Talk delivered at the Stockholm Workshop on Emergent Control Topics (https://sites.google.com/view/ieee-css-societal-challenges22/home?authuser=0)

1. Controlling complexity
challenges and opportunities
MARIO DI BERNARDO
University of Naples Federico II, Italy
& Scuola Superiore Meridionale
IEEE CSS Pre-Workshop, Stockholm – 15th June 2022

2. • Controlling complex systems: why and how
• Controlling across scales
• Self-organizing control of complex
networks
• Conclusions
Outline

3. • A complex system is:
a) A collection of objects or agents with high cardinality..
b) ..which interact with one another in a nontrivial way..
c) ..such that the collective behaviour of the system is unexpected or
different from, or not immediately predictable from, the aggregation
of the behaviour of its individual parts
What is a complex system?
L. Torres, A. Blevins, D. Bassett, T. Eliassi-Rad, The Why, How and When of Representations for Complex Systems, SIAM Review 2021

4. • How can we orchestrate in real-time the emerging collective
behaviour of a large-scale multiagent system?
1. Whom do we sense?
2. Whom do we control?
3. What do we compute?
4. How do we certify convergence?
Controlling complexity
Feedback Control = Sense + Compute + Actuate

5. • To achieve this goal we can act on
1. the agents
2. the links interconnecting them
3. the topology of the network structure itself
• Each of these approaches yields different types of problems
• e.g. controllability, observability, proving stability and
convergence, control design etc.
How to control

6. • We need to “close the loop” across different scales
A multi-scale problem
NATURE.COM/NATURE
12 May 2011
ANTIBIOTICS
A SPOONFUL
OF SUGAR
Carbohydrate‘helpers’boost
antibacterialefficacy
PAGE216
POLICY
WHO NEEDS
CHANGE
AWorldHealthOrganization
fortoday’sworld
PAGE143
GEOLOGY
WAITING FOR
VESUVIUS
WhattodowhenEurope’s
mostdeadlyvolcano blows
PAGE140
THE INTERNATIONAL WEEKLY JOURNAL OF SCIENCE
TAMING
COMPLEXITY
The mathematics of network control — from
cell biology to cellphones PAGES158&167
Cover 12 May US.indd 1 05/05/2011 10:35

7. • Crucial question:
Can agents at the microscopic level self-determine and
control the value of some macroscopic network observable?
• As an example take a network of nonlinear oscillators
Self-organizing control of networks

8. • We can prove that the synchronous solution is locally
transversally stable if
• Goal: find a strategy making the agents able to self-organize
to maximise the algebraic connectivity of a network
Optimal Synchronizability
<latexit sha1_base64="QVYd/6YqGj083VregncFyqdmTl8=">AAAB/3icbVDLSsNAFL3xWesrKrhxM1gEVyUpoi5ECm5cVrAPaEKYTCbt0MkkzEyEUrvwV9y4UMStv+HOv3HaZqGtBwYO59zDvXPCjDOlHefbWlpeWV1bL22UN7e2d3btvf2WSnNJaJOkPJWdECvKmaBNzTSnnUxSnISctsPBzcRvP1CpWCru9TCjfoJ7gsWMYG2kwD70FOslGHncZCIc1NA18kQe2BWn6kyBFolbkAoUaAT2lxelJE+o0IRjpbquk2l/hKVmhNNx2csVzTAZ4B7tGipwQpU/mt4/RidGiVCcSvOERlP1d2KEE6WGSWgmE6z7at6biP953VzHl/6IiSzXVJDZojjnSKdoUgaKmKRE86EhmEhmbkWkjyUm2lRWNiW4819eJK1a1T2vundnlfpVUUcJjuAYTsGFC6jDLTSgCQQe4Rle4c16sl6sd+tjNrpkFZkD+APr8weIapUf</latexit>
2 > ⌫

9. • This problem could be solved via semi-definite
programming..
• ..assuming the entire network structure is known
• Can a network self-organize into the globally optimal
structure through nodes acting according to a local rule?
Self-organization
Local rules; Only interactions with neighbours, and all nodes
follow the same rules
Limited knowledge - Nodes have a one-hop horizon of knowledge
Boyd, S. (2006, August). Convex optimization of graph Laplacian eigenvalues. In Proceedings of the International Congress of
Mathematicians (Vol. 3, No. 1-3, pp. 1311-1319).

10. • A solution could be to use an adaptive law to select the
weights
• Specifically, to solve
• We could choose
• But this requires agents to possess global information…
Adaptive weights
tion problems and discuss their common structures. We
demonstrate our approach with several numerical examples,
highlighting some of the most interesting features.
II. WEIGHT ADAPTATION
We begin by proposing a weight adaptation method that
finds a local optimum of the constrained continuous op-
timization problem with objective function f(w) and p
inequality constraints gk(w), where w is the vector of edge
weights in the network of interest:
minimize
w
f(w) (2,a)
subject to gk(w)  0 8k 2 1, . . . , p (2,b)
The weight adaptation method we use is the set of ordinary
differential equations (ODEs),
ẇ{i,j} =
@f(w)
@w{i,j}
X
k
@gk(w)
@w{i,j}
⌫k (3,a)
˙
⌫k = gk(w)⌫k (3,b)
which is based on the first order necessary Karush-Kuhn-
Tucker (KKT) conditions [22]. In this equation, the variables
⌫k behave as KKT multipliers, the dual variables, where
the w{i,j} are the primal variables. The first order necessary
KKT conditions for a local minimum w⇤
are simply:
⇤
P ⇤ ⇤
obtained throu
ȧ = k
where k1, k2,
with k1 > k2
can then be c
position a⇤
of
Further to thi
pletely distribu
the arithmetic
components o
made using tw
consensus esti
estimator each
of the compo
vector 11>
a
n . A
variables, and
[21]. Specifica
˙
⇡a = k
ża =
In the second
tion problems and discuss their common structures. We
demonstrate our approach with several numerical examples,
highlighting some of the most interesting features.
II. WEIGHT ADAPTATION
We begin by proposing a weight adaptation method that
finds a local optimum of the constrained continuous op-
timization problem with objective function f(w) and p
inequality constraints gk(w), where w is the vector of edge
weights in the network of interest:
minimize
w
f(w) (2,a)
subject to gk(w)  0 8k 2 1, . . . , p (2,b)
The weight adaptation method we use is the set of ordinary
differential equations (ODEs),
ẇ{i,j} =
@f(w)
@w{i,j}
X
k
@gk(w)
@w{i,j}
⌫k (3,a)
˙
⌫k = gk(w)⌫k (3,b)
which is based on the first order necessary Karush-Kuhn-
Tucker (KKT) conditions [22]. In this equation, the variables
⌫k behave as KKT multipliers, the dual variables, where
the w{i,j} are the primal variables. The first order necessary
⇤
obtained through the ad
ȧ = k1
11>
a
n
where k1, k2, and k3 a
with k1 > k2 2, k3 >
can then be computed
position a⇤
of the syste
2 =
Further to this, it was
pletely distributed throu
the arithmetic mean of
components of a, a>
a
n
made using two separa
consensus estimators [2
estimator each agent m
of the components of
vector 11>
a
n . Another ve
variables, and is not use
[21]. Specifically, we h
˙
⇡a = kg(a ⇡a
ża = kIL(w)

11. A multilayer strategy

12. • To estimate the algebraic connectivity we started from the
strategy by [Yang et al, 2010]
Estimation strategy
4. EIGENVALUE ESTIMATION
onnectivity estimator
[21], the algebraic connectivity of a weighted undirected network can be estimated
P
a D !
k1
n
11>
a ! k2La ! k3
!
a>
a
n
! 1
"
a (55)
ode follows
P
ai D !k1hai C k2
X
j2Ni
w¹i;jº.aj ! ai / ! k3.haı2
i ! 1/ai (56)
s. International Journal of Robust and
published by John Wiley & Sons Ltd.
Int. J. Robust Nonlinear Control 2017; 27:1499–1525
DOI: 10.1002/rnc
4. EIGENVALUE ESTIMATION
ebraic connectivity estimator
ribed in [21], the algebraic connectivity of a weighted undirected network can be estimated
e ODE :
P
a D !
k1
n
11>
a ! k2La ! k3
!
a>
a
n
! 1
"
a (55
every node follows
P
ai D !k1hai C k2
X
j2Ni
w¹i;jº.aj ! ai / ! k3.haı2
i ! 1/ai (56
he Authors. International Journal of Robust and
r Control published by John Wiley & Sons Ltd.
Int. J. Robust Nonlinear Control 2017; 27:1499–152
DOI: 10.1002/rn
UTED OPTIMISATION AND CONTROL OF LAPLACIAN EIGENVALUES 1509
i being the arithmetic mean of the components of a and haı2
i D 1
n
P
i a2
i being
m of squared components. It can be shown that if the algebraic connectivity is
ary point a!
is defined as
a!
D ˙v2.w/
s
n.k3 ! k2!2.w//
k3
(57)
t eigenvector associated with !2.w/. This allows the following estimates to be
1
!2.w/ D
k3
k2
!
1 !
jjajj2
2
n
"
(58)
2
@!2.w/
D
.ai ! aj /2
(59)
OPTIMISATION AND CONTROL OF LAPLACIAN EIGENVALUES 1509
ng the arithmetic mean of the components of a and haı2
i D 1
n
P
i a2
i being
squared components. It can be shown that if the algebraic connectivity is
oint a!
is defined as
a!
D ˙v2.w/
s
n.k3 ! k2!2.w//
k3
(57)
envector associated with !2.w/. This allows the following estimates to be
1
!2.w/ D
k3
k2
!
1 !
jjajj2
2
n
"
(58)
2
@!2.w/
@w¹i;j º
D
.ai ! aj /2
jjajj2
2
(59)
DISTRIBUTED OPTIMISATION AND CONTROL OF LAPLACIAN EIGENVALUE
with hai D 1
n
P
i ai being the arithmetic mean of the components of a and haı2
i D
the mean of the sum of squared components. It can be shown that if the algebrai
distinct, the stationary point a!
is defined as
a!
D ˙v2.w/
s
n.k3 ! k2!2.w//
k3
where v2 is the unit eigenvector associated with !2.w/. This allows the following
performed [21]:
1
!2.w/ D
k3
k2
!
1 !
jjajj2
2
n
"
2
@!2.w/
@w¹i;jº
D
.ai ! aj /2
jjajj2
2
The equilibrium point a!
is locally exponentially stable, provided that k1 > k2!2
k2 > 0. Moreover, the rate constant of this local exponential convergence is given
P. Yang, R.A. Freeman, G. J. Gordon, K.M. Lynch, S.S. Srinvasa, R. Sukthankar Decentralized estimation and control of graph connectivity for mobile
sensor networks- Automatica 46 (2010) 390-396

13. • PI consensus layers are used
to estimate the global variables
giving rise to a multilayer strategy
• Layer 1: PI Consensus
• Layer 2: Estimation of and
its sensitivities
• Layer 3: Weight adaptation law
(based on gradient descent)
Local estimation and control
2
3
(its own weight wi,j, the weighted degrees of its parents
li,i and lj,j and their maximum allowed weighted degrees
ki and kj). Only the sensitivity of the chosen objective
function, @f(w)
@wi,j
, with respect to the edge remains as a
global parameter, which will be computed locally via a
set of distributed estimators (in Case 1, by the 2 Es-
timator layer, shown in Figure 2). This will permit the
optimization algorithm to be fully distributed.
In particular, to estimate the sensitivities of the alge-
braic connectivity to variation of the edge weights, we
use the distributed strategy by Yang et al. [19] to eval-
uate the algebraic connectivity of a weighted undirected
network in a distributed fashion. The strategy can be
implemented as two additional layers, the Proportional-
Integral (PI) Consensus layer and the 2-Estimator layer
shown in Fig. 2. The dynamics of these two layers can
be described by the following set of di↵erential equations
inspired by power iteration (see [19] for further details) :
ȧ = k1'a k2La k3( a 1) a (8)
˙
'a = (a 'a) kP L'a kIL a (9)
˙ a = kIL'a
˙
a = (a2
a) kP L a kIL!a (10)
˙
!a = kIL a
For concise notation, component-wise product of vec-
tors is signified by , and squaring a vector is taken
component-wise also, so that a2
= a a. Here a is an
estimate of the eigenvector associated with 2, which re-
quires two further global variables: the arithmetic mean
P
ai
kP kI
2
Estimator
k1 k2 k3
PI
Consensus
a,i = d
ha2i
Weight
Optimiser
ai
aj, 8 j 2 Ni
w{i,j}, 8 j 2 Ni
@ 2
@w{i,j}
w{i,j}, 8 j 2 Ni
Estimators
'a,i = c
hai
L(w)
'a,j, a,j 8 j 2 Ni
Figure 2. Schematic diagram of the distributed multilayer
approach for 2 maximisation proposed in the paper, with
faster processes at the bottom, and slower processes built on
tributed Adaptive Optimization and Control of Network Structures
Louis Kempton, Guido Herrmann and Mario di Bernardo
ract— In this paper we present a generic distributed
adaptation framework to optimize some network ob-
es of interest. We focus on the algebraic connectivity 2,
ctral radius n, the synchronizability n/ 2, or the total
e graph resistance ⌦ of undirected weighted networks,
scribe distributed systems for the estimation of these
ns and their derivatives for on-line adaptation of the
eights.
I. INTRODUCTION
n controlling a multi-agent network, the structure of
twork plays a vitally important role in determining
formance of the system. Functions of the eigenvalues
Graph Laplacian matrix L, 0 = 1  2  · · · 
ave been shown to be instrumental in determining
operties and performance of a wide range of multi-
network systems. For example, the second smallest
alue of L, the algebraic connectivity 2, governs the
gence rate in many consensus algorithms [1], [2]. In
ear synchronization applications, the local transversal
y of the synchronous solution is dependent on 2 or
Distributed
ẇ(t)
L(w(t))
Constraints G(V, E)
@f(L(w))
@w
Weight
Adaptation
Distributed
Gradient
Estimation
Consensus
Fig. 1. General overview of the distributed estimator-optimizer system. Dis-
tributed estimation of the gradient of the objective function is accomplished
in the blue box. Typically this system will require agents to reach consensus
on some global variables, and distributed Proportional-Integral consensus
is employed to achieve this [21]. The estimated gradient is fed into the
optimizer system which formulates a control law for the weight update,
using gradient descent whilst enforcing the constraints. The network adapts
according to this control law, with the current network structure influencing
the gradient estimation.
with [15] proposing an adaptive method for estimating the
algebraic connectivity in a completely decentralized manner.
EE 55th Conference on Decision and Control (CDC)
esort & Casino
ber 12-14, 2016, Las Vegas, USA

14. • The network self-organizes so as to maximise its algebraic
connectivity
Results

15. • The adaptive distributed strategy can be used to control the
connectivity to some desired value
Controlling the graph connectivity
entralized Control of 2
pposed to maximizing or minimizing these spectral
ns of the graph Laplacian, control can also be achieved
inor changes to the weight adaptation law. To show
e demonstrate control of the algebraic connectivity to
d reference value ⇢ using the weight adaptation law:
ẅi,j = ka
@g(w)
@wi,j
✓
⇢ c
2
(i)
◆
c1ẇi,j (27)
eights are simply forced in the direction of increasing
n the current estimated algebraic connectivity is less
e desired value, and forced in the opposite direction in
e that the current estimate of the algebraic connectivity
the desired value.
n example, we use the case described in Section IV-A,
w force the 2 towards a value of ⇢ = 0.25, Figure 8.
a feasible reference algebraic connectivity for the
d network as it is less than the constrained maximum
08, but the solution is clearly not unique. To demon-
ne of the benefits of using an adaptive method, we
e the loss of an edge between two nodes, by setting
ght to 0 at t = 2000 seconds. At this point, algebraic
ivity is temporarily reduced, but through the action of
troller, the set point of 2 = 0.25 is quickly recovered.
V. CONCLUSION
have shown that a network of agents may cooperate
r, exchanging only local information, and only main-
a small number states in local memory (9 for 2
0 500 1000 1500 2000 2500 3000 3500 4000
Time (t)
0.0
0.2
0.4
0.6
0.8
1.0
Edge
weights
Figure 8. With a small change to the weight adaptation law, Equation (27),
As opposed to maximizing or minimizing these spectral
functions of the graph Laplacian, control can also be achieved
with minor changes to the weight adaptation law. To show
this, we demonstrate control of the algebraic connectivity to
a desired reference value ⇢ using the weight adaptation law:
ẅi,j = ka
@g(w)
@wi,j
✓
⇢ c
2
(i)
◆
c1ẇi,j (27)
As an example, we use the case described in Section IV-A, but
now force the 2 towards a value of ⇢ = 0.2, Figure 8. This
is a feasible reference algebraic connectivity for the weighted
network as it is less than the constrained maximum of 0.3708,
but the solution is clearly not unique.
V. CONCLUSION
We have shown that a network of agents may cooperate
together, exchanging only local information, and only main-

16. • We can extend the strategy to estimate other functions of
the Laplacian spectrum, e.g.
• Laplacian eigenratio
• Total effective graph resistance
Extensions
estimating eigenvalues close to non-uniqueness compared with [25], allowing the graph Laplacian
eigenvalues to be optimised past the first point when !2 ! !3 or !n ! !n!1, which can lead to
significant improvements in performance.
The distributed optimisation and control strategy that we propose consists of multiple layers of
distributed estimation. The edge weight dynamics rely on estimates of the graph Laplacian eigen-
values, and these estimators themselves require further estimated variables to function correctly. An
outline of this three-layer structure is illustrated in Figure 1. In Sections 3 to 5 of the paper, we will
first present each layer independently, and then in Section 6, we specify how the three layers are
interconnected, and derive sufficient conditions for convergence of the distributed strategy, using
the framework of multilevel singular perturbation theory [30]. In Section 7, we will illustrate the
theoretical results on a set of representative numerical examples, confirming the effectiveness of the
approach.
0 500 1000 1500 2000
0
2
4
6
8
10
Spectrum
(L(w))
Time (t)
2
n
Fig. 4. Edge weights are adapted using a totally distributed method to
minimize n (highlighted in blue) whilst maintaining the constraint 2 2.
The initial network (inset) has n ⇡ 10.52 and 2 ⇡ 2.64 with unit edge
weights. At t = 2000 the network has decreased its spectral radius to
n ⇡ 5.66, whilst maintaining an algebraic connectivity of 2 ⇡ 1.99.
distributed estimation of 2 and n, but now all modes
excited with equal energy using additive white noise (d
is a vector of length n of independent Wiener processes
intensity 1).
The second PI consensus estimator ⇡a2 , za2 is used
estimate the variance of the vector a (it has expected m
of 0 due to deflation on the consensus mode), and this is
into a simple first order low pass filter with time cons
1
kL
to remove some of the noise. Each node can then m
a distributed estimate of the total effective graph resistan
b
⌦(i)
= 2k2n
✓
nyi
1
2k1
◆
(
To estimate the partial derivatives of the total effec
graph resistance, an extension to the system (25) is requi
Specifically, the following further equations need to be ad
to (25). For further details on this extended system see [2
0 5000 10000 15000 20000
10
12
14
16
18
20
22
Total
Effective
Graph
Resistance
⌦
0 5000 10000 15000 20000
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Edge
weights
(w)
Time (t)
Fig. 5. The resultant total effective graph resistance is controlled towards
a desired value of 10. Some stochasticity remains in the system, but the
[3
[4
[5
[6
[7
[8
[9
[10
[11
<latexit sha1_base64="A66RjX4aMAgKPH7KRBlQAyYZER8=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiRdWAUXBTfiQirYBzQhTCaTduhkEmYmQglZuHHjh7hxoYhbP8KdX+EvOH0g2npg4HDOPdy5x08YlcqyPo3C0vLK6lpxvbSxubW9Y+7utWWcCkxaOGax6PpIEkY5aSmqGOkmgqDIZ6TjD8/HfueWCEljfqNGCXEj1Oc0pBgpLXlm2QkFwpnDdCRA3lX+Q2u5Z1asqjUBXCT2jFQa9Yev06J32fTMDyeIcRoRrjBDUvZsK1FuhoSimJG85KSSJAgPUZ/0NOUoItLNJkfk8FArAQxjoR9XcKL+TmQoknIU+XoyQmog572x+J/XS1V44maUJ6kiHE8XhSmDKobjRmBABcGKjTRBWFD9V4gHSLeidG8lXYI9f/Iiadeq9nHVvtZtnIEpiqAMDsARsEEdNMAFaIIWwOAOPIJn8GLcG0/Gq/E2HS0Ys8w++APj/RvqNZsw</latexit>
N
2
Fig. 4. Edge weights are adapted using a totally distributed meth
minimize n (highlighted in blue) whilst maintaining the constraint 2
The initial network (inset) has n ⇡ 10.52 and 2 ⇡ 2.64 with unit
weights. At t = 2000 the network has decreased its spectral rad
n ⇡ 5.66, whilst maintaining an algebraic connectivity of 2 ⇡
Note that this is slightly in the infeasible region due to oscillation abo
solution.
matrix, the total effective graph resistance ⌦, given by:
⌦(w) = n
n
X
i=2
1
i(w)
We could estimate all n 1 eigenvalues by succe
deflation. For example, we could estimate 3 in a sim
manner to the algebraic connectivity estimator, but defl
on the consensus mode v1 and also on v2, using the esti
for v2 from Equation (8). This process could be conti
for all n 1 eigenvalues, but would rapidly become unw
for even modest sized networks.
Thus, if we want to provide a distributed estimator
a relatively few number of local variables at each
(that does not grow as the network becomes arbitr
large) another method must be employed. A solution

17. • Suppose we wish to drive the total graph resistance of an
arbitrary network to some desired value
• We could solve the problem by choosing
• But how to carry out the estimation?
Controlling the graph resistance
with
ode
rily
to
We
hite
ion
stic
25].
heir
tor,
.
otal
sult
e of
nal
we
{i,j}
Example 3: We now use these two stochastic systems
(25) and (27) with the filtered gradient estimation (28) to
illustrate a set point control problem. Given an arbitrary
connected undirected weighted network, we wish to devise
a distributed weight control law that will drive the total
effective resistance of the network ⌦ to a desired value. In
this example, we choose to drive ⌦ to 10, formulating the
unconstrained minimization problem:
minimize
w
(⌦(L(w)) 10)2
(29)
which will most likely have multiple solutions. Nevertheless
we can formulate the distributed weight control law:
ẇ{i,j} =
X
k2{i,j}
⇣
b
⌦(k)
10
⌘
@⌦
@w{i,j}
(k)
(30)
Figure 5 shows the edge weights in a randomly chosen
network of n = 8 nodes and m = 14 edges, adapting under
the control law to drive the total effective resistance of the
minimize n (highlighted in blue) whilst maintaining the constraint 2 2.
The initial network (inset) has n ⇡ 10.52 and 2 ⇡ 2.64 with unit edge
weights. At t = 2000 the network has decreased its spectral radius to
n ⇡ 5.66, whilst maintaining an algebraic connectivity of 2 ⇡ 1.99.
Note that this is slightly in the infeasible region due to oscillation about the
solution.
matrix, the total effective graph resistance ⌦, given by:
⌦(w) = n
n
X
i=2
1
i(w)
(24)
We could estimate all n 1 eigenvalues by successive
deflation. For example, we could estimate 3 in a similar
manner to the algebraic connectivity estimator, but deflating
on the consensus mode v1 and also on v2, using the estimate
for v2 from Equation (8). This process could be continued
for all n 1 eigenvalues, but would rapidly become unwieldy
for even modest sized networks.
Thus, if we want to provide a distributed estimator with
a relatively few number of local variables at each node
(that does not grow as the network becomes arbitrarily
large) another method must be employed. A solution is to
graph resistan
Specifically, t
to (25). For f
db
dc
d⇡b
dzb
Each node is
of the partial
by means of
@
Example 3
(25) and (27
illustrate a s
connected un
nwieldy
or with
h node
bitrarily
on is to
ise. We
f white
entation
ochastic
od [25].
to their
nerator,
ages.
he total
e result
noise of
ortional
uch we
@⌦
@w{i,j}
= 2k2q{i,j}
Example 3: We now use these two stochastic systems
(25) and (27) with the filtered gradient estimation (28) to
illustrate a set point control problem. Given an arbitrary
connected undirected weighted network, we wish to devise
a distributed weight control law that will drive the total
effective resistance of the network ⌦ to a desired value. In
this example, we choose to drive ⌦ to 10, formulating the
unconstrained minimization problem:
minimize
w
(⌦(L(w)) 10)2
(29)
which will most likely have multiple solutions. Nevertheless
we can formulate the distributed weight control law:
ẇ{i,j} =
X
k2{i,j}
⇣
b
⌦(k)
10
⌘
@⌦
@w{i,j}
(k)
(30)
Figure 5 shows the edge weights in a randomly chosen
network of n = 8 nodes and m = 14 edges, adapting under
the control law to drive the total effective resistance of the

18. • Note that this is now a function of all nonzero Laplacian
eigenvalues
• To solve this problem we used a stochastic approach
injecting noise in all network nodes
Estimation law
d with equal energy using additive white noise (dW
ector of length n of independent Wiener processes of
ity 1).
e second PI consensus estimator ⇡a2 , za2 is used to
ate the variance of the vector a (it has expected mean
ue to deflation on the consensus mode), and this is fed
simple first order low pass filter with time constant
remove some of the noise. Each node can then make
ributed estimate of the total effective graph resistance:
b
⌦(i)
= 2k2n
✓
nyi
1
2k1
◆
(26)
timate the partial derivatives of the total effective
resistance, an extension to the system (25) is required.
fically, the following further equations need to be added
). For further details on this extended system see [26].
db = ( k1⇡b k2Lc)dt + dW
dc = k (Lb c)dt
d⇡b = (k (b ⇡b) kP L⇡b + kILzb)dt
dzb = kIL⇡bdt
(27)
where we approximate the Ito integral of the Stochastic
Differential Equation (SDE) using the Milstein method [25].
Then we need only assume that each agent has access to their
own normally distributed pseudo-random number generator,
available in the vast majority of programming languages.
The SDE for making a distributed estimate of the total
effective graph resistance can be formulated using the result
that the linear consensus system with additive white noise of
intensisty has expected variance at long time, proportional
to the total effective graph resistance [6]. As such we
introduce the SDE [26]:
da = ( k1⇡a k2La)dt + dW
d⇡a = (k (a ⇡a) kP L⇡a + kILza)dt
dza = kIL⇡adt
d⇡a2 = (k (a2
⇡a2 ) kP L⇡a2 + kILza2 )dt
dza2 = kIL⇡a2 dt
dy = kL(⇡a2 y)dt (25)
This has a very similar form to the systems used for the
unconstraine
m
which will m
we can form
ẇ{i,
Figure 5 sh
network of n
the control l
network to th
An import
weights nev
inherently st
persistent ex
problem desc
so edge weig
Nevertheless
whose total

19. • Random network of 8 nodes and 14 links
Validation
0 5000 10000 15000 20000
10
12
14
16
18
20
22
Total
Effective
Graph
Resistance
⌦
0 5000 10000 15000 20000
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Edge
weights
(w)
Time (t)
Fig. 5. The resultant total effective graph resistance is controlled towards
a desired value of 10. Some stochasticity remains in the system, but the
[3] F. Com
J. Phys.
[4] L. Huan
of maste
Phys. R
[5] L. M.
synchro
[6] G. F. Y
consens
Control
[7] A. Gho
of a gra
[8] S. Boyd
Proc. In
[9] Y. Kim
eigenva
Control
[10] A. Simo
algebrai
2013.
[11] M. C.
connect

20. • We discussed the problem of
controlling complex systems
• Devising strategies at the
microscopic level able to induce
desired collective properties at the
macroscopic level
• As an example we looked at a self-
organizing strategy to control
functions of the Laplacian spectrum
• The crucial problem is how to close
the loop across these two scales to
achieve other desired emergent
properties..
Concluding Remarks

21. • Using tools from statistical mechanics such as mean-field
methods might be the key to close the loop across scales…
From ODEs to PDEs and back
Nikitin, D., de Wit, C.C. and Frasca, P., A continuation method for large-scale modeling and control: from ODEs to PDE, a round trip, IEEE TAC, 2021

22. • Finding ways to merge PDEs/Stochastic Control with
methods from statistical Physics to achieve the control of
large-scale complex systems
• But…
• Controlling some of the agents
in real-time does precisely that…
• Lots of opportunites for exciting
research
A crucial problem
“if particles could think Physics would be much harder”
M. Gell-Man (1969 Physics Nobel Laureate)

23. Acknowledgements
• Di Meglio, P. De Lellis, M. di Bernardo, "Decentralized gain adaptation for optimal pinning controllability of complex networks", IEEE Control System
Letters, 4(1), pp 253-258, 2019
• L. Kempton, G. Herrmann, M. di Bernardo, "Self-Organisation of Weighted Networks for Optimal Synchronizability", IEEE Transactions on Control of
Network Systems, 5(4), pp. 1541-1550, 2018
• L. Kempton, G. Herrmann, M. di Bernardo, "Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an
adaptive multi-layer strategy", International Journal of Robust and Nonlinear Control, 27(9), pp. 1499-1545, 2017
• M. Coraggio, P. De Lellis, M. di Bernardo, "Convergence and Synchronization in Networks of Piecewise-Smooth Systems via Distributed
Discontinuous Coupling", Automatica, 129, p. 109596, 2021
Pietro De Lellis Mario Coraggio
Anna Di Meglio
Louis Kempton

24. • As an application take the problem of controlling a group of
mobile agents so that a formation with high connectivity
emerges
Example
in the network is located at a position xi, with the collection
of all robots positions being denoted x. The strength of
communication between two neighboring robots i and j
is represented by the edge weight w{i,j}, which is a non-
increasing function of the distance between robots.
We aim to maximize the algebraic connectivity of the
state-dependent weighted graph Laplacian, whilst ensuring
that the network remains reasonably spaced and avoids
collisions, by enforcing an upper bound on the weighted
degree of each node, which is simply the diagonal elements
of the graph Laplacian li,i. Specifically we tackle the non-
convex optimization problem:
minimize
x
2(L(w(x)))
subject to li,i(x)  1 8i 2 1, . . . , n
(11)
Unlike the convex problem of maximising 2 over w [8], this
formulation is non-convex due to the fact that edge weights
are now functions of the positions xi of the robots. In this
example we choose to model this relation by an inverse-
square law reflecting the intensity of signal strength of a
radio transmitter, assuming that the antenna is radiating in
all directions equally in three dimensions,
w{i,j} = (x) =
1
||xi xj||2
2
(12)
Other monotonic non-increasing weight functions may be
used, for example sigmoids, ramps, and hyperbolae [12],
[10]. One advantage of using an inverse-square law is that
the edge weight grows unboundedly as the distance between
neighboring robots tends to zero. As the maximum weighted
degree of each robot is bounded, this provides an effective
are shown in Figure 2, clearly showing an increase
algebraic connectivity over time.
0 500 1000 1500 2000 2500 3000 3500
0.0
0.5
1.0
1.5
2.0
Spectrum
(L(w))
Time (t)
2
Fig. 2. The spectrum of the weighted graph Laplacian as the
evolves in time. It can be seen that the algebraic connectivity (hig
in red) is increased over time, however, in the limit, the network sys
into a persistent oscillation as 2 = 3 at the locally optimal pos
x x
y
Fig. 3. The network adapts from the initial positions (xi, yi
agents, chosen uniformly at random in the interval ([0, 20], [0, 20
A complete communication graph is chosen, with edge weights de
by the Euclidean distance between agents, and illustrated in the figu
the thickness of the edges. After a simulated 4000 seconds, th
degree of each node, which is simply the diagonal elements
of the graph Laplacian li,i. Specifically we tackle the non-
convex optimization problem:
minimize
x
2(L(w(x)))
subject to li,i(x)  1 8i 2 1, . . . , n
(11)
Unlike the convex problem of maximising 2 over w [8], this
formulation is non-convex due to the fact that edge weights
are now functions of the positions xi of the robots. In this
example we choose to model this relation by an inverse-
square law reflecting the intensity of signal strength of a
radio transmitter, assuming that the antenna is radiating in
all directions equally in three dimensions,
w{i,j} = (x) =
1
||xi xj||2
2
(12)
Other monotonic non-increasing weight functions may be
used, for example sigmoids, ramps, and hyperbolae [12],
[10]. One advantage of using an inverse-square law is that
the edge weight grows unboundedly as the distance between
neighboring robots tends to zero. As the maximum weighted
degree of each robot is bounded, this provides an effective
action for preventing collisions.
The distributed weight adaptation law is formulated ac-
cording to Equations (3,a) and (3,b), using local estimates
0 500 1000 1500 2000
0.0
0.5
Spect
Time (t
Fig. 2. The spectrum of the weighted
evolves in time. It can be seen that the a
in red) is increased over time, however, in
into a persistent oscillation as 2 = 3 a
x
y
Fig. 3. The network adapts from the
agents, chosen uniformly at random in th
A complete communication graph is chos
by the Euclidean distance between agents,
the thickness of the edges. After a sim
have arranged themselves into a locally o
value of 2(w(x, y)). For this specific p
themselves into a ring of 13, with a seco
(9)
of
he
10)
a
on
t i
on
of
j
on-
he
ng
ds
ed
nts
on-
11)
follows a system of 8 coupled ODEs. Crucially, this remains
fixed even for arbitrarily large networks, and the number of
variables in each ODE scales linearly with the number of
neighbors of each node. Therefore, the memory and compu-
tational requirements of each robot in the network scales
well even for large networks. The initial network shown
in Figure 3 evolves according to this networked system of
ODEs and results in a locally optimal formation. As the
robots’ positions change over time, the strength of the edge
weights change, resulting in a dynamically weighted graph
Laplacian L(t). The eigenvalues of this dynamic network
are shown in Figure 2, clearly showing an increase in the
algebraic connectivity over time.
0 500 1000 1500 2000 2500 3000 3500 4000
0.0
0.5
1.0
1.5
2.0
Spectrum
(L(w))
Time (t)
2
Fig. 2. The spectrum of the weighted graph Laplacian as the network
0 500 1000 1500 2000 2500 3000 3500 4000
0.0
0.5
1.0
1.5
2.0
Spectrum
(L(w))
Time (t)
2
Fig. 2. The spectrum of the weighted graph Laplacian as the network
evolves in time. It can be seen that the algebraic connectivity (highlighted
in red) is increased over time, however, in the limit, the network system falls
into a persistent oscillation as 2 = 3 at the locally optimal positions.
x x
y
Fig. 3. The network adapts from the initial positions (xi, yi) of 16
agents, chosen uniformly at random in the interval ([0, 20], [0, 20]) (left).
A complete communication graph is chosen, with edge weights determined