As the number of sensors is often limited and the underlying network may be
sparse, detecting all attacked nodes may not be always possible. An
alternative approach is to place the available sensors on key nodes in the
network in order to maximize $r$ in Lemma 15, i.e., detecting maximum number
of attacks. The sensor placement problem for optimal attack detection is
discussed in Section 10.2.

For the cases where the initial condition of the system is unknown, an
undetectable attack $\boldsymbol{\zeta}$ is characterized by the existence of
a pair of initial states $\boldsymbol{\psi}_{1}[0]$ and
$\boldsymbol{\psi}_{2}[0]$ such that
$\boldsymbol{y}(\boldsymbol{\psi}_{1}[0],0,t)=\boldsymbol{y}(\boldsymbol{\psi}_{2}[0],\boldsymbol{\zeta},t)$
for all $t\geq 0$. In such cases, one needs to first recover the initial
conditions of the system in order to detect the attack. This demands the
system to be strongly observable. Recall from Section 4 and Theorem 3 that
system (8) is strongly observable if the graph is $(2f+1)$-connected. This
extra level of graph connectivity, compared to Theorem 16, is a price paid for
the lack of the knowledge of the initial states.

######  Theorem 18.

Suppose that dynamics (8) with measurement (9) is subject to a set of $f$
attacked nodes and unknown initial conditions. To prevent a perfect attack, it
is sufficient for graph $\mathcal{G}$ to be $(2f+1)$-connected.

###  8.2 Attack Identification Procedure

In order to identify the attacked nodes, first, node $i$ must find the true
initial value of all other nodes (we assume that it satisfies the condition in
Theorem 3). After obtaining the vector of initial states,
$\boldsymbol{\psi}[0]$, and assuming that the interaction matrix,
$\boldsymbol{W}$, is known to $i$, it can apply dynamics (8) to obtain
$\boldsymbol{\psi}[1]-\boldsymbol{W}\boldsymbol{\psi}[0]=\boldsymbol{B}_{\mathcal{F}}\boldsymbol{\zeta}[0]$.
Every nonzero component in the vector on the left hand side of this equation
indicates an additive error injected by the corresponding node. Thus, every
node that is malicious during time-step $0$ can be identified by this method.
The same process can be repeated to find all nodes that were malicious during
the first $L$ time steps from the transmitted values $y_{i}[0:L]$ in (10).
Note that using iteration policy (8) to identify the attacks requires the
system to be free of noise or external disturbances, as otherwise, they
mislead us in the above procedure. Further details on centralized and
distributed attack detection and identification techniques can be found in
[35].

###  8.3 Other Attack Detection and Identification Approaches

Our discussion in this section has thus far focused on the attack model in Eq.
(8) for distributed function calculation. Before closing this section, we
briefly summarize certain other attack detection mechanisms that are relevant
in other contexts. For resilient distributed parameter estimation, attack
detection constitutes a key component of the $\mathcal{FRDE}$ algorithm in
[127] that we discussed in Section 7.1. For state estimation, agents under
attack are identified based on robust control techniques in [150], and a
saturating adaptive gain method in [128]; see Section 7.2. At a high-level,
the approaches in [127], [150], and [128] share a common principle for
detecting whether an agent $i$ is under attack: they involve computing an
appropriate innovation signal that captures the extent to which the
estimates/measurements of agent $i$ differ from those of its neighbors. For
distributed optimization, [115] proposed a heuristic gradient based method to
detect the misbehaving agents. The insight is that the attackers’ biasing
actions can result in a large gradient value. Hence, a regular agent can
attempt to detect such attacks by approximating the gradient of each neighbor
and track it over time relative to the mean of the gradients of the remaining
neighbors. Finally, we note that in the context of multi-robot coordination,
the authors in [169, 170] propose methods to tackle the so called “Sybil
attack”, where an attacker spoofs or impersonates the identities of existing
agents to gain a disproportionate advantage in the network. The key idea in
these works is to detect such spoofing attacks by exploiting the physics of
wireless signals. For specific details, we refer the reader to the respective
papers.

##  9 Graph-Theoretic Interpretations of the Attack Impact

In the previous sections, we discussed how to use network connectivity to
withstand the adversarial actions in several distributed algorithms. In some
cases, however, it is not possible to completely nullify the attacker’s
actions. In those cases, an alternative approach is to mitigate its impact (as
was done in the local filtering approaches for consensus and optimization). In
this section, we quantify the attacker’s impact on networked control systems
in terms of the topology of the underlying network.

###  9.1 Controllability of Networks Under Attack

One way to define the attacker’s impact is via the largest subset of nodes
which can be controlled (or reached) by the set of the attacked nodes. There
is a vast literature that studies the controllability of networks with a
limited number of actuators [171, 172, 83, 173, 174]. In this section, our
focus is on structural controllability as a discrete measure of
controllability, and Gramian-based methods as continuous measures.

####  9.1.1 Structural and Strong Structural Controllability

As mentioned in Section 2.4, a structured system is said to be controllable if
this property holds for at least one numerical choice of free (nonzero)
parameters in the system. Theorem 1 provided graph-theoretic conditions for
structural controllability. Condition (i) in Theorem 1 is called the
reachability condition. The reachability, by itself, can be a measure of the
attack impact. In particular, when the attacker’s goal is to disseminate a
signal throughout the network and infect as many nodes as possible, maximizing
reachability is beneficial to the attacker. On the other hand, when the
attacker’s objective is to steer the states towards its desired direction,
controllability is the appropriate measure. We present the following
definition.

######  Definition 9.18.

The reachable set, $\mathcal{S}^{r}(\mathcal{F})$ (respectively controllable
set, $\mathcal{S}^{c}(\mathcal{F})$) for a set of attacked nodes
$\mathcal{F}$, is a subset of nodes in graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$ whose elements can all be reached
(controlled) by the set of attacked nodes $\mathcal{F}$ and is maximal in the
sense that no additional node can be added to $\mathcal{S}^{r}(\mathcal{F})$
($\mathcal{S}^{c}(\mathcal{F})$) without breaking this property. Elements of
the attacked set, $\mathcal{F}$, are called the source nodes of
$\mathcal{S}^{r}(\mathcal{F})$ (or of $\mathcal{S}^{c}(\mathcal{F})$).

Based on Theorem 1, it is clear that
$|\mathcal{S}^{c}(\mathcal{F})|\leq|\mathcal{S}^{r}(\mathcal{F})|$. Fig. 10
(a) shows a graph whose largest reachable set, for $|\mathcal{F}|=1$, is
larger than the largest controllable set. In graph (b) the largest reachable
and controllable sets are identical. Note that $\mathcal{S}^{c}(\mathcal{F})$
is not necessarily a subset of $\mathcal{S}^{r}(\mathcal{F})$ as shown in
graph (c). The source nodes are shown with a darker color.

Figure 10: Graphs and their largest reachable and controllable subsets.

Due to resource constraints, the attacker naturally tries to solve either of
the following problems:

  * (i)

Controlling (or reaching) the largest possible subset of nodes in the network
with a given number of attacked nodes, or

  * (ii)

Controlling (or reaching) the whole network with minimum number of attacked
nodes.

The latter has been investigated under the context of minimal structural
controllability problems [175]. It is shown that the problem of finding the
minimum number of control input (attacked) nodes is in general NP hard and in
specific cases (such as dedicated inputs) can be solved in polynomial time.
When reachability is the objective, the problem is related to estimating the
reachable set of nodes which can be solved in polynomial time [176].

A system is called strong structurally controllable if ${\rm
rank}(\boldsymbol{W},\boldsymbol{B})=n$ for all (nonzero) choice of free
parameters in $\boldsymbol{W}$ and $\boldsymbol{B}$. The dimension of the
strong structurally controllable subspace is the minimum rank of the
controllablility matrix,
$\mathcal{C}\left(\boldsymbol{W},\boldsymbol{B}\right)$. There are graph-
theoretic bounds on this quantity for consensus dynamics as stated below.

######  Theorem 19 ([177]).

For set $\mathcal{F}$ chosen by the attacker, the dimension of the strong
structurally controllable subspace is lower bounded by

${\rm rank}\hskip
2.84526pt\mathcal{C}\left(\boldsymbol{W},\boldsymbol{B}\right)\geq\max_{i\in\mathcal{V},j\in\mathcal{F}}{\rm
dist}(i,j)+1,$ (33)

where ${\rm dist}(i,j)$ is the shortest distance between nodes $i$ and $j$.
Tighter lower bounds for the dimension of the strong structurally controllable
subspace can be found in [177]. According to Theorem 33, from the attacker’s
perspective, the optimal decision is to select a node with maximum distance
from the rest of the nodes in the graph. Another interpretation from the above
result, which had been discussed before in [171], is the reverse effect of
network connectivity on the controllability. Specifically, sparse networks may
contain pairs of nodes that are far apart, and consequently may have large
controllability subspaces, while well-connected networks may have smaller
controllability subspaces.

####  9.1.2 Gramian-Based Controllability

Unlike the (discrete) structural or rank-based controllability measures of
dynamical systems [178], the controllability Gramian provides a continuous
measure for this property, in the form of the energy required to drive the
dynamical system towards specific directions in the state space. The $T$-step
controllability Gramian is defined as

$\mathcal{W}_{\mathcal{F},T}\triangleq\sum_{\tau=0}^{T-1}\boldsymbol{W}^{\tau}\boldsymbol{B}_{\mathcal{F}}\boldsymbol{B}_{\mathcal{F}}^{\prime}(\boldsymbol{W}^{\prime})^{\tau}$
(34)

where $\boldsymbol{B}_{\mathcal{F}}$ corresponds to the set of attacked nodes
and defined in (8). The controllability Gramian $\mathcal{W}_{\mathcal{F},T}$
is positive definite if and only if the system is controllable in $T$ steps
[179]. However, even if a system is controllable, certain directions of the
state space may be hard to reach [180]. The smallest eigenvalue of the
Gramian, $\lambda_{1}(\mathcal{W}_{\mathcal{F},T})$, is inversely related to
the amount of energy required to move the system in the direction that is the
most difficult to control, i.e., the eigenvector corresponding to
$\lambda_{1}(\mathcal{W}_{\mathcal{F},T})$. Other controllability metrics,
such as ${\rm trace}(\mathcal{W}_{\mathcal{F},T})$ and ${\rm
trace}(\mathcal{W}_{\mathcal{F},T}^{-1})$, quantify the energy needed on
average to move the system around on the state space. From the attacker’s
perspective, the system should be easily controllable. Thus, it targets nodes
in which one of the above mentioned spectra is optimized. In particular, the
attacker attempts to minimize its effort to steer the system by maximizing
$\lambda_{1}(\mathcal{W}_{\mathcal{F},T})$ and ${\rm
trace}(\mathcal{W}_{\mathcal{F},T})$ or minimizing ${\rm
trace}(\mathcal{W}_{\mathcal{F},T}^{-1})$.

The selection of control nodes to optimize the spectrum of the Gramian does
not generally admit a closed-form solution. The exception is ${\rm
trace}(\mathcal{W}_{\mathcal{F},T})$ for which [181]

${\rm
trace}(\mathcal{W}_{\mathcal{F},T})=\sum_{i\in\mathcal{F}}\bigg{(}\sum_{\tau=0}^{T-1}\boldsymbol{W}^{2\tau}\bigg{)}_{ii}.$

When $\boldsymbol{W}$ is Schur stable, then
$\sum_{\tau=0}^{\infty}\boldsymbol{W}^{2\tau}=(I-\boldsymbol{W}^{2})^{-1}$.
For continuous time systems, if $\boldsymbol{W}$ is Hurwitz, this closed form
solution becomes ${\rm
trace}(\mathcal{W}_{\mathcal{F}})=\sum_{i\in\mathcal{F}}\big{(}\boldsymbol{W}^{-1}\big{)}_{ii}$.
For other metrics, selecting optimal nodes, from the attacker’s perspective,
is a combinatorial problem and (in general) hard to solve. However, recent
studies on submodularity and monotonicity of some of those metrics indicate
that greedy algorithms for selecting the control nodes result in a sub-optimal
solution with a guaranteed performance bound [182, 183, 184, 185].

In the following subsection, we outline a graph-theoretic interpretation of
the attacker’s strategy to optimize ${\rm trace}(\mathcal{W}_{\mathcal{F},T})$
in consensus dynamics.

####  9.1.3 Case Study: Consensus Dynamics

We consider Gramian-based controllability on two types of consensus dynamics
on undirected graphs, namely average consensus and leader-following consensus.

1\. Edge Attack in Average Consensus: Assume that the attack happens in the
form of a flow which enters one node and exists from another node. In
particular, we say that $ik\in\mathcal{F}$ if the pair $i,k$ are chosen by the
attacker and their dynamics are

$\displaystyle\dot{\psi}_{i}$ $\displaystyle=\sum_{j\in\mathcal{N}_{i}^{\rm
in}}(\psi_{j}-\psi_{i})+\zeta_{ik},$ $\displaystyle\dot{\psi}_{k}$
$\displaystyle=\sum_{j\in\mathcal{N}_{k}^{\rm
in}}(\psi_{j}-\psi_{k})-\zeta_{ik},$ (35)

where $\zeta_{ik}$ is the attack flow. This type of input signal, as
schematically shown in Fig. 11 (a), happens in power systems (DC input links)
and distribution networks [186, 187]. If we write the dynamics in vector form,
it becomes

$\displaystyle\dot{\boldsymbol{\psi}}$
$\displaystyle=-L\boldsymbol{\psi}+\mathcal{B}\boldsymbol{\zeta},$ (36)

where $\mathcal{B}$ is the incidence matrix of the graph induced by the
attacked edges. The controllability Gramian is

$\mathcal{W}_{c}=\int_{0}^{\infty}\mathbf{e}^{-L\tau}\mathcal{B}\mathcal{B}^{\prime}\mathbf{e}^{-L^{\prime}\tau}d\tau.$
(37)

Since $L$ is marginally stable, the infinite integral does not exist. However,
the eigenvector of the marginally stable eigenvalue belongs to the subspace
corresponding to the consensus value, which is of little interest to the
attacker (as otherwise no attack would be needed). We remove this subspace by
grounding one node (removing a row and column corresponding to that node)
which makes the Laplacian non-singular. The grounded Laplacian matrix induced
by the grounded node $v$ is denoted by $L_{fv}$ (or simply $L_{f}$). Then we
have

$\displaystyle{\rm tr}(\mathcal{W}_{c})={\rm
tr}\left(\mathcal{B}^{\prime}\int_{0}^{\infty}\mathbf{e}^{-2L_{f}\tau}d\tau\mathcal{B}\right)$
$\displaystyle=\frac{1}{2}{\rm
tr}\left(\mathcal{B}^{\prime}L^{-1}_{f}\mathcal{B}\right).$ (38)

It was shown in [188] that the above value is independent of the choice of the
grounded node, i.e., we have

$\displaystyle{\rm
tr}\left(\mathcal{B}^{\prime}L^{-1}_{f}\mathcal{B}\right)={\rm
tr}\left(\mathcal{B}^{\prime}L^{\dagger}\mathcal{B}\right)=\sum_{ij\in\mathcal{F}}\mathfrak{R}_{ij},$
(39)

where $L^{\dagger}$ is the Moore–Penrose inverse of $L$ and
$\mathfrak{R}_{ij}$ is the effective resistance between nodes $i$ and $j$.
Equation (39) indicates that the trace of the controllability Gramian is the
summation of effective resistances between node pairs chosen by the attacker.
Thus, if the attacker seeks to maximize (39), i.e., minimize the attack
energy, by choosing $m$ node pairs, it should choose $m$ pairs with the
largest effective resistance in the graph. Figure 11 (a) is an example of an
optimal attack.

Figure 11: (a) Attacker’s optimal decisions based on (39), (b) attacker’s
optimal decision based on (41).

2\. Node Attack in Leader-Follower Consensus: We consider a leader-follower
dynamical system on undirected graphs which is widely studied in formation
control problems [171, 189, 190]. Based on this model, there is a leader,
which has access to the control input or determines the set-point, and a set
of followers, which follow the state of the leader. The dynamics of leader and
followers are given by

$\begin{bmatrix}\dot{\boldsymbol{\psi}}_{f}(t)\\\\[3.00003pt]
\dot{\boldsymbol{\psi}}_{\ell}(t)\end{bmatrix}=-\underbrace{{\begin{bmatrix}L_{f}&L_{\ell
f}\\\\[3.00003pt]
L_{f\ell}&L_{\ell}\end{bmatrix}}}_{L}\begin{bmatrix}{\boldsymbol{\psi}}_{f}(t)\\\\[3.00003pt]
{\boldsymbol{\psi}}_{\ell}(t)\end{bmatrix}+\begin{bmatrix}\boldsymbol{B}_{\mathcal{F}}\\\\[3.00003pt]
0\end{bmatrix}\boldsymbol{\zeta}_{\mathcal{F}}(t),$ (40)

where $\boldsymbol{\psi}_{f}(t),\boldsymbol{\psi}_{\ell}(t)$ and
$\boldsymbol{\zeta}_{\mathcal{F}}(t)$ are the state of followers, the state of
the leader, and the attack vector, respectively. The leader is not affected by
communication attacks. It also keeps its state constant and does not
incorporate the states of the followers, i.e.,
$\dot{\boldsymbol{\psi}}_{\ell}(t)=0$. Thus, we have $L_{f\ell}=L_{\ell}=0$.
Matrix $\boldsymbol{B}_{\mathcal{F}}$, formed from indicator vectors as in
(8), determines the nodes that are under attack and $L_{\ell f}$ determines
the connection of the leader to the followers. Matrix $L_{f}$ is the grounded
Laplacian matrix discussed before. One of the key properties of this matrix,
which is used in this analysis, is that
$[L_{f}^{-1}]_{ii}=\mathfrak{R}_{i\ell}$, where $\mathfrak{R}_{i\ell}$ is the
effective resistance between node $i$ and the leader. Similar to the case of
edge attack, the objective of the attacker is to maximize the trace of the
controllability Gramian. Calculating the Gramian integral yields

${\rm
trace}(\mathcal{W}_{\mathcal{F}})=\sum_{i\in\mathcal{F}}\big{(}L_{f}^{-1}\big{)}_{ii}=\sum_{i\in\mathcal{F}}\mathfrak{R}_{i\ell}.$
(41)

Based on (41), if the attacker wants to minimize the average energy by
attacking $m$ nodes, it must select $m$ nodes with the largest effective
resistance from ${\ell}$. Figure 11 (b) is an example of an optimal attack.

###  9.2 System Norm Approaches

Another way to quantify the attacker’s impact is through the use of system
norms from the attack signal to the output of interest, e.g., the state of the
agents. Two widely used system norms in robust control are $\mathcal{H}_{2}$
and $\mathcal{H}_{\infty}$ norms. Since these methods were initially developed
on continuous time systems, we focus on such systems in this section.

Suppose that the evolution of a network of agents is described by (4) where
$\boldsymbol{D}=0$ and $\boldsymbol{B}=\boldsymbol{B}_{\mathcal{F}}$ which
corresponds to the set of attacked nodes defined in (8). The transfer function
of the error dynamics from attack input $\boldsymbol{\zeta}_{\mathcal{F}}(t)$
to output $\boldsymbol{y}(t)$ in the Laplace domain is
$G(s)\triangleq\boldsymbol{C}(sI-\boldsymbol{W})^{-1}\boldsymbol{B}_{\mathcal{F}}$.
The system $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ norms are defined as

$\displaystyle\|G\|_{2}\triangleq\left(\frac{1}{2\pi}{\rm
trace}\int_{0}^{\infty}G^{*}(j\omega)G(j\omega)d\omega\right)^{\frac{1}{2}},$
$\displaystyle\|G\|_{\infty}\triangleq\sup_{\omega\in\mathbb{R}}{\sigma_{\rm
max}(G(j\omega))},$ (42)

where $\sigma_{\rm max}(.)$ is the maximum singular value of a matrix. The
system $\mathcal{H}_{2}$ norm can also be calculated based on the
observability Gramian $\mathcal{W}_{o}$, which is the solution of the
following Lyapunov equation

$\displaystyle\boldsymbol{W}^{\prime}\mathcal{W}_{o}+\mathcal{W}_{o}\boldsymbol{W}=-\boldsymbol{C}^{\prime}\boldsymbol{C},$
$\displaystyle\|G\|_{2}={\rm
trace}(\boldsymbol{B}_{\mathcal{F}}^{\prime}\mathcal{W}_{o}\boldsymbol{B}_{\mathcal{F}}).$
(43)

Unlike the approach of the structured system theory to resilient NCSs, which
was based on nonzero patterns of system matrices and not the magnitude of the
elements, the system norm approach depends on the exact value of the matrix
elements. Thus, having knowledge about the nonzero patterns of dynamic
matrices is not sufficient and one has to specify the type of matrices which
describe the interactions between agents. We revisit the leader-follower
consensus dynamics discussed in Section 9.2.1 to further explain this fact.

######  Remark 9.19.

($\mathcal{H}_{2}$ versus $\mathcal{H}_{\infty}$ norm) From a security
perspective, either the system $\mathcal{H}_{2}$ or $\mathcal{H}_{\infty}$
norm can be used to quantify the attack impact. If the frequency content of
the attack signal is unknown, using a $\mathcal{H}_{2}$ norm is a more
reasonable choice as it is calculated over all frequencies. However, if the
objective is to find the worst-case attack impact over all frequencies, the
system $\mathcal{H}_{\infty}$ norm is an appropriate choice.

####  9.2.1 Case Study: Consensus Dynamics Revisited

1\. Edge Attack in Average Consensus: Consider the consensus dynamics under an
attack flow as in (36). Suppose that the attacker measures the difference
between node values under attack, i.e.,
$\boldsymbol{y}=\mathcal{B}^{\prime}\boldsymbol{\psi}$. Dynamics (36) together
with this measurement form a symmetric system, i.e., $L$ is symmetric and
$\boldsymbol{B}=\boldsymbol{C}^{\prime}$. Thus, the $\mathcal{H}_{\infty}$
norm is equal to the DC gain of the system [191], i.e., for the transfer
function from $\boldsymbol{\zeta}$ to $\boldsymbol{y}$ we have

${\|G\|_{\infty}}=\sigma_{\rm
max}\big{(}\mathcal{B}^{\prime}L^{\dagger}\mathcal{B}\big{)}$ (44)

Unlike the trace of $\mathcal{B}^{\prime}L^{\dagger}\mathcal{B}$, in (39),
interpreting its largest singular value is hard. We consider the simple case
where only one node pair, $i$ and $j$, is under attack, i.e.,
$\mathcal{B}=\mathbf{e}_{ij}$. In this case, (44) becomes scalar and we have
$\|G\|_{\infty}=\sigma_{\rm
max}\big{(}\mathbf{e}_{ij}^{\prime}L^{\dagger}\mathbf{e}_{ij}\big{)}=\mathfrak{R}_{ij}$.
Thus, in order to have a large impact, the attacker must choose node pairs
with the largest effective resistance in the network.

2\. Node Attack in Leader-Follower Consensus: Here, a single attacker $i$
targets a set of nodes and $\boldsymbol{B}_{\mathcal{F}}=C^{\prime}$. As
before, the $\mathcal{H}_{\infty}$ norm is equal to the DC gain of the system
[82] and we have $\|G\|_{\infty}=\sigma_{\rm
max}\big{(}\boldsymbol{B}_{\mathcal{F}}^{\prime}L_{f}^{-1}\boldsymbol{B}_{\mathcal{F}}\big{)}$.
For the case of a single node under attack, we have
$\boldsymbol{B}_{\mathcal{F}}=\mathbf{e}_{i}$ and
$\|G\|_{\infty}=\mathbf{e}_{i}^{\prime}L_{f}^{-1}\mathbf{e}_{i}=\mathfrak{R}_{i\ell}$.
Hence, to have a large impact on a target node, the attacker must choose a
node in the network with the largest effective resistance from the leader.

##  10 Related Problems in Resilient Networked Control Systems

In this section, we briefly discuss other problems on the resilience of
networked control systems which use graph theory as a tool in the analysis.

###  10.1 Resilience to Actuator/Sensor and Link Removals

So far, we discussed the case where the agents or the communications between
the agents are under attack. In that context, we assume that the control
inputs are not affected by the attacker. In some situations, however, the
attacker may choose to remove certain sensors, actuators, or communication
links entirely. In those cases, the main concern is to retain the
controllability of the system. This can be written in terms of a robust
structural controllability problem. In particular, the objective is to
maintain a system’s controllability despite the removal of a subset of
actuators.

Consider the linear time invariant system (3) where a subset of control inputs
are removed (potentially due to adversarial actions). In this case, the
minimum number of actuators which retain the controllability of the system is
determined by the following problem

$\displaystyle\underset{\boldsymbol{B}\in\mathbb{R}^{n\times m}}{\text{arg
min}}$ $\displaystyle\|\boldsymbol{B}\|_{0}$ (45) such that
$\displaystyle(\boldsymbol{W},{\boldsymbol{B}}_{\mathcal{U}\setminus{\mathcal{F}}}){\text{
is struc. cont. $\forall\mathcal{F}\subset\mathcal{U}$}},$ $|\mathcal{F}|\leq
m$,

where ${\boldsymbol{B}}_{\mathcal{U}\setminus{\mathcal{F}}}$ corresponds to
the structure of the input matrix $\boldsymbol{B}$ whose columns corresponding
to set $\mathcal{F}$ are removed (i.e., actuators are failed). The number of
actuator faults are upper bounded by $m$. It is shown that the above problem
is NP-hard and polynomial time algorithms to approximate the solution of those
problems have been proposed [192, 193].

In other set of problems, the attacker targets a set of links in the network
to remove. Several performance measures may be affected by such an action.
When robust controllability (or observability) is of interest, the problem can
be written as follows [192]

$\displaystyle\underset{\boldsymbol{B}\in\mathbb{R}^{n\times m}}{\text{arg
min}}$ $\displaystyle\|\boldsymbol{B}\|_{0}$ (46) such that
$\displaystyle(\boldsymbol{W}_{\mathcal{E}_{\mathcal{X},\mathcal{X}}\setminus{\mathcal{E}_{\mathcal{F}}}},{\boldsymbol{B}}){\text{
is struc. cont.
$\forall\mathcal{E}_{\mathcal{F}}\subset\mathcal{E}_{\mathcal{X},\mathcal{X}}$}},$
$|\mathcal{E}_{\mathcal{F}}|\leq\bar{m}$,

where $\mathcal{E}_{\mathcal{F}}$ is the set of edges affected by the
attacker, upper bounded by $\bar{m}$. Similar to (45), the above problem is
NP-hard in general.

###  10.2 Strategic Sensor and Actuator Placement on Graphs

Our focus in the previous sections was to find graph conditions which ensure
resiliency to attacks for certain distributed algorithms. However, in many
situations, the underlying network topology is sparse and cannot be changed.
Furthermore, the number of sensors/actuators is limited. In these cases, an
alternative approach is to place those limited number of sensors (or
actuators) on specific nodes in the network in order to optimally detect the
attack or mitigate its impact.

There is a vast literature on sensor (or actuator) placement to enhance the
observability (or controllability) in terms of the rank of the observability
(or controllability) matrix, Gramian-based metrics (as discussed previously),
or the error variance of the Kalman filters [194, 195]. In all these problems,
there is a single decision maker which deploys sensors (or place actuators) on
a set of nodes. However, in security problems, the adversary plays as another
decision maker which tries to optimize its own cost function, e.g., maximize
impact or minimize visibility. This introduces a strategic sensor (or
actuator) placement problem, taking the attacker’s actions into account. In
this direction, game theory can be used as a powerful tool to address this set
of problems; see [51, 196, 197, 198, 199] and references therein.

Strategic sensor placement in the network to detect cyber-attacks has been
recently studied [200, 201, 202, 203]. In this setting, the attacker seeks to
apply attack inputs while being stealthy and the detector tries to detect the
attack. Several approaches have been adopted to characterize the equilibria of
the security games. Nash equilibrium is used to model simultaneous decision
making and Stackelberg game model is used for the case where the defender must
act before the attacker. In design problems, the Stackelberg game is a popular
approach to defend against cyber-attacks. In particular, the detector acts as
the game leader and places sensors on nodes considering the worst case attack
strategies. The applicability of each method, based on the nature of the
attack and the structure of the cyber physical system, is discussed in [196].

In addition to the strategic attack detection, a defense mechanism can help
mitigate the impact of the attack via certain control actions [204, 205, 206,
51]. The attacker’s impact can be quantified by either of the methods
discussed in Section 9. A comprehensive defense strategy must include both
strategic detection and mitigation mechanisms. This is a potential avenue for
further research.

###  10.3 Network Coherence as a Measure of Resiliency

In the network control systems literature, the notion of network coherence is
used to quantify the ability of a network to reject communication disturbances
while performing a formation control or a consensus algorithm in large scale
systems [207]. It is usually described in terms of system $\mathcal{H}_{2}$ or
$\mathcal{H}_{\infty}$ norms from the disturbance signal to the output of
interest, e.g., position (phase) or velocity (frequency) [208, 209, 210].
Interpreting the disturbances as attack inputs, some of the approaches and the
results in this line of research, e.g., scalability of algorithms and leader
selection schemes, can be readily used in resilient distributed algorithms, as
we will briefly discuss in the following paragraphs.

####  10.3.1 Leader Selection

The objective is to choose the optimal nodes in the network as leaders, i.e.,
nodes which receive the control signal, such that the network coherence is
maximized, i.e., system $\mathcal{H}_{2}$ or $\mathcal{H}_{\infty}$ norms from
the disturbance input to the output of interest are minimized. It is shown in
[211] that the optimal leader to minimize the $\mathcal{H}_{2}$ norm in
consensus dynamics is the information central node in the network: a node in
which the summation of effective resistances to the rest of the nodes in the
network is minimized. It is also shown that the leader which optimizes the
$\mathcal{H}_{\infty}$ norm is not necessarily the graph’s information center
[189]. Moreover, a graph-theoretic condition for the leader to co-optimize
both metrics is discussed in [189].

####  10.3.2 Scalability

Another problem of interest is the scalability of the network coherence in
graphs with various structures. In [207], it is discussed that how the network
coherence scales with the network size for regular lattices in 1, 2 and higher
dimensions. The scalability of these metrics is also discussed for random
graphs. In particular, for Erdős-R$\acute{e}$nyi Random Graphs and random
regular graphs, tight characterizations of the network coherence are discussed
in [189, 212].

##  11 Resilience in Certain Classes of Graphs

In this section, we summarize some pertinent topological properties of certain
classes of graphs. We focus on the properties of graph connectivity and graph
robustness provided in Section 2.1.2, since they are particularly relevant to
the resilience of distributed algorithms against adversarial actions. Our
focus is on undirected networks, unless indicated. We start from simpler
cases.

###  11.1 Paths, Cycles, Trees, and Complete Graphs

An undirected path of length $n$ is the simplest connected graph with
connectivity $\kappa=1$ and robustness $r=1$. A cycle is a path of length $n$
whose start and end nodes are connected. For a cycle graph, the connectivity
is $\kappa=2$ and the robustness is $r=1$. Trees are connected acyclic graphs
with connectivity $\kappa=1$ and robustness $r=1$. A complete graph is
$(n-1)$-connected and $\lceil\frac{n}{2}\rceil$-robust.

###  11.2 Circulant Networks, $k$-Nearest Neighbor Paths, and 1-D Random
Geometric Graphs

######  Definition 11.20 ($k$-Nearest Neighbor Paths).

A $k$-nearest neighbor path, $\mathcal{P}(n,k)$, is a network comprised of $n$
nodes in a path, where the nodes are labeled as $1,2,...,n$ from one end of
the path to the other and each node $i$ can communicate with its $k$ nearest
neighbors behind it and $k$ nearest neighbors ahead of it, i.e.,
${i-k},{i-k+1},...,{i-1},{i+1},{i+2},...,{i+k}$, for some $k\in\mathbb{N}$. An
example of $\mathcal{P}(n,k)$ is shown in Fig. 12 (a).

Figure 12: (a) A 2-nearest neighbor path (b,c) an undirected and a directed
circulant graph, (d) A connected formation on a 2-dimensional lattice.

Such $k$-nearest neighbor paths are relevant for modelling vehicle platoons,
due to the limited sensing and communication range for each vehicle. It is
shown in [213] that a $k$-nearest neighbor path, $\mathcal{P}(n,k)$, is a
$k$-connected graph. We will discuss the robustness of $\mathcal{P}(n,k)$
later in this section. A similar structure to $\mathcal{P}(n,k)$ is a
1-dimensional geometric random graph which captures edges between nodes that
are in close (spatial) proximity to each other.

######  Definition 11.21 (Geometric Random Graphs).

A geometric random graph
$\mathcal{G}_{n,\rho,l}^{d}=\\{\mathcal{V},\mathcal{E}\\}$ is an undirected
graph generated by first placing $n$ nodes (according to some mechanism) in a
$d$-dimensional region $\Omega_{d}=[0,l]^{d}$, where $d\in\mathbb{Z}_{\geq
1}$. We denote the position of node $i\in\mathcal{V}$ by $x_{i}\in\Omega_{d}$.
Nodes $i,j\in\mathcal{V}$ are connected by an edge if and only if
$\|x_{i}-x_{j}\|\leq\rho$ for some threshold $\rho$, where $\|.\|$ is some
appropriate norm (often taken to be the standard Euclidean norm). When the
node positions are generated randomly (e.g., uniformly and independently) in
the region, one obtains a geometric random graph.

In the more general models of $\mathcal{G}_{n,\rho,l}^{d}$, the length $l$ is
also allowed to increase and the density $\frac{n}{l^{d}}$ can converge to
some constant, making it suitable for capturing both dense and sparse random
networks. The following result holds for 1-dimensional geometric random
graphs.

######  Proposition 20.

In $\Omega_{1}=[0,l]$ with fixed $l$, if $\mathcal{G}_{n,\rho,l}^{1}$ is
$k$-connected, then it is at least $\lfloor\frac{k}{2}\rfloor$-robust.

Based on Definition 11.21, the $k$-nearest neighbor path can be seen as a
geometric graph $\mathcal{G}_{n,\rho,l}^{d}$ with $\rho=\frac{lk}{n-1}$ and
placing the nodes as follows: the first node, $1$, is placed on one end of the
line and the $i$-th node is placed at distance $\frac{(i-1)l}{n-1}$ from $1$.
Thus, based on Proposition 20 and the fact that the $k$-nearest neighbor path
$\mathcal{P}(n,k)$ is $k$-connected, we conclude that it is at least
$\lfloor\frac{k}{2}\rfloor$-robust.

######  Definition 11.22 (Circulant Graphs).

An undirected graph of $n$ nodes is called circulant if the $n$ vertices of
the graph can be numbered from 0 to $n-1$ in such a way that if some two
vertices numbered $x$ and $(x\pm d)$ $\rm mod$ $n$ are adjacent, then every
two vertices numbered $z$ and $(z\pm d)$ $\rm mod$ $n$ are adjacent. A
directed graph is circulant (with the above labeling) if some two vertices
numbered $x$ and $(x+d)$ $\rm mod$ $n$ are adjacent, then every two vertices
numbered $z$ and $(z+d)$ mod $n$ are adjacent.

Informally speaking, an undirected $k$-circulant graph is a $k$-nearest
neighbor cycle graph. Thus, with the same reasoning, an undirected
$k$-circulant graph is $2k$-connected and at least
$\lfloor\frac{k}{2}\rfloor$-robust. The result is extended to directed
circulant graphs where it is shown that it is at least
$\lfloor\frac{k+2}{4}\rfloor$-robust [214]. An example of an undirected and a
directed circulant graph is shown in Fig. 12 (b,c).

###  11.3 Formation Graphs on 2-Dimensional Lattice

A specific type of geometric graph is the two dimensional lattice which has
been widely used in formation control of autonomous robots [215, 216]. A
lattice is a set of linear combinations with integer coefficients of the
elements of a basis of $\mathbb{R}^{2}$. The elements of the set are lattice
points. Let $\boldsymbol{v}_{1}$ and $\boldsymbol{v}_{2}$ be bases of a
2-dimensional lattice with
$\|\boldsymbol{v}_{1}\|=\|\boldsymbol{v}_{2}\|=\ell$ where $\ell$ is the
lattice length. Every point $x$ on the lattice can be described by
$x=a_{i}\boldsymbol{v}_{1}+b_{j}\boldsymbol{v}_{2}$ where
$a_{i},b_{j}\in\mathbb{Z}$. In a lattice, two nodes $i$ and $j$ are connected
if $\|x_{i}-x_{j}\|\leq\ell$ where $\|.\|$ is the Euclidean norm. Given a set
of nodes $\mathcal{V}$ and a distance $\ell$, the graph
$\mathcal{G}_{\ell}=\\{\mathcal{V},\mathcal{E}_{\ell}\\}$ with edge set
$\mathcal{E}_{\ell}\triangleq\\{(i,j)|\|x_{i}-x_{j}\|\leq\ell\\}$ is called
the proximity graph of the set $\mathcal{V}$. We describe the communication
range by a function $R:\mathbb{Z}_{\geq 1}\to\mathbb{R}$ that maps the number
of robots $m$ to a distance where $m$ robots are ensured to be reached. A
formation of $n$ robots is said to be connected if its associated proximity
graph $\mathcal{G}_{\ell}$ is connected. An example of a lattice and a
connected formation is shown in Fig. 12 (d). It was shown in [215] that in a
connected formation of $n$ robots, every robot has at least $1\leq m\leq n-1$
robots within a distance $m\ell$. Based on this, one can compute a minimum
communication range for the robots in a formation to guarantee resilience in
the communication network.

######  Proposition 21 ([215]).

Given a set $\mathcal{V}$ of $4f+1$ nodes in a connected formation, if the
communication range of every node satisfies $R\geq 3f\ell$, then the
associated graph of the formation is $(2f+1)$-robust.

###  11.4  Random Graphs

A common approach to modelling complex networks is via the framework of random
graphs, i.e., by drawing a graph from a certain probability distribution over
the set of all possible graphs on a given set of nodes. Such random graph
models have diverse applications [217, 218], including in modeling cascading
failures in large scale systems [219, 220]. Here, we summarize the
connectivity and robustness properties of certain commonly studied random
graph models.

####  11.4.1 Erdős-R$\acute{e}$nyi Random Graphs

An Erdős-R$\acute{e}$nyi (ER) random graph $\mathcal{G}(n,p)$ is a graph on
$n$ nodes, where each edge between two distinct nodes is presented
independently with probability $p$ (which could be a function of $n$). We say
that a graph property holds asymptotically almost surely if the probability of
drawing a graph with that property goes to $1$ as $n\rightarrow\infty$. The
following theorem shows the probability threshold for which a graph
$\mathcal{G}\in\mathcal{G}(n,p)$ is $r$-connected and $r$-robust.

######  Theorem 22 ([149]).

For any constant $r\in\mathbb{Z}_{\geq 1}$,

$t(n)=\frac{\ln n+(r-1)\ln\ln n}{n}$

is a threshold function for the ER random graph $\mathcal{G}$ to have minimum
degree $r$, to be $r$-connected, and to be $r$-robust.

According to the example graph in Fig. 3 (a), graph robustness is a much
stronger property than the graph connectivity and the minimum degree. However,
Theorem 22 indicates that the above threshold function for $r$-connectivity
(and minimum degree $r$) is also a threshold function for the stronger
property of $r$-robustness in ER random graphs.

####  11.4.2 Random Regular Graphs

Let $\Omega_{n,d}$ be the set of all undirected graphs on $n$ nodes where
every node has degree $d$ (note that this assumes that $nd$ is even). A random
$d$-regular graph (RRG), denoted $\mathcal{G}_{n,d}$ is a graph drawn
uniformly at random from $\Omega_{n,d}$. For $d\geq 3$, it is shown that
$\mathcal{G}_{n,d}$ is asymptotically almost surely $d$-connected [217]. Based
on [221], for any $\epsilon>0$, the algebraic connectivity of a random
$d$-regular graph satisfies

$\lambda_{2}(L)\geq d-2\sqrt{d-1}-\epsilon,$ (47)

asymptotically almost surely. As discussed in [216], if the algebraic
connectivity of a graph is bigger than $r-1$, then the network is at least
$\lfloor\frac{r}{2}\rfloor$-robust. Hence, according to (47), an RRG is at
least $\lfloor\frac{d-2\sqrt{d-1}}{2}\rfloor$-robust asymptotically almost
surely.

####  11.4.3 Random Interdependent Networks

An interdependent network $\mathcal{G}$ is denoted by a tuple
$\mathcal{G}=(\mathcal{G}_{1},\mathcal{G}_{2},\allowbreak\ldots,\mathcal{G}_{k},\mathcal{G}_{p})$
where $\mathcal{G}_{l}=(\mathcal{V}_{l},\mathcal{E}_{l})$ for $l=1,2,\ldots,k$
are called the subnetworks of the network $\mathcal{G}$, and
$\mathcal{G}_{p}=(\mathcal{V}_{1}\cup\mathcal{V}_{2}\cup\ldots\cup\mathcal{V}_{k},\mathcal{E}_{p})$
is a $k$-partite network with $\mathcal{E}_{p}\subseteq\cup_{l\neq
t}\mathcal{V}_{l}\times\mathcal{V}_{t}$ specifying the interconnection (or
inter-network) topology. Applications of interdependent networks in modelling
communication networks and power grid are discussed in [222]. Define the
sample space $\Omega_{n}$ to consist of all possible interdependent networks
$(\mathcal{G}_{1},\mathcal{G}_{2},\ldots,\mathcal{G}_{k},\mathcal{G}_{p})$ and
the index $n\in\mathbb{N}$ denotes the number of nodes in each subnetwork. A
random interdependent network is a network
$\mathcal{G}=(\mathcal{G}_{1},\mathcal{G}_{2},\ldots,\mathcal{G}_{k},\mathcal{G}_{p})$
drawn from $\Omega_{n}$ according to a given probability distribution.

We assume that
$|\mathcal{V}_{1}|=|\mathcal{V}_{2}|=\cdots=|\mathcal{V}_{k}|=n$ and that the
number of subnetworks $k$ is at least 2. Similar to Theorem 22 for ER random
graphs, there exists a sharp threshold for connectivity and robustness of
random interdependent networks.

######  Theorem 23 ([223]).

Consider a random interdependent network
$\mathcal{G}=(\mathcal{G}_{1},\mathcal{G}_{2},\ldots,\mathcal{G}_{k},\mathcal{G}_{p})$.
Then, for any positive integers $r$ and $k\geq 2$,

$t(n)=\frac{\ln n+(r-1)\ln\ln n}{(k-1)n}$

is a threshold for $r$-connectivity and $r$-robustness of $\mathcal{G}$.

####  11.4.4 Random Intersection Networks

Random intersection graphs belong to class of random graphs for which every
node is assigned a set of objects selected by some random mechanism. They have
applications in wireless sensor networks, frequency hopping spread spectrum,
spread of epidemics, and social networks [224].

Given a node set $\mathcal{V}=\\{1,2,...,n\\}$, each node $i$ is assigned an
object set $S_{i}$ from an object pool $\mathcal{P}$ consisting of $P_{n}$
distinct objects, where $P_{n}$ is a function of $n$. Each object $S_{i}$ is
constructed using the following two-step procedure: (i) The size of $S_{i}$,
$|S_{i}|$, is determined according to some probability distribution
$\mathcal{D}:\\{1,2,...,P_{n}\\}\to[0,1]$ in which
$\sum_{x=1}^{P_{n}}\mathbb{P}\left(|S_{i}|=x\right)=1$. (ii) Conditioning on
$|S_{i}|=s_{i}$, set $S_{i}$ is chosen uniformly among all $s_{i}$-size
subsets of $\mathcal{P}$. Finally, an undirected edge is assigned between two
nodes if and only if their corresponding object sets have at least one object
in common. There are variations of the general random intersection graph such
as binomial random intersection graphs and uniform random intersection graphs,
each of which focuses on a certain probability distribution $\mathcal{D}$. The
following theorem discusses the connectivity and robustness of random
intersection graphs.

######  Theorem 24 ([225]).

Consider a general random intersection graph
$\mathcal{G}(n,P_{n},\mathcal{D})$. Let $X$ be a random variable following
probability distribution $\mathcal{D}$. With a sequence $\alpha_{n}$ for all
$n$ defined through $\frac{\\{\mathbb{E}[X]\\}^{2}}{P_{n}}=\frac{\ln
n+(r-1)\ln\ln n+\alpha_{n}}{n}$, if $\mathbb{E}[X]=\Omega(\sqrt{\ln n})$,
$Var[X]=o(\frac{\\{\mathbb{E}[X]\\}^{2}}{n(\ln n)^{2}})$ and $\alpha_{n}=o(\ln
n)$, and $\lim_{n\to\infty}\alpha_{n}=\infty$, then the graph is
asymptotically almost surely $r$-connected and $r$-robust.

##  12 Future Directions

This paper provided an overview of the existing graph-theoretic tools which
can be used to analyze the resilience of distributed algorithms in networked
control systems. Comparing with system-theoretic approaches to the robustness
and fault tolerance of control systems, graph-theoretic approaches are
relatively new and demand more development, primarily in the following three
directions: (i) Developing graph-theoretic methods to facilitate analyzing a
wider range of distributed algorithms and more complex adversarial actions.
(ii) Reinterpreting the known system-theoretic notions of resilience and
robustness of dynamical systems from a graph-theoretic perspective. (iii)
Investigating the resilience of a wider range of distributed algorithms using
available graph-theoretic tools.

Here, we propose a few research avenues which are worth investigating in the
future.

Spectral Approach to Network Structures: One of the necessary steps towards
reconciling system-theoretic approaches and graph theory is to find algebraic
interpretations of certain network structures. Algebraic graph theory is an
active topic of research in mathematics [81]. However, specific structural
properties of networks that are widely used in analyzing the resilience of
networked control systems, e.g., network robustness, are quite new notions
defined within the field of systems and control. Hence, their algebraic
interpretations have not been well studied.

An example is a work on the relation of the algebraic connectivity and the
network robustness. Defining the edge-boundary of a set of nodes
$S\subset\mathcal{V}$ is given by $\partial{S}=\\{(i,j)\in\mathcal{E}\mid i\in
S,j\in\mathcal{V}\setminus{S}\\}$. The isoperimetric constant of $\mathcal{G}$
is defined as [226]

$i(\mathcal{G})\triangleq\min_{S\subset\mathcal{V},|S|\leq\frac{n}{2}}\frac{|\partial
S|}{|S|}.$ (48)

Based on the above definition and the definition of the network robustness we
conclude that if $i(\mathcal{G})>r-1$, then the graph is at least $r$-robust.
Moreover, we have $\lambda_{2}(L)\leq 2i(\mathcal{G})$ [226]. Based on this,
if $\lambda_{2}(L)>r-1$, then the network is at least
$\lfloor\frac{r}{2}\rfloor$-robust [223, 216]. However,
$\lfloor\frac{r}{2}\rfloor$ provides a loose lower bound for the network
robustness. An example is a star graph which is 1-robust with
$\lambda_{2}(L)=1$. Further research is needed to be done in this direction.

Resilience with Minimum Communication: Due to the fact that communications
between nodes can be costly in many applications, the problem of reaching a
certain level of resilience with minimum communication, i.e., edges between
nodes, is worth investigating. In specific dynamical systems, e.g., consensus
dynamics, adding edges may also degrade the controllability of the system. Few
recent works have focused on minimizing the number of edges while reaching a
certain level of security [227] or maximizing the connectivity without
violating the controllability of the system [228]. While reaching a certain
level of connectivity with the minimum number of edges has been well-studied
in the literature, the same problem for network robustness remains open.

Graph-Theoretic Approach to Attack Energy: In Section 9 some graph-theoretic
interpretations of the attack’s impact was discussed. Among those, the attack
energy has not been well studied. The objective of the attacker, other than
impact and detectability, can be to access the system with minimum energy. One
way to quantify the attack energy is via using the spectra of the
controllability Gramian $\mathcal{W}_{\mathcal{F}}$. An interesting research
avenue is to design the network to maximize the attack energy using an
appropriate spectrum of the controllability Gramian.

Resilience of other Classes of Random Graphs: The study of the connectivity
and network robustness of random graphs with the various probability
distribution of edge formation is another important future research line.
Among them are scale-free networks in which their degree distributions follow
a power-law. Many networks have been reported to be scale-free, including
preferential attachment and the fitness model [229]. Analyzing the structural
properties of these classes of networks enables us to quantify the resilience
of distributed algorithms on various natural and man-made large-scale systems.

Resilience of Networks with Nonlinear Interactions: Throughout this survey
paper, the focus was on NCSs in which the interactions between agents are
linear. In some cases of NCSs, the local interactions are nonlinear, e.g.,
synchronization of Kuramoto Oscillators with applications to power systems
[230] and attraction-repulsion functions in swarm robotics [231]. Extension of
the methods discussed in this survey to those classes of nonlinear systems
requires further investigations.

Resilience of Multi-Agent Reinforcement Learning Algorithms: Another emerging
area of interest pertains to enabling a team of agents to cooperatively learn
optimal policies for interacting with their environment, particularly when the
dynamics of the environment are not initially known to the agents. Such multi-
agent reinforcement learning settings share some common features with the
distributed consensus, optimization, and estimation problems we described
earlier in the paper, in that adversarial agents can provide incorrect
information about their observations and costs to the other agents. However,
these problems also introduce additional lines of complexity for resilience,
in that adversarial agents can also affect the underlying system with their
inputs. There have been recent initial explorations of resilient algorithms in
such settings [232, 233, 234, 235], but much work remains to be done to
understand how to mitigate adversaries that can not only send incorrect
information but can also take destructive actions.

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