# Knowledge-Based Prediction of Network Controllability Robustness

Yang Lou  Yaodong He  Lin Wang  Kim Fung Tsang  and Guanrong Chen  Y. Lou, Y.
He, K.F. Tsang and G. Chen are with the Department of Electrical Engineering,
City University of Hong Kong.L. Wang is with the Department of Automation,
Shanghai Jiao Tong University and also with the Key Laboratory of System
Control and Information Processing.Supported by NSFC 62002249 and 61873167,
and HK ITF ITP/058/17LP.(Corresponding author: Guanrong Chen.)This paper has
been published in IEEE Transactions on Neural Networks and Learning Systems,
https://doi.org/10.1109/TNNLS.2021.3071367.

###### Abstract

Network controllability robustness reflects how well a networked system can
maintain its controllability against destructive attacks. Its measure is
quantified by a sequence of values that record the remaining controllability
of the network after a sequence of node-removal or edge-removal attacks.
Traditionally, the controllability robustness is determined by attack
simulations, which is computationally time consuming or even infeasible. In
the present paper, an improved method for predicting the network
controllability robustness is developed based on machine learning using a
group of convolutional neural networks (CNNs). In this scheme, a number of
training data generated by simulations are used to train the group of CNNs for
classification and prediction, respectively. Extensive experimental studies
are carried out, which demonstrate that 1) the proposed method predicts more
precisely than the classical single-CNN predictor; 2) the proposed CNN-based
predictor provides a better predictive measure than the traditional spectral
measures and network heterogeneity.

###### Index Terms:

Complex network, convolutional neural network, controllability, robustness,
knowledge-based prediction.

##  I Introduction

Complex networks as an interdisciplinary research field has gained growing
popularity since the late 1990s, encompassing network science, systems
engineering, applied mathematics, statistical physics, and biological as well
as social sciences [1, 2, 3, 4]. Scientific studies are trying to understand
the essence and characteristics of complex networks while engineering studies
are trying to control them for beneficial applications. In the pursuit of
networked systems control, whether or not they can be controlled is a
fundamental issue, which leads to the basic concept of network controllability
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

The concept of controllability refers to the ability of a networked system in
changing from any initial state to any desired state under a feasible control
input in finite time [15]. In retrospect, it was shown that identifying the
minimum number of external control inputs (recalled driver nodes), needed to
achieve the structural controllability of a directed network, which requires
searching for a maximum matching of the network [5]. Thereafter, in [6], an
efficient measure is introduced for assessing the state controllability, based
on the rank of the network controllability matrix, for both directed and
undirected networks.

It took quite a long time for people to understand the intrinsic relation
between topology and controllability of a general, mostly directed network. It
was found that clustering and modularity have no prominent impact on the
network controllability, but degree correlation has a certain effect [7]. It
was revealed [8] that random networks of any topology are controllable by an
extremely small number of driver nodes if both of its minimum in- and out-
degrees are greater than two. A control centrality was introduced in [16] to
measure the importance of nodes regarding their roles against random attacks.
The network controllability of some canonical graph models is studied and
compared in [17]. For growing networks, the evolution of network
controllability is investigated in [18]. Moreover, the controllability of
multi-input/multi-output networked systems is studied in [10, 19], with
necessary and sufficient conditions derived. Recently, it was realized that
some special motifs such as loops and chains are beneficial for enhancing the
robustness of network controllability against attacks [20, 21, 22]. A
comprehensive survey of the subject is presented in [15].

Regarding the controllability robustness against attacks, which includes
random failures and malicious destructions, a large number of studies have
been reported [23, 24, 25, 16, 26, 27, 28]. For node- or edge-removal attacks,
the main issue is to develop a measure that reflects how well the networks can
maintain their controllability after the attacks took place. One measure for
the network controllability is quantified by the number of driver nodes needed
to recover or retain the network controllability after the occurrence of an
attack, while its robustness is quantified by a sequence of values that record
the remaining levels of the network controllability after a sequence of
attacks [21]. To optimize the network robustness, one usually aims to enhance
and maintain a highest possible connectedness of the network against attacks
[25]. Given a degree-preserving constraint (i.e., the degree of each node
remains unchanged through the process of optimization), an edge-rewiring
method is proposed in [29], which increases the number of edges between high-
degree nodes, so as to generate a new network with a largest $k$-shell
component. In [30], the structure of a network is modified by degree-
preserving edge-rewiring, where a spectral measure is employed. By optimizing
a specified spectral measure of the network through random edge-rewiring, the
robustness of the resultant network is enhanced consequently. However, it was
noted that the correlation between spectral measures and the robustness is
indeed unclear [31]. Nevertheless, given a reliable predictive measure or
indicator of the network robustness, optimization algorithms can be applied
[32, 33, 34, 35]. In the case that there are more than one predictive measure,
multi-objective optimization schemes can be adopted [36, 28]. In [37], it is
shown that network robustness against edge- and node-removals can be enhanced
simultaneously. A common observation is that heterogeneous networks with
onion-like structures are robust against attacks [25, 38, 39, 40]. The
evolution of alternative attack and defense is studied in [41], where attack
refers to edge-removal and defense means edge-replenishment. The connectedness
of the largest-sized cluster is a commonly-used measure for such robustness
[25]. It is noted that, although the connectedness robustness has a certain
positive correlation with the controllability robustness, they have very
different characteristics and measures.

Although the correlation between network topology and network controllability
has been investigated, no specific theoretical indicator or performance index
was found that can precisely quantify the general network controllability
robustness. The nature of the attack methods leads to different measures of
the importance of a node (or an edge) in a network. In the literature, degree
and betweenness are two commonly-used measures for the importance of nodes and
edges, respectively [42].

It was observed that a power-law degree distribution does not necessarily
imply a fragile controllability robustness against targeted node-removals;
what really contributes to enhance the network controllability robustness is
the multi-chain structure [43] and multi-loop structure [20, 21]. Later, it
was observed [22] that it is particularly beneficial to the network
controllability robustness if the multi-loops are across the entire networks
rather than only within local communities. Lately it was empirically observed
[44] that to achieve optimal controllability robustness against random node
attacks, both in- and out-degree distributions of a directed network should be
extremely homogeneous.

On the other hand, in the field of machine learning, deep neural networks have
shown powerful capability in performing classification and regression tasks in
image processing. Convolutional neural network (CNN) is one kind of effective
deep neural network [45]. CNN is able to automatically analyze inner features
of a dataset without human interference. But, if the user has some prior
knowledge and it can be ensured that such prior knowledge would not mislead
machine learning, then CNN will become even much powerful for data analysis
and processing. Successful real-world applications of CNNs include text
recognition and classification [46, 47, 48], face recognition and detection
[49], image segmentation [50], etc.

(a) BA unshuffled

(b) ER unshuffled

(c) QSN unshuffled

(d) SW unshuffled

(e) BA shuffled

(f) ER shuffled

(g) QSN shuffled

(h) SW shuffled

Figure 1: An example of filtering out the generation-based features. The
network size $N=200$ with average degree $\langle k\rangle=5.12$.

Traditionally, for large-scale complex networks, their controllability
robustness is evaluated by attack simulations, which however are extremely
computationally time consuming. The major computational cost includes: 1) to
search for the node to attack, e.g., the maximum betweenness node; 2) the
calculation of controllability, e.g., Eq. (2). Both have to be calculated
iteratively. To improve the efficiency of prediction for controllability
robustness, this paper takes a machine learning-based approach to designing a
knowledge-based predictor for the controllability robustness (iPCR), which is
an improved version of the single CNN-based predictor of the controllability
robustness (PCR) developed in [51], taking advantage of available prior
knowledge.

One unique feature of this iPCR is that it can be applied to both directed and
undirected networks, since there is no essential difference for the CNN to
process an image converted from a directed or an undirected network, where the
symmetry in the graph-converted image does not affect the learning of the CNN.
As such, the proposed iPCR has a much wider application range than other
traditional methods.

Another improved mechanism in iPCR is that the graph-converted images are
updated independently of the generation process. Given an unweighted network,
its adjacency matrix can be converted to a black-white image, where a black
pixel represents a $0$ element and a white pixel represents a $1$. Given a
weighted network with real-valued elements in the adjacency matrix, a gray-
scale image is plotted. As shown in Fig. 1, the upper row shows the intrinsic
features of the images pertaining to the generation process. These biased
features are filtered out by shuffling the rows and columns of the image, as
shown in the lower row of the figure. For example, in a Barabási–Albert (BA)
scale-free network, the preferential attachment mechanism gives the ‘old’
nodes higher degrees, which are usually allocated near each other therefore
have small numbers in the adjacency matrix. As a result, there are always some
sparks in the BA-converted image, as shown in Fig. 1 (a). The generation-based
features necessarily mitigate the task of a CNN in classification and
regression. Therefore, in the iPCR, these special features are filtered out by
shuffling the rows and columns of the images, as shown in the lower row of
Fig. 1.

To briefly summarize, the proposed design of the iPCR is based on the
following observations: 1) there is no clear correlation between the
topological features and the controllability robustness of a general network,
directed or undirected, 2) the adjacency matrix of a network can be
equivalently represented as a gray-scale image, 3) the CNN technique has
proved successful in image processing, and 4) prior knowledge at hand could be
sufficiently utilized as preprocessing and filtering tools. In the iPCR, a
number of training data generated by simulations are used to train the group
of CNNs for classification and prediction, respectively.

Extensive experimental studies are carried out, which demonstrate that 1) the
proposed method predicts more precisely than the single-CNN predictor; 2) the
CNN-based prediction method provides a better predictive measure than the
traditional spectral measures and network heterogeneity.

The rest of the paper is organized as follows: Section II reviews the network
controllability and its robustness against various destructive attacks.
Section III introduces the proposed iPCR. In Section IV, experimental study is
performed with analysis. Finally, Section V concludes the investigation.

##  II Network Controllability and Its Robustness

Consider a linear time-invariant networked system described by $\dot{{\bf
x}}=A{\bf x}+B{\bf u}$, where $A$ and $B$ are constant matrices of compatible
dimensions, $\bf x$ is the state vector, and $\bf u$ is the control input. The
system is state controllable if and only if the controllability matrix $[B\
AB\ A^{2}B\ \cdots A^{N-1}B]$ has a full row-rank, where $N$ is the dimension
of $A$, also the size of the networked system. If a system is state
controllable, then its state vector $\bf x$ can be driven from any initial
state to any desired state in the state space by a suitable control input $\bf
u$ within finite time. The concept of structural controllability is a slight
generalization dealing with two parameterized matrices $A$ and $B$, in which
the parameters characterize the structure of the underlying networked system:
if there are specific parameter values that can ensure the parametric system
be state controllable, then the system is structurally controllable.

The controllability of a network, or networked system, is measured by the
density of the controlled nodes, $n_{D}$, defined by

$n_{D}\equiv\frac{N_{D}}{N},$ (1)

where $N_{D}$ is the number of driver nodes needed to retain the network
controllability, and $N$ is the network size. This measure $n_{D}$ allows
networks with different sizes can be compared. In comparison, the smaller the
$n_{D}$ value is, the better the network controllability will be.

For a directed network, the number $N_{D}$ can be calculated according to the
minimum inputs theorem derived based on maximum matching [5]. A maximum
matching is a matching that contains the largest possible number of edges,
which cannot be further extended in the network. A node is matched if it is
the end of an edge in the matching; otherwise, it is unmatched. When a maximum
matching is found, the number $N_{D}$ of driver nodes is determined by the
number of unmatched nodes, i.e. $N_{D}=\text{max}\\{1,N-|E^{*}|\\}$, where
$|E^{*}|$ is the number of edges in the maximum matching $E^{*}$.

As for an undirected network, its controllability can be calculated according
to the exact controllability theorem derived based on the controllability
matrix [6]. Given an undirected network, its number $N_{D}$ of driver nodes is
calculated by

$N_{D}=\text{max}\\{1,N-\text{rank}(A)\\}.$ (2)

The measure of controllability robustness is calculated by

$n_{D}(i)\equiv\frac{N_{D}(i)}{N-i}\,,\ \ i=1,2,\ldots,N-1,$ (3)

where $N_{D}(i)$ is the number of driver nodes needed to retain the network
controllability after a total of $i$ nodes have been removed, and $N$ is the
original network size. When these values are plotted, a curve is obtained,
called the controllability curve.

To compare the controllability robustness of two networks against the same
attack sequence, their controllability curves are plotted against each other
for better visualization. Numerically, a controllability curve $c$ is given by
an $(N-1)$ vector
$n_{D}^{c}=[n_{D}^{c}(1),n_{D}^{c}(2),\cdots,n_{D}^{c}(N-1)]$. Thus, given two
controllability curves, $c_{1}$ and $c_{2}$, the difference (error) of the two
curves, when the same number of $i$ nodes are removed, is calculated by

$\sigma(i)=|n_{D}^{c_{1}}(i)-n_{D}^{c_{2}}(i)|\,.$ (4)

The average error $\bar{\sigma}$ is then calculated by

$\bar{\sigma}=\frac{1}{N-1}\sum_{i=1}^{N-1}{\sigma(i)}\,.$ (5)

The vector $\sigma(i)$ is used to measure the error between the predicted
controllability curve against the true curve; while the scalar $\bar{\sigma}$
measures the overall error of prediction.

The overall network controllability robustness $R_{c}$ is defined as [52, 27]

$R_{c}=\frac{1}{N-1}\sum_{i=1}^{N-1}n_{D}(i)\,,$ (6)

where, as an extension of the robustness measure defined in [25], $n_{D}(i)$
represents the controllability of the network when a total of $i$ nodes have
been removed from the network. Given two complex networks under the same
attack, the one with a lower $R_{c}$ value is considered having better
controllability robustness.

In the following, for convenience in description, sometimes the integer index
sequence $i=1,2,\ldots,N-1$ will be replaced by the fractional index sequence
$p=\frac{1}{N},\frac{2}{N},\ldots,\frac{N-1}{N}$, thereby equivalently
replacing $n_{D}(i)$ ($i=1,2,\ldots,N-1$) with $n_{D}(p)$
($p=\frac{1}{N},\frac{2}{N},\ldots,\frac{N-1}{N}$).

##  III Predictor for Network Controllability Robustness

###  III-A Framework of Predictor

Figure 2: The framework of PCR [51], where a single CNN is used for
controllability robustness (CR) prediction. The input is an image converted
from the adjacency matrix; the output is the corresponding controllability
curve.

The framework of PCR is shown in Fig. 2, where a single CNN is trained for
prediction, referred to as a predictor. This framework straightforwardly
performs fairly good predictions, with an overall error less than the standard
deviation of the testing samples [51]. However, there are two main issues
about this framework. First, many of the PCR predicted controllability curves
are vibrating (especially in the initial stage of the attacks), while the real
controllability curves are generally smooth. Second, PCR ignores all available
prior knowledge, and trains the single CNN using the raw data without any
preprocessing. The proposed iPCR addresses the above two issues by employing a
multi-CNN structure, with an extra filter.

To address the first issue, in the new framework a data processer called
filter is installed after the prediction and before the output. Using
available prior knowledge about the dynamics, upper and lower boundaries of
the controllability curve can be pre-set. Specifically, during a node-removal
attack process, where $i$ ($i=1,2,\ldots,N-1$) nodes are removed, the upper
and lower bounds of the controllability curve at position $i$ are pre-set as
follows:

$ub(i)=\frac{\text{min}(N_{D}^{0}+i,N-i)}{N-i}\,,$ (7)

and

$lb(i)=\frac{1}{N-i}\,,$ (8)

where $ub(i)$ and $lb(i)$ represent the upper and lower bounds of $n_{D}(i)$,
respectively; $N_{D}^{0}$ means the minimum required number of driver nodes
for the original network before being attacked, which can be calculated by Eq.
(2).

The following boundary processor is designed:

$n_{D}(i)=\left\\{\begin{array}[]{lr}ub(i),&\text{if }n_{D}(i)>ub(i),\\\
lb(i),&\text{if }n_{D}(i)<lb(i).\\\ \end{array}\right.$ (9)

Based on this, a median filter with a mask length $L$ is implemented.

To address the second issue, although human-intervention-free is one of the
most attractive properties of deep learning, some available knowledge and
common sense may be used if such human knowledge would not mislead the machine
learning process. Such prior knowledge of the network data will be
preprocessed before prediction. For instance, if the network topology is known
beforehand, then the prediction work can be passed to a CNN that is
specialized for such a topology, which can have better prediction performance.

Figure 3: The framework of iPCR, where a CNNc is used for network
classification. If the input data can be clearly classified as one specific
group, then iPCR uses a specifically trained CNN$i$, $i=1,2,\ldots,nc$, where
$nc$ is the number of clusters; otherwise, if the input data is non-
classifiable based on the current knowledge, then iPCR degenerates to PCR
using a single CNNall.

iPCR consists of a group of CNNs, including a classifier CNNc and several
predictors CNN$i$ ($i=1,2,\ldots,nc$), as shown in Fig. 3. All the predictors
have the same CNN structure but are trained by different datasets. Each CNN$i$
($i=1,2,\ldots,nc$) is trained by the specific cluster of data, such that it
is specialized in predicting a cluster, although probably not suitable for
another. CNNall is trained by all the training data. CNNc is trained by
applying the prior knowledge of the user.

In the experimental study, two types of prior knowledge are tested, namely the
network topology (presented in Sec. IV) and the node degree (presented in
Supplementary Information (SI)111https://fylou.github.io/pdf/ipcrsi.pdf due to
space limitation in the paper). Experimental results show that the former
provides helpful prior knowledge, while the later is misleading and
consequently the prediction results are degenerated. Finally, before
outputting the predicted results, iPCR operates a filter that includes a
boundary processor (as shown in Eq. (9)) and a median filter.

###  III-B Convolutional Neural Network

The iPCR framework, which includes a classifier, several predictors and a
filter, is now introduced along with its configuration and parameter settings.

Figure 4: [Color online] The architecture of the CNN used for networks classification and controllability robustness prediction, where FM is an abbreviation for feature map, and FC for fully connected. The data sizes $N_{i}=\lceil N/(i+1)\rceil$, $i=1,2,\ldots,7$. The concatenation layer rearranges the matrix into a vector, from FM $7$ to FC $1$, i.e., $L_{0}=N_{7}\times N_{7}\times 512$. $L_{1}\in(L_{0},N-1)$ is a hyperparameter. Set $L_{1}=4096$ in this paper. For prediction, another fully-connected layer FCp is used as the output layer, yielding an $(N-1)$ vector in the output. For classification, a fully-connected layer FCc followed by a softmax layer is used. The output is labeled according to the input data. TABLE I: Parameter settings of the seven groups of convolutional layers. Group | Layer |  | Kernel
---
size
Stride |  | Output
---
channel
Group 1 | Conv7-64 | 7x7 | 1 | 64
Max2 | 2x2 | 2 | 64
Group 2 | Conv5-64 | 5x5 | 1 | 64
Max2 | 2x2 | 2 | 64
Group 3 | Conv3-128 | 3x3 | 1 | 128
Max2 | 2x2 | 2 | 128
Group 4 | Conv3-128 | 3x3 | 1 | 128
Max2 | 2x2 | 2 | 128
Group 5 | Conv3-256 | 3x3 | 1 | 256
Max2 | 2x2 | 2 | 256
Group 6 | Conv3-256 | 3x3 | 1 | 256
Max2 | 2x2 | 2 | 256
Group 7 | Conv3-512 | 3x3 | 1 | 512
Max2 | 2x2 | 2 | 512

Fig. 4 shows the detailed CNN structure. The detailed parameter settings of
the $7$ groups of convolutional layers are given in Table I. Here, the CNN
architecture follows the Visual Geometry
Group222http://www.robots.ox.ac.uk/~vgg/ architecture [53]. The number of
feature map (FM) groups is set to $7$, since the input size is $1000\times
1000$ in the following experiments. Note that this number should be set to be
greater for a larger input dataset. Each FM consists of a convolution layer, a
ReLU, and a max pooling layer. A ReLU provides a commonly-used activation
function $f(x)=\text{max}(0,x)$ [54]. The output of each hidden layer, i.e., a
multiplication of weights, is summed up and then rectified by the ReLU for the
next layer. The max pooling layer reduces the dimension of the dataset for the
input to the next layer.

For prediction, the output is an $(N-1)$ vector that represents the predicted
controllability curve; while for classification, the output is a vector of
$nc$ real numbers, $pl_{i}$ ($\sum_{i=1}^{nc}pl_{i}=1$) that represents the
probability of the input image belonging to cluster $i$, ($i=1,2,\ldots,nc$).
The output layer of the classifier is implemented by adding an extra softmax
[55] layer in the end.

A threshold $\eta$ is defined for the classifier such that, only if there
exists $pl_{i}\geq\eta$ ($i=1,2,\ldots,nc$), it returns a result indicating
that the input image is classifiable (belonging to cluster $i$); otherwise,
the input is recognized as non-classifiable. A too-low threshold will decrease
the successful rate, while a too-high threshold may result in many non-
classifiable cases. In simulations, $\eta$ is set to $0.8$, which yields a
successful rate higher than $0.8333$ in classification (see Tables II and
III). The results will not be sensitively influenced when $\eta$ is slightly
changed.

Note that for different purposes, the internal weights of the CNN will be set
differently. Here, for illustration, the structures of predictors and
classifiers are plotted together. But a predictor and a classifier do not
share any internal weight, and each CNN works independently in iPCR.

The loss function used in the classifier is the cross entropy. Given the
predicted and the true probability distribution of an instance, denoted by
$pl$ and $tl$ respectively, the cross entropy of this instance is calculated
as follows:

$H=-\sum_{i=1}^{nc}tl(i)\cdot\text{log}[pl(i)]\,.$ (10)

The loss function used in the predictor is equal to the mean-squared error
between the predicted controllability curve $pv$ and the true curve $tc$,
which is calculated as follows:

$\mathcal{L}=\frac{1}{N-1}\sum_{i=1}^{N-1}||pv(i)-tv(i)||\,,$ (11)

where $||\cdot||$ is the Euclidean norm.

The training process for the classifier and predictor aims at adjusting the
weights of CNNs, with the objectives of minimizing the cross entropy in Eq.
(10) and the mean-squared error in Eq. (11), respectively.

It is worth mentioning that resizing is commonly used for CNNs to process
inputs with different sizes. However, therein the application scenario is
different. A row and a column represent a node together with all its connected
edges, therefore resizing will change the original topology. Nevertheless, the
experimental study presented in Sec. IV-D shows that, if the information loss
is very small, the prediction results are still acceptable. There are a few
works that deal with different network sizes by using the same CNN, with
additional assumptions, for example, [56]. In this paper, CNNs are used to
process raw data of complex networks, without any assumption or knowledge on
the network structures, and thus the input size can be fixed.

Source codes of this work are available for the
public333https://fylou.github.io/sourcecode.html.

##  IV Experimental Studies

###  IV-A Experimental Settings

Four typical undirected synthetic networks are adopted for simulation, namely
the Barabási–Albert (BA) scale-free network [57], Erdös–Rényi (ER) random-
graph network [58], q-snapback network (QSN) [20, 22], and Newman–Watts (NW)
small-world network [59].

In the following subsections, the generation methods and parameter settings of
the above four networks are introduced, respectively.

Note that, given the network size $N$ and average degree $\langle k\rangle$,
there are $M=\lfloor N\cdot\langle k\rangle\rfloor$ edges in total. Standard
notation $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and
ceiling functions, respectively.

####  IV-A1 Barabási–Albert (BA) Scale-Free Network

A BA network is generated as follows:

  * •

Start with $n_{0}$ fully-connected nodes (i.e., an $n_{0}$-clique).

  * •

For nodes $i$ ($i=n_{0}+1,\ldots,N$), each of them connects to each of nodes
$j$ ($j=1,\ldots i-1$) with a probability of
$p_{BA}=\frac{k_{j}}{\sum_{l}k_{l}}$, where $k_{j}$ denotes the degree of node
$j$. At each step, there are $e_{BA}$ edges being added preferentially.

Set $n_{0}=\lceil\langle k\rangle\rceil+1$ and
$e_{BA}=\frac{M-\binom{n_{0}}{2}}{N-n_{0}}$. To exactly control the number of
the generated edges to be $M$, proportionally adding or removing edges can be
performed.

####  IV-A2 Erdös–Rényi (ER) Random-Graph Network

An ER network is generated as follows:

  * •

Start with $N$ isolated nodes.

  * •

Pick up all possible pairs of nodes from the $N$ given nodes, denoted as $i$
and $j$ ($i\neq j$, $i,j=1,2,...,N$), once and once only. Connect each pair of
nodes with a probability $p_{ER}\in[0,1]$.

Let $p_{RG}=\frac{\langle k\rangle}{N-1}$. To exactly control the number of
the generated edges to be $M$, uniformly-randomly adding or removing edges can
be performed.

####  IV-A3 q-Snapback Network

The q-snapback network (QSN) was originally constructed as a directed network
[20] but is converted to be an undirected one here, with only one layer
$r_{QSN}$ for simplicity. It is generated as follows:

  * •

Start with a chain of $N$ nodes, where each node $i$ ($i=2,...,N-1$) has two
edges connecting to its neighboring nodes $i-1$ and $i+1$.

  * •

For each node $i=r_{QSN}+1,\,r_{QSN}+2,\ldots,N$, it connects backward to the
previously-appeared nodes $i-l\times r_{QSN}$
($l=1,2,\ldots,\lfloor\frac{i}{r_{QSN}}\rfloor$), with the same probability
$q\in[0,1]$.

The probability parameter $q$ can be calculated from the given $N$, $M$, and
$r_{QSN}$. For $r_{QSN}=1$,
$q=\frac{(M-N)\cdot{r_{QSN}}}{\sum_{j=2+r_{QSN}}^{N-2}{j}}=\frac{M-N}{\sum_{j=3}^{N-2}{j}}$.
To exactly control the number of the generated edges to be $M$, uniformly-
randomly adding or removing edges can be performed.

####  IV-A4 Newman–Watts (NW) Small-World Networks

An NW network is generated as follows:

  * •

Start with an $N$-node loop having $K$ connected nearest-neighbors on each
side.

  * •

Some edges are added without removing any existing edges, until totally $M$
edges have been added.

Set $K=2$ in the following; that is, a node $i$ is connected to its two
nearest neighbors on each side, i.e., with nodes $i-1$, $i+1$, $i-2$ and
$i+2$.

Since the above generation methods will generate networks with some strong
visible features (as illustrated by Fig. 1), the rows and columns of the
resulting adjacency matrices are shuffled and random isomorphs are generated,
so as to filter out these undesirable features.

The training data are generated by performing attack simulations on the
generated networks, such that the controllability curves (see Eq. (3)) can be
obtained. These graph-converted images and controllability curves are used for
iPCR training. Given an input graph-converted image, the trained iPCR can be
used to predict its controllability curve, skipping the time-consuming attack
simulation process.

Next, the prediction performances of PCR and iPCR are compared on 1)
unweighted networks against random node-removal attacks (see Sec. IV-B), 2)
weighted networks against targeted node-removal attacks (see Sec. IV-C); 3)
real-world networks under random attacks (see Sec. IV-D). In these three
experimental studies, PCR and iPCR aim to predict precise controllability
curves. In Sec. IV-E, PCR and iPCR are compared on predicting the ordinal
ranks of the network controllability robustness, with respect to $6$ spectral
measures and the heterogeneity. Finally, the computational costs are briefly
discussed in Sec. IV-F. In all the following comparisons, a filter consisting
of a boundary processor and a median filter (with $L=9$) are installed in both
PCR and iPCR.

###  IV-B Unweighted Networks Under Random Attacks

The controllability robustness prediction on unweighted networks with size
$N=1000$ and average degree $\langle k\rangle=3$, $4$, and $5$, under random
node-removal attacks, is studied.

There are $12$ network configurations in total. For each configuration, $500$
training samples are used. Each sample includes an adjacency matrix (as the
input) and its controllability curve obtained from simulation (as the output).
CNNall of iPCR is trained by $12\times 500=6000$ training samples; while each
of CNN1,2,3,4 is trained by $3\times 500=1500$ samples. Each CNNk
($\text{k}=1,2,3,4$) is specifically trained for one of the four network
types, namely BA, ER, QSN, and NW; while CNNall is trained by the ensemble of
all the networks. PCR is trained in the same way as CNNall of iPCR. Given a
random attack, in simulation the result is averaged from $10$ independent
runs, so as to balance mitigating and randomness influences, which also
reduces the burden of computation.

Another set of $100$ testing samples for each network configuration are
generated independently. The classification results of CNNc are shown in Table
II. As illustrated by Fig. 1, shuffling filters out the method-generated
features in the resultant images, resulting images indistinguishable by eyes,
which makes the classification task becoming tougher. It can be seen from
Table II that CNNc correctly classifies the four types of networks at a
successful rate higher than $0.8333$. Since the threshold is set to $\eta=0.8$
in the softmax layer of the classifier, an input is non-classifiable if it
generates a result with the probability of success less than $0.8$, and in
this case the input will be passed to CNNall for prediction. As can be seen
from Table II, the rate of non-classifiable (NC) data is low, indicating the
effectiveness of the classifier which uses prior knowledge.

Note that if an input is incorrectly classified, it will be passed to a wrong
predictor that is specialized for a different network type. This will totally
mislead the prediction, and therefore is harmful. In iPCR, according to Table
II, BA and QSN may be mis-classified as ER at rates $0.0922$ and $0.0165$,
respectively; NW may be mis-classified QSN at a rate $0.0042$; ER will not be
mis-classified to other networks, but becomes non-classifiable at rate
$0.0254$. Overall, the classification error rates are quite low, so iPCR is
proved working well.

TABLE II: Confusion matrix of the CNNc classifier for classifying unweighted networks. NC means the input is non-classifiable; (pre) represents the predicted type and (act) represents the actual type of the network. |  | BA
---
(pre)
| ER
---
(pre)
| QSN
---
(pre)
| NW
---
(pre)
NC
| BA
---
(act)
0.8333 | 0.0922 | 0 | 0 | 0.0745
| ER
---
(act)
0 | 0.9746 | 0 | 0 | 0.0254
| QSN
---
(act)
0 | 0.0165 | 0.9342 | 0 | 0.0494
| NW
---
(act)
0 | 0 | 0.0042 | 0.9833 | 0.0126
Figure 5: [Color online] Comparison of PCR and iPCR on unweighted networks
under random attacks. $P_{N}$ represents the portion of nodes having been
removed from the network; $n_{D}$ is calculated by Eq. (1). Green curves: the
true value (tVal) from simulation; red curves: predicted by PCR; black curves:
predicted by iPCR. The shaded shadow in the same color represents the range of
standard deviation.

The performance comparison between PCR and iPCR is shown in Fig. 5. In each
subplot, in a unique network configuration, the green curve shows the true
value (tVal) generated by simulation; the red dashed curve shows the predicted
values by PCR; the black dotted curve represents the predicted results of
iPCR. The shadow in the same color represents the range of standard deviation.
As can be seen from the plots, the black curves are obviously closer to the
green curves, better than the red curves, meaning that iPCR predicts the
controllability more accurately than PCR, in all $12$ cases. The results
confirm that prior knowledge of the network topology is indeed helpful if
correctly used. It is notable that the predicted curves are not as oscillatory
as those obtained in [51], thanks to the filters used in both PCR and iPCR.

(a) $N$=$1000$, the inset gives a zoom-in for $P_{N}\in[0.9,1]$.

(b) $N$=$600$

(c) $N$=$1200$

(d) weighted, targeted attacks

Figure 6: [Color online] The average errors ($\sigma$) comparison of PCR and
iPCR: unweighted networks with (a) $N=1000$, (b) $N=600$, and (c) $N=1200$,
under random attacks; (d) weighted networks ($N=1000$) under targeted attacks.
The green curve (Std) represents the standard deviation of the testing
samples.

The average error calculated according to Eq. (4) is plotted in Fig. 6 (a),
where the black curve shows that iPCR has a lower average error ($\sigma$)
than the red dashed curve of PCR, through the entire attacking process. The
inset shows a clearer plot of the comparison for $P_{N}\in[0.9,1]$. The
average error is taken from all $12$ configurations, namely, a total of $1200$
testing samples. It is noticeable that both PCR and iPCR gain average errors
with standard deviations much lower than that of the testing set (i.e., the
$1200$ testing samples), throughout the entire attack process.

To verify the scaling property, given the same conditions except for the
network size, the prediction performances of iPCR and PCR are compared. Figs.
6 (b) and (c) show the average errors ($\sigma$) comparisons when $N=600$ and
$1200$, respectively. The detailed predicted controllability curves are shown
in SI.

###  IV-C Weighted Networks Under Targeted Attacks

The controllability robustness prediction on weighted networks with size
$N=1000$ and average degree $\langle k\rangle\in[3,5]$, under targeted node-
removal attacks, is studied.

For each network instance, its average degree $\langle k\rangle$ is a real
random number generated from the range of $[3,5]$; its edge weights are
uniformly-randomly assigned from the range of $(0,1)$. Again, PCR contains a
single CNN, while iPCR uses a CNNall with four specialized CNN1,2,3,4 for the
four types of networks respectively, if classifiable. The targeted attack
performs node-removals according to the degrees of nodes, from high to low
sequentially.

TABLE III: Confusion matrix of the CNNc classifier on classifying weighted networks. NC means the input is non-classifiable; (pre) represents the predicted type and (act) represents the actual type of the network; initial ‘w’ is for ‘weighted’. |  | wBA
---
(pre)
| wER
---
(pre)
| wQSN
---
(pre)
| wNW
---
(pre)
UC
| wBA
---
(act)
0.9913 | 0 | 0 | 0 | 0.0087
| wER
---
(act)
0 | 0.9549 | 0.0150 | 0 | 0.0301
| wQSN
---
(act)
0 | 0 | 0.9915 | 0 | 0.0085
| wNW
---
(act)
0 | 0 | 0.0074 | 0.9815 | 0.0111

The confusion matrix shown in Table III suggests that the precision of the
CNNc classifier is high. Slightly different from Table II, here the CNNc can
either correctly classify the weighted BA and QSN respectively, or return a
result of non-classifiable, without any mis-classification. The weighted ER
and NW have very low probabilities to be classified as weighted QSN. Shuffling
is also performed on these weighted networks. The overall precision on
classifying weighted networks is slightly higher than that on unweighted
networks.

In the experiments reported in Sec. IV-B, the average degree $\langle
k\rangle$ is set to integers $3,4,5$, respectively. Each column in Fig. 5
shows the same type of networks, with increasing $\langle k\rangle$ from $3$
to $5$. Although PCR and iPCR are trained without any information about the
average degrees, both PCR and iPCR return different predictions, when the
input is of the same network type with different average degrees. However,
this does not imply that average degree is a good feature or useful prior
knowledge. In contrast, the average degree is known to be not suitable for
preprocessing when used as prior knowledge. An example is given in the SI,
where three network clusters are defined, namely ‘$\langle k\rangle=3$’,
‘$\langle k\rangle=4$’, and ‘$\langle k\rangle=5$’. The prediction results is
distorted due to the low precision of classification. This demonstrates that
the prior knowledge used should be correct and appropriate, as common sense,
otherwise misleading could happen.

Figure 7: [Color online] Comparison of PCR and iPCR on weighted networks under
targeted attacks. $P_{N}$ represents the portion of nodes having been removed
from the network; $n_{D}$ is calculated by Eq. (1). Green curves: the true
value (tVal) from simulation; red curves: predicted by PCR; black curves:
predicted by iPCR. The shaded shadow in the same color represents the range of
standard deviation.

Fig. 7 shows the prediction results of PCR and iPCR on weighted networks under
targeted attacks. Again, it is clear that, in each subplot, the black dotted
curve is closer to the green curve than the red dashed curve. The higher
precision of iPCR in prediction is partially due to the high classification
rate presented in Table III. Another reason is that an specialized CNN
predictor is always better than a mixed one, as is intuitively so.

Fig. 6 (d) shows that the average prediction error of iPCR (black curve) is
lower than that of PCR (red dashed curve), throughout the entire attack
process. Note that both PCR and iPCR gain average errors with standard
deviations much lower than that of the testing samples through a long period.
Differing from random attacks, in a targeted attack, when the portion of
removed nodes is somewhat greater than $0.7$, the network requires
$n_{D}\approx 1$ to gain a full controllability. Although PCR and iPCR gain
lower predictive errors during this stage (when $P_{N}$ is somewhat greater
than $0.7$), the standard deviation of the testing sample actually becomes
nearly zero.

###  IV-D Real-world Networks Under Random Attacks

The PCR and iPCR trained in Sec. IV-B are used for predicting the
controllability robustness of $6$ real-world networks with $N\approx 1000$.
Basic information of these networks is given in Table IV; the network data are
from Network Repository 444http://networkrepository.com/.

TABLE IV: Basic information of the real-world networks. | abbr.
---
name
file name | brief description | $N$ | $M$
DDG | DD-g79 | protein | 1022 | 2889
DEL | delaunay-n10 | DIMACS10 problem | 1024 | 3056
DW5 | dwt-1005 |  | 1005 | 3808
DW7 | dwt-1007 |  symmetric connection from Washington | 1007 | 3784
LSH | lshp1009 |  | Alan George’s
---
L-shape problem
1009 | 2928
ORS | orsirr-1 |  | oil reservoir
---
simulation
1030 | 2914

(a) Prediction Results

(b) Prediction Errors

Figure 8: [Color online] Performance comparison of PCR and iPCR on real-world
networks under random attacks: (a) prediction results; (b) prediction errors.

Since the sizes of the real-world networks are slightly larger than $1000$
(see Table IV), resizing is performed on the graph-converted images, i.e., a
pair of row and column is randomly picked and removed until it reaches
$N=1000$. For each network, the random resizing is repeated for $20$ times,
and the prediction results and errors are then averaged. As can be seen from
Fig. 8 (a), iPCR predicts the controllability curves closer to the true curves
than PCR does, especially in the early stage. Fig. 8 (b) shows that iPCR
obtains clearly lower prediction errors than PCR for networks DEL, DW5, DW7
and LSH; while for DDG and ORS, PCR obtains lower errors than iPCR only in a
limited period at the middle stage. Both PCR and iPCR predict the
controllability performances of these real-world networks with low errors.

###  IV-E Comparison of Prediction Measures

Spectral measures have long been used to quantify the connectedness robustness
of complex networks against node- and edge-removal attacks. It has certain
positive correlation to controllability, but they cannot be treated equally.

Here, six commonly-used spectral measures, namely spectral radius (SR),
spectral gap (SG), natural connectivity (NCo), algebraic connectivity (ACo),
effective resistance (ERe), and spanning tree count (STC) are compared in
measuring the controllability robustness. Definitions and computational
formulas for these measures can be found in, e.g., [30]. Recently, it was also
found that heterogeneity (HE) reflects the controllability robustness [22]. In
this paper, PCR and iPCR are used to predict the entire controllability curves
in Secs. IV-B and IV-C. Noticed that a predicted curve (a vector) can also be
converted to a measure (a scalar) through Eq. (6). Thus, in the following, the
above $9$ prediction measures will be compared, namely, the $6$ common
spectral measures (SR, SG, NCo, ACo, ERe, STC), HE, PCR and iPCR.

The above prediction measures are used to predict the ordinal ranks of a total
of $1200$ networks, for four network types with three different average
degrees. These ranks are listed in a descending order in terms of
controllability robustness, from the best to the worst. Specifically, each
prediction measure returns a predicted rank list of the $1200$ networks. Then,
the $9$ rank lists returned by the $9$ prediction measures are compared to the
true rank list generated by simulation. The rank error information is
summarized in Table V, where the rank error $\sigma_{rank}$ is calculated by

$\sigma_{rank}=|rl_{pm}-rl_{t}|\,,$ (12)

where $rl_{pm}$ represents the rank list predicted by a prediction measure,
and $rl_{t}$ represents the true rank obtained from simulation.

For example, given two predicted rank lists, $rl_{pm1}=[3,5,2,4,1]$ and
$rl_{pm2}=[2,1,4,5,3]$, and a true rank list, $rl_{t}=[1,2,3,4,5]$, the rank
errors are obtained as $\sigma_{rank1}=[2,3,1,0,4]$,
$\sigma_{rank1}=[1,1,1,1,2]$, respectively. The mean, maximum, and minimum of
the rank error can be calculated accordingly. The number of ‘$0$’ elements in
a rank error list $\sigma_{rank}$ is counted and then included in the ‘correct
rank’ column. Finally, the number of networks, which are predicted top $10\%$
and then confirmed top $10\%$ by simulation, in terms of controllability
robustness, is counted and included in the ‘top $10\%$’ column. The number in
the ‘bottom $10\%$’ column is similarly calculated. The detailed rank values
of the $9$ prediction measures are given in SI.

TABLE V: Rank error information for the $9$ predictive measures. Bold number means the best performing prediction measures. |  | average
---
rank
error
| max
---
rank
error
| min
---
rank
error
| correct
---
rank
| top
---
10%
| bottom
---
10%
SR | 387.52 | 912 | 0 | 3 | 0 | 5
SG | 370.09 | 933 | 1 | 0 | 0 | 7
NCo | 394.58 | 993 | 0 | 3 | 0 | 5
ACo | 496.98 | 1095 | 0 | 1 | 0 | 0
ERe | 160.64 | 704 | 0 | 2 | 52 | 100
STC | 547.14 | 1192 | 2 | 0 | 0 | 0
HE | 187.13 | 910 | 0 | 2 | 13 | 91
PCR | 112.72 | 481 | 0 | 2 | 46 | 104
iPCR | 103.73 | 590 | 0 | 6 | 45 | 97

It can be seen from Table V that iPCR receives the minimum average rank error
$103.73$, followed by PCR. PCR obtains the minimum max rank error, followed by
iPCR. Seven out of nine predictive measures receive a min rank error $0$,
meaning that these measures predict at least one rank exactly as the true
rank. iPCR predicts $6$ ranks correctly. ERe predicts $52$ top $10\%$ networks
that are truly top $10\%$ networks, although their exact ranks may be
different, which are followed by PCR and iPCR that correctly predict $46$ and
$45$ networks respectively. Finally, PCR well predicts $104$ bottom $10\%$
networks, followed by ERe and iPCR. The test dataset contains $1200$ networks,
hence there are $120$ networks ranked as top and bottom $10\%$, respectively.

It is thus clear that PCR and iPCR return better prediction measures than the
spectral measures and the heterogeneity. More importantly, PCR and iPCR return
not only the predictive measures, but also the entire controllability changing
process of a network against the node-removal attack; while the spectral
measures and heterogeneity return only a single quantitative value for the
controllability robustness.

However, it is notable that PCR and iPCR require a substantial amount of
training data, while the spectral measures and heterogeneity do not.
Nevertheless, as demonstrated in [51], the overhead in training a CNN is quite
low which, compared to the exhaustive attack simulation, is negligible.

###  IV-F Computational Costs

Compared to PCR, iPCR employs an extra CNN for classification, while the rest
computation costs of PCR and iPCR are similar. Thus, the computational cost of
iPCR is around two times of that of PCR. As discussed in [51], the cost of
simulations to measure the controllability robustness is non-trivial. Compared
to simulations, PCR accelerates the prediction speed by hundreds of times.

Given a PC with a 64-bit operation system, installed Intel i7-6700 (3.4 GHz)
CPU, GeForce GTX 1080 Ti GPU, to collect a controllability curve for an ER
network with $N=1000$ and $\langle k\rangle=5$ under random node attack, the
elapsed time is about $162$ seconds for simulation, $0.42$ seconds for PCR,
and $1.22$ seconds for iPCR. An example of run time comparison (including PCR,
iPCR, and attack simulation) is available on web with source
codes555https://fylou.github.io/sourcecode.html.

##  V Conclusions

Network controllability robustness, which reflects how well a networked system
can maintain its controllability after destructive attacks, is usually
measured via attack simulations. Such an exhaustive simulation approach can
return the true value of the controllability robustness, but is
computationally costly and very time consuming. The predictor of
controllability robustness (PCR) employs a single convolutional neural network
(CNN) to successfully and efficiently achieve the prediction. In this paper,
an improved multi-CNN and knowledge-based PCR (iPCR) is designed and
evaluated, which takes advantage of prior knowledge from the given data.
Extensive experimental studies, with thorough comparisons to seven other
comparable measures, demonstrate that 1) iPCR predicts more precisely than
PCR; 2) iPCR provides a better predictive measure than the traditional
spectral measures and network heterogeneity.

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