Graphon Control of Large-scale Networks of Linear Systems
To achieve control objectives for extremely complex and very large scale networks using standard methods is essentially intractable. In this work, we propose and develop a methodology for the approximate control of complex network systems by the use of graphon theory and the theory of infinite dimensional systems.
First, the graphon dynamical system models are formulated in an appropriate infinite dimensional space in order to represent arbitrary-size networks of linear dynamical systems, and to define the convergence of sequences of network systems with limits in the space. The exact controllability and the approximate controllability of graphon dynamical systems are then investigated.
Second, the minimum energy state-to-state control problem and the linear quadratic regulator (LQR) problem for systems on complex networks are considered. The control problem for the graphon limit system is solved in each case and the respective control laws are then approximated to obtain control laws for the finite network system. In each of the two cases, the convergence properties of the approximation schemes are established.
Finally, numerical examples of complex networks with randomly sampled weightings are presented to illustrate the effectiveness of the graphon control methodology.
Complex network systems such as Internet of Things (IoT), electric, neuronal, food web, epidemic, stock market and social networks, are ubiquitous, and the study of large scale networks has been the focus of much research over the recent past 20 years. In particular, researchers have been studying networks of interacting dynamical systems to learn which collective behaviours may emerge from system interactions over complex networks ([1, 2, 3, 4]). Furthermore, in addition to the structural properties of networks, system theoretic notions such as controllability, observability, consensus dynamics and synchronization have been widely applied ([5, 6, 7, 8, 9, 10, 11, 12, 13]). In fact, to achieve general control objectives for extremely complex and very large scale networks (henceforth, complex networks) using standard methods above is an intractable task.
Graphon theory, introduced and developed in recent years by L. Lovász, B. Szegedy, C. Borgs, J. T. Chayes, V. T. Sós, and K. Vesztergombi among others (see [14, 15, 16, 17, 18]), provides a theoretical tool to characterize complex graphs and graph limits. This work draws on graph theory, measure theory, probability, and functional analysis, and has been applied in different areas such as games [19, 20], signal processing  and crowd-sourcing .
We propose a graphon based control methodology for controlling complex network systems. The preliminary results are presented in [23, 24, 25]. The general graphon control strategy consists of the following steps:
Identify the graphon limit of the sequence of networks as the number of nodes goes to infinity.
Solve the corresponding control problem for the limit graphon dynamical system.
Approximate the control law for the limit system so as to generate approximate control laws for finite network systems.
Apply the resulting control laws to the networks of systems along the sequence .
In this paper, the minimum energy state-to-state control problem and the linear quadratic regulator problem are solved for complex network systems using the graphon control strategy. Main contributions of this work include:
the formulation of graphon differential equations and graphon dynamical systems, which form fundamental components of this study of arbitrary-size networks of linear systems.
the development of graphon state-to-state control methodology to solve state-to-state control problem on complex networks.
the proposed graphon linear quadratic regulation methodology to solve linear quadratic regulator problems for complex networks.
The paper is organized as follows: In Section II, the fundamentals of graphon theory are presented, followed by the development of the graphon unitary operator algebra and graphon differential equations. Section III introduces the network system model and and its equivalent representation by the graphon dynamical system. In Section IV, we study the properties of graphon dynamical systems, including existence and uniqueness of the solution and controllability. In Section V and Section VI, the graphon control strategies for state-to-state control problem and linear quadratic regulator problem are presented respectively. For each problem, the approximation method is developed with convergence properties proved. Section VII contains numerical examples to illustrate the effectiveness of the graphon control methodology.
Functions, graphons, and operators are represented by bold face letters to differentiate them from vectors, graphs, and matrices.
Ii-a Graphs, Adjacency Matrices and Pixel Pictures
The underlying structure of a network can be described by a graph specified by a vertex set and an edge set which represents the connections between vertices. An equivalent representation of a graph by a matrix called an adjacency matrix is defined to be the square matrix such that an element is one when there is an edge from vertex to vertex , and zero otherwise. If the graph is a weighted graph where edges are associated with weights, then the adjacency matrix has corresponding weighted elements.
Another representation of the adjacency matrix is given by a pixel diagram where the 0s are replaced by white squares and the 1s by black squares. The whole pixel diagram is presented in a unit square, so the square elements have sides of length , where is the number of vertices.
In the literature (see e.g. ), a meaningful convergence with respect to the cut metric is defined for sequences of dense and finite graphs. Graphons are then the limit objects of converging graph sequences. This concept is illustrated by a sequence of half graphs  represented by a sequence of pixel diagrams on the unit square converging to its limit in Fig. 2.
The set of finite graphs endowed with the cut metric gives rise to a metric space, and the completion of this space is the space of graphons. Graphons are represented by bounded symmetric Lebesgue measurable functions , which can be interpreted as weighted graphs on the vertex set . We note that in some papers, for instance , the word ”graphon” refers to symmetric, integrable functions from to . In this paper, unless stated otherwise, the term ”graphon” is used to refer to functions and denotes the space of graphons. Let represent the space of all graphons satisfying ; let denote the space of all symmetric measurable functions .
The cut norm of a graphon is then defined as
with the supremum taking over all measurable subsets and of . The inequalities between the different norms on a graphon are
Denote the set of measure preserving bijections from to by . The cut metric between two graphons and is then given by
where . We see that the cut metric is given by measuring the maximum discrepancy between the integrals of two graphons over measurable subsets of , then minimizing the maximum discrepancy over all possible measure preserving bijections.
Strictly speaking the cut metric is not a metric since the distance between two distinct graphons under the cut metric can be zero. (See [16, 27]). However, by identifying functions and for which , we can construct the metric space which denotes the image of under this identification. Similarly we construct from and from .
We define the metric for any graphons and as
and the metric as
similarly, we define the metric as
and the metric as
For any two graphons and the following inequalities hold immediately:
The (or ) metric and metric share the same equivalence classes under the measure preserving transformations [18, Corollary 8.14]. Hence the (or ) metric is also well defined on .
Ii-C Compactness of the Graphon Space
Theorem 1 ().
The space is compact.
This remains valid if is replaced by any uniformly bounded subset of closed in the cut metric .
Theorem 2 ().
The space is compact.
Sets in (or ) compact with respect to the metric are compact with respect to the cut metric. It follows immediately from (8) and Theorem 2 (or Theorem 1), if a graphon sequence is Cauchy in the metric then it is also a Cauchy sequence in the cut metric and under both metrics, the limits are identical in (or ).
Define the closed ball in with radius as .
Theorem 3 ().
The space with is compact.
By compactness, infinite sequences of graphons will necessarily possess one or more sub-sequential limits under the cut metric.
Henceforth we only consider the topology on the space of graphons. Consequently the convergence of a sequence of graphons will henceforth be interpreted as convergence in the complete space of graphons in the metric. By the ordering of metrics given in (8) this further implies convergence in the compact space (of equivalence classes) of graphons under in the weaker cut metric topology.
Examples of sets of graphons which have common limits in the metric and cut metric topologies are given by the so-called monotone families of graphons (see Appendix A). These correspond to graphs which recursively add nodes or edges at each of an infinite set of discrete time instants.
Ii-D Step Functions in the Graphon Space
Graphons generalize weighted graphs in the following sense. A function is called a step function if there is a partition of into measurable sets such that is constant on every product set . The sets are the steps of . For every weighted graph (on node set ), a step function is given as follows: partition into measurable sets of measure , then for and , we let , where denotes the node weight of node, and denotes the weight of the edge from node to node (i.e., is the entry in the adjacency matrix of ). Evidently the function depends on the labelling of the nodes of . We define the uniform partition of by setting and . Then , . Under the uniform partition, the step functions can be represented by the pixel diagram on the unit square. (See ).
Ii-E Graphons as Operators
A graphon can be interpreted as an operator The operation on is defined as follows:
The operator product is then defined by
where See  for more details.
Note that if and , then , since for all
Consequently, the power of an operator is defined as
with . is formally defined as the identity operator on functions in , but we note that is not a graphon.
Ii-F The Graphon Unitary Operator Algebra
It is evident that the operator composition defined in (10) above yields an operator algebra with a multiplicative binary operation possessing the associativity, left distributivity, right distributivity properties and compatibility with the scalar field , that is, for any in the vector space and
Thus we have an operator algebra over the field acting on elements of with operator multiplication as given in (9). By adjoining the identity element to the algebra (see e.g. ) we obtain a unitary algebra . The identity element is defined as follows: for any
where is the measure satisfying for all , and in particular .
The graphon unitary operator algebra will be used in the definition of the controllability Gramian and the input operator. More specifically, we use the subset where is the subset of that corresponds to .
Ii-G Graphon Differential Equations
Let be a Banach space. A linear operator is closed if is closed in the product space (see ). denotes the Banach algebra of all linear continuous mappings denotes the Banach space of equivalence classes of strongly measurable (in the Böchner sense) mappings that are -integrable, , with norm A mapping is said to be a strongly continuous semigroup on if the following properties hold:
for all , is continuous on .
A uniformly continuous semigroup is a strongly continuous semigroup such that with as the operator norm on a Banach space. The infinitesimal generator of a strongly continuous semigroup is the linear operator in defined by
Let be a graphon and hence a bounded and closed linear operator from to . Following , is the infinitesimal generator of the uniformly (hence strongly) continuous semigroup
Therefore, the initial value problem of the graphon differential equation
has a solution given by
Theorem 4 (Appendix C).
Let be a sequence of graphons such that in the metric. Then for all , as in the metric where the convergence is pointwise in time and uniform on any time interval .
Iii Network Systems and Their Limit Systems
Iii-a Network System Model
Consider an interlinked network of linear (symmetric) dynamical subsystems , each with an dimensional state space. The subsystem at the node in the network has interactions with specified as below:
with , the (symmetric) block-wise adjacency matrices of and of the input graph, where if has no connection to and similarity for . Then the (symmetric) linear dynamics for the network system can be represented by
where denotes the so called averaging operator given by . Let where For simplicity, we require the elements of and to be in for each (note that in general and have elements that are bounded real numbers for which case we would achieve similar results). In addition, we note that if we take the supremum norm on vectors in , i.e. , and the corresponding operator norm of , i.e. , then
Iii-B Network Systems Described by Step Functions
Let be a sequence of systems with the node averaging dynamics each of which is described according to (15). Let and for all . Let be the step functions corresponding one-to-one to and ; these are specified using the uniform partition of by the following matrix to step function mapping : for all ,
and similar for .
Define a piece-wise constant (PWC) function on to be any function of the form where are complex numbers and each is a bounded interval (open, closed, or half-open). Let denote the space of piece-wise constant functions under the uniform partition .
Let correspond one-to-one to via the following vector to PWC function mapping also denoted by : for all ,
and similarly correspond one-to-one to .
Iii-C Limits of Sequences of Network Systems
Now the sequence of network systems with the node averaging dynamics can be described by the sequence of step function operators as Let the graphon sequences and be Cauchy sequences of step functions in (under the same measure preserving bijection). Due to the completeness of , the respective graphon limits and exist and these will then necessarily be the limits in the cut metric (see ).
In fact, we can generalized the control input operator to , i.e., can consists of the identity operator part and the graphon part as .
Consider a sequence of systems . Decompose the input operator into the identity part and the graphon part as .
A sequence of systems is convergent if
there exist such that
there exist such that converges to in the metric, i.e. and under the same sequence of measure preserving bijections in the metric.
Then the limit system is represented by where . With an abuse of notation, in the following sections we use and to represent input operators in .
Iv The Limit Graphon System and Its Properties
Iv-a Limit Graphon Systems
We follow  and specialize the Hilbert space of states and the Hilbert space of controls appearing there to the space . We formulate an infinite dimensional linear system as follows:
where , , and hence bounded operators on , is the system state at time and is the control input at time .
Iv-B Uniqueness of the Solution
where the Hilbert space (control space) in the present case is .
The graphon system in (19) has a unique solution for all and all .
Since as a graphon operator generates a uniformly continuous semigroup, is satisfied. Moreover as a linear operator is bounded and hence is a continous linear mapping from control space to the state space satisfying . Therefore, is satisfied and following  the system (19) has a unique solution for all and all . ∎
A system is exactly controllable on if for any initial state and any target state , there exists a control driving the system from to , i.e. with .
A system is approximately controllable on if for any initial state , any target state and any , there exists a control driving the system from to points in the state space within a -distance from , i.e.
The controllability Gramian operator is defined as
A necessary and sufficient condition for exact controllability on is the uniform positive definiteness of :
for all , where and is the norm (see [29, 31]). The positive definiteness of the controllability Gramian operator as a kernel is equivalent to the approximate controllability of the corresponding system (see [29, 31]).
Theorem 6 (Appendix D).
Let be a graphon in and let be a bounded linear operator. The linear system is exactly controllable if and only if all eigenvalues of are lower bounded by a positive constant .
Proposition 1 (Appendix D).
Let be a graphon in and let be a bounded linear operator. Then exactly controllable implies is a non-compact operator.
V Graphon State-to-state Control of Large-scale Networks
V-a Approximation of Input Functions via Piece-wise Constant Functions
The following basic result will be employed in the analysis.
Theorem 7 ( p.198).
Let be any measure on and let . Then piece-wise constant functions on form a dense subset of .
Piece-wise constant functions can approximate functions arbitrarily. In this paper we wish to approximate the control input , through a piece-wise constant function in denoted by . Specifically, the approximation of an input via the function with the partition of is given as follows: for all ,
where denotes the measure of .
V-B Limit Control for Network Systems with General Graphon Input Mappings
Theorem 8 (Appendix E).
Consider a sequence of network systems converging to a graphon system in the following sense: in the metric and in the metric as . Consider the problem of driving the systems from the origin to some target state. Then for any :
there exists a control for approximating the control for such that
furthermore, for any there exists such that each ,
where represents the terminal state of under control , represents the terminal state of under control , and the control approximation is given in the following: for all , , with the uniform partition .
According to the mapping, the control law for the finite network system is given by
V-C Limit Control for Network Systems with the Identity Input Mapping
In general, the control input mapping is not limited to be a graphon mapping. As long as the control input map is a continous mapping from to , the existence and uniqueness of solutions are guaranteed. Since the identity operator is a continous mapping from to , the system has a unique solution. We note that while the identity operator may be represented by a positive measure on the diagonal in and hence may formally be treated as an element of , it is not an element of and hence not in .
Consider a sequence of finite dimensional system with node averaging dynamics and as its equivalent step function system sequence according to (16).
Theorem 9 (Appendix E).
Suppose in the metric as . Consider the problem of driving the systems from the origin to some target state. Then for each , there exists a control for approximating the control for such that
where , represents the terminal state of under control , represents the terminal state of under control , and the control approximation is given by for all , with the uniform partition . Furthermore, for any , there exists such that for each ,
V-D The Graphon State-to-state Control (GSSC) Strategy
Consider the control problem of steering the states of each member of to each of a sequence of desired states . The Graphon State-to-state Control (GSSC) Strategy consists of four steps:
Let be the sequence of graphon dynamical systems equivalent to under the mapping and assume that it converges to the graphon system . Let be the image of under , which is assumed to converge to some in the norm.
Specify the corresponding state to state control problem for with as the target terminal state and choose a tolerance .
Find a control law solving .