂E
∂z = 0,
eigenvectors matrix such that P̃P̃T = P̃T P̃ = I, and z ∗ can be obtained, where z ∗ is the opinion of the zealot
D̃ = diag{λ1 , λ2 , ..., λN } is a diagonal matrix contain- node which yields the lowest energy cost when controlling
the system toward consensus c. Further, the network
ing the eigenvalues of Ã, where they are ordered as-
structure Ã, as well as the nodes which are influenced
cendingly: λ1 ≤ λ2 ≤ ... ≤ λN . Equivalently, Ã is
by the zealot node can also cause a change in the control
obtained by removing the n-th row/column off the full
energy.
network matrix, A. The full network matrix, A, in-
cludes the directed links of the zealot node to normal
agents in the last column, ain , is non-symmetric and A. Analytical equations of energy cost
thus eigen-decomposed as n × n dimensional matrices
A = PDP−1 and AT = VDV−1 , where P and V are
the eigenvectors matrices of A and AT respectively, and Depending on the final control time, tf , and driver
D = diag{λ1 , λ2 , ..., λN , 0} comprises D̃ as its reduced nodes placement, M̃−1 changes and Eqn. (8) changes ac-
N -dimensional principal submatrix, with the n-th diag- cordingly, leading to different behaviors. Subsequently,
onal entry being 0 (zealot node does not have self-link similar to Refs.12 and32 , the energy cost is analyzed in
since it has fixed z opinion). Most complex systems tend terms of differing number of driver nodes through con-
to operate near a stable state, so for the continuous- trol input matrix B̃, which is encoded in M̃, and different
time model, the eigenvalues λi (for i = 1, 2, ..., N ) are all final control time regimes (small tf and large tf ).
negative27,31 , and network à is negative definite (ND). Controlling the system with N driver nodes such that
Note that matrix terms containing ∼ symbols refer to all N normal agents each receive a control signal, B̃ =
the N ×N reduced matrix without the zealot node, while IN , Q̃ = IN , and M̃ becomes a diagonal matrix with its
those without refer to the n × n full matrix. main diagonal entries being M̃ii = F̃ii = 2λ1 i [e2λi tf − 1].
For controlling all N normal agents, (CWCT ) = W̃, M̃−1 is also a diagonal matrix, with [M̃−1 ]ii = M̃ 1
=
Rt ii
T 2λi
where W̃ = 0 f eÃ(tf −t) B̃B̃T eà (tf −t) dt is the control- [e2λi tf −1]
, which in the small tf limit, using Taylor ex-
lability Gramian of the reduced network connection ma- pansion e2λi tf ≈ 1 + 2λi tf , and in the large tf limit, as
trix, Ã, and B̃ ∈ RN ×M is the input control matrix of the eigenvalues are negative [e2λi tf − 1] ≈ −1, leading to
the reduced network matrix, with b̃ij = 1 if control signal (
j attaches to node i (b̃ij = 0 otherwise). The controlla- −1 −2λi , large tf ,
bility Gramian, W̃, can be expressed as the Hadamard M̃ (i, i) ≈ −1 (9)
tf , small tf .
product analytical form12,27,32
When controlling the system with one driver node,
W̃ =P̃M̃P̃T = P̃[Q̃ ◦ F̃]P̃T , (5) where there is a single control signal u1 (t) attached to5
arbitrary node h, then B̃ is a N × 1 matrix with b̃h1 = 1, are influenced by the zealot node (the n-th node),
leading to B̃B̃T = J̃hh , where J̃hh is a N × N single-
entry matrix33 such that [J̃hh ]ij = 1 when i = j = large tf
E1 =
h, and zero otherwise. Consequently, Q̃ij = p̃hi p̃hj , X −4λi λj YN N
λi + λk Y λj + λk
(λi +λj )tf c2 p̃li p̃mj
M̃ij = p̃hi p̃hj [ e λi +λj −1 ],
and in the large tf limit, i,j,l,m
p̃hi p̃hj (λi + λj ) k=1 λi − λk k=1 λj − λk
k6=i k6=j
the exponential terms vanish because all λi are nega-
−p̃ p̃hj ∗ N N
−1
tive, and M̃ij = λihi
+λj . Using M̃ = |M̃
M̃|
, where M̃∗ − 2cz
X
[V−1 ]nl p̃li p̃mj
−4λi λj Y λi + λk Y λj + λk
p̃hi p̃hj (λi + λj ) λi − λk k=1 λj − λk
and |M̃| are the adjoint matrix and determinant of M̃ i,j,l,m k=1
k6=i k6=j
respectively,32 N
X −4λi λj Y λi + λk
N N + z2 [V−1 ]nl p̃li p̃mj pmn [P−1 ]nn
−4λi λj Y λi + λk Y λj + λk i,j,l,m
p̃hi p̃hj (λi + λj ) k=1
λi − λk
−1
M̃ (i, j) = . k6=i
p̃hi p̃hj (λi + λj ) λi − λk λj − λk N
k=1 k=1 λj + λk
k6=i k6=j
Y
× ,
(10) λ − λk
k=1 j
k6=j
For small tf , neither first-order nor second-order Taylor
(14)
expansion of F̃ij yields an invertible M̃, and M̃−1 (i, j) P N P
P N P
N P
N
has to be estimated with32 |M̃| ∼ tN ∗ Nij where = .
f and M̃ (i, j) ∼ tf
0
i,j,l,m i=1 j=1 l=1 m=1
such that
For small tf regime, one driver energy cost, al-
N −N0
M̃−1 (i, j) ∼ tf ij , (11) though Eqn. (11) is a valid approximation, owing
to the coupling terms [VeDtf V−1 ]ni [P̃M̃−1 P̃T ]ij and
where the integer exponents Nij (for i, j = 1, 2, ..., N ) or [VeDtf V−1 ]ni [P̃M̃−1 P̃T ]ij [PeDtf P−1 ]jn in Eqn. (8), it
N0 refer to the Nij -th or N0 -th order Taylor expansion
is difficult to express small tf M̃−1 (i, j) terms in ap-
of F̃ij where invertibility is satisfied, and are computed
proximate form. Therefore, numerical M̃−1 is used in-
numerically.
stead. Furthermore, since d drivers energy cost also re-
Using d number of drivers to control the network,
quire numerical M̃−1 , all three analytical energy cost
where 1 < d < N , B̃ is a N × d matrix with b̃ij = 1
equations should thus be expressed by Eqn. (8). Letting
if control signal j (where j = 1, 2, ..., d) attaches to node T
d VeDtf V−1 = eA tf , and PeDtf P−1 = eAtf ,
i (where i = 1, 2, ..., N ), leading to B̃B̃T = Jdk dk ,
P
k=1
where dk = {1, 2, ..., N } refers to the arbitrary k-th small tf
d
P E1
N
N X N
N X
driver node. Thereafter, Q̃ij = p̃dk i p̃dk j , and M̃ij =
AT tf
large tf 2 −1 −1
X T
X T
Ed =c [P̃M̃ P̃ ]ij − 2cz [e ]ni [P̃M̃ P̃ ]ij
k=1
i=1 j=1 i=1 j=1
d (λi +λj )tf small tf
−1] Ed
p̃dk i p̃dk j [e
P
λi +λj . Owing to the summation, it
N
N X
k=1 2
X AT tf −1 T Atf
+z [e ]ni [P̃M̃ P̃ ]ij [e ]jn ,
is difficult factor the terms to derive M̃−1 analytically. i=1 j=1
Thus, d drivers energy cost results have to be computed (15)
with numerical M̃−1 . where the driver nodes placement are encoded in M̃−1
Substituting Eqs. (9) and (10) into Eqn. (8), the an- through the inverse of Eqn. (6), and the time regimes are
alytical energy costs are (the superscripts denote the tf set by tf .
regime, and the subscripts denote the number of drivers From the presented analytical energy cost equations,
used to control the network) note that the choice of zealot-influenced nodes affects
the energy cost. In the large tf regime, such as Eqns.
large tf
X X (12) and (14), the choice of zealot-influenced nodes en-
EN = − 2c2 p̃ik p̃jk λk + 4cz [V−1 ]ni p̃ik p̃jk λk
ters the equations through [V−1 ]ni and pjn , which are
i,j,k i,j,k
X respectively the final row and final column of the eigen-
2 −1 −1
− 2z [P ]nn [V ]ni p̃ik p̃jk pjn λk , vectors matrix of the full (transposed) network, AT and
i,j,k A. Depending on which nodes are being influenced by
(12) the zealot node (through ain = {0, 1}), [V−1 ]ni and pjn
P N P
P N P
N change accordingly, and the choice of influenced nodes is
where = , consequential to the energy cost. One the other hand,
i,j,k i=1 j=1 k=1
in the small tf regime, the N drivers energy cost is in-
small tf variant to the choice of zealot-influenced nodes. In Eqn.
EN ≈ c2 t−1 2
f N − 2czr + z rtf , (13)
(13), ceteris paribus, only the r number (and not choice)
N
P of zealot-influenced nodes tunes the energy cost.
where r = ain is the integer number of nodes which
i=1 Taking ∂E∂z = 0, the minima are (subscripts denote6
number of drivers and tf regime) node directly receives a control signal ui (t) are presented
P −1 in Figs. 2 and 3 for large tf and small tf respectively.
c [V ]ni p̃ik p̃jk λk As expected, the analytical energy cost E(z), as a func-
∗ i,j,k
zN, large tf = −1 P −1 , (16) tion of varying fixed zealot opinion z (Eqns. (12) and
[P ]nn [V ]ni p̃ik p̃jk pjn λk (13)), validated against numerical computations, show a
i,j,k
quadratic behavior with respect to z in Figs. 2(a), 2(g),
3(a), 3(b), 3(g), and 3(h). Therefore, depending on what
∗
zN, small tf ≈ ctf , −1
(17) the zealot’s fixed opinion z is, the energy cost needed to
control a complex network to consensus c = 5 follows a
∗
z1, large tf = quadratic curve, which has a turning point at z ∗ (black
N N
dotted line) that assists in lowering the energy cost, as
−4λ λj λi +λk Q λi +λk
[V−1 ]nl p̃li p̃mj p̃ p̃ (λi +λ compared to the energy cost needed for controlling the
P Q
c λi −λk λi −λk
hi hj i j)
i,j,l,m k=1 k=1
k6=i k6=i complex network in a situation where there are no zealots
, present, denoted by the horizontal green dashed line in-
−4λ λ N N
λ +λ λ +λ
[V−1 ]nl p̃li p̃mj pmn [P−1 ]nn p̃ p̃ (λi +λ j
P Q i k
Q i k
hi hj i j) λi −λk λi −λk tersecting with the quadratic curve at z = 0. Away from
i,j,l,m k=1 k=1
k6=i k6=i minima z ∗ , the energy cost increases at a z 2 rate, which
(18)
when above the green dashed line, opinion z becomes
and detrimental to the controlling of complex networks, and
the zealots’ presence increases energy cost.
∗
N N T
For the large tf regime results, Figs. 2(a) and (g) cor-
z1, [eA tf ]ni [P̃M̃−1 P̃T ]ij
P P
small tf
c respond to the energy costs (Eqn. (12)) needed to con-
∗ i=1 j=1
zd, large tf = N N . trol ER and SF networks when a particular set of nodes
∗
P P AT t
[e f] −1 P̃T ] [eAtf ] have been influenced by the zealot node, with strength
zd, ni [P̃M̃
small tf
ij jn
i=1 j=1 of zealot opinion z varying in the range [−10, 10]. In any
(19) one instance of a numerical experiment, the zealot node
When the zealots hold unwavering optimal opinion z ∗ , can influence r number of normal agent nodes, which are
their presence assist in driving the network opinions to- fixed, once chosen, for the entirety of that instance. De-
ward c by lowering the energy cost, as compared to the pending on the configuration of zealot-influenced nodes,
situation where there were no zealots present. In the the parabolas as shown in Figs. 2(a) and (g) vary accord-
small tf regime, when using N drivers to control the ingly with different turning points z ∗ , and their associ-
∗ large tf
complex network, the optimal zealot zN, small tf opinion, ated minimum energy costs EN (z ∗ ), and steepness.
is unaffected by network properties, and is proportional To understand how the selection of r number of zealot-
to consensus c and t−1 f . In contrast, all other turning
influenced nodes affects the energy cost, their configura-
points z ∗ are affected by network properties. In those tions were varied with increasing r, from r = 1, r = 2,
cases, it is difficult to analyze z ∗ by inspection owing to r = 3, r = 4, up to r/N = 60% of the network nodes, and
large tf
the many coupled terms present, and numerical experi- the turning points z ∗ , minima energy costs EN (z ∗ ),
ments are needed to gain more insight. and associated state space trajectory lengths were mea-
sured.
The state space trajectory was first introduced in Fig.
B. Numerical experiments 1, for example, when the zealot node (holding fixed zealot
opinion z = −5) is introduced into the system (for con-
In the results that follows, it can be assumed that trolling network node states toward consensus c = 2),
tf = 20 and tf ≤ 0.1 respectively for large and small tf the length of the state space trajectory elongates. Math-
regimes, and the complex networks are being driven to- ematically, the length l of the state space trajectory is
ward consensus c = 5, which is fixed throughout, while z, computed as follows:
the zealot’s unwavering opinion is tuned, and the choice v
of nodes influenced by the zealot node (normal agent X 99 uX
uN h i2
nodes which receive directed links from the zealot node) l= t xj (ti ) − xj (ti+1 ) , (20)
is varied. i=1 j=1
where it should be noted that although the time vari-
1. N drivers able t ∈ [0, tf ] is a continuous variable, it is compu-
tationally sampled at 100 evenly spaced ti values (for
The results of the numerical experiments when con- i = 1, 2, ..., 100). Thus, the length l of the state space
trolling random Erdős–Rényi34 (ER) and scale-free (SF; trajectory is computationally approximated as the sum
static model35 ) complex networks topologies toward con- of the lengths of 99 pieces of straight line Euclidean dis-
sensus c = 5, with N = 300 normal agents, and average tances between x(t1 ) and x(t2 ), x(t2 ) and x(t3 ), ..., x(t99 )
degree hki = 6, using N number of drivers, such that each and x(t100 ).7
(a) (b) (c)
15000 4000 30
20
10
3000 0
10000 20 0 5 10
2000
5000 10
1000
0 0 0
-10 -5 0 5 10 0 10 20 30 0 5 10 15
ER
(d) (e) (f)
10 4
6 5000 10000 4000
4000
4500
4000
3000
4 3500
100 120 90 100 110
5000 2000
2
1000
0 0 0
0 100 200 300 400 0 100 200 300 90 95 100 105 110
(g) (h) (i)
15000 4000 150
3000
10000 100
2000
3000
5000 2000 50
1000 1000
2 4 6 8
0 0 0
-10 -5 0 5 10 0 50 100 0 20 40 60 80
SF
(j) (k) (l)
10 4
6 5000
4000
10000 4000
4500
4000
3000
4 3500
100 120 90 100 110
5000 2000
2
1000
0 0 0
0 100 200 300 400 0 100 200 300 90 95 100 105 110
FIG. 2: Continuous-time linear dynamics: Large tf regime, N drivers results in ER and SF networks with varying
configurations of zealot-influenced nodes. (a)—(f) are results from a ER network, while (g)—(l) are results from a SF
network. (a) and (g) validate Eqn. (12). (b) and (h) show that as the turning points z ∗ increases, their associated minima
large tf
energy costs EN (z ∗ ) decreases. (c) and (i) show that the average node degree hki of a configuration of zealot-influenced
nodes correlate to its turning point z ∗ . Inset in (c) show the linear fits of all the data points. (d) and (j) show that length l is
proportional to energy cost for different configurations of nodes influenced by a zealot of fixed z = −5 opinion (black dashed
large tf
lines in both are fitted by EN (l) ≈ l1.77 ). (e) and (k) are the same, except that the zealot has fixed opinion z = 5.
Likewise, (f) and (l) are the same, except that the zealot has fixed opinion z = z ∗ . The insets in (d), (e), (j), and (k) show the
data points associated with r = 1, r = 2, r = 3, r = 4 which are clustered closely. The different markers denote the varying r
number of zealot-influenced normal agents according to legend in (b) or (i), and green dashed line indicates the energy cost in
the absence of any zealot.
One may wish to ask if a higher z ∗ is more beneficial assisting optimal zealot opinion z ∗ , the less the control
or less beneficial to the energy cost needed in controlling signals have to work to drive the network state vector
large tf
the complex network. Plotting z ∗ against EN (z ∗ ) toward consensus. Note that the optimal zealot opinion
in Figs. 2(b) and (h), it can be seen that within the z ∗ is not necessarily always the same as consensus c = 5,
same r value, an increase in turning point z ∗ leads to and as r value increases, z ∗ decreases owing to the fact
a decrease in minima energy cost EN
large tf
(z ∗ ). Further, that the network node states follow linear dynamics. In
increasing r values leads to reduction of minima energy other words, because linear dynamics is always adding
cost, as the more normal agents that are influenced by or subtracting node states based on the coupled com-8
(a) (b) (c)
105 106 104
5 4
5 5
8
10 10
4 8 4 8 7
r=1
7.5 7.5 r=2
3 -100 0 100 3 -1000 0 1000 6 r=3
r=4
10 4
r/N=10%
2 2 5 r/N=20% 7.6
r/N=40% 7.5
1 1 4 r/N=60% 7.4
(Any color) Analytical 45 50 55
W/o Zealots
0 0 3
-100 -50 0 50 100 -1000 -500 0 500 1000 45 50 55
ER
(d) (e) (f)
105 104 105
5 -20 8 100 5 200
10 4 10 4
8.4 8.4
8.2 8.2
4 8 7 4 8
7.8 -40 80 7.8 180
7.6 7.6
3 86.6 86.62 6 3 86.6 86.7
-60 60 160
5 10 4
2 7.7 50 2
7.65
-80 4 40 140
1 7.6 1
86.6 86.62
0 -100 3 20 0 120
86.6 86.7 86.8 86.9 87 86.6 86.8 87 87.2 87 87.5 88 88.5 89
(g) (h) (i)
105 106 104
5 5 8
104 105
4 8 4 8 7 r=1
7.5 7.5 r=2
3 -100 0 100 3 -1000 0 1000 6 r=3
r=4
r/N=10% 10 4
2 2 5 r/N=20% 7.6
r/N=40% 7.5
r/N=60%
1 1 4 (Any color) Analytical
7.4
45 50 55
W/o Zealots
0 0 3
-100 -50 0 50 100 -1000 -500 0 500 1000 45 50 55
SF
(j) (k) (l)
105 104 105
5 -20 8 100 5 200
10 4 10 4
8.4 8.4
8.2 8.2
4 8 7 4 8
7.8 -40 80 7.8 180
7.6 7.6
3 86.6 86.62 6 3 86.6 86.65 86.7
-60 60 160
10 4
2 5 7.7 50 2
7.65
-80 40 140
1 4 7.6 1
86.6 86.65
0 -100 3 20 0 120
86.6 86.8 87 87.2 86.6 86.8 87 87.2 87.4 86.5 87 87.5 88 88.5 89
FIG. 3: Continuous-time linear dynamics: Small tf regime, N drivers results in ER and SF networks. Unlike the large tf
regime results, varying the choice of zealot-influenced normal agents is inconsequential to the energy cost, and only the r
number of zealot-influenced normal agents matters. (a)—(f) are results from a ER network, while (g)—(l) are results from a
SF network. (a) and (g) ((b) and (h)) validate Eqn. (13) at tf = 0.1 (tf = 0.01) for different r values, where the insets show
the analytical energy cost curves associated with r = 1, r = 2, r = 3, and r = 4. Note that the turning points z ∗ is invariant
to the r number of zealot-influenced nodes. (c) and (i) show the turning points z ∗ and associated minima energy costs (insets
show the data points associated with r = 1, r = 2, r = 3, r = 4, where there are slight deviations between analytical and
numerical results due to the first-order approximation eAtf ≈ In + Atf used in deriving analytical results). (d) and (j) display
the energy cost and associated state space trajectory length l when different r number of zealot-influenced nodes is varied
(the symbol markers correspond to r value according to legends in (c) or (i)), as well as the opinion z of the zealot is varied
from [−100, −20] (color represent the associated z values). (e) and (k) are the same, except that z ∈ [20, 100]. Likewise, (f)
and (l) are the same, except that z ∈ [120, 200]. Green dashed line indicates the energy cost in the absence of any zealot.
plex networked interactions, having more normal agents opinion z ∗ need to be a milder z value below 5 in order for
influenced by the zealot node leads to the saturation of the zealot’s presence to be energy cost-assisting. Finally,
node states (bias toward the direction of z) in the long (not shown in Fig. 2), increasing r value also leads to the
time. Thus, when r value is relatively large, such as when large tf
EN (z) quadratic curve having a steeper parabola,
r/N = 40% or r/N = 60%, a zealot opinion of z = c = 5 and the effective assisting z range where energy cost is
will not necessarily assist in lowering energy cost due to lowered (relative to green dashed line) is shortened.
the saturation of node states. Instead, the optimal zealot
Which network properties of the zealot-influenced9
nodes control the turning points z ∗ and associated min- energy cost increases as compared to the no-zealot en-
ima energy costs E(z ∗ )? To investigate, various com- ergy cost (when z = 0). Further, at this z value, an
mon network properties, such as the eigenvalues, degree increase in r number of zealot-influenced nodes leads to
hki, centralities (betweenness, closeness, and eigenvec- high saturation of zealot-bias node states owing to the
tors), and PageRank36 of each node are measured and networked linear dynamics and the driver nodes would
used to distinguish independent configurations of zealot- thus need to consume higher energy cost to overcome
influenced nodes by selecting them in descending or- the zealot’s influence. In the converse situation, in Figs.
der based on these features. From these measurements, 2(e) and (h), when z = 5, supportive to the consensus
it was found that PageRank and node degree hki of goal, most of the energy cost are lowered, compared to
the zealot-influenced nodes are correlated to the turning the no-zealot energy cost. When z = 5, for configura-
points z ∗ , where an increase in average PageRank or hki tions near the associated turning point z ∗ , the zealot’s
of the zealot-influenced nodes leads to an increase in z ∗ , presence is beneficial for control, decreasing energy cost.
as shown in Figs. 2(c) and (i) (only hki results displayed). For the magenta triangles, corresponding to data points
To ensure the sampling of zealot-influenced nodes are of r/N = 60%, a z value of 5 is far away from their associ-
well-represented along a diverse z ∗ range, the configu- ated turning points z ∗ , and in this situation, the zealot’s
rations of zealot-influenced nodes were chosen according presence is adversarial for control, increasing energy cost.
to descending order of nodes degree hki. For example, At z = z ∗ , all configurations are at their respective turn-
for r/N = 10%, the top 10% nodes with highest degree ing points, and the zealots’ presence assist in controlling
hki were selected, followed by the next 10%, and so on. the network, reducing energy cost.
For r/N = 20%, r/N = 40%, and r/N = 60%, the se- In the small tf regime, using N drivers, the energy
lection is similar, except that between sets, there could costs (Eqn. 13) quadratic curves are plotted in Figs.
be overlapping choices of nodes. Using any other selec- 3(a), (b), (g), and (h) respectively for ER and SF net-
tion strategy other than distinguishing hki (or PageRank) works with small tf = 0.1 and tf = 0.01. In these
would lead to a tight cluster of data points around similar figures, the color-coded plots correspond to the r num-
z ∗ values in Figs. 2(b) and (h). Since node degrees hki ber of zealot-influenced nodes in accordance to legends in
correlate to turning points z ∗ , the network topological Figs. 3(c) or (i). As predicted analytically (Eqn. (17)),
differences between ER and SF networks in degree dis- ∗ −1
the turning point zN , small tf ≈ ctf is independent of
tributions P (k)34 explain the disparity of z ∗ data points
the r number, as well as the choice, of zealot-influenced
in Figs. 2(b) and (h), where the z ∗ of the SF network
nodes. From these figures, the theoretical prediction is
tend to be larger, due to the presence of hubs with large
validated, and all turning points are the same, regardless
average degree hki. Physically, this means that when
of r value, and affected only by small tf value, where
hubs in SF networks have become influenced by the zealot ∗ ∗
node with assisting z ∗ opinion, the energy cost is most
zN , small tf ≈ 50 and zN , small tf ≈ 500 respectively for
reduced. Conversely, when hubs are influenced by zealot tf = 0.1 and tf = 0.01, given that c = 5 is fixed. Its
nodes with z opinion far away from its turning point, the invariant turning point z ∗ and associated minima energy
small tf
energy cost is most increased as compared to the situa- cost EN (z ∗ ) are plotted in Figs. 3(c) and (i) for vary-
tion where there are no zealots present. ing r values, where an increase in r leads to a decrease in
Figs. 2(d)–(f) and (j)–(l) reveal how the zealots sway minima energy cost. Unlike its large tf counterparts, no
the state space trajectory lengths l and associated energy topological effects between ER and SF networks are ob-
costs. For each configuration of zealot-influenced nodes, served, as in the small tf regime, there is barely enough
driving the state vector toward consensus c = 5, the en- time for the network topological effects such as node de-
ergy cost and length l are measured respectively for the gree hki to take effect, and only the r number (and not
zealot’s z value as z = −5 in Figs. 2(d) and (j), z = 5 in choice) of influenced nodes matters to the energy cost.
Figs. 2(e) and (k), and z = z ∗ in Figs. 2(f) and (l). In all The energy cost EN
small t
f
(l) as a function of state space
of these figures, the computations show that the energy trajectory length l (Eqn. (20)) are computed and plotted
cost is proportional to length l: in Figs. 3(d), (e), (f), (j), (k), and (l). Since the small
large tf tf regime energy cost is invariant to the choice of zealot-
EN (l) ∝ l. (21)
influenced nodes, each r value (r denoted by different
With the exception of z = z ∗ results for r = 1, r = 2, symbols) only has one data point. To generate more data
small tf
r = 3, and r = 4 configurations, the data points show points to analyze EN (l) and l dependency, z values
that a configuration that leads to a control action that were varied (z values denoted by color). Respectively,
takes a longer l path requires a higher energy cost. De- z ∈ [−100, −20] for Figs. 3(d) and (j), z ∈ [20, 100] for
pending on the zealot’s z value relative to a specific con- Figs. 3(e) and (k), and z ∈ [120, 200] for Figs. 3(f) and
figuration’s z ∗ value, the zealots’ presence may reduce (l). z ∈ [20, 100] corresponds to the z range where the
or increase energy cost relative to the energy cost in the zealot’s presence assists in lowering the energy cost (rel-
absence of zealots. For example, in Figs. 2(d) and (j), ative to the no-zealot energy cost), while z ∈ [−100, −20]
when z = −5, contrarian to the consensus goal, and far and z ∈ [120, 200] are all z values where the zealot’s pres-
away from their respective r values turning point z ∗ , all ence would increase energy cost. In Figs. 3(d), (f), (j),10
and (l), state space trajectory l correlates to energy cost, chain network, where the driver node is located at the
small tf root node (node 1), it was found that the average path
and an increase in l leads to an increase in EN (l).
Further, increasing r value or |z| values also leads to in- distances from driver node to zealot-influenced nodes
creased l and energy cost, suggesting that when more nor- anti-correlate with log10 (|z ∗ |), where configurations with
mal agents are influenced by the zealot node with strong zealot-influenced nodes further away from the driver node
opinions, more effort is needed to steer the network node 1 tend to have lower log10 (|z ∗ |), and lower minima energy
large tf
states toward consensus. In the assisting z ∈ [20, 100] cost log10 (E1 (z ∗ )). This is not surprising, consider-
small tf ing that it has already been reported that for a chain
range (Figs. 3(e) and (k)), l anti-correlates to EN (l),
where an increase in l leads to a decrease in energy cost. network, energy cost increases exponentially as the path
When r value is the highest, and z value is closest to distances increases linearly37 . Thus, when zealots influ-
z ∗ = 50, energy cost is most reduced. ence nodes that are furthest away from the driver node
with assisting z ∗ opinion, the energy cost can be most
reduced.
For more complicated simple network topologies in
2. One driver
Figs. 12–14, none of the network properties are predic-
large tf
tive of |z ∗ | nor E1 (z ∗ ). However, in all of these,
When using only one control signal to control the state space trajectory length l remains a strong predic-
complex network, the various node states are indirectly tor of energy cost, yielding the scaling law
driven through various paths throughout the network
from the sole control signal, and the resulting state space log10 (E1
large tf
(l)) ∼ log10 (l), (22)
trajectory is highly circuitous12 . Owing to the highly
circuitous state space trajectory, it is difficult to analyze as evidenced in Figs. 4(f)–(h), and Figs. 13–15(d)–(f), re-
the effects the choice of zealot-influenced nodes have on spectively for z = −5, z = 5, and z = z ∗ . Thus, when a
the energy cost in a complex network. Therefore, in the particular configuration of zealot-influenced nodes leads
results that follows, simple network topologies such as to low length l, energy cost is most reduced. When
chain, ring, and star networks25 will be studied instead. z = −5 or z = 5, most of the configurations result in
The one driver large tf regime results from the nu- increased energy cost relative to no-zealot energy cost
merical experiments with chain, star, and ring networks (green dashed lines) as the z values are far from their
are presented in Figs. 4, and 12–14 (Figs. 12–14 in Ap- turning points z ∗ . When z = z ∗ , all configurations lead
pendix A). In these figures, the energy costs needed in to reduced energy cost. In star or ring network, the most
controlling these networks when there is a zealot node optimal configurations can reduce the energy cost by 3-4
present and influencing r = 1, r = 2, r = 3, and r = 4 orders of magnitude. Finally, note that most of the scat-
number of normal agents are measured. For each r value, ter data points overlap, and there is no clear distinction
the measurement is repeated for an exhaustive N Cr num- between r values, indicating that for one driver, large
ber of times, each time with different unique sets of tf regime, the zealot’s influencing of 1 normal agent has
zealot-influenced nodes (for small N , this search space as much potential in swaying the control action as influ-
is feasible). As predicted from Eqn. (14), the energy cost encing 4 normal agents. This suggests that, because the
is quadratic with respect to the zealot’s fixed z opinion, one driver state space trajectory is highly circuitous12 , r
which are all validated in Fig. 4(b) and Figs. 12–14(b). number of nodes being influenced by the zealot is not pre-
Unlike its large tf regime, N drivers counterpart, there dictive of how much the energy cost will change since the
is no clear relationship between z ∗ and E(z ∗ ), as shown circulation of the zealot’s influence may move the state
in Figs. 4(c), and 12–14(c). However, in all of these, the space trajectory around through different indirect ways.
statistical trend suggests that when |z ∗ | is small, minima The numerical experiments are repeated for the one
large tf
energy costs E1 (z ∗ ) is most reduced relative to the driver node calculations in the small tf regime. Respec-
no-zealot energy cost (horizontal green dashed line). Fur- tively, Fig. 5, and Figs. 16–18 (Figs. 16–18 are found in
ther, it was noticed during computation that when |z ∗ | Appendix B) correspond to chain network with node 1
is large, the parabola of the E1
large tf
(z) curve stretches, being the driver node, chain network with node 4 be-
∗
and when |z | is small, it steepens. Finally, some of the ing the driver node, star network, and ring network.
turning points z ∗ can lie in the negative z region, indicat- Similar to its large tf counterparts, there is no clear
ing that despite holding contrarian opinion to the goal of trend between turning points log10 (|z ∗ |) and minima
small tf
driving the network toward consensus c = 5, a contrarian log10 (E1 (z ∗ )), except that it appears that a smaller
small tf
zealot surprisingly assists in the reducing control energy. |z ∗ | value tend to have a lower E1 (z ∗ ). Further, the
∗
The various network properties of configurations of turning points z can also lie in the negative z region,
zealot-influenced nodes were measured to find out if indicating that a contrarian zealot can actually assist in
any of them can explain the turning points z ∗ . Un- lowering the energy cost. Overall, the physical behav-
like its (large tf regime) N drivers counterparts, node ior of the one driver node computations in the small tf
degree hki of zealot-influenced nodes do not predict z ∗ regime is similar to its large tf regime counterparts, ex-
value for the one driver result. Instead, in Fig. 4, for a cept that the required energy is much larger, which is not11
(a) (b) (c)
1 u1(t)
2
3
4
5
(d) (e)
6
7
8 z
(f) (g) (h)
FIG. 4: (a) Continuous-time linear dynamics: Large tf regime, one driver results in a chain network, where the driver node is
located at node 1, and the configurations of zealot-influenced nodes are varied. (b) validates Eqn. (14). (c) shows how the
large tf
turning points log10 (|z ∗ |) relate to its associated minima energy costs log10 (E1 (z ∗ )). Note that unlike the results from N
∗
drivers, the turning points z could lie on the negative z region, where a contrarian zealot opinion assists the driver node by
reducing energy cost. (d) shows that there is a trend where the turning points z ∗ tend to decrease when the average path
distances from driver node 1 to the zealot-influenced normal agent nodes increase. (e) displays a similar trend, showing that
when nodes far away from the driver nodes are influenced by the zealot holding z ∗ opinion, the energy cost is reduced.
Errorbars of r = 2 (red right triangles) indicate standard deviations, while the errorbars of r = 3 (black squares), and r = 4
(purple crosses) are not shown in plots for the purpose of visibility. (f), (g), and (h) respectively indicate that the energy cost
scales with length l according to log10 (E(l)) ∼ log10 (l) for various configurations of zealot-influenced nodes when z = −5,
z = 5, and z = z ∗ when controlling the chain network toward consensus c = 5. When z = −5 or z = 5, almost all
configurations of zealot-influenced nodes increase the energy cost (relative to the energy cost needed for controlling the
network in the absence of zealots) because the z values are far from their respective turning points z ∗ . When z = z ∗ , all
configurations of zealot-influenced nodes reduce the energy cost
surprising, given that the required energy cost decreases similar results hold for SF networks) with continuous-
as tf increases12 . The scaling behavior time linear dynamics using 80 control signals. The re-
sults, both large tf and small tf regimes are plotted in
small tf Fig. 6. Figs. 6(a) and (f) validate Eqn. (15), showing that
log10 (E1 (l)) ∼ log10 (l) (23)
the energy cost is quadratic with respect to the zealot’s
are corroborated by Fig. 5(f)–(h), and Figs. 16–18(d)–(f). fixed z opinion. Furthermore, Fig. 6(f) shows that the
turning point z ∗ lie in the negative z region, although
the goal is to drive the state vector towards c = 5, indi-
3. d drivers cating that a contrarian zealot is beneficial for control in
some zealot-influenced nodes configurations. There is no
clear relationship between turning points z ∗ and their as-
The numerical experiments were repeated for control- sociated minima energy costs E(z ∗ ), as displayed in Figs.
ling a N = 200, hki = 6 ER network (it is expected that12
(a) (b) (c)
1 u1(t)
2
3
4
5 (d) (e)
6
7
8 z
(f) (g) (h)
FIG. 5: (a) Continuous-time linear dynamics: Small tf regime, one driver results in a chain network, where the driver node is
located at node 1, and the configurations of zealot-influenced nodes are varied. (b) validates Eqn. (15). (c) shows the
small tf
relationship between turning points log10 |z ∗ | and associated minima energy costs log10 (E1 (z ∗ )). (d) shows that as the
average path distances from driver node 1 to the zealot-influenced normal agent nodes increase, the turning points z ∗
decrease. (e) shows that when normal agents furthest away from the driver node are influenced by the zealot holding z ∗
opinion, the energy cost is most reduced. (f), (g), and (h) respectively indicate that the energy cost scales with its associated
length l as log10 (E(l)) ∼ log10 (l), when z = −5, z = 5, and z = z ∗ for various configurations of zealot-influenced nodes.
6(b) and (h). This is in contrast to large Tf N drivers, configuration costs is relative to their respective turn-
where an increase in z ∗ leads to a decrease in E(z ∗ ), and ing point z ∗ and the steepness of the parabola. For the
the one driver results, where a decrease in z ∗ tend to large tf results, z = −5 and z = 5 configurations are far
decrease E(z ∗ ). from the minima and all configurations have higher en-
Nevertheless, the energy cost of each zealot-influenced ergy costs compared to the no-zealot energy cost (green
nodes configuration can be explained by the state space dashed line). At z = z ∗ , all configurations are exactly at
trajectory length l. l was calculated, along with its asso- their minima, and all configurations reduce energy cost,
ciated energy cost, for fixed z = −5, z = 5, and z = z ∗ , although by less than one order of magnitude. In the
where for each r number of zealot-influenced nodes, from small tf regime, for z = −5, z = 5, and z = z ∗ , all
r = 2 to r/N = 60%, 20 independent configurations of configurations are close to or at the minima, and the
nodes were randomly selected. For r = 1, all independent zealot’s influence reduces the energy cost as compared to
configurations from node 1 to node N were chosen. From no-zealot energy cost (not shown in Figs. 6(h)–(j)), which
Figs. 6(c)–(e) and (h)–(j), it is evident that the energy was around log10 (E) ≈ 11.6.
cost scales as
log10 (Ed (l)) ∼ log10 (l), (24)
where a configuration that causes an increased l would
cost the drivers more energy cost. The energy that each13
10
4 (a) (b)
4.4
3.4
3.2
4.35
3
2.8
4.3
2.6
2.4
4.25
-50 0 50 -2 -1 0 1 2
(c) (d) (e)
6.5 6.5 4.4
6 6
4.35
5.5 5.5
5 5
4.3
4.5 4.5
4 4 4.25
2.4 2.6 2.8 3 3.2 2.4 2.6 2.8 3 3.2 2.26 2.28 2.3 2.32
FIG. 6: Continuous-time linear dynamics: Large tf d drivers results in a N = 200, hki = 6 ER network with 80 control
signals. (a) validates Eqn. (15), showing that the energy cost is quadratic with respect to zealot’s z opinion. (b) shows the
relationship between turnings z ∗ and their associated minima energy cost Ed (z ∗ ). (c)–(e) plot the state space trajectory
length l that each zealot-influenced nodes configuration takes the system and the corresponding energy cost at z = −5, z = 5,
and at z = z ∗ . They show that the energy cost of each configuration can be explained by length l, where the energy cost
scales as log10 (Ed (l)) ∼ log10 (l). Legend displays the marker symbols corresponding to each r number of zealot-influenced
normal agents.
1011 (a) (b)
8 11.6
7
11.4
6
5 11.2
4
11
3
-100 0 100 0 1 2 3 4
(c) (d) (e)
11.1 11.1 10.896
11.05 10.894
11
11
10.892
10.95 10.9
10.89
10.9
10.8 10.888
4.9 4.95 5 4.86 4.88 4.9 4.92 4.94 4.96 4.98 4.894 4.895 4.896 4.897 4.898
FIG. 7: Continuous-time linear dynamics: Small tf d drivers results in a N = 200, hki = 6 ER network with 80 control
signals. (a) validates Eqn. (15), showing that the energy cost is quadratic with respect to zealot’s z opinion. (b) shows the
relationship between turnings z ∗ and their associated minima energy cost Ed (z ∗ ). (c)–(e) plot the state space trajectory
length l that each zealot-influenced nodes configuration takes the system and the corresponding energy cost at z = −5, z = 5,
and at z = z ∗ . They show that the energy cost of each configuration can be explained by length l, where the energy cost
scales as log10 (Ed (l)) ∼ log10 (l). In the small tf regime, all configurations reduce the energy cost relative to no-zealot energy
cost (not shown in Figs. (c)–(e)), which was log10 (E) ≈ 11.6. Legend displays the marker symbols corresponding to each r
number of zealot-influenced normal agents.14
III. DISCRETE-TIME LINEAR DYNAMICS WITH is the weighted connection between nodes i and j, and
CONFORMITY BEHAVIOR MODEL ñi
P
s̃i = ãij is the strength of node i, obtained by sum-
j=1
Next, how zealots affect the energy cost in controlling ming over all nearest neighbors’ weighted connections.
a complex network with discrete-time linear dynamics Introducing input control signal terms into Eqn. (25),
and conformity behavior26 is studied. The dynamics of the zealot’s connections, and rewriting in vector nota-
this model incorporates conformity features, where each tional form, the system-level dynamics is
member of the network adapts their node state to follow
the average of their nearest neighbors over time. Ar-
x(τ + 1) =S−1 Ax(τ ) + Bu(τ )
guably, it is more realistic as it captures conformity be- (26)
havior, which has been reported in various social sys- =Ax(τ ) + Bu(τ ),
¯
tems. For example, evolutionary games38,39 , learning
behaviors40,41 , and collective movements42–46 of animals where x(τ ) = [x1 (τ ), x2 (τ ), ..., xN (τ ), z]T is the n × 1
in groups. state vector of all N normal agents, and the zealot
An example is given in Fig. 8 to introduce this network node (the n-th node, where n = N + 1), u(τ ) =
dynamics, where the node states xi (t) may represent dif- [u1 (τ ), u2 (τ ), ..., uM (τ )]T is the M ×1 control signals vec-
ferent opinions on a particular subject, with a positive tor, A is the full n × n network matrix, which comprises
xi (t) value indicating support of an idea, and a negative à as its first N × N block, where ãij = ãji is non-zero
xi (t) value representing opposition. A minimum of three if normal agents i and j have interactions, otherwise it is
nodes are needed to showcase the conformity dynamics zero, and the n-th column describes the zealot’s directed
because otherwise, with two nodes, they will just mimic link to normal agents, where ain = 1 if zealot node in-
each other’s node states in perpetuity without reaching fluences node i, otherwise it is zero, B is the control
conformity. In Fig. 8(a), there are N = 3 normal agents input matrix which describes which nodes are directly
who start with different node states xi (τ = 0) drawn from controlled by a control signal such that bij = 1 if nor-
random uniform [0, 1]. Over time, each normal agent mal agent node i is controlled by control signal j, oth-
mimics their nearest neighbors’ node states and the net- erwise it is zero, S−1 = diag{ s11 , s12 , ..., s1N , 1} is a n × n
work node states reach conformity as demonstrated in
diagonal matrix which holds s1i (for i = 1, 2, ..., N ) on
Fig. 8(b). The 3-dimensional state space trajectory of
its main diagonals, with the n-th entry being 1, and
this time evolution of node states is presented in Fig. ni
P
8(c). In Fig. 8(d), a single control signal u1 (τ ) attaches si = aij sums over the full A matrix, inclusive of
j=1
to node 1 to control the network, driving the node states
toward consensus c = 2, as shown in Fig. 8(e). The state the zealot node’s directed connections. A = S−1 A is
space trajectory of this control action is plotted in Fig. the full network matrix, where the normal¯agents evolve
8(f). Finally, in Fig. 8(g), a zealot node is introduced in time with conformity, yet the zealot node remains
into the system, where it holds a fixed z = −5 opinion. fixed with state z. Note that unlike the continuous-
Fig. 8(h) shows the node states evolution of the normal time dynamics, which requires self-links to model sys-
agents being driven toward c = 2, along with the zealot’s tem stability, the discrete-time dynamics with confor-
fixed opinion at z = −5. Correspondingly, in Fig. 8(i), mity model requires that ãii = 0 to ensure system sta-
the state space trajectory of this control action, under bility (see Appendix D). Further, the zealot’s self-loop
the influence of the zealot, elongates as the control sig- [S−1 ]nn = ann = ann = 1 is necessary for modelling
nal u1 (τ ) now has to consume more energy to overcome zealotry, leading to¯ xn (τ + 1) = xn (τ ) = z, and the
the zealot’s contrarian opinion. Note that it suffices to zealot node’s opinion remains fixed at z against time.
display the state space trajectory in 3 dimensions as the Finally, the non-symmetric full network matrix can be
zealot’s fixed node state, associated with the fourth di- eigen-decomposed as A = PDP−1 and AT = VDV−1 ,
¯ ¯ ¯ ¯matrices ¯of A (A
where P (V) is the eigenvectors ¯ T¯),¯and
mension, would simply shift and constraint x1 (τ ), x2 (τ ), ¯ ¯ ¯ ¯
and x3 (τ ) onto the fixed x4 = z hyperplane. D is the n × n diagonal matrix containing the eigenval-
¯ of A such that D = diag{Λ , Λ , ..., Λ , Λ }, where
ues
Networked conformity dynamics is achieved when, at 1 2 N n
Λ1 ≤ Λ ¯ ≤ ... ≤ Λ¯ ≤ Λ = 1. From computation,
the individual level, each normal agent node updates 2 N n
their node states in the next discrete-time round, τ + 1, regardless of network size or topology, the eigenvalues of
by taking the average of their nearest neighbors’ in the all normal agents |Λi | < 1, while the zealot node has
current round τ 26 . Therefore, for each normal agent node eigenvalue Λn = 1.
i (for i = 1, 2, ..., N ),
ñi
1 X
xi (τ + 1) = ãij xj (τ ), (25) A. Analytical equations of energy cost
s̃i j=1
where node i has ñi nearest normal agent neighbors, The energy cost required to control the network with
xj (τ ) is the opinion of neighbor j at round τ , ãij = ãji discrete-time linear dynamics and conformity behaviorYou can also read
