# AGE RANGE ESTIMATION USING MTCNN AND VGG-FACE MODEL

Dipesh Gyawali1, Prashanga Pokharel2, Ashutosh Chauhan3, Subodh Chandra
Shakya4 (1dipesh9393, 2pokharel997, 3Cashutosh6711<EMAIL_ADDRESS>
InfoDevelopers Pvt. Ltd.
Lalitpur, Nepal

###### Abstract

The Convolutional Neural Network has amazed us with its usage on several
applications. Age range estimation using CNN is emerging due to its
application in myriad of areas which makes it a state-of-the-art area for
research and improve the estimation accuracy. A deep CNN model is used for
identification of people’s age range in our proposed work. At first, we
extracted only face images from image dataset using MTCNN to remove
unnecessary features other than face from the image. Secondly, we used random
crop technique for data augmentation to improve the model performance. We have
used the concept of transfer learning in our research. A pretrained face
recognition model i.e VGG-Face is used to build our model for identification
of age range whose performance is evaluated on Adience Benchmark for
confirming the efficacy of our work. The performance in test set outperformed
existing state-of-the-art by substantial margins.

###### Index Terms:

Age Range Estimation, CNN, MTCNN, Transfer Learning, VGG-Face

##  I INTRODUCTION

The face images are used for different applications recently in face
recognition[1], surveillance system[15], emotion identification[16], and smart
attendance for multifarious purposes. Identifying the age range using CNN has
become a challenging task as the age range among the adults seems to be
similar than the child and aged people[8]. The facial features among the child
and old people vary drastically from the adult people which makes it easier to
identify the age of child and old people[7]. But the identification among the
adult age becomes cumbersome due to the presence of almost the same facial
features.

By far, CNN has become most effective for processing image data in object
detection, face recognition, and image classification. Image data consists of
a lot of parameters which are difficult for a basic neural network to process.
A lot of matrix calculation needs to be handled for 2D image data where CNN is
a most effective neural network as different convolutional layers in CNN are
used to extract the feature map of images and the convolutional process with
the filter make it easy to process image data.

In our research, we have identified the age range among various ages of people
using novel approach. Transfer learning has been considered more effective
than building our own custom CNN model for a limited number of dataset. The
model which was pre-trained for image classification and face recognition[3]
is used for estimating the age range.

Figure 1: Face images of different age groups on Adience Benchmark[11]

##  II RELATED WORKS

A dataset including one million faces of celebrities are prepared which
eventually improved face recognition accuracy[1]. Face recognition using new
dataset across various identities minimizing the label noise[2],using either a
single image or set of faces[3], analysis and survey[4] is performed. Age
Estimation is done using VGG-Face model[5], the ranking SVM[6] and finding
sufficient embedding space by applying manifold learning methods which models
data with multiple linear regression function[7] effectively. A craniofacial
growth model is proposed which models growth related variations in shape of
human faces[8]. Different faces are divided into several minute regions for
the extraction of Local Binary Pattern(LBP) to use as a face descriptor by
concatenating into feature vectors[9]. Two new approaches, Ranking-CNN[10]
based on rank relationship and deep CNN architecture for low quality face
images[24] are proposed for identifying age from different perspectives. Using
unfiltered faces on Adience Benchmark[11] and in the presence of limited
learning data using deep CNN[12], identification of gender and age is
performed. Various research challenges in addition with recent survey are
delineated for research in face recognition along with age estimation[13]. The
effects of aging on performance of age invariant face recognition are
surveyed[14] and analyzed using deep features analysis[21].

The above works showed that a handful of research tasks have already been
carried out using different metrics and methods. Although a plethora of works
have been done, a significant improvement in the identification of age range
across adult age group due to similar facial features had not been achieved.
In this experiment, we have initially extracted only the facial features from
joint face alignment using MTCNN network and later identified age range using
VGG-Face model which significantly reduced overfitting and improved
identification among adult age group as compared to previous works. Also, a
comparative analysis has been done across multiple face models to analyze
efficiency of our work using unconstrained face images.

##  III DATA PREPARATION

MTCNN (Multi-task Cascaded Convolutional Networks) is a network[17] which is
used in our proposed work to extract the facial features. MTCNN contains 3 CNN
steps, each step called as Proposal Network (P-net), Refine Network (R-net)
and Output Network (O-net). All the input images are fed to Proposal Network
(P-Net), which is a CNN model. The candidate window inside the image with
their bounding box regression vector is received as output from P-net. We
create an image pyramid, in order to detect faces of all different sizes as
shown in Fig. 2.

Figure 2: Image pyramids and NMS and bounding box regression (P-Net)

After feeding those candidates to Refined Network (R-Net), we have identified
regression vector of bounding box and used Non-Max Suppression (NMS) to find
and integrate densely overlapped candidates as shown in Fig. 3. The outcomes
from this network are passed to O-Net.

Figure 3: NMS and bounding box regression (R-Net)

Finally, Output Network (O-Net) provides three output, coordinate of bounding
box, the coordinate of five facial landmarks, and the confidence level of each
box as shown in Fig. 4. This bounding box image is saved as a new image which
is then passed into the proposed model.

Figure 4: Output-network

Images in the dataset are rescaled to 256x256 pixels. These images are
extracted of size 224x224 pixels using random crop[18]. Horizontal flip and
rotation is applied to the extracted images for data augmentation purpose. All
age groups were divided into 8 classes. For each class, images were splitted
80-20 for training and validation. Finally this dataset is fed to the network.

##  IV PROPOSED SYSTEM

A large number of images are required for building our own custom model to
ameliorate its performance. In order to prevent overfitting, we have used
transfer learning for building our CNN network. The model is well validated
and tested after optimizing it using appropriate hyperparameters. The VGG-Face
model performs well on Adience Benchmark[11] due to its less number of CNN
layers which can be useful for small dataset.

###  IV-A Architecture

For the purpose of age estimation, the pre-trained model used for face
recognition task[3] is considered extremely useful. The age estimation is
generally performed by observing the facial features of different age
groups[7]. Due of this, for the estimation of age robustly, model previously
trained on face recognition can be used. By the use of pre-trained models,
there will be less chance of overfitting. Also, we have used MTCNN for
extracting the facial features while training in our dataset.

Figure 5: CNN Architecture

To identify a better age range, we have used pre-trained model VGG-Face. The
model performed pretty well using VGG-Face due to its less number of CNN
layers and appropriate tuning of hyperparameters. Eight convolutional layers
comprising of Conv2D & Pooling and three fully connected layers makes up a
overall of eleven layers of VGG-Face model. All the layers except fully
connected softmax layers were used for training purposes. The layers from VGG-
Face are kept frozen first and then additional layers are added to the VGG-
Face model as per our requirements. Two fully connected layers are added in
corresponding with one dropout layer and one softmax output layer to identify
the age range for 8 classes. The layers and the respective dimensions used in
our network are represented as in the Table I.

Table I: Our proposed CNN architecture using VGG-Face Model

Layer type name 0 input - 1 conv conv1_1 2 relu relu1_1 3 conv conv1_2 4 relu
relu1_2 5 mpool pool1 6 conv conv2_1 7 relu relu2_1 8 conv conv2_2 9 relu
relu2_2 10 mpool pool2 11 conv conv3_1 12 relu relu3_1 support - 3 1 3 1 2 3 1
3 1 2 3 1 filt dim - 3 - 64 - - 64 - 128 - - 128 - num filts - 64 - 64 - - 128
- 128 - - 256 - stride - 1 1 1 1 2 1 1 1 1 2 1 1 pad - 1 0 1 0 0 1 0 1 0 0 1 0
Layer type name 13 conv conv3_2 14 input relu3_ 2 15 conv conv3_3 16 relu
relu3_3 17 mpool pool3 18 conv conv4_1 19 relu relu4_ 1 20 conv conv4_ 2 21
relu relu4_2 22 conv conv4_ 3 23 relu relu_3 24 mpool pool4 25 conv conv5_1
support 3 1 3 1 2 3 1 3 1 3 1 2 3 filt dim 256 - 256 - - 256 - 512 - 512 - -
512 num filts 256 - 256 - - 512 - 512 - 512 - - 512 stride 1 1 1 1 2 1 1 1 1 1
1 2 1 pad 1 0 1 0 0 1 0 1 0 1 0 0 1 Layer type name 26 relu relu5_1 27 conv
conv5_2 28 relu relu5_2 29 conv conv5_3 30 relu conv5_3 31 mpool pool5 32 conv
fc6 33 relu relu6 34 dropout dropout7 35 conv fc8 36 relu relu7 37 softmax
prob support 1 3 1 3 1 2 7 1 1 1 1 1 filt dim - 512 - 512 - - 512 - - 1000 - -
num filts - 512 - 512 - - 1000 - - 100 - - stride 1 1 1 1 1 2 1 1 1 1 1 1 pad
0 1 0 1 0 0 0 0 0 0 0 0

###  IV-B Loss Function

We have multiple classes of age-range ranging from 0-2 to 60-100 with a total
of eight classes. The actual and predicted probability distributions for all
eight classes are identified whose average differences are summarized with a
score calculated using cross entropy[19]. A total of eight nodes are present
at the output layer for each age range in which ’softmax’ activation is used
to predict the probability across each classes. Due to a multiclass
classification problem and effectiveness, we have used categorical cross
entropy as a loss function[19].

$CE=-\sum_{c=1}^{M}t_{o,c}log\left(P_{o,c}\right)$ (1)

where t is either 0 or 1 if label class c is the correct classification for
observation o. M is a total number of classes. P is predicted probability of
class c.

###  IV-C Optimizer

Due to efficient computation, little memory requirements and presence of
intuitive interpretations of hyperparameters, we have used Adam optimizer[20]
in our research which is considered more effective comparatively with other
optimizers.

###  IV-D Training

We used transfer learning(VGG-Face) to train our model. The convolutional
layer parameters of VGG-Face are not changed and kept frozen. We optimize the
fully connected layers parameters by adding additional two fully connected
layers, one dropout layer and a final output layer to the VGG-Face model.
Number of filters in the first of two fully connected layers are 1000 and 100
consecutively. In these fully connected layers we use ‘relu’ activation
function. We use 0.3 (30% chance of setting a neuron’s output value to zero)
values for one dropout layer. ’Softmax’ activation is used in the final layer
to classify the age range. ‘Adam optimizer’ is used as an optimizer with
‘categorical cross entropy’ as a loss function in our model.

###  IV-E Prediction

After training the model, we have used the Adience Benchmark to evaluate the
model performance. A test image is rescaled to 256x256 pixels. Then five
images are extracted of size 224x224 pixels. The four images are obtained from
four corners of the image and the final image is obtained from the center of
the original test image as shown in Fig. 6. These five images are fed into our
trained neural network to calculate the softmax probability output vector of
each of the five images. These output scores vector of these five images were
averaged. Average result helps to reduce the impact of poor quality and low-
resolution images.

Figure 6: Image Rescaling to 256x256 pixels initially. Five images are
extracted from each four corners and centre which are resized to 224x224 for
feeding into the network.

##  V EXPERIMENTS AND RESULTS

A wide known and flexible deep learning library i.e Keras consisting of
Tensorflow as a backend engine is used in our proposed work for effectiveness.
Training was done on NVIDIA GTX 2080 GPU with 4352 CUDA cores. Our training
time took approximately four hours. We trained VGG-Face Model for 50 epochs
with a batch size of 64. The model performance is evaluated using Adience
benchmark. The Adience benchmark is a large-scale face dataset which consists
of 19487 multiple face images with 8 classes of age range covering large
variation in facial expression, pose, occlusion, resolution and illumination.
The exact accuracy on Adience benchmark came out to be 70.96%. We have used
this Adience benchmark for appraising the efficacy of our work. Different age
range’s label used in our research with the corresponding number of images per
label on Adience Benchmark are shown in Table II.

Table II: The Adience Benchmark

0-2 4-6 8-13 15-20 25-32 38-43 48-53 60+ Total Male 745 928 934 734 2308 1294
392 442 8192 Female 682 1234 1360 919 2589 1056 433 427 9411 Both 1427 2162
2294 1653 4897 2350 825 869 19487

Consequently, our proposed model outperformed the previous methods reported in
[5], [11], and [12].The accuracy was compared with the previous work done for
age estimation in which our model performs better as compared to other
performances. Table III shows exact accuracy and 1-off accuracy from the
previous work and our proposed work using VGG-Face architecture. Our proposed
work outperforms the previous work and confirm the efficiency of our proposed
work. The use of MTCNN approach has significantly improved accuracy on Adience
Benchmark. Fig. 7 shows the confusion matrix on Adience Benchmark.

Table III: Age estimation result on the Adience Benchmark

Method Exact Accuracy 1-off Accuracy [11] 45.1 79.5 [12] using single crop
49.5 84.6 [12] using over-sample 50.7 84.7 [5] 59.9 90.57 Proposed work 70.96
92.7

Figure 7: Normalized Confusion Matrix for Adience Benchmark. The label is
marked from a-h to make ease while building confusion matrix. The label is
defined as a(0-2),b(4-6),c(8-13),d(15-20),e(25-32),f(38-43),g(48-53),h(60+).

The true label and the predicted label are denoted from indices of each rows
and each columns respectively in the confusion matrix. The occurences of
prediction is given by number showing on each cell. In the leading diagonal of
confusion matrix, the true label and predicted label are equal and other off-
diagonal elements represent occurence mislabeled by classifier. There are more
correct predictions if there are higher values in the leading diagonal. The
confusion matrix in Fig. 7. clearly shows probability of prediction. Our
result from the confusion matrix depicts that the highest accuracy is 98.90%
which of (0-2) age range as shown in Fig. 7. The highest accuracy is due to
the distinctive features which enable the classifier to distinguish easily.
The images of middle age groups consist of almost similar features which
resulted in less accuracy as compared to (0-2) age range and (60+) age range.
In our work, the age group of (48-53) and (25-32) is highly misclassified with
41.67% and 47.30%. The classification report on Adience Benchmark using VGG-
Face model is shown in the Table IV.

Table IV: Classification Report on Adience Benchmark

Age Range From Adience Dataset Precision Recall F1-Score Support 0-2 0.95 0.99
0.97 1427 4-6 0.86 0.80 0.83 2162 8-13 0.81 0.75 0.78 2294 15-20 0.60 0.68
0.64 1653 25-32 0.58 0.47 0.52 4897 38-43 0.49 0.67 0.57 2350 48-53 0.66 0.42
0.51 825 60+ 0.70 0.80 0.75 869 Accuracy 0.71 19487 macro avg 0.71 0.70 0.69
19487 weighted avg 0.72 0.71 0.71 19487

We have also used VGG-Face2 Model to analyze the efficacy of our task. Our
proposed work using VGG-Face model proved to be far better than VGG-Face2
model which is shown in the Table V. There are more numbers of CNN layers on
VGG-Face2 as compared to VGG-Face. As we had not enough face image data, the
VGG-Face2 model was subjected to overfitting and could not perform well so
that the performance across VGG-Face2 model seems very poor as compared to
VGG-Face model.

Figure 8: Some of the correctly classified images on Adience Benchmark Figure
9: Some of the misclassified images on Adience Benchmark

Fig. 8 and Fig. 9 shows some of correctly classified and misclassified images
on Adience Benchmark with their corresponding confidence level. The
misclassified classes are mainly of adults due to same facial features. The
age range of child and old age were almost classified correctly. The number of
misclassified class could be improved further if we have high number of images
for training. Different models can be used if there is copious amount of data
in addition with proper pre-training process which might improve our proposed
work.

Table V: Exact Accuracy on Different Face Models

Label Proposed model using VGG-Face Model using VGG-Face2 0-2 98.90 80.40 4-6
79.69 36.52 8-13 75.19 29.64 15-20 68.12 52.92 25-32 47.30 10.37 38-43 67.48
13.08 48-53 41.67 49.89 60+ 79.75 77.86 Overall Exact Accuracy 70.96 42.02

##  VI CONCLUSION

We have used the VGG-Face model and MTCNN to extract the facial features for
estimating the age range for our research work. MTCNN helped to extract only
the facial features from the image data which helps to determine the most
germane features from the face. Our research work outperforms the previous
work by almost 12% on the Adience Benchmark. Due to the small number of
dataset for training and large number of layers for feature extraction, the
VGG-Face2 model is subjected to overfitting which does not perform well while
evaluating its performance. Usage of MTCNN and fine tuning the VGG-Face model
significantly improved the network’s performance for age range estimation.

Furthermore, there are still some misclassified images with high confidence
because of the same facial features, lightness, occlusions and multiple person
in an image. If the number of images in various conditions are increased and
pre-training tasks of CNN network are performed significantly, the model
performance can increase further. Also, VGG-Face2 model performance can be
improved by using high number of face images and fine-tuning the network. The
exact accuracy is still low due to the small number of dataset available for
age estimation which could be improved in future works.

## ACKNOWLEDGMENT

Our research is based upon work done at InfoDevelopers Pvt. Ltd. We would like
to express gratitude towards our supervisor Mr. Sandesh Pandey for supervising
and evaluating our research work. We are also thankful to Info Developers
Private Limited for providing us the research platform and materials regarding
our research procedures.

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Annamaria Ficara, Giacomo Fiumara, Pasquale De Meo 11institutetext: University
of Messina, MIFT Department, 98166 Messina, Italy,
11email<EMAIL_ADDRESS>22institutetext: University of
Messina, DICAM Department, 98168 Messina, Italy,
22email<EMAIL_ADDRESS>
# Classical and Quantum Random Walks to Identify Leaders in Criminal Networks

Annamaria Ficara${}^{\textrm{({\char 0\relax})}}$ 11 0000-0001-9517-4131
Giacomo Fiumara 11 0000-0003-1528-7203 Pasquale De Meo 22 0000-0001-7421-216X
Salvatore Catanese 11 0000-0002-0369-8235

###### Abstract

Random walks simulate the randomness of objects, and are key instruments in
various fields such as computer science, biology and physics. The counter part
of classical random walks in quantum mechanics are the quantum walks. Quantum
walk algorithms provide an exponential speedup over classical algorithms.
Classical and quantum random walks can be applied in social network analysis,
and can be used to define specific centrality metrics in terms of node
occupation on single-layer and multilayer networks. In this paper, we applied
these new centrality measures to three real criminal networks derived from an
anti-mafia operation named Montagna and a multilayer network derived from
them. Our aim is to (i) identify leaders in our criminal networks, (ii) study
the dependence between these centralities and the degree, (iii) compare the
results obtained for the real multilayer criminal network with those of a
synthetic multilayer network which replicates its structure.

###### keywords:

Classical random walks, quantum random walks, centrality, criminal networks

##  1 Introduction

The term “random walk” was firstly introduced in 1905 by Pearson [20]. Random
walks can be useful to solve practical problems. In fact, the randomness of
objects can be analyzed and simulated through them. Also the correlation among
objects can be computed using random walks. They are fast becoming key
instruments in various fields such as physics, economics, computer science,
biology, or chemistry [25].

A simple random walk model in the mathematical space is a random process on a
regular lattice where at each step one point jumps to another position
according to a certain probability distribution [25]. Random walks can be
applied to a network (or graph) defined as a set of nodes with edges between
them. Graphs can be organized into multiple layers, which represent different
types of nodes or edges. In this kind of application, the transition
probability between nodes is greater when the strength of the association
between nodes is stronger. After an appropriate number of steps, a random path
able to describe the structure of a network can be obtained. In computer
science, the PageRank is the most known algorithm based on random walks [19].
This algorithm randomly walks among web pages to compute their importance.

Aharonov, Davidovich and Zagury were the first to propose quantum walks in
1993 [1]. These walks can be considered as the counter part of classical
random walks in quantum mechanics. The main difference between quantum walks
and classical random walks is that the former do not converge to some limiting
distributions [25]. Quantum interference can cause significantly faster or
slower spreading of quantum walks compared to classical random walks.
Algorithms based on quantum walks have lower time complexity with respect to
the ones based on classical random walks, and they provide an exponential
speedup over any classical algorithm [11].

Random walks can find an application in computer vision, semi-supervised
learning, network embedding, and complex social network analysis [17, 22].
Some researchers also studied the application of random walks on knowledge
discovery, graphs, science of science, and text analysis [22, 2, 12]. Quantum
walks can be used instead to element distinctness, search problems, and
decision trees [23, 3]. Moreover, classical algorithms are often accelerated
by using quantum walks.

Wald and Böttcher [24] at first introduced a framework for classical, quantum,
and hybrid random walks with stochastic resetting on networks. Then, Böttcher
and Porter [6] used continuous-time classical and quantum random walks to
formulate occupation, PageRank, betweenness, and closeness centralities for
multilayer networks.

In this paper, we decided to apply classical and quantum occupation
centralities on very specific kinds of real networks, i.e. Mafia networks.
These networks are different from traditional social networks because first of
all they consist entirely of suspected criminals, and then they are considered
as organizations of criminal groups and activities. For this reason, research
on criminal networks is focused on their structure, efficiency, resilience and
capacity to achieve criminal objectives. To the best of our knowledge,
classical and quantum walks have never used to identify leaders in criminal
networks. Criminal leaders are strategic brokers within criminal networks [8].
Moreover, to reinforce our finding, we want to compare the results of
classical and quantum walks on Mafia networks to the one obtained on Barabási-
Albert (BA) [5] models, which according to our previous work [9], are the one
to better replicate criminal networks.

Mafia networks are in fact a good example of real-world networks with respect
to the geometry of connections [16]. Indeed, these connections are the
building blocks of the entire Mafia, and more generally of organized crime.
The Sicilian Mafia is one of the major native mafia-like organizations. It is
composed by loose confederations of about one hundred families (or cosche).
Since at least the 19th century, the social and economic life especially in
Southern Italy is affected by these families [18]. Social network analysis and
graph theory can be applied to Mafia networks to describe their structure and
functioning, to identify leaders [15], to evaluate police interventions aimed
at disrupting them [10], or to construct crime prevention systems [7].

##  2 Classical and Quantum Random Walk Centrality Measures

###  2.1 Classical Random Walks

Random walks on graphs are sequences of nodes which start from a source node
by selecting an edge, crossing the edge to reach a new node, and repeating the
process. An equivalent concept can be found in the Markov chain which is a
stochastic process that assumes values in a discrete set, where the next state
of the chain only depends on the current state and not on the past states
[21]. The Markov chain can be viewed as a graph where nodes are represented by
the states and edges by the possible next states, which are randomly
determined. The set of states is discrete, but their evolution in terms of
time can be discrete or continuous.

In a discrete-time Markov chain, each step is associated with a probability
distribution, i.e. a set of probabilities that a walker is on a node or in a
state. This probability distribution can be described with a vector, after
choosing an order for the states. Given a graph $G(V,E)$ with set of nodes
$V=\\{v_{1},\ldots,v_{n}\\}\,(\lvert V\rvert=n)$ and set of edges $E$, the
probability distribution is described by the vector $[p_{1}(t)\cdots
p_{n}(t)]$, where $p_{1}(t)$ is the probability that a walker is on node
$v_{1}$ at time $t$ and so on. If the process begins with the walker on the
first node, $p_{1}(0)=1$ and $p_{i}(0)=0$ for $i=2,\ldots,n$.

The future position of the walker cannot be precisely identified, but it is
possible to determine a probability distribution knowing the transition matrix
$M$. By knowing the probability distribution at the time $t$, the distribution
at time $t+1$ can be obtained by the following formula:

$\displaystyle p_{i}(t+1)=\sum_{j=1}^{n}M_{ij}p_{j}(t)\,.$ (1)

In vector form, we have $\vec{p}(t+1)=Mp(t)$.

The entry $M_{ij}$ of the transition matrix is the probability of the walker,
who is on node $v_{j}$, to go to node $v_{i}$. The transition matrix is
related to the adjacency matrix $A$ of the graph, and it can be defined as:

$\displaystyle M_{ij}=\frac{A_{ij}}{k_{j}}\,,$ (2)

where $k_{j}$ is the degree of node $v_{j}$, $A_{ij}=1$ when two nodes $v_{i}$
and $v_{j}$ are connected or $A_{ij}=0$ if they are not. If there is no edge
from $v_{j}$ to $v_{i}$, the walker goes to one of the adjacent nodes and
transition probability is the same for all of them.

When time is a continuous variable, the problem of a walker who can go from
node $v_{j}$ to a neighbour $v_{i}$ at any time can be seen as a liquid
seeping from $v_{j}$ to $v_{i}$. At the beginning, it is likely that the
walker is in the node $v_{j}$. As time goes by, the probability that the
walker is in $v_{j}$ decreases, but the probability he is in one of its
neighbors increases. The transition between neighbors occurs with a
probability $\gamma$ per unit time, which is a transition rate presumably
constant for all times and for all nodes. An infinitesimal time interval
$\epsilon$ is generally used to set up and solve the differential equation of
the problem with continuous variables. Then, the probability that a walker
goes from node $v_{j}$ to $v_{i}$ is given by $\gamma\epsilon$. If node
$v_{j}$ has a degree $k_{j}$, it means it has $k_{j}$ neighbors. Therefore,
the probability that the the walker is in one of the neighbors of $v_{j}$
after time $\epsilon$ is $k_{j}\gamma\epsilon$, while the probability of
finding him in $v_{j}$ is $1-k_{j}\gamma\epsilon$. At this point, the entry
$M_{ij}(t)$ of the transition matrix at time $t$ is defined as the probability
that the walker on node $v_{j}$ goes to node $v_{i}$ in the time interval $t$:

$\displaystyle
M_{ij}(\epsilon)=\begin{cases}1-k_{j}\gamma\epsilon+O(\epsilon^{2}),\quad\textrm{if}\
i=j;\\\ \gamma\epsilon+O(\epsilon^{2}),\,\,\,\,\quad\qquad\textrm{if}\ i\neq
j.\end{cases}$ (3)

An auxiliary matrix called generating matrix $H$, is also defined as:

$\displaystyle H_{ij}(\epsilon)=\begin{cases}k_{j}\gamma,\quad\textrm{if}\
i=j;\\\ -\gamma,\,\quad\textrm{if}\ i\neq j\ \textrm{and adjacent};\\\
0,\qquad\textrm{if}\ i\neq j\ \textrm{and non-adjacent.}\end{cases}$ (4)

The next state of a Markov chain only depends on the current configuration of
the chain. The transition matrix can be multiplied at different times as
follows:

$\displaystyle M_{ij}(t+\epsilon)=\sum_{k}M_{ik}(t)M_{kj}(\epsilon)\,.$ (5)

The index $k$ runs over all the neighbours of node $v_{j}$. In fact, if the
walker is in node $v_{j}$, the probability that he goes to $v_{k}$ in the time
interval is $M_{kj}(\epsilon)$, independently of the value of $k$. If there is
no edge between $v_{j}$ and $v_{k}$ for a specific $k$, $M_{kj}(\epsilon)=0$.

By isolating the term $k=j$, using the Eqs. 3 and 4, and then by moving the
first term on the right-hand side to the left-hand side and dividing it by
$\epsilon$, the following differential equation is obtained:

$\displaystyle\frac{dM_{ij}(t)}{dt}=-\sum_{k}H_{kj}M_{ik}(t)\,$ (6)

The solution of Eq. 6 with initial condition $M_{ij}(0)=\delta_{ij}$ is:

$\displaystyle M(t)=e^{-Ht}\,.$ (7)

After the definition of the transition matrix, the probability distribution at
time $t$ can be easily obtained. If the initial distribution is $\vec{p}(0)$,
$\vec{p}(t)=M(t)\vec{p}(0)$.

###  2.2 Quantum Random Walks

The electron spin is described by quantum mechanics as a unit vector in the
Hilbert space $\mathbb{C}^{2}$ [21]. The spin up and the spin down are
respectively described by the vectors $\lvert 0\rangle$ and $\lvert 1\rangle$:

$\displaystyle\setstackgap{L}{1.2}\lvert
0\rangle=\bracketVectorstack{10}\,,\quad\lvert
1\rangle=\bracketVectorstack{01}\,.$ (8)

The notions of spin up and spin down refer to $\mathbb{R}^{3}$. Quantum
mechanics describes in fact the behavior of the electron before entering the
magnetic field. In this case and if the electron is somehow isolated from the
macroscopic environment, its spin state is described by a linear combination
of vectors $\lvert 0\rangle$ and $\lvert 1\rangle$. Therefore,
$\lvert\psi\rangle=a_{0}\lvert 0\rangle+a_{1}\lvert 1\rangle$, where the
coefficients $a_{0}$ and $a_{1}$ are complex numbers which satisfy the
constraint $\lvert a_{0}\rvert^{2}+\lvert a_{1}\rvert^{2}=1$.

The time evolution of an isolated quantum system is described by a unitary
transformation. If the state of the quantum system at time $t_{1}$ is
described by vector $\lvert\psi_{1}\rangle$, the system state
$\lvert\psi_{2}\rangle$ at time $t_{2}$ is obtained from
$\lvert\psi_{1}\rangle$ by a unitary transformation $U$, which depends only on
$t_{1}$ and $t_{2}$. Therefore, we have
$\lvert\psi_{2}\rangle=U\lvert\psi_{1}\rangle$.

The evolution postulate is to be written in the form of a differential
equation which provides a method to obtain operator $U$ once given the
physical context. It is called Schrödinger equation and it is as follows:

$\displaystyle\frac{d\lvert\psi\rangle}{dt}=-iH\lvert\psi\rangle\,.$ (9)

In the passage from the continuous-time classical walk model to the
continuous-time quantum walk model, the transition matrix is simply converted
to an equivalent unitary operator, and the vector which describes the
probability distribution to a state vector. Matrix $H$, given by Eq. 4 is
Hermitian, and therefore matrix $M$ given by Eq. 7 is not unitary. $M$ can be
easily made unitary within the context of Hilbert spaces by multipling $h$
with the imaginary unit, that is by replacing $H$ with $iH$. Then, the
evolution operator of the continuous-time quantum walk can be defined as
follows:

$\displaystyle U(t)=e^{-iHt}\,.$ (10)

If the initial condition is $\lvert\psi(0)\rangle$, the quantum state at time
$t$ is:

$\displaystyle\lvert\psi(t)\rangle=U(t)\lvert\psi(0)\rangle\,,$ (11)

and the probability distribution is:

$\displaystyle p_{k}=\lvert\langle k\lvert\psi(t)\rangle\rvert^{2}\,,$ (12)

where $k$ runs over all nodes of the graph (or states of the Markov chain) and
$\lvert k\rangle$ is the state of the computational basis which correspond to
the node $v_{k}$. The computational basis of space $\mathbb{C}^{2}$ is the set
$\\{\lvert 0\rangle,\lvert 1\rangle\\}$.

###  2.3 Random Walk Occupation Centralities

Recently continuous-time classical and quantum random walk centrality measures
was defined for single-layer and multilayer networks [24, 6].

Starting from Eq. 7, the explicit Euler integration scheme can be used to
simulate a classical random walker:

$\displaystyle p_{t+1}=1-H_{c}\,\Delta t\,p_{t}\,,$ (13)

where $H_{c}$ is the classical Hamiltonian defined as $H_{c}=LD^{-1}$, where
$L=D-A$ is the Laplacian matrix and $D$ is the degree matrix with
$D_{ii}=k_{i}$ if $i=j$ and $D_{ij}=0$ if $i\neq j$.

The classical random walks approaches for connected networks and sufficiently
long times a stationary probability distribution $p^{*}$ such that:

$\displaystyle
p^{*}_{i}=\sum\limits_{j=1}^{n}\frac{A_{ij}}{k_{j}}p^{*}_{j}=\frac{k_{i}}{\sum\limits_{i=1}^{n}k_{i}}\,.$
(14)

The stochastic vector $p^{*}_{i}$ indicates the stationary classic random walk
occupation probability $OP_{c}$ for a node $v_{i}$, and it can be used to
define a random walk centrality measure for graphs.

The Schrödinger equation (see Eq. 9) can be instead solved using the Crank-
Nicholson integration scheme [4, 24]. Expanding the evolution operator in Eq.
10 into a Taylor series, we have:

$\displaystyle\lvert\psi_{t+1}\rangle=-2i\Delta
tH_{q}\lvert\psi_{t}\rangle+\lvert\psi_{t-1}\rangle\,,$ (15)

where $H_{q}$ is the Hermitian quantum Hamiltonian defined as
$H_{q}=\mathcal{L}$, where $\mathcal{L}=D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$ is
the normalized version of the Laplacian matrix $\mathcal{L}$.

For continuous-time quantum walks, unitary time evolution does not lead to a
stationary state. Instead, the long-time behavior of a quantum walk is
characterized by its long-time mean:

$\displaystyle
q^{*}_{i}=\lim\limits_{T\to\infty}\frac{1}{T}\int_{0}^{T}\langle
i\rvert\lvert\psi(t)\rangle\langle\psi(t)\rvert\lvert i\rangle dt$ (16)

where $dt$ is an infinitesimal time step,
$\lvert\psi(t)\rangle\langle\psi(t)\rvert$ is a density operator, and $\lvert
i\rangle\in\mathbb{C}^{\mathcal{N}}$ is an orthonormal basis vector.

Even in this case, $q^{*}_{i}$ denotes the continuous-time quantum random walk
occupation probability $OP_{q}$ of a node $v_{i}$, and therefore it gives a
type of random walk centrality.

##  3 Methods and Results

We used the classical and quantum occupation centralities described in
Subsect. 2.3 to study the importance of nodes in three real single-layer
criminal networks (Meetings, Phone Calls and Crimes) related to an anti-mafia
operation called Montagna [7, 10, 13], from which we derived a fourth network
in the form of a flattened multilayer network. The Montagna operation focused
on the Mistretta and Batanesi clans who monopolized the sector of public
contracts in the Tyrrhenian strip and in the nebroidal district of the
province of Messina, by working with some entrepreneurs associated to the
Sicilian Mafia. Meetings contains $256$ physical meetings between $101$
suspected criminals observed during the police physical surveillance. Phone
Calls contains $124$ phone calls between $100$ suspects wiretapped during the
police audio surveillance. Crimes contains $74$ connections between $25$
suspected criminals who committed crimes together. The three networks share
$20$ nodes. We also created an undirected and weighted multilayer network with
$226$ actors, $454$ intralayer edges and $3$ layers corresponding to the three
single-layer criminal networks Meetings, Phone Calls and Crimes. In this work,
a single-layer network (flattened) is created by merging the three layers of
the multilayer network. In the flattened network, there is one node for each
actor of the starting multilayer network and an edge between two nodes if the
corresponding actors are linked in one layer. Meeting, wiretap and co-
offending networks provide fundamentally different pictures of the criminal
organization, and therefore are all particularly useful in finding leaders.
Also the flattened multilayer network can be useful to have a more complete
picture of the organization. Figure 1 shows Meetings, Phone Calls, Crimes and
the flattened multilayer network.

Figure 1: Meetings, Phone Calls, Crimes and the flattened multilayer network
from the Montagna anti-mafia operation.

Our analysis was conducted using the mathematical framework111Available at:
https://gitlab.com/ComputationalScience/ctqw˙centrality of continuous-time
classical and quantum centrality measures created by Böttcher and Porter [6].

The top $20$ nodes ranked by their classical and quantum occupation
probabilities are shown in Fig. 2. In the analysis of central nodes in a Mafia
network, we have to consider the hierarchical structure of a Mafia family in
which individuals can be categorized as boss, underboss, consigliere (i.e., an
advisor to the boss), messaggero (i.e., a messenger among families),
caporegime (i.e., who heads a local crew of soldiers), soldier (i.e., average
type criminal) and associate (e.g., entrepreneurs, pharmacists, police
officers, politician etc.). The results for the Meetings and Phone Calls
networks confirm the one obtained through the application of the degree as we
did in [15], revealing that the most important actors are nodes $18$ and $47$.
These nodes are respectively Caporegime of the Mistretta family and deputy
Caporegime of the Batanesi family. The application of $OP_{C}$ and $OP_{q}$ on
the Crimes network reveals instead different central nodes who are not key
members of the Mafia hierarchy but are associates such as entrepreneurs (i.e.,
nodes $54$ and $64$), construction workers (i.e., node $72$) or soldiers
(i.e., nodes $48$ and $49$). Differently from the application of degree on the
multilayer network, in this case, we obtain different central nodes such as
$103$, $136$ and $111$. Nodes $103$ and $111$ were intercepted because they
were in phone contact respectively with a Caporegime of the Mistretta family
(i.e., node $18$) and a non-existent entrepreneur, created for the purpose of
fraud (i.e., node $109$). Node $136$ is instead a cohabitant of a Caporegime
of the Batanesi family (i.e., node $27$). The role of these figures in the
mafia business in unclear but their centrality can direct investigations and
lead to deeper investigation of them. Classical and quantum occupation
probabilities identify in most cases the same leaders in criminal networks.
Therefore, when a micro approach is adopted in SNA application to organized
crime focusing, as in our case, on one organization or network with a
relatively low number of nodes, there is no clear advantage to using quantum
random walks instead of classical ones. However, quantum walks can be useful
to speed up the investigations, and to find leaders when a macro approach is
adopted to study crime through SNA. In this case, in fact, larger networks
focusing on specific national or regional criminal markets or offender
categories are analyzed.

Figure 2: 20 top ranked nodes computed with classical $OP_{c}$ and quantum
$OP_{q}$ random walk occupation probabilities in the single layer networks
Meetings, Phone Calls and Crimes, and the flattened Montagna multilayer
network.

In Fig. 3, we show the probabilities that a node with a certain degree is
occupied by a classical and quantum random walker. As explained in [6], we can
observe a linear dependence between $OP_{c}$ and $OP_{q}$ which is due to the
fact that quantum walks satisfy Eq. 14 because they do not approach a
stationary state. Their long-time behavior is instead characterized by the
long-time mean as shown in Eq. 16. Quantum walks are not proportional to the
degree which is not able to capture node occupation properties of this kind of
walks. These properties, in fact, also depend on other structural features of
the underlying networks. The minimum and maximum degrees of the Meetings,
Phone Calls, Crimes networks are respectively $1$ and $24$, $1$ and $25$, $1$
and $13$, as shown in the plot of the degree distribution in [14]. The
differences between these classical and quantum centralities seem to be
greater in the flattened multilayer network.

Figure 3: Classical and quantum random walk occupation probabilities for the
single layer networks Meetings, Phone Calls and Crimes, and the flattened
Montagna multilayer network that a node of degree $k$ is occupied by a
classical and quantum random walker, respectively.

In [6], the authors asserted that the differences between the classical and
quantum occupation centralities were (i) larger in BA single-layer networks
[24] or in synthetic multilayer networks which include BA layers, and (ii)
smaller than those observed for synthetic networks. The second results is not
valuable for real criminal networks, and this can be explained just through
the first finding. In our previous work [9], we used some popular network
models such as Erdös-Rényi, Watts-Strogatz and different configurations of BA
to replicate the topology of a criminal network. Our experiments identified
the BA model as the one which better represents a criminal network. To prove
our thesis, we decided to build a flattened multilayer network with three BA
layers. Each layer is created with the same number of nodes of the Meetings,
Phone Calls, Crimes networks (i.e., $101$, $100$, and $25$, respectively) and
a number of edges as close as possible to that of these networks (i.e., $256$,
$124$, and $74$, respectively). BA models are built based on a specific
parameter $m$, which indicates the number of edges that are preferentially
attached to existing nodes with high degree. For the first layer, we tried two
different configurations of the BA graphs with $m=2$ (obtaining $198$ edges)
and $m=3$ (obtaining $294$ edges). For the second layer, we used $m=1$
(obtaining $99$ edges) and $m=2$ (obtaining $196$ edges). For the third layer,
we choose $m=3$ (obtaining $66$ edges) and $m=4$ (obtaining $84$ edges).
Figure 4 shows classical and quantum occupation centralities versus the node
degrees on the described synthetic multilayer network.

Figure 4: Classical and quantum random walk occupation probabilities for a
flattened multilayer network composed by three Barabási-Albert layers with
$101$ nodes and $198$ edges, $100$ nodes and $99$ edges, $25$ nodes and $66$
edges (left column); $101$ nodes and $294$ edges, $100$ nodes and $196$ edges,
$25$ nodes and $84$ edges (right column).

##  4 Conclusions and Discussion

In this paper, we used classical and quantum continuous-time random walk
centrality measures to identify leaders in (i) three single-layer criminal
networks related to an anti-Mafia operation called Montagna, and (ii) a
flattened multilayer network created from them.

In the formulation of these measures, different Hamiltonians describe the
evolution of classical and quantum walks. Other classical and quantum
continuous-time random walk centralities, such as random walk betweenness or
closeness, can be derived by modifying these Hamiltonians, and tracking
different properties of absorbing random walks. In our future works, we want
to apply these other measures on our criminal networks to identify leaders,
and to define a new disruption framework based on random walks, comparing the
targeted attacks based on the classical centrality measures with those based
on their random walk versions.

Moreover, we are planning to extend the mathematical framework by Böttcher and
Porter [6] for whole multilayer networks considering the layered structure,
and not just the flattening function. In fact, we can obtain a complete
picture of the activities and the structure of criminal organizations only
considering the various layers not as single-layer networks but as a whole
multilayer network [14]. Then, we want to explore the possibility of
developing a disruption framework for multilayer networks.

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