Quantifying the Controllability of Coarsely Characterized Networked Dynamical Systems

1 INTRODUCTION . In this paper we study controllability for networked dynamical systems when our knowledge of system structure is limited to coarse summaries . We are motivated by myriad real-world settings where system identification must be performed based upon measurements taken by low-resolution instruments unable to probe fine-scale structure . Our motivating example is the human brain . While efforts are under way to produce a canonical human brain map , our knowledge of the brain as an interconnected , network system is not yet to the level of the whole-brain individual neuron ( Betzel and Bassett , 2017 ) . And yet , motivated by emerging medical technologies , there are important control tasks we would like to tackle . For example , novel brain implants designed for epilepsy patients aim to “ steer ” the brain away from states that correspond to seizures ( Heck et al. , 2014 ; Muldoon et al. , 2016 ) . Our goal is to quantify the controllability of a fine-scale networked dynamical system given access only to coarse knowledge of network structure . Generally , without parametric structure , this is impossible . But real networks do have structure and so in our model we assume the fine-scale network has a connectivity induced by an underlying stochastic block model ( SBM ) . Approximation of high-dimensional ( fine-scale ) dynamical systems by lower-dimensional ( coarsescale ) ones is known as “ model order reduction ” ( MOR ) in the controls literature . There is a key difference in assumptions that differentiate our setting from that literature . In MOR the starting point is a complete description of the high-dimensional system . The task is to formulate a lower-dimension system , the dynamics of which well-approximate those of the full system . In contrast , we start from coarse summaries of the fine-scale system . We do not have access to the fine-scale dynamics and must exploit parametric knowledge ( via the assumption of a generative SBM ) . One might think of the distinction as akin to “ active ” versus “ passive ” MOR . Traditional MOR is active in that it actively decides how to coarsen the system to yield the best reduction . But for us , our knowledge is limited by the precision of our instrumental observations , so passively collected data is our starting point . Controllability is a function both of system dynamics and how we actuate the system ( Pasqualetti et al. , 2014 ; Yuan et al. , 2013 ) . Herein we assume that we both measure and actuate a system only coarsely . A control question we study is which coarse-level actuations are most “ influential ” in controlling the underlying fine-scale system . Such knowledge can assist with actuation selection ; e.g. , in our motivating epilepsy application , where best to position devices to be able to collapse the unstable brain-state oscillations that lead to seizures . ( O ’ Leary et al. , 2018 ; Pazhouhandeh et al. , 2019 ; Kassiri et al. , 2017 ; Shulyzki et al. , 2015 ) To accomplish our goal we characterize the average controllability of a vector of systems , each corresponding to a different coarse-scale actuation input . By comparing these vectors , and because these vectors well approximate the corresponding vectors for the fine-scale system , we aim ( in the long term ) to produce clinically-usable information for the neurologist . Contribution : Our work is the first of its kind that proposes a learning-based framework for inferring the controllability of fine networks from coarse measurements , and characterizes the mismatch between the controllability of the coarse and fine-scale networks . We study two approaches . 1 . In Section 5 , we build from MOR . We define an auxiliary , fictitious , reduced-order system based on the coarse data , and use the average controllability vector of this system to approximate that of the fine-scale system . We derive a tight upper bound on the “ approximationerror ” which is the sum of two terms . One term goes to zero as the coarse network size increases and the network becomes dense . The second term is a function of the synchronization between the coarse summary data and the underlying community structure . If synchronization is not sufficiently high , this term may not approach zero even as the network size increases . 2 . In Section 6 we learn the fine-scale system ’ s average controllability vector directly from the coarse data . This learning-based algorithm builds on the mixed-membership algorithm of Mao et al . ( 2017 ) for unsupervised learning of the parameters and the community structure of a SBM . We derive a tight upper bound on estimation error and characterize its convergence . Although the error bound implicitly depends on synchronization , unlike in the MOR-based approach , the error of this approach converges to zero as the coarse network size and its density increases . 2 BACKGROUND / RELATED WORK . Coarsened SBM as a generative process : The study of extracting community structure from coarse summaries is recent . The authors in ( Ghoroghchian et al. , 2021 ) used the stochastic block model ( SBM ) , developed in the community detection literature ( Abbe , 2017 ) , to lay out a framework for a coarsened and weighted variant of the SBM . We build off those results in this paper . The structure of many real-world networks , including brain networks is , at least empirically , known to have community structure across various spatial scales ( Sporns and Betzel , 2016 ; Pavlović et al. , 2020 ) . The SBM and its variants provide a powerful modeling framework to facilitate fundamental understanding of graph community organization and have found applications in many domains , including social and power networks . ( Dulac et al. , 2020 ; Funke and Becker , 2019 ; Abbe , 2017 ) . Complex networks controllability : The development of control methods for complex networks is a major effort in network science ( Scheid et al. , 2020 ) . Coupling traditional notions of controllability with graph theory reveals several insights into the role of network structure ( e.g. , presence of communities , diameter , and sparsity ) , size , and edge weight strength in controlling large-scale networks ( Wu-Yan et al. , 2018 ; Kim et al. , 2018 ; Constantino et al. , 2019 ; Sun , 2015 ) . Further one may want to understand which group of nodes , when actuated as inputs , can be used to steer the network to an arbitrary target state , and at what cost ( Cortesi et al. , 2014 ; Gu et al. , 2015 ) . Recent works in network neuroscience Gu et al . ( 2015 ) have popularized the notion of average controllability . This scalar metric associate a measure the relative control influence of a group of nodes . In this paper we consider a vector of such scalar measures to study the comparative influence of different sets of nodes . To the best of our knowledge ours is the first work that characterizes this type of error bounds for the controllability of coarse graphs . 3 PRELIMINARY NOTIONS . Notation : We denote vectors and matrices using bold faced small and upper case letters . The n dimensional all-ones and -zero vectors are denoted by 1n and 0n . For M = [ Muv ] ∈ Rn×m , define ‖M‖∞ = max1≤u≤n ∑m v=1 |Muv| ; ‖M‖max = maxu , v |Muv| ; and ‖M‖2 = √ λmax ( MTM ) . Let m = n , then define the spectral radius by ρ ( M ) = maxi { |λi| } ; diag ( M ) = [ M11 , . . . , Mnn ] T ∈ Rn ; and Diag ( M ) sets the off-diagonal entries of M to zero . For matrices M′is with arbitrary dimensions , BlkDiag ( M1 , . . . , Md ) denotes the block diagonal matrix . The inequality M1 ≤M2 implies element wise inequality . We write f ( n ) = O ( h ( n ) ) iff there exist positive reals c0 and n0 such that |f ( n ) | ≤ c0h ( n ) for all n ≥ n0 . The support of a vector , supp ( m ) , is the set of indices i such that mi 6= 0 . The cardinality of a set V is denoted by |V| . For a positive integer m , we denote [ m ] , { 1 , . . . , m } . 1 ( m ) returns a vector of same size with non-zero replaced by 1 . Networks : A network is defined by an un-directed graph G , ( V , E ) , where the node set V , { 1 , . . . , n } and edge set E ⊆ V × V . For an edge ( u , v ) ∈ E , assign the weight Auv = Avu ∈ R , and define the weighted symmetric adjacency matrix of G as A , [ Auv ] , where Auv = Auv = 0 whenever Auv /∈ E . A random network is an un-directed graph with a random adjacency matrix . 3.1 LINEAR DYNAMICAL SYSTEM ON RANDOM NETWORK . For a network G with n nodes and the symmetric adjacency matrix A , associate a state xi [ k ] ∈ R to the i-th node , and let the nodes evolve with the linear and time-invariant ( LTI ) dynamics 1 : x [ t+ 1 ] = 1 c · tr ( A ) Ax [ t ] + Bu [ t ] , ∀ t = 0 , 1 , . . . . ( 1 ) The state x [ t ] = [ x1 [ t ] , . . . , xn [ t ] ] T is steered to an arbitrary value by an input u [ t ] ∈ Rn . Here , the input matrix B = Diag ( b ) ∈ Rn×n , where b ∈ { 0 , 1 } n determines which components of u [ t ] enters the network2 . For e.g. , for B = Diag ( 1n1 ,0n−n1 ) , the input enters the network through control nodes set K = { 1 , . . . , n1 } . The normalization c · tr ( A ) factor , with appropriately chosen constant c > 0 , ensures that system in Eq . 1 is asymptotically stable . Finally , we define Anom , 1c·tr ( A ) A for the normalized matrix , and use this convention throughout the paper . For fixed system matrix Anom , a necessary and sufficient condition for the asymptotic stability3 of Eq . 1 is that ρ ( Anom ) ≤ 1 . For random Anom , we consider the probabilistic stability : P [ ρ ( Anom ) ≤ 1 ] —the greater the value , the greater the chance that Anom is stable . For SBM generated random symmetric matrices , we provide sharp non-asymptotic lower bounds on P [ ρ ( Anom ) ≤ 1 ] . The networked LTI system in Eq . 1 is T-step controllable if x [ 0 ] = 0 can be steered to any target state x ∈ Rn for some inputs : u [ 0 ] , . . . , u [ T − 1 ] . The T -step controllability Gramian of Eq . 1 given below , among other things , allows us to study if Eq . 1 is controllable or not . CT ( Anom , B ) = ∑T−1 t=0 ( Anom ) tBBT ( Anom ) t. ( 2 ) By definition CT ( Anom , B ) 0 , and it is well known that G with n nodes is T -step controllable if n-step controllable ; or equivalently , CT ( Anom , B ) 0 . For other interesting properties of Eq . 2 we refer to ( Chen , 1999 ) . For the simplicity of exposition , we let T →∞ and consider the infinite time horizon Gramian : C ( Anom , B ) = limT→∞ CT ( Anom , B ) , which exists with 1 − P [ ρ ( Anom ) ≥ 1 ] ; see also Pasqualetti et al . ( 2014 ) . We drop the notation ( Anom , B ) in C when the context is clear . Average energy : A widely used metric to measure how hard or easy it is to control the network is average energy : ∫ ‖x‖2=1 x TC†x dx/ ∫ ‖x‖2=1 dx , which evaluates to n −1tr ( C† ) ( Cortesi et al. , 2014 ) . Here , where C† is the pseudo inverse , and xTC†x is the minimum control energy needed to steer x [ 0 ] = 0 to an arbitrary target state x ∈ Rn . Thus , average energy measures the minimum control energy required to steer x [ 0 ] = 0 to an arbitrary state uniformly distributed over the unit sphere . 1 One may think our LTI model as the linearized system of an underlying non-linear system . Controllability of non-linear systems require a case by case analysis and we leave this topic for future research . 2Alternatively , Bu [ t ] = BKuK [ t ] , where BK is the sub-matrix of B whose columns are indexed byK ⊂ [ n ] . However , we stick with notation in Eq . 1 to make our analysis less cumbersome . 3The LTI system Eq . 1 is asymptotically stable if ‖x [ t ] ‖2 → 0 as t→∞ , for u [ t ] = 0 and x [ 0 ] 6= 0 . Average controllability : Numerical computation of C† for large-scale networks is demanding . Owing to the fact that tr ( C† ) ≥ 1/tr ( C ) , one uses tr ( C ) —called the average controllability—as a proxy for average energy Gu et al . ( 2015 ) . The higher the average controllability is for a given set of control nodes defined by B , the smaller their average energy , thus higher their influence on the network .

Summary: The paper studies the problem of controlling the dynamics of a networked dynamical system, under partial observations. Such systems arise in a number of applications, and therefore the control problem is important and well motivated. The authors consider a reduced order system from coarse data, and derive bounds on the convergence and approximation error for the original dynamical systems model. These are evaluated empirically on a small synthetic dataset