Controlling COVID-19 transmission with isolation of influential nodes

Controllability scores for selecting control nodes of large-scale network systems

“…Using the duality of controllability and observability [30], we can define observability scores, which express the importance of each state node for observing the whole state, by considering ẋ(t) = Ax(t) y(t) = Cx(t) with a diagonal matrix C ∈ R n×n as well as the controllability scores using system (1). That is, our approach provides a novel method for determining sensor placements based on the concept of observability [23], [40]- [42].…”

“…In this section, we summarize the concepts of controllability ellipsoid and average of minimum energy defined by using the controllability Gramian of system (1). Moreover, we introduce average and volumetric control energy centralities proposed in [19] to compare with novel centralities in this paper.…”

“…. , n) so that a volume of controllability ellipsoid E c is made as large as possible, under given a matrix A of system (1). Because each input node has a one-to-one correspondence with each state node, p * contains information on the importance of the controllability of each state node.…”

“…To control large-scale network systems such as infection networks [1], [2] and brain networks [3], the selection of control nodes is a crucial step and seriously affects the system performance. That is, when we want to control a large-scale linear system ẋ(t) = Ax(t) with state x(t) ∈ R n and constant matrix A ∈ R n×n , it is important to appropriately design a matrix B ∈ R n×m with some restrictions such that desired properties of ẋ(t) = Ax(t) + Bu(t) (1) are satisfied. For the reason, the selection problems have been extensively studied from the various perspectives as follows.…”

“…denotes the standard vector which has 1 at i-th position and zeros at other positions. Moreover, the authors in [19], [20] formulated the selection problems as discrete optimization problems which choose a column vector of B in (1) to optimize the metrics from a given candidate set. Furthermore, [16], [17], [20] have studied the problems using continuous optimization approaches.…”