{\sigma}\big{(}\bm{G}(\text{j}w)\big{)},$ (S26)

where ${\sigma}(\cdot)$ is the largest singular value and
$\mathcal{H}_{\infty}$ denotes the associated normed space of all linear maps
[3]. Furthermore, the subspaces of $\mathcal{H}_{2}$ and
$\mathcal{H}_{\infty}$ consisting of proper rational matrix functions (i.e.,
matrices whose elements are rational functions in the complex variable $s$)
are denoted by $\mathcal{RH}_{2}$ and $\mathcal{RH}_{\infty}$, respectively.
In addition, the strictly proper rational function subspaces of
$\mathcal{RH}_{2}$ and $\mathcal{RH}_{\infty}$ are denoted by
$\frac{1}{s}\mathcal{RH}_{2}$ and $\frac{1}{s}\mathcal{RH}_{\infty}$,
respectively.

By the Laplace transform, the optimal control problem in Eq. 9 of the main
text,

$\displaystyle\underset{\bm{u}\in L_{2}[0,+\infty)}{\text{min}}\
J=\int_{0}^{\infty}\bm{x}(\tau)^{T}\bm{Q}\bm{x}(\tau)+\bm{u}(\tau)^{T}\bm{R}\bm{u}(\tau)d\tau$
(S27) $\displaystyle\text{s.t. }\dot{\bm{x}}=\bm{C}\bm{x}+\bm{B}\bm{u},\
\bm{x}(0)=\bm{x}_{0},$

can be equivalently written in the $s$-domain as

$\displaystyle\underset{\bm{u}\in\mathcal{H}_{2}}{\text{min}}\
\left\lVert\begin{bmatrix}\bm{Q}^{1/2}&\\\
&\bm{R}^{1/2}\end{bmatrix}\begin{bmatrix}\bm{x}(s)\\\
\bm{u}(s)\end{bmatrix}\right\rVert_{\mathcal{H}_{2}}^{2}$ (S28)
$\displaystyle\text{s.t.}\begin{array}[]{l}\bm{x}(s)=(s\bm{I}-\bm{C})^{-1}(\bm{B}\bm{u}(s)+\bm{x}_{0}).\end{array}$

Given a feedback controller $\bm{K}(s)$, the transfer function from the
initial state $\bm{x}_{0}$ to the state response $\bm{x}(s)$ is given by
$\bm{\Phi}(s)=\big{(}s\bm{I}-\bm{C}-\bm{B}\bm{K}(s)\big{)}^{-1}$, and the
response of the controller output $\bm{u}(s)$ is given by the transfer
function
$\bm{H}(s)=\bm{K}(s)\big{(}s\bm{I}-\bm{C}-\bm{B}\bm{K}(s)\big{)}^{-1}$. The
parameterization of all $\bm{\Phi}(s)$ and $\bm{H}(s)$ that are achievable by
some stabilizing controller $\bm{K}(s)$ is given by the following theorem.

######  Theorem 1 (System-Level Parameterization for State Feedback Systems
[18])

The following are true for any linear system
$\dot{\bm{x}}=\bm{C}\bm{x}+\bm{B}\bm{u}$: 1) The affine space defined by

$\begin{bmatrix}s\bm{I}-\bm{C}&-\bm{B}\end{bmatrix}\begin{bmatrix}\bm{\Phi}(s)\\\
\bm{H}(s)\end{bmatrix}=\bm{I},\ \
\bm{\Phi},\bm{H}\in\frac{1}{s}\mathcal{RH}_{\infty}$ (S29)

parameterizes all system responses from $\bm{x}_{0}$ to $\bm{x}(s)$ and
control responses from $\bm{x}_{0}$ to $\bm{u}(s)$ that are achievable by an
internally stabilizing state feedback controller; 2) For all transfer matrices
$\\{\bm{\Phi}(s),\bm{H}(s)\\}$ satisfying Eq. S29, the controller
$\bm{K}(s)=\bm{H}(s)\bm{\Phi}(s)^{-1}$ is internally stabilizing and achieves
the desired responses $\bm{x}(s)=\bm{\Phi}(s)\bm{x}_{0}$ and
$\bm{u}(s)=\bm{H}(s)\bm{x}_{0}$.

This theorem shows that there is a bijection between the stabilizing
controllers and the responses in the affine space defined by Eq. S29.
Therefore, instead of directly designing the feedback control law $\bm{K}(s)$,
we can equivalently design the transfer matrices $\bm{\Phi}(s)$ and
$\bm{H}(s)$ under the affine constraint given by Eq. S29. With this
parameterization, the optimal control problem can be rewritten as

$\displaystyle\underset{\bm{\Phi},\bm{H}\in\frac{1}{s}\mathcal{RH}_{\infty}}{\text{min}}\
J=\left\lVert\begin{bmatrix}\bm{Q}^{1/2}&\\\
&\bm{R}^{1/2}\end{bmatrix}\begin{bmatrix}\bm{\Phi}(s)\\\
\bm{H}(s)\end{bmatrix}\bm{x}_{0}\right\rVert_{\mathcal{H}_{2}}^{2}$ (S30)
$\displaystyle\text{s.t.}\begin{array}[]{l}\begin{bmatrix}s\bm{I}-\bm{C}&-\bm{B}\end{bmatrix}\begin{bmatrix}\bm{\Phi}(s)\\\
\bm{H}(s)\end{bmatrix}=\bm{I}.\end{array}$

The solution of this problem for any given $\bm{x}_{0}$ is given by

$\bm{\Phi}(s)=(s\bm{I}-\bm{C}+\bm{B}\bm{R}^{-1}\bm{B}^{T}\bm{P})^{-1},$ (S31)
$\bm{H}(s)=-\bm{R}^{-1}\bm{B}^{T}\bm{P}(s\bm{I}-\bm{C}+\bm{B}\bm{R}^{-1}\bm{B}^{T}\bm{P})^{-1},$
(S32)

where $\bm{P}$ is the solution of the Riccati equation, Eq. S7. The optimal
controller thus becomes a static feedback law and is given by

$\bm{K}(s)=\bm{H}(s)\bm{\Phi}(s)^{-1}=-\bm{R}^{-1}\bm{B}^{T}\bm{P}.$ (S33)

We note that $\bm{\Phi}(s)$ and $\bm{H}(s)$ in Eqs. S31 and S32, respectively,
are independent of $\bm{x}_{0}$ and hence serve as the “fundamental solution”
of the optimal control problem in Eq. S30 for arbitrary initial conditions.
The columns of $\bm{\Phi}(s)$ and $\bm{H}(s)$ can be computed independently.
To see this, we set $\bm{x}_{0}=\bm{e}_{j}$ in Eq. S30, rewrite the problem as
the minimization over $\bm{\phi}_{j}(s)=\bm{\Phi}(s)\bm{e}_{j}$ and
$\bm{h}_{j}(s)=\bm{H}(s)\bm{e}_{j}$, and eliminate the unnecessary constraints
corresponding to all but the $j$th columns of $\bm{\Phi}(s)$ and $\bm{H}(s)$.
This leads to Eq. 10 in the main text.

##  S7 Disturbance-Oriented Localization

By projecting the original optimal control problem in Eq. 10 of the main text
onto the information neighborhood $\mathcal{N}_{j}$ of node $j$, we have the
following projected optimization problem:

$\displaystyle\underset{\widetilde{\bm{\phi}}_{j},\widetilde{\bm{h}}_{j}\in\frac{1}{s}\mathcal{RH}_{\infty}}{\text{min}}\
\left\lVert\begin{bmatrix}\widetilde{\bm{Q}}^{1/2}_{j}&\\\
&\widetilde{\bm{R}}^{1/2}_{j}\end{bmatrix}\begin{bmatrix}\widetilde{\bm{\phi}}_{j}(s)\\\
\widetilde{\bm{h}}_{j}(s)\end{bmatrix}\widetilde{\bm{e}}_{j}^{T}\bm{x}_{0}\right\rVert_{\mathcal{H}_{2}}^{2}$
(S34)
$\displaystyle\text{s.t.}\begin{array}[]{l}\begin{bmatrix}s\bm{I}-\widetilde{\bm{C}}_{j}&-\widetilde{\bm{B}}_{j}\end{bmatrix}\begin{bmatrix}\widetilde{\bm{\phi}}_{j}(s)\\\
\widetilde{\bm{h}}_{j}(s)\end{bmatrix}=\widetilde{\bm{e}}_{j}.\end{array}$

To solve this projected problem, we invoke the inverse Laplace transform to go
back to the time domain, which transforms the problem in Eq. S34 into

$\displaystyle\underset{\widetilde{\bm{u}}_{j}\in
L_{2}[0,+\infty)}{\text{min}}\
\widetilde{J}=\int_{0}^{\infty}\big{[}\widetilde{\bm{x}}_{j}(\tau)^{T}\widetilde{\bm{Q}}_{j}\widetilde{\bm{x}}_{j}(\tau)+\widetilde{\bm{u}}_{j}(\tau)^{T}\widetilde{\bm{R}}_{j}\widetilde{\bm{u}}_{j}(\tau)\big{]}d\tau$
(S35)
$\displaystyle\text{s.t.}\begin{array}[]{l}\dot{\widetilde{\bm{x}}}_{j}=\widetilde{\bm{C}}\widetilde{\bm{x}}_{j}+\widetilde{\bm{B}}\widetilde{\bm{u}}_{j},\
\widetilde{\bm{x}}_{j}(0)=\widetilde{\bm{e}}_{j}^{T}\bm{x}_{0}.\end{array}$

This is simply a projected version of the optimal control problem defined by
Eq. S27. The optimal control law is given by the static feedback matrix

$\widetilde{\bm{K}}_{j}=-\widetilde{\bm{R}}_{j}^{-1}\widetilde{\bm{B}}_{j}^{T}\widetilde{\bm{P}}_{j},$
(S36)

where $\widetilde{\bm{P}}_{j}$ is the solution to the Riccati equation

$\widetilde{\bm{C}}_{j}^{T}\widetilde{\bm{P}}_{j}+\widetilde{\bm{P}}_{j}\widetilde{\bm{C}}_{j}-\widetilde{\bm{P}}_{j}\widetilde{\bm{B}}_{j}\widetilde{\bm{R}}_{j}^{-1}\widetilde{\bm{B}}_{j}^{T}\widetilde{\bm{P}}_{j}+\widetilde{\bm{Q}}_{j}=\bm{0}.$
(S37)

The $s$-domain solution corresponding to the problem in Eq. S34 is then given
by

$\widetilde{\bm{\phi}}_{j}(s)=(s\bm{I}-\widetilde{\bm{C}}_{j}-\widetilde{\bm{B}}_{j}\widetilde{\bm{K}}_{j})^{-1}\widetilde{\bm{e}}_{j},$
(S38)
$\widetilde{\bm{h}}_{j}(s)=\widetilde{\bm{K}}_{j}\widetilde{\bm{\phi}}_{j}(s).$
(S39)

After solving the projected problem for all $1\leq j\leq N$, we can construct

$\widetilde{\bm{\Phi}}(s)=\begin{bmatrix}{\bm{N}}_{1}^{T}\widetilde{\bm{\phi}}_{1}(s)&{\bm{N}}_{2}^{T}\widetilde{\bm{\phi}}_{2}(s)&\cdots&{\bm{N}}_{n}^{T}\widetilde{\bm{\phi}}_{n}(s)\end{bmatrix},$
(S40)
$\widetilde{\bm{H}}(s)=\begin{bmatrix}{\bm{T}}_{1}^{T}\widetilde{\bm{h}}_{1}(s)&{\bm{T}}_{2}^{T}\widetilde{\bm{h}}_{2}(s)&\cdots&{\bm{T}}_{n}^{T}\widetilde{\bm{h}}_{n}(s)\end{bmatrix}.$
(S41)

From Theorem 1, it then follows that the overall optimal control law takes the
form

$\bm{u}(s)=\widetilde{\bm{K}}(s)\bm{x}(s)=\widetilde{\bm{H}}(s)\widetilde{\bm{\Phi}}(s)^{-1}\bm{x}(s).$
(S42)

Crucially, we now show that the controller $\widetilde{K}(s)$ is guaranteed to
be stabilizing and its associated control objective value approaches that of
the global controller when the system is sufficiently localized. That is, the
controller designed based on the projected model, $\widetilde{\bm{K}}(s)$,
will enjoy certain stability and near-optimal performance guarantees when
implemented on the original system. Due to the locality of the system, in view
of Eq. S29, the following equality shows that the projected problem can be
regarded as a perturbed version of the original one:

$\displaystyle(s\bm{I}-\bm{C})\bm{N}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s)-\bm{B}\widetilde{\bm{T}}_{j}^{T}\widetilde{\bm{h}}_{j}(s)-\bm{e}_{j}=(\bm{I}-\bm{N}_{j}^{T}\bm{N}_{j})(-\bm{C})\bm{N}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s):=\bm{\epsilon}_{j}(s).$
(S43)

The perturbation term $\bm{\epsilon}_{j}(s)$ is expected to be very small in
magnitude when $\mathcal{N}_{j}$ is sufficiently large. This is the case
because: 1) the non-zeros elements of vector
$\bm{N}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s)$ are limited to those
corresponding to the neighbourhood $\mathcal{N}_{j}$, and their magnitude
decays as $\mathcal{O}\big{(}v(\tau)^{-1}\big{)}$ with the information
distance $\tau$; 2) the multiplication by $-\bm{C}$ preserves the decay
pattern due to the locality of the system; and 3) the further multiplication
by $(\bm{I}-\bm{N}_{j}^{T}{\bm{N}}_{j})$ turns the elements inside the
information neighborhood ${\mathcal{N}}_{j}$ into zeros and leaves only
negligible elements outside ${\mathcal{N}}_{j}$. That is, the solution of the
projected problem (Eq. S34 with mismatch $\bm{\epsilon}_{j}(s)$) when lifted
to the original space is an approximate solution of the original problem in
Eq. S30. Concatenating Eq. S43 for all $1\leq j\leq n$ and accounting for Eq.
S40 and Eq. S41, we have

$(s\bm{I}-\bm{C})\widetilde{\bm{\Phi}}(s)-\bm{B}\widetilde{\bm{H}}(s)=\bm{I}+\bm{\Sigma}(s),$
(S44)

where
$\bm{\Sigma}(s)=[\bm{\epsilon}_{1}(s),\bm{\epsilon}_{2}(s),\cdots,\bm{\epsilon}_{n}(s)]$.
If the perturbation $\bm{\Sigma}(s)$ is sufficiently small such that

$(\bm{I}+\bm{\Sigma}(s))^{-1}\in\frac{1}{s}\mathcal{RH}_{\infty},$ (S45)

then the matrices

$\widetilde{\bm{\Phi}}^{\prime}(s)=\widetilde{\bm{\Phi}}(s)\big{(}\bm{I}+\bm{\Sigma}(s)\big{)}^{-1},$
(S46)
$\widetilde{\bm{H}}^{\prime}(s)=\widetilde{\bm{H}}(s)\big{(}I+\bm{\Sigma}(s)\big{)}^{-1}$
(S47)

would satisfy exactly the condition given by Eq. S29 in Theorem 1. Hence, Eqs.
S46 and S47 constitute achievable system and control responses with the
controller
$\widetilde{\bm{K}}(s)=\widetilde{\bm{H}}^{\prime}(s)\widetilde{\bm{\Phi}}^{\prime}(s)^{-1}=\widetilde{\bm{H}}(s)\widetilde{\bm{\Phi}}(s)^{-1}$,
which stabilizes the original system even though this controller is designed
based on the projected model.

A sufficient condition for Eq. S45, according to the small-gain theorem [19,
20], is given by any of the following:

$\left\lVert\bm{\Sigma}\right\rVert_{\mathcal{H}_{\infty}}<1,$ (S48)
$\left\lVert\bm{\Sigma}\right\rVert_{\mathcal{L}_{1}}<1,$ (S49)
$\left\lVert\bm{\Sigma}^{T}\right\rVert_{\mathcal{L}_{1}}<1,$ (S50)

where $\left\lVert\cdot\right\rVert_{\mathcal{L}_{1}}$ denotes the
$\mathcal{L}_{1}$ system norm. The $\mathcal{L}_{1}$ system norm is defined
through the impulse response in the time domain [21] as

$\left\lVert\bm{\Sigma}^{T}(t)\right\rVert_{\mathcal{L}_{1}}=\underset{1\leq
j\leq
n}{\text{max}}\sum_{i=1}^{n}\int_{0}^{\infty}\lvert\epsilon_{ij}(t)\rvert dt,$
(S51)

where $\epsilon_{ij}(t)$ is the $i$th component of the perturbation (column)
vector $\bm{\epsilon}_{j}(s)$ in Eq. S43 under inverse Laplace transform. The
computation required to verify the small-gain condition in Eq. S50 can be
distributed over the columns of the impulse response matrix $\bm{\Sigma}(t)$,
since Eq. S50 is equivalent to

$\sum_{i=1}^{n}\int_{0}^{\infty}\lvert\epsilon_{ij}(t)\rvert dt<1,\ \ \forall\
1\leq j\leq n.$ (S52)

Once the controller defined by Eq. S42 stabilizes the system, the next
question is to determine the extent to which the dynamical performance of this
controller compares with that of the theoretical global optimal control. Let
$J^{*}$ be the optimal objective value for the original optimal control
problem in Eq. S30, and define

$\widetilde{J}^{*}=\left\lVert\bm{Q}^{1/2}\widetilde{\bm{\Phi}}(s)\right\rVert_{\mathcal{H}_{2}}+\left\lVert\bm{R}^{1/2}\widetilde{\bm{H}}(s)\right\rVert_{\mathcal{H}_{2}},$
(S53)

i.e., the optimal objective value obtained from the solutions of the projected
problems in Eq. S34. Since $\widetilde{\bm{\Phi}}^{\prime}(s)$ and
$\widetilde{\bm{H}}^{\prime}(s)$ form a feasible solution of the optimal
control problem in Eq. S30, we have an upper bound for $J^{*}$:

$\displaystyle J^{*}$
$\displaystyle\leq\left\lVert\bm{Q}^{1/2}\widetilde{\bm{\Phi}}^{\prime}(s)\right\rVert_{\mathcal{H}_{2}}+\left\lVert\bm{R}^{1/2}\widetilde{\bm{H}}^{\prime}(s)\right\rVert_{\mathcal{H}_{2}}$
(S54)
$\displaystyle\leq\left\lVert\bm{Q}^{1/2}\widetilde{\bm{\Phi}}(s)\right\rVert_{\mathcal{H}_{2}}\left\lVert\bm{I}+\bm{\Sigma}(s)\right\rVert_{\mathcal{H}_{\infty}}+\left\lVert\bm{R}^{1/2}\widetilde{\bm{H}}(s)\right\rVert_{\mathcal{H}_{2}}\left\lVert\bm{I}+\bm{\Sigma}(s)\right\rVert_{\mathcal{H}_{\infty}}$
$\displaystyle\leq\widetilde{J}^{*}\left\lVert\bm{I}+\bm{\Sigma}(s)\right\rVert_{\mathcal{H}_{\infty}}$
$\displaystyle\leq\frac{\widetilde{J}^{*}}{1-\left\lVert\bm{\Sigma}(s)\right\rVert_{\mathcal{H}_{\infty}}}.$

To derive a lower bound for $J^{*}$, we interchange the roles of the original
and projected systems in Eq. S43 and obtain

$\displaystyle(s\bm{I}-{\bm{C}})\bm{N}_{j}{\bm{\phi}}_{j}(s)-{\bm{B}}{\bm{T}}_{j}{\bm{h}}_{j}(s)-\widetilde{\bm{e}}_{j}=-\bm{N}_{j}\bm{C}(\bm{N}_{j}^{T}\bm{N}_{j}-\bm{I}){\bm{\phi}}_{j}(s):=\bm{\xi}_{j}(s),$
(S55)

which implies

$(s\bm{I}-{\bm{C}}){\bm{\Phi}}(s)-{\bm{B}}{\bm{H}}(s)=\bm{I}+\bm{\Xi}(s),$
(S56)

where we define $\Xi(s)=[\xi_{1}(s),\xi_{2}(s),\cdots,\xi_{N}(s)]$. Due to the
locality, the perturbation term can be small enough such that
$(\bm{I}+\bm{\Xi}(s))^{-1}\in\frac{1}{s}\mathcal{RH}_{\infty}$, which is
guaranteed by any of the inequalities in Eqs. S48–S50. Under this condition,
${\bm{\Phi}}(s)(\bm{I}+\bm{\Xi}(s))^{-1}$ and
${\bm{H}}(s)(\bm{I}+\bm{\Xi}(s))^{-1}$ form a feasible solution of the
projected problem in Eq. S34. This implies

$\displaystyle\widetilde{J}^{*}$
$\displaystyle\leq\left\lVert\bm{Q}^{1/2}{\bm{\Phi}}(s)(\bm{I}+\bm{\Xi}(s))^{-1}\right\rVert_{\mathcal{H}_{2}}+\left\lVert\bm{R}^{1/2}{\bm{H}}(s)(\bm{I}+\bm{\Xi}(s))^{-1}\right\rVert_{\mathcal{H}_{2}}$
(S57)
$\displaystyle\leq\big{(}\left\lVert\bm{Q}^{1/2}{\bm{\Phi}}(s)\right\rVert_{\mathcal{H}_{2}}+\left\lVert\bm{R}^{1/2}{\bm{H}}(s)\right\rVert_{\mathcal{H}_{2}}\big{)}\left\lVert(\bm{I}+\bm{\Xi}(s))^{-1}\right\rVert_{\mathcal{H}_{\infty}}$
$\displaystyle=J^{*}\left\lVert(\bm{I}+\bm{\Xi}(s))^{-1}\right\rVert_{\mathcal{H}_{\infty}}$
$\displaystyle\leq\frac{{J}^{*}}{1-\left\lVert\bm{\Xi}(s)\right\rVert_{\mathcal{H}_{\infty}}}.$

Combining the inequalities in Eqs. S54 and S57, we have

$\widetilde{J}^{*}(1-\left\lVert\bm{\Xi}(s)\right\rVert_{\mathcal{H}_{\infty}})\leq{J}^{*}\leq\frac{\widetilde{J}^{*}}{1-\left\lVert\bm{\Sigma}(s)\right\rVert_{\mathcal{H}_{\infty}}}.$
(S58)

This shows that, when the system is more localized,
$\widetilde{J}^{*}\rightarrow J^{*}$ as $\bm{\Sigma}(s)\rightarrow 0$ and
$\bm{\Xi}(s)\rightarrow 0$. Thus, when the residual terms $\bm{\Sigma}(s)$ and
$\bm{\Xi}(s)$ are small in magnitude, the optimal objective value for the
projected problem is close to that of the original problem, meaning that
solving the projected problem provides a near-optimal solution to the original
problem.

##  S8 Controller-Oriented Localization

Although the controller in Eq. S42 obtained by the disturbance-oriented
localization is guaranteed to be stabilizing and near-optimal, this controller
is a dynamic feedback controller, whereas the global optimal controller given
by Eq. S33 is static. As a consequence, the controller given by Eq. S42 is a
dynamical system for each driver node with the same state-space dimension as
the entire network system, which makes the controller impractical for large
networks. Here, we show how to convert the dynamical controller into static
ones that are scalable to large networks.

To understand why the localized solutions in Eqs. S40 and S41 lead to
dynamical controllers and how they can be converted into static ones, we take
the viewpoint of the controller installed at each node individually. Recall
that the projected problem in Eq. S34 at node $j$ is obtained by projecting
the original problem onto the information neighborhood $\mathcal{N}_{j}$. Also
recall that, since the controller at node $i$ responds to disturbances at all
node $j$ for which $i\in\mathcal{N}_{j}$, we define the set of all such nodes
$j$ as the control neighborhood of $i$:

$\mathcal{C}_{i}=\\{1\leq j\leq N\ |\ i\in\mathcal{N}_{j}\\}.$ (S59)

By using the projection matrix $\bm{f}_{i}^{T}$ defined in the main text, the
controller installed at the $i$th node is
$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s)$, which is a linear map from the entire
state space to the input of that node. The nonzero columns of
$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s)$ correspond to nodes belonging to
$\mathcal{C}_{i}$. From Eq. S42, we have

$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s)\widetilde{\bm{\Phi}}(s)=\bm{f}_{i}^{T}\widetilde{\bm{H}}(s),$
(S60)

which is equivalent to

$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s){\bm{N}}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s)=\bm{f}_{i}^{T}{\bm{T}}_{j}^{T}\widetilde{\bm{K}}_{j}\widetilde{\bm{\phi}}_{j}(s),\
\forall\ j\in\mathcal{C}_{i},$ (S61)

as verified using Eqs. S39–S41. Suppose that the set of feedback matrices
$\\{\widetilde{\bm{K}}_{j}\\}_{j=1}^{N}$ for the projected problem satisfies

$\bm{f}_{i}^{T}{\bm{T}}_{j}^{T}\widetilde{\bm{K}}_{j}\bm{N}_{j}\bm{e}_{p}=\bm{f}_{i}^{T}{\bm{T}}_{k}^{T}\widetilde{\bm{K}}_{k}\bm{N}_{k}\bm{e}_{p},\
\forall\,j,k\in\mathcal{C}_{i},\ p\in\mathcal{N}_{j}\cap\mathcal{N}_{k}.$
(S62)

That is, for any given node $p$, the feedback matrix from the state of node
$p$ to the control input of node $i$ take the same value for all node pairs
$(j,k)$ in $\mathcal{C}_{i}$ whose information neighborhoods $\mathcal{N}_{j}$
and $\mathcal{N}_{k}$ both contains node $p$. We denote this common matrix
$\bm{f}_{i}^{T}{\bm{T}}_{j}^{T}\widetilde{\bm{K}}_{j}\bm{N}_{j}\bm{e}_{p}$ in
Eq. S62 as $\widetilde{\bm{K}}_{ip}$. It follows that

$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s)=\sum_{j\in\mathcal{C}_{i}}\widetilde{\bm{K}}_{ij}\bm{e}_{j}^{T}$
(S63)

satisfies Eq. S60. Thus, when the condition in Eq. S62 holds, the controller
at node $i$ becomes a static feedback matrix.

However, since $\widetilde{\bm{K}}_{j}$ is determined independently for each
$j\in\mathcal{C}_{i}$ by solving the projected Riccati equation (Eq. S37), the
condition in Eq. S62 is not guaranteed to hold. To ensure that condition is
satisfied by the set of feedback matrices
$\\{\widetilde{\bm{K}}_{j}\\}_{j=1}^{N}$, we modify the way in which the
projected problems defined by Eq. S34 are formulated. To do this, we merge
$\mathcal{N}_{j}$ for all $j\in\mathcal{C}_{i}$ to form the set

$\mathcal{\widehat{N}}_{i}=\bigcup\limits_{j\in\mathcal{C}_{i}}\mathcal{N}_{j},$
(S64)

which is a superset of the information neighborhood of any
$j\in\mathcal{C}_{i}$. Now, if we project the original problem onto a superset
of the information neighborhood, the quality of the approximation of the
projected problem would not be compromised. This gives an extra degree of
freedom to construct the projected problems, so that the condition in Eq. S62
can be satisfied. Let $\widehat{\bm{N}}_{i}$ be the projection matrix from the
entire state space to the state subspace corresponding to nodes in
$\mathcal{\widehat{N}}_{i}$. In analogy with $\mathcal{T}_{i}$ and
${\bm{T}}_{i}$ in the main text, we introduce

$\mathcal{\widehat{T}}_{i}=\big{\\{}1\leq k\leq r\ |\ [\bm{B}]_{jk}\neq
0\text{ for some }j\in\mathcal{\widehat{N}}_{i}\big{\\}}$ (S65)

and define $\widehat{\bm{T}}_{i}$ as the projection matrix from the entire
input space $\mathbb{R}^{r}$ to the input subspace $\mathbb{R}^{\widehat{L}}$
associated with $\mathcal{\widehat{T}}_{i}$, where
$\widehat{L}=\lvert\mathcal{\widehat{T}}_{i}\rvert$. We can then project the
original problem onto $\mathcal{\widehat{N}}_{i}$ to obtain a new set of
projected problems, one for each $j\in\mathcal{C}_{i}$:

$\displaystyle\underset{\widehat{\bm{\phi}}_{j},\widehat{\bm{h}}_{j}\in\frac{1}{s}\mathcal{RH}_{\infty}}{\text{min}}\
\left\lVert\begin{bmatrix}\widehat{\bm{Q}}^{1/2}_{j}&\\\
&\widehat{\bm{R}}^{1/2}_{j}\end{bmatrix}\begin{bmatrix}\widehat{\bm{\phi}}_{j}(s)\\\
\widehat{\bm{h}}_{j}(s)\end{bmatrix}\right\rVert_{\mathcal{H}_{2}}^{2}$ (S66)
$\displaystyle\text{s.t.}\begin{array}[]{l}\begin{bmatrix}s\bm{I}-\widehat{\bm{C}}_{j}&-\widehat{\bm{B}}_{j}\end{bmatrix}\begin{bmatrix}\widehat{\bm{\phi}}_{j}(s)\\\
\widehat{\bm{h}}_{j}(s)\end{bmatrix}=\widehat{\bm{e}}_{j},\end{array}$

where
$\widehat{\bm{C}}_{i}=\widehat{\bm{N}}_{i}\bm{C}\widehat{\bm{N}}_{i}^{T}$,
$\widehat{\bm{B}}_{i}=\widehat{\bm{N}}_{i}\bm{B}\widehat{\bm{T}}_{i}^{T}$,
$\widehat{\bm{Q}}_{i}=\widehat{\bm{N}}_{i}\bm{Q}\widehat{\bm{N}}_{i}^{T}$,
$\widehat{\bm{R}}_{i}=\widehat{\bm{T}}_{i}\bm{R}\widehat{\bm{T}}_{i}^{T}$, and
$\widehat{\bm{e}}_{j}=\widehat{\bm{N}}_{i}\bm{e}_{j}$. We call this set of
projected problems the _controller-oriented localization_ of the original
problem, since each is a local version of the original problem around a driver
node. The solution to Eq. S66 is obtained by solving the Riccati equation

$\widehat{\bm{C}}_{i}^{T}\widehat{\bm{P}}_{i}+\widehat{\bm{P}}_{i}\widehat{\bm{C}}_{i}-\widehat{\bm{P}}_{i}\widehat{\bm{B}}_{i}\widehat{\bm{R}}_{i}^{-1}\widehat{\bm{B}}_{i}^{T}\widehat{\bm{P}}_{i}+\widehat{\bm{Q}}_{i}=\bm{0},$
(S67)

which yields the optimal feedback
$\widehat{\bm{K}}_{i}=-\widehat{\bm{R}}_{i}^{-1}\widehat{\bm{B}}_{i}^{T}\widehat{\bm{P}}_{i}$.
Hence, for each $j\in\mathcal{C}_{i}$,

$\widehat{\bm{\phi}}_{j}(s)=(s\bm{I}-\widehat{\bm{C}}_{i}-\widehat{\bm{B}}_{i}\widehat{\bm{K}}_{i})^{-1}\widehat{\bm{e}}_{j},$
(S68)
$\widehat{\bm{h}}_{j}(s)=\widehat{\bm{K}}_{i}\widehat{\bm{\phi}}_{j}(s).$
(S69)

Compared to Eqs. S38 and S39, the optimal response in Eqs. S68 and S69 is
obtained by projecting the original problem onto a superset of the information
neighborhood $\mathcal{N}_{j}$, and hence this solution provides at least the
same accuracy as the solution in Eqs. S38 and S39. The resulting controller is
static because

$\widetilde{\bm{K}}_{j}=\bm{T}_{j}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}\bm{N}_{j}^{T}$
(S70)

satisfies the condition in Eq. S62. The corresponding responses in
$\mathcal{N}_{j}$ achieved by $\widetilde{\bm{K}}_{j}$ are given by

$\widetilde{\bm{\phi}}_{j}(s)=\bm{N}_{j}\widehat{\bm{N}}_{i}^{T}\widehat{\bm{\phi}}_{j}(s),$
(S71)
$\widetilde{\bm{h}}_{j}(s)=\bm{T}_{j}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{h}}_{j}(s).$
(S72)

It follows that

$\displaystyle\big{(}\bm{f}_{i}^{T}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}\big{)}{\bm{N}}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s)=\big{(}\bm{f}_{i}^{T}\bm{T}_{j}^{T}\bm{T}_{j}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}\big{)}{\bm{N}}_{j}^{T}\widetilde{\bm{\phi}}_{j}(s)=\bm{f}_{i}^{T}\bm{T}_{j}^{T}\widetilde{\bm{K}}_{j}\widetilde{\bm{\phi}}_{j}(s),$
(S73)

i.e.,
$\bm{f}_{i}^{T}\widetilde{\bm{K}}(s)=\bm{f}_{i}^{T}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}$
satisfies Eq. S61. This implies that
$\bm{f}_{i}^{T}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}$
is the appropriate feedback matrix for the controller at node $i$, and thus

$\bm{f}_{i}^{T}\bm{u}(t)=\bm{f}_{i}^{T}\widehat{\bm{T}}_{i}^{T}\widehat{\bm{K}}_{i}\widehat{\bm{N}}_{i}\bm{x}(t).$
(S74)

Therefore, the design of the feedback law of each controller only requires
solving the projected Riccati equation locally around that driver node and the
computation can be performed in parallel for all drivers across the network.
This provides a decentralized method for designing a near-optimal control
strategy for the entire network.

##  S9 Basic Control Tasks and Linearization Methods

In scientific and engineering applications, one is often required to actively
control the dynamics of complex networks so that they exhibit certain
desirable behaviors and functionalities. Different applications require
addressing different control tasks, and the most often encountered in practice
are equilibrium stabilization, trajectory tracking, and command following. All
these control tasks can be accomplished by proper design of feedback control
laws, as described next. To proceed, we consider a general system

$\dot{\bm{x}}=\bm{f}(\bm{x})+\bm{B}\bm{u}$ (S75)

in which the function $\bm{f}(\cdot)$ can be nonlinear.

_Equilibrium stabilization_ refers to the control task in which the system is
driven from a given initial condition to a desired equilibrium and held stably
there. In power grids, for example, each power flow solution corresponds to an
equilibrium of the system. A major task of power system controllers is to
bring the system towards the most efficient and reliable power flow
equilibrium and to maintain the system at the equilibrium in the presence of
disturbances. Consider the system in Eq. S75 with a desired equilibrium
$(\bm{x}^{*},\bm{u}^{*})$, i.e., $\bm{f}(\bm{x}^{*})+\bm{B}\bm{u}^{*}=\bm{0}$.
The problem is to design a feedback law of the form
$\bm{u}(t)=\bm{K}(\bm{x}(t)-\bm{x}^{*})+\bm{u}^{*}$ that drives the system
from the initial state $\bm{x}_{0}$ to the target equilibrium $\bm{x}^{*}$. In
general, the map $\bm{K}(\cdot)$ can be nonlinear but is assumed to be
homogeneous, i.e., $\bm{K}(\bm{0})=\bm{0}$. By defining
$\Delta\bm{x}=\bm{x}-\bm{x}^{*}$ and $\Delta\bm{u}=\bm{u}-\bm{u}^{*}$, we have
$\Delta\dot{\bm{x}}=\bm{f}(\bm{x}^{*}+\Delta\bm{x})-\bm{f}(\bm{x}^{*})+\bm{B}\Delta\bm{u}$
(throughout the Article we use $\Delta\dot{\bm{x}}$ to denote the time
derivative of variable $\Delta\bm{x}$ even in cases where $\bm{x}^{*}$ is time
dependent). This reduces the original problem to the problem of designing a
control $\Delta\bm{u}$ to stabilize the system at the origin. To apply the
linear-quadratic optimal control theory, we replace the nonlinear term
$\bm{f}(\bm{x}^{*}+\Delta\bm{x})-\bm{f}(\bm{x}^{*})$ with
$\bm{C(\bm{x},\bm{x}^{*})}\Delta\bm{x}$, which is linear in $\Delta\bm{x}$.
Linearization methods to obtain $\bm{C(\bm{x},\bm{x}^{*})}$ are presented at
the end of this section. Accordingly, the problem further reduces to
$\Delta\dot{\bm{x}}=\bm{C(\bm{x},\bm{x}^{*})}\Delta\bm{x}+\bm{B}\Delta\bm{u}$,
which shares the same structure as the linear system in Eq. 3 of the main
text. Thus, Algorithm 3 (in _Materials and Methods_) can be used to design
$\bm{K}$ for this problem, which can be time independent or time varying,
depending on the linearization method employed.

_Trajectory tracking_ aims to drive the system to a given feasible trajectory
and then force it to follow the trajectory stably. A feasible trajectory of
the system is a pair of curves $(\bm{x}^{*}(t),\bm{u}^{*}(t))$ that satisfy
Eq. S75. For example, it can be the set of planned optimal trajectories for a
group of autonomous spacecrafts in formation, or a selected common orbit for a
group of periodic oscillators in a synchronous state. The task is to design a
feedback control
$\bm{u}(t)=\bm{K}\big{(}\bm{x}(t)-\bm{x}^{*}(t)\big{)}+\bm{u}^{*}(t)$ such
that the system starting from an given initial condition converges to and then
stays on the target trajectory. The design approach using Algorithm 3 is
essentially the same as for equilibrium stabilization, except that
$(\bm{x}^{*}(t),\bm{u}^{*}(t))$ is a time-varying target.

_Command following_ differs from the two tasks above in that a desired
equilibrium or feasible trajectory is not known a priori. The goal is to
design a control law in which a subset $\bm{x}_{1}$ of the state vector
elements will quickly follow any (possibly time-varying) control command
$\bm{r}$ given in real time. Thus, $\bm{r}$ is unspecified at the design stage
and the state vector can be written as
$\bm{x}=[\bm{x}_{1}^{T},\bm{x}_{2}^{T}]^{T}$, where $\bm{x}_{2}$ represents
the other elements of the state vector. In order to completely track the
command $\bm{r}$, there must exist an equilibrium $(\bm{x}^{*},\bm{u}^{*})$
such that $\bm{f}(\bm{x}^{*})+\bm{B}\bm{u}^{*}=\bm{0}$,
$\bm{x}_{1}^{*}-\bm{r}=\bm{0}$. Because $\bm{r}$ is unknown a priori,
$(\bm{x}^{*},\bm{u}^{*})$ is also not available in advance and therefore the
controller cannot be designed against a known $(\bm{x}^{*},\bm{u}^{*})$ as
done above. This problem is solved by augmenting the system with an internal
state of the controller $\bm{z}$, representing the integral of the error in
following $\bm{r}$. The augmented system reads
$\dot{\bm{x}}=\bm{f}(\bm{x})+\bm{B}\bm{u}$, $\dot{\bm{z}}=\bm{x}_{1}-\bm{r}$.
We seek to design a feedback control law in the form
$\bm{u}=\bm{K}_{1}(\bm{x}_{1}-\bm{r})+\bm{K}_{2}\bm{x}_{2}+\bm{K}_{3}\bm{z}$,
which can be seen as a high-dimension generalization of the proportional-
integral control widely used in industrial applications [22]. The closed-loop
system is then given by
$\dot{\bm{x}}=\bm{f}(\bm{x})+\bm{B}\bm{K}_{1}(\bm{x}_{1}-\bm{r})+\bm{B}\bm{K}_{2}\bm{x}_{2}+\bm{B}\bm{K}_{3}\bm{z}$,
$\dot{\bm{z}}=\bm{x}_{1}-\bm{r}$, and the equilibrium equation is rewritten as
$\bm{f}(\bm{x}^{*})+\bm{B}\bm{K}_{2}\bm{x}_{2}^{*}+\bm{B}\bm{K}_{3}\bm{z}^{*}=\bm{0}$,
$\bm{x}_{1}^{*}-\bm{r}=\bm{0}$. Defining $\Delta\bm{x}_{1}=\bm{x}_{1}-\bm{r}$,
$\Delta\bm{x}_{2}=\bm{x}_{2}-\bm{x}_{2}^{*}$, and
$\Delta\bm{z}=\bm{z}-\bm{z}^{*}$, we have
$\Delta\dot{\bm{x}}=\bm{f}(\bm{x})-\bm{f}(\bm{x}-\Delta\bm{x})+\bm{B}\bm{K}_{1}\Delta\bm{x}_{1}+\bm{B}\bm{K}_{2}\Delta\bm{x}_{2}+\bm{B}\bm{K}_{3}\Delta\bm{z}$,
$\Delta\dot{\bm{z}}=\Delta\bm{x}_{1}$. By linearizing
$\bm{f}(\bm{x})-\bm{f}(\bm{x}-\Delta\bm{x})$ as $\bm{C}(\bm{x})\Delta\bm{x}$,
the system becomes

$\begin{bmatrix}\Delta\dot{\bm{x}}_{1}\\\ \Delta\dot{\bm{x}}_{2}\\\
\Delta\dot{\bm{z}}\end{bmatrix}=\left[\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$\bm{C}(\bm{x})$}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup$\bm{0}$}\hfil\lx@intercol\\\
&&\lx@intercol\hfil\hbox{\multirowsetup$\bm{0}$}\hfil\lx@intercol\\\
\cline{1-3}\cr\\\
\lx@intercol\hfil\hbox{\multirowsetup$\bm{I}$}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup$\bm{0}$}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup$\bm{0}$}\hfil\lx@intercol\end{array}\right]\begin{bmatrix}\Delta{\bm{x}_{1}}\\\
\Delta{\bm{x}_{2}}\\\ \Delta{\bm{z}}\end{bmatrix}+\begin{bmatrix}\bm{B}\\\
\bm{0}\\\ \bm{0}\end{bmatrix}\begin{bmatrix}\bm{K}_{1}\ \ \bm{K}_{2}\ \
\bm{K}_{3}\end{bmatrix}\begin{bmatrix}\Delta{\bm{x}_{1}}\\\
\Delta{\bm{x}_{2}}\\\ \Delta{\bm{z}}\end{bmatrix},$ (S76)

independently of the unknown equilibrium. The feedback law $[\bm{K}_{1}\
\bm{K}_{2}\ \bm{K}_{3}]$ can then be designed according to Algorithm 3.

All three control tasks just described rely on linearization of
$\bm{f}(\bm{x}^{*}+\Delta\bm{x})-\bm{f}(\bm{x}^{*})$ or, equivalently, of
$\bm{f}(\bm{x})-\bm{f}(\bm{x}-\Delta\bm{x})$. The most basic linearization
approach is the _Jacobian linearization_ , in which the first-order Taylor
expansion around a known equilibrium or feasible trajectory is used as an
approximation:
$\bm{f}(\bm{x}^{*}+\Delta\bm{x})-\bm{f}(\bm{x}^{*})\approx\frac{\partial\bm{f}}{\partial\bm{x}}\bigr{|}_{\bm{x}^{*}}\Delta\bm{x}.$
When the linearized system is stable, the approach guarantees that the
original nonlinear system has a non-empty basin of attraction around
$\bm{x}^{*}$, which can be enlarged by using high-gain feedback. Moreover, the
feedback matrix does not change while the system is controlled (unless
$\bm{x}^{*}$ varies in time), which is computationally appealing. Another
common approach is the so-called _extended linearization_ , in which the
nonlinear function is algebraically factorized as
$\bm{f}(\bm{x}^{*}+\Delta\bm{x})-\bm{f}(\bm{x}^{*})=\bm{C}(\bm{x}^{*},\Delta\bm{x})\Delta\bm{x}$.
The specific form of $\bm{C}(\bm{x}^{*},\Delta\bm{x})$ depends on the problem
formulation and is in general not unique. An existence condition and a
complete parameterization of all such factorizations can be found in [23, 24].
Due to the dependence of the linear coefficient matrix on the state, different
Riccati equations are solved at different time steps to obtain a time-varying
feedback law (owing to this, the method is also called the state-dependent
Riccati equation approach). The appeal of this approach is that it not only
guarantees local stability but also generates a system trajectory that
satisfies the Hamilton–Jacobi–Bellman equation, which is a necessary condition
for the optimality of the trajectory [25]. For the command following problem,
which requires a linearization coefficient matrix that does not depend on the
unknown equilibrium $\bm{x}^{*}$, a common practice is to first use the
extended linearization to obtain $\bm{f}(\bm{x})=\bm{C}(\bm{x})\bm{x}$ and
then approximate the nonlinear term as
$\bm{f}(\bm{x})-\bm{f}(\bm{x}-\Delta\bm{x})\approx\bm{C}(\bm{x})\Delta\bm{x}.$
Despite a certain lack of theoretical justification, this approximation has
been successfully used in many practical problems [26, 27].

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\begin{overpic}[width=227.62204pt,tics=5]{local_scaling_b.pdf} \end{overpic}
Fig. S1: The scaling of the off-diagonal decay constant for the inverse of
matrix $\bm{C}$. Here, we consider $\bm{C}=-\bm{L}-\bm{I}$, where $\bm{L}$ is
the Laplacian matrix of an ER random network with $\bar{d}=10$ and $\bm{I}$ is
a identity matrix to ensure that matrix $\bm{C}$ is invertible. According to
our construction of information distance, given a characteristic function
$v(\cdot)$, we can identify a information distance $\rho(\cdot,\cdot)$ such
that $\left\lVert\bm{C}_{ij}\right\rVert\leq\kappa\cdot
v\big{(}\rho(i,j)\big{)}^{-1}$ for all $i,j=1,2,\ldots,N$. The constant
$\kappa$ can be chosen as the spectral radius $\lambda_{r}$ of matrix
$\bm{C}$. We now consider the off-diagonal decay of $\bm{C}^{-1}$ and seek to
identify a number $c^{\prime}$ such that
$\left\lVert(\bm{C}^{-1})_{ij}\right\rVert\leq c^{\prime}\lambda_{r}\cdot
v\big{(}\rho(i,j)\big{)}^{-1}$ for all $i,j=1,2,\ldots,N$. When we use a
characteristic function $v(\cdot)$ that satisfies the GRS condition (e.g., the
one we use in this paper), the constant $c^{\prime}$ can be chosen to be
independent of the network size $N$ (blue curve). In contrast, if we use a
characteristic function $v(\cdot)$ that violates the GRS condition (e.g.,
$v(z)=e^{z}$), the constant $c^{\prime}$ may necessarily grow with $N$ (red
curve).

Table S1: Relative approximation error∗ for the smallest eigenvalue of the
controllability Gramian.

model networks† | ER | BA | WS
---|---|---|---
$(9.3\pm 7.7)\times 10^{-3}$ |  $(2.2\pm 0.7)\times 10^{-2}$ |  $(6.1\pm 5.0)\times 10^{-3}$
empirical networks‡ | power grid | air transportation | human brain
$2.13\times 10^{-6}$ |  $7.27\times 10^{-4}$ |  $1.21\times 10^{-2}$

  * •

∗Computed as
$\lvert\lambda_{\text{min}}(\widetilde{\bm{W}}_{\text{c}}^{\infty})-\lambda_{\text{min}}(\bm{W}_{\text{c}}^{\infty})\rvert/\lambda_{\text{min}}(\bm{W}_{\text{c}}^{\infty})$,
with each node chosen as a driver and assigned an information neighborhood of
size $L=\lceil N/100\rceil$.

  * •

†Mean and standard deviation of the relative approximation error over $100$
network realizations for the same parameters as in Fig. 3.

  * •

‡Relative approximation error for the empirical networks and dynamics used in
Fig. 5D–F.