# Towards a Better Understanding of the Characteristics of Fractal Networks

EZPEnikő Zakar-Polyák  MNMarcell Nagy  RMRoland Molontay  Department of
Stochastics, Institute of Mathematics, Budapest University of Technology and
Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  ELKH-BME Stochastics
Research Group, Műegyetem rkp. 3., H-1111 Budapest, Hungary

###### Abstract

The fractal nature of complex networks has received a great deal of research
interest in the last two decades. Similarly to geometric fractals, the
fractality of networks can also be defined with the so-called box-covering
method. A network is called fractal if the minimum number of boxes needed to
cover the entire network follows a power-law relation with the size of the
boxes. The fractality of networks has been associated with various network
properties throughout the years, for example, disassortativity, repulsion
between hubs, long-range-repulsive correlation, and small edge betweenness
centralities. However, these assertions are usually based on tailor-made
network models and on a small number of real networks, hence their ubiquity is
often disputed.

Since fractal networks have been shown to have important properties, such as
robustness against intentional attacks, it is in dire need to uncover the
underlying mechanisms causing fractality. Hence, the main goal of this work is
to get a better understanding of the origins of fractality in complex
networks. To this end, we systematically review the previous results on the
relationship between various network characteristics and fractality. Moreover,
we perform a comprehensive analysis of these relations on five network models
and a large number of real-world networks originating from six domains. We
clarify which characteristics are universally present in fractal networks and
which features are just artifacts or coincidences.

fractal networks,

network models,

structural properties,

small world,

assortativity,

betweenness centrality,

###### keywords:

Research

## Introduction

Network science has received a great deal of research interest in the past two
decades since networks can efficiently model numerous real-world structures
and phenomena, including the Internet, the WWW, cellular networks, and social
networks [1]. The primary goal of network science is to better understand the
structure, origin, and evolution of real networks. For example, if we aim to
efficiently stop or prevent a pandemic, it is important to explore the
biological structure of the virus [2], the mechanisms underlying the disease
[3] and the social interactions of communities [4, 5].

The breakthrough in network science dates back around the millennium since the
rapid and large-scale development of computer science made it possible to
store and efficiently analyze complex networks [6]. An observation that there
are properties, which are generally present in a large number of networks
regardless of their origin, was also made in these years. The most important
features of networks include the scale-free property [7] and the small-world
property [8].

The fractality of networks is another well-studied characteristic. While the
notion of fractal scaling was originally introduced in geometry, it has been
extended to complex networks as well [9]. The book of Rosenberg [10] and the
survey of Wen and Cheong [11] give an extensive overview of fractal networks.
Fractal scaling was verified in various real-world networks [9, 12, 13] and
has been associated with numerous important properties, such as robustness
against intentional attacks [14] and accelerated flow [15]. Consequently, it
is in dire need to uncover the underlying mechanisms causing fractality.
Several studies have been published throughout the years that focus on the
exploration of the origins of fractality [14, 16, 17, 18] without a clear
consensus.

In this work, we investigate which network characteristics influence the
emergence of fractality in complex networks. To this end, we review the most
influential studies, and we also extend the methodological approaches in the
literature. Furthermore, we propose a completely different technique to gain a
better understanding of the origins of fractality, namely, we utilize the
tools of machine learning. To make our findings as universal as possible, all
of the aforementioned analyses rely on our large collection of real-world and
model-generated networks.

The investigated characteristics that have been connected to the fractality of
networks are the following:

  1. 1.

Yook et al. [12] and Song et al. [14] argued that fractality originates from
the disassortativity of the network and the repulsion (disconnectedness) of
the hubs.

  2. 2.

Fujiki et al. [16] and Rybski et al. [19] demonstrated, using different
approaches, that there is a connection between long-range anti-correlation and
fractality.

  3. 3.

Wei et al. [18] demonstrated that the distribution of edge betweenness
centrality ($BC$) influences fractality and even a few edges with high $BC$
can destroy the fractal structure of a network.

  4. 4.

Csányi and Szendrői [20] were the first to draw attention to the opposing
relationship between fractality and small-worldness, however, among many
others, they also mention that the transition between fractal and small-world
is smooth and these two properties can also be present simultaneously.

  5. 5.

Finally, Kitsak et al [17] argued that in fractal networks there is a weaker
correlation between the degree and the betweenness centrality of the nodes
than in non-fractal networks.

In the Foundations and preliminaries section, we first lay the foundation of
our analyses by showing how fractality can be determined in networks,
presenting different fractal network models, and describing our dataset, which
forms the basis of the analyses. In the Analysis of network characteristics
section, we put under the microscope the aforementioned characteristics, which
have been associated with fractality, one by one, and in A machine learning
approach section, we use machine learning algorithms to study how the
composite of the network characteristics influence fractality. Finally, in the
Summary section, we summarize our findings and propose further research
questions.

## Foundations and preliminaries

In this section, we introduce the concept of fractal network, we lay the
foundation of our analyses including the determination of fractality and the
description of the used mathematical network models, and finally, we describe
our collection of network data in detail.

### Fractality of networks

Similarly to the case of geometric fractals, the fractality of networks can
also be defined by the so-called box-covering method, using the length of the
shortest path between two nodes as the distance metric. The method can be
summarized as follows [9]: The nodes of the network are partitioned into boxes
of size $l_{B}$ in such a way that any two nodes of a box are less than
$l_{B}$ far from each other. The minimum number of boxes needed to cover the
entire network with boxes of size $l_{B}$ is denoted by $N_{B}(l_{B})$. A
network is defined to be fractal, if the relation of $N_{B}(l_{B})$ and
$l_{B}$ follows a power law, i.e.:

$N_{B}(l_{B})\sim l_{B}^{-d_{B}}.$

The $d_{B}$ exponent is called the box dimension or fractal dimension of the
network.

Box-covering is proved to be an NP-hard problem [21], therefore, there is no
efficient algorithm, which could find the exact solution, i.e. the optimal
$N_{B}(l_{B})$ number of boxes. However, numerous approximating methods have
been proposed, for a collection and comparative analysis of box-covering
algorithms, we refer to [22]. Here, we present only one of the most widely
used methods, the Compact Box Burning (CBB) algorithm, which we use later for
the boxing of our networks. The method works as follows [21]:

  1. 1.

Let $C$ be the set of uncovered nodes.

  2. 2.

Randomly choose a $c\in C$ node, and remove it from $C$.

  3. 3.

Remove every node from $C$, which is at distance at least $l_{B}$ from $c$.

  4. 4.

Repeat steps 2 and 3 until $C$ becomes empty. At this point, the chosen $c$
nodes form a compact box, thus no other nodes could be added to this box.

  5. 5.

Repeat steps 1-4 until the whole network is covered.

### Determination of fractality

The identification of the fractal nature of networks is of great importance,
however, it is a very challenging task, since most of the solutions rely on
visual evaluations. To avoid the uncertainty of these techniques, we apply a
more automated method to determine the presence of fractality in networks.

In theory, the determination of the fractality of a network can be done by
testing whether the minimal number of boxes $N_{B}(l_{B})$ – determined by the
box-covering method – scales as a power of the box size. A statistical
framework for the detection of power law behavior in empirical data was
developed by Clauset et al. [23], however, this framework is rarely used to
quantify the fractality of networks due to the special nature of the problem.
First of all, due to the NP-hard nature of the box-covering method, the use of
approximation algorithms is necessary which makes the results less suitable
for statistical analysis. Furthermore, for smaller networks or for those with
small average distances, the number of points resulting from box covering is
not large enough to obtain reliable information by these statistical tests.
Moreover, the presence of different properties in networks is usually not
pure, especially in real networks. It is a common phenomenon that fractal
scaling holds only in an $(l_{B,MIN},l_{B,MAX})$ range of $l_{B}$ [23, 24].
Often for small $l_{B}$ values, the power law relation prevails, while for
large $l_{B}$ values exponential relation holds. Consequently, one has to
choose a range of $l_{B}$ values to run the statistical tests in, which is
itself a challenging task and it also reduces the sample size.

Due to the aforementioned difficulties, in practice, the most common technique
for detecting the fractal nature of a network is to plot the
$(l_{B},N_{B}(l_{B}))$ data points on a log-log plot, fit a straight line, and
decide about the goodness-of-fit by the mean squared error, the coefficient of
determination or by simply looking at the plots [16, 25]. Obviously, these
methods and the conclusions drawn from their results are highly influenced by
personal decisions as it is also pointed out by Kovács et al. [22].
Furthermore, considering a large number of networks, the visual evaluation of
plots becomes impracticable. Therefore, we will use a more automated way to
decide about the fractality of networks.

We use a method introduced by Takemoto [26], which takes advantage of the
observation that while in fractal networks the $N_{B}(l_{B})\sim
l_{B}^{-d_{B}}$ relation holds, for non-fractal networks $N_{B}(l_{B})\sim
e^{-d_{e}l_{B}}$ is true [24]. Here, we apply a modified version presented by
Akiba et al. [27]. Namely, we fit both a power law and an exponential curve in
the form of the mentioned relations to the normalized $(l_{B},N_{B}(l_{B})/N)$
points, where $N$ is the number of nodes in the network. The fitting is done
by excluding the point corresponding to $l_{B}=1$ because it is usually an
outlier. The fractality can be measured by the ratio of the root-mean-square
errors of the two curves:

$R=\frac{RMSE_{\text{power law}}}{RMSE_{\text{exponential}}}.$

The idea of normalization of data points and the use of RMSE allows us to
compare the goodness-of-fit of different networks.

[width=0.45]figures/fractality_real_007.pdf
[width=0.45]figures/fractality_real_3.pdf
[width=0.45]figures/fractality_real_09.pdf

Figure 1: Results of fitting power law and exponential curves to some real
networks, illustrated on log-log plots. (a) Brain network with $R=0.07$ (b)
Social network with $R=3.3$ (c) Metabolic network with $R=0.9$

One might say that if $R<1$, then the network is fractal since in this case,
the power law curve fits better than the exponential one, otherwise it is non-
fractal. However, as was also mentioned earlier, a network is not necessarily
purely fractal, but it can still possess the fractal property for a given
range. This metric also allows us to measure fractality on a continuous scale,
the closer the $R$ ratio is to $0$, the more fractal the network is. However,
in order to compare the characteristics of fractal and non-fractal networks,
we still need to create a cut-off point. We observed in both real- and model-
generated networks that $R=0.65$ is a reasonable choice. Since the boundary is
fuzzy, one could make a stricter partition, but for our analyses, it would not
make a significant difference. Here, we say that the investigated networks
with $R<0.65$ are rather fractal than non-fractal and vice versa. Figure 1
shows a few illustrative examples.

It is also important to note that the described method cannot be used for
networks with a small diameter (e.g. smaller than 6). However, for these
networks, the fractal nature can hardly be interpreted anyway. Furthermore,
this method may also not give appropriate results for some mathematical
network models where the fractal scaling only asymptotically holds. For this
reason, we use this method for the identification of fractality only for real
networks, while we stick to the theoretical findings in the case of model-
generated networks.

### Network models

Mathematical models play a crucial role in understanding the properties of
networks. Numerous models have been introduced throughout the years to capture
fractal scaling in networks and to better understand the relation between
fractality and other network characteristics. In this section, we describe
five such network models, with a special emphasis on the connection of their
parameters with fractality.

#### Song-Havlin-Makse model

One of the most well-known fractal network models is the Song-Havlin-Makse
model (SHM) [14]. The network grows dynamically and the degree correlation
(hub repulsion/attraction) of the emerging graph is driven by a predefined
parameter $p$. The model is defined as follows:

  1. 1.

The initial graph is a simple structure, e.g. two nodes connected via a link.

  2. 2.

In the iteration step $t+1$ we connect $m$ offspring to both endpoints of
every edge, i.e. an $x$ node gains $m\cdot deg_{t}(x)$ offspring, where $m$ is
a predefined parameter and $deg_{t}(x)$ is the degree of node $x$ at the end
of step $t$.

  3. 3.

In iteration step $t+1$ every $(x,y)$ edge is removed independently with
probability $p$, where $p$ is a predefined parameter. When an edge is removed,
it is replaced by a new edge between the offspring of $x$ and $y$.

Figure 2 illustrates two realizations that can be generated with the described
model. The fractality is influenced by the choice of parameter $p$, namely, it
can be shown that the generated network is fractal for $p=1$, and non-fractal
for $p=0$ [14, 28]. The intermediate values develop mixtures between the two
properties. Our observation is that networks with $p>0.6$ can be considered
fractal, while those with $p<0.4$ are clearly non-fractal, which is
illustrated in Figure 3. It can also be seen that the transition from fractal
to non-fractal is smooth, hence in the $0.4\leq p\leq 0.6$ range, it is
questionable to assign the networks to any of the two categories. Later in the
section called A machine learning approach, where a binary classification is
carried out, we are still creating a cut-off point at $p=0.5$, because we do
not want to exclude the intermediate networks.

[width=0.95]figures/shm.pdf

Figure 2: Illustration of the Song-Havlin-Makse model with parameter settings
(a) $m=2,p=0$, (b) $m=2,p=0.6$. In both cases, three iteration steps were
made.

[width=0.8]figures/boxing_shm_w05_2.pdf

Figure 3: Illustration of the fractality of the Song-Havlin-Makse model for
different parameter settings. The iteration number is set to 5, parameter $m$
is fixed to 2, and only parameter $p$ is taking different values.

The SHM model was introduced to show that fractal networks exhibit strong
repulsive relations between their hubs, which conjecture is reviewed later in
the Disassortativity and hub repulsion section.

#### Hub attraction dynamical growth model

Kuang et al. modified the Song-Havlin-Makse model (SHM) in such a way that the
new mechanism can generate fractal networks with strong hub attraction [29].
The hub attraction dynamical growth (HADG) model is based on the previously
described SHM model, with the following modification applied: first, the
rewiring probability of the model is flexible, more precisely, it depends on
the degree of the endpoints of the links. The other modification is what they
call the within-box link-growth method, which means that after an edge is
rewired, the model adds additional edges between the newly added offspring, in
order to increase the clustering coefficient of the network. The evolution of
the HADG model is defined as follows [29]:

  1. 1.

The initial condition and the growth of the model are the same as in step 1
and 2 of the Song-Havlin-Makse model.

  2. 2.

We rewire the $(x,y)$ edge at time $t+1$ with probability

$\displaystyle p_{xy}$ $\displaystyle=\begin{dcases}a,&\text{ if
}\frac{deg_{t}(x)}{deg_{t}^{max}}>T\text{ and
}\frac{deg_{t}(y)}{deg_{t}^{max}}>T\\\ b,&\text{ otherwise, }\end{dcases}$

where $deg_{t}(x)$ is the degree of node $x$ and $deg_{t}^{max}$ is the
maximum degree in the network at time $t$ and $a,b,T\in[0,1]$ are predefined
parameters. Thus, if we define $a<b$, then hubs will have a higher probability
to be connected than non-hubs.

  3. 3.

At step $t+1$, for each old $y$ node, we add $deg_{t}(y)$ edges between the
newly generated offspring of $y$.

It should be mentioned that in the original paper, Kuang et al. used the
notations $a$ and $b$ for the probabilities that an edge is not rewired [29],
consequently, as a slight abuse of notation, the probabilities we use here are
equivalent to $1-a$ and $1-b$ with regard to the original article.

[width=0.95]figures/hadgm.pdf

Figure 4: Illustration of the Hub attraction dynamical growth model for (a)
$m=2,a=0,b=0.1,T=0.9$, (b) $m=2,a=0,b=1,T=0.9$. In both cases, three iteration
steps were made.

[width=0.8]figures/boxing_hadgm_2.pdf

Figure 5: Illustration of the fractality of the Hub attraction dynamical
growth model for different parameter settings. The iteration number is fixed
to 4, parameter $m$ to 3, and $T=0.9$.

Figure 4 illustrates two networks that can be generated with the model using
different parameter settings. Kuang et al. concludes that there are fractal
networks with assortative behavior, i.e. where the most connected nodes can be
connected since this model can generate such graphs with appropriate parameter
settings [29]. We can support the observation of the authors, namely, we found
that choosing $b>0.1$ (with our notation) results in fractal networks, and
with $b\leq 0.1$ we can generate non-fractal networks, independently of
parameter $a$ given that $a<b$. This is well-illustrated in Figure 5.

####  $(u,v)$-flower

The family of $(u,v)$-flowers was introduced by Rozenfeld, Havlin, and Ben-
Avraham [30]. Similarly to most of the previous models, this model also
generates networks through iterations, but the edge replacement procedure is
quite different. The model is defined as follows:

  1. 1.

The initial graph is a cycle consisting of $w=u+v$ nodes and edges, where $u$
and $v$ are predefined parameters, and we can assume that $u\leq v$.

  2. 2.

In the iteration step $t+1$ every $(x,y)$ edge is replaced by two paths
connecting $x$ and $y$, one with length $u$ and one with length $v$.

[width=0.95]figures/uv.pdf

Figure 6: Illustration of the $(u,v)$-flower with three iteration steps and
(a) $u=1,v=3$, (b) $u=2,v=3$.

Two networks generated with different parameter settings are shown in Figure
6. Rozenfeld et al. showed that the model generates fractal networks when
$u>1$, and non-fractal ones when $u=1$ [30]. This statement is illustrated in
Figure 7. Furthermore, it was also shown in [30] that in the $u=1$ case the
resulting networks are small-world, which supports the idea that fractal and
small-world are conflicting properties. This statement is investigated in the
Small-world property section.

[width=0.8]figures/boxing_uv_2.pdf

Figure 7: Illustration of the fractality of the $(u,v)$-flower for different
parameter settings. In all of the cases, the iteration number is set to four,
and $w=7$ is also fixed.

#### Repulsion based fractal model

In [31] we introduced the repulsion-based fractal (RBF) model, which is also
based on the SHM model [14] and adapts some concepts of the HADG model as well
[29]. The model evolves through time and rewires edges with probability based
on the degree of the endpoints to create repulsion among nodes. The within-box
link-growth method of Kuang et al. [29] is also adapted by the model to
increase the clustering coefficient, hence creating more realistic networks.
The growing mechanism of the repulsion-based fractal model is as follows:

  1. 1.

The initial condition and the growth of the model are the same as in step 1
and 2 of the Song-Havlin-Makse model.

  2. 2.

In iteration step $t+1$ we remove every edge $(x,y)$ with probability

$p_{xy}^{Y}=1-\left|Y-\frac{\deg_{t}(x)+\deg_{t}(y)}{2\cdot\deg_{t,\max}}\right|,$

where $Y\in[0,1]$ is a predefined parameter, $\deg_{t}(x)$ is the degree of
node $x$, $\deg_{t,\max}$ is the maximum degree at step $t$. When an edge is
removed, it is replaced with a uniformly randomly chosen new edge between the
offspring of its endpoints.

  3. 3.

We add $\deg_{t}(y)$ edges among the newly generated offspring of every old
node $y$. In order to avoid creating self-loops, this step is only executed,
when $m>1$.

[width=0.95]figures/rbfm.pdf

Figure 8: Illustration of the Repulsion based fractal model for (a) $Y=0$, (b)
$Y=1$. In both cases, three iteration steps were made, and $m=2$ was also
fixed.

Parameter $Y$ influences which group of nodes should repel each other (within
the group). Figure 8 illustrates the two extreme cases of the model. This
model generates fractal networks for all $Y\in[0,1]$, as it can be seen in
Figure 9, and hence suggests that the property, which gives rise to fractality
is repulsion, but the repulsion does not necessarily have to be among hubs
[31].

[width=0.8]figures/boxing_rbfm_2.pdf

Figure 9: Illustration of the fractality of the Repulsion based fractal model
for different parameter settings. The iteration number is chosen to be five,
and $m=1$.

#### Lattice small-world transition model

The lattice small-world transition model (LSwTM) was also introduced in [31].
It utilizes the fractal nature of grid-like structures and also adapts the
preferential attachment mechanism to work against fractal scaling. The model
is defined as follows:

  1. 1.

We start with a $d$-dimensional (practically $d=2$) grid graph with
$n_{1}\times n_{2}\times\dots\times n_{k}$ vertices.

  2. 2.

With probability $p$, every edge of $(x,y)$ is replaced by $(x,z)$, where $z$
is chosen with a probability that is proportional to $p_{z}$:

$p_{z}=\frac{1}{1+\exp\left(-a\cdot\left(\frac{\deg(z)}{\deg_{\max}}-\frac{1}{2}\right)\right)},$

where $a$ is a positive constant, $\deg(z)$ is the degree of node $z$ and
$\deg_{\max}$ is the maximum degree of the current graph. By default, $y$ is
replaced with $z$ during the rewiring process, however, if in this way the
graph becomes disconnected, $x$ is replaced instead.

[width=0.95]figures/mixture.pdf

Figure 10: Illustration of the LSwTM for (a) $p=0$, (b) $p=0.1$, using a
$10\times 10$ lattice as the initial graph. Subfigure (a) is from [31]

Even a small probability of rewiring results in a network that differs greatly
from the initial grid graph, as illustrated in Figure 10. The fractality of
the generated network depends on the choice of $p$. For $p=0$ the network is
purely fractal, and as $p$ grows the model shows a transition from fractal to
non-fractal networks [31]. As Figure 11 demonstrates, it is reasonable to
choose $p=0.01$ as a cutpoint. It was also shown in [31] that this model
demonstrates a fractal—small-world transition.

[width=0.8]figures/boxing_mixture_2.pdf

Figure 11: Illustration of the fractality of the LSwTM for different choices
of parameter $p$ with $n_{1}=50,n_{2}=100$.

### Data

To gain a complete understanding of the relationship between fractality and
other network properties, it is essential to consider a diverse and large-
scale collection of real-world and model-generated networks as the basis of
our analysis. Although mathematical models give insight into the evolution of
networks and some distinguished network properties, they usually cannot
capture every characteristic of real networks. To be as comprehensive as
possible, we generated networks with the models introduced in the Network
models section, with various parameter settings, in addition, we collected a
large number of real networks originating from six different domains. The
analyses were performed in Python, and all network-related calculations,
including the network generation process, were done using the NetworkX package
[32].

#### Model-generated networks

We selected the parameters of the different models to get a representative
sample of the space spanned by the network models while keeping the number of
networks reasonably low for computational purposes. For this reason, we
limited ourselves to networks with at most around 10,000 nodes. Our choices of
the parameter values are summarized in Table 1 and 2.

SHM | HADGM | RBFM
---|---|---
$n$ | 1, 2, 3, 4, 5 | $n$ | 1, 2, 3, 4, 5 | $n$ | 1, 2, 3, 4, 5
$m$ | 1, 2, 3, 5, 10, 20, 50 | $n=1$ | $m$ | 1, 2, 3, 5, 10, 20, 50 | $n=1$ | $m$ | 1, 2, 3, 5, 10, 20, 50 | $n=1$
1, 2, 3, 5, 10, 20 | $n=2$ | 1, 2, 3, 5, 10, 20 | $n=2$ | 1, 2, 3, 5, 10, 20 | $n=2$
1, 2, 3, 5, 10 | $n=3$ | 1, 2, 3, 5, 10 | $n=3$ | 1, 2, 3, 5, 10 | $n=3$
1, 2, 3 | $n=4$ | 1, 2, 3 | $n=4$ | 1, 2, 3 | $n=4$
1, 2 | $n=5$ | 1, 2 | $n=5$ | 1, 2 | $n=5$
$p$ | 0, 0.2, 0.4, 0.6, 0.8, 1 | $b$ | 0, 0.1, 0.5, 1 | $Y$ | 0, 0.5, 1
$a$ | 0 | $b=0,0.1$
0, 0.4 | $b=0.5$
0, 0.4, 0.9 | $b=1$
$T$ | 0.9

Table 1: Parameter settings for the Song-Havlin-Makse, Hub attraction
dynamical growth, and Repulsion-based fractal models, with which the analyzed
networks were generated. The parameter $n$ denotes the number of iterations,
for the meaning of the other parameters, see the Network models section. The
resulting number of networks: 138 (SHM), 161 (HADGM), 69 (RBFM).

$(u,v)$-flower | LSwTM
---|---
$u,v$ |  | $1\leq u\leq v$
---
$w=u+v=2,3,\dots,10$
$N=n_{1}\cdot n_{2}$ |  | 10, 50, 100, 200, 500, 800,
---
1000, 1500, 2000, 3000, 5000
$n$ | $1,2,\dots,9$ | $w=2$ | $p$ |  0, 0.00001, 0.0001, 0.001, 0.0025, 0.005, 0.0075, 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 1
$1,2,\dots,8$ | $w=3$
$1,2,\dots,6$ | $w=4$
$1,2,\dots,5$ | $w=5,6$
$1,2,3,4$ | $w=7,8,9,10$

Table 2: Parameter settings for the $(u,v)$-flower and the Lattice small-world
transition model, with which the analyzed networks were generated. $n$ denotes
the number of iterations, for the meaning of the other parameters, see the
Network models section. In the case of the LSwTM, the values of the
$n_{1},n_{2}$ parameters are chosen in such a way that $|n_{1}-n_{2}|$ is
minimal. The resulting number of networks: 118 ($(u,v)$-flower), 154 (LSwTM).

For those analyses, where the evaluation is done on a network-by-network basis
by observing plots, we restricted ourselves to a smaller number of networks.
We created three size categories of networks with approximately 800, 2000, and
5000 nodes. For every model, three to seven networks per size category were
chosen including both fractal and non-fractal networks (except for the RBFM,
where only fractal networks can be generated).

#### Real networks

Real networks were collected from various online repositories [33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45]. Table 3 gives a short description of the
different domains from which we collected the networks together with the
number of networks. In total, we work with 275 real-world networks. Some of
their main features are listed in Table 4, aggregated by domains. For those
analyses, which require visual evaluation, we selected four to six networks
from every domain, bearing in mind to have both fractal and non-fractal
networks from all size categories presented in the domain.

Domain | Description |  Number of networks
---|---|---
Brain |  | Human and animal connectomes
---
(neural connections in the brain)
45
Metabolic |  | Protein-protein interactions of organisms
---
47
Cheminformatics | Graph structure of enzymes | 45
Infrastructural |  | Transportation and distribution networks
---
30
Foodweb | What-eats-what in an ecological community | 69
Social |  | Facebook, Twitter and collaboration networks
---
39

Table 3: Description of the network domains and the number of collected
networks.

We decided on the fractality of the networks as we described in the
Determination of fractality section. In order to eliminate the randomness of
the box-covering algorithm, we repeated the procedure 15 times and averaged
their outcomes. The resulting class distribution of model-generated networks,
real networks, and all combined networks is shown in Figure 12. It can be seen
that there are much more fractal networks amongst both the model-generated and
the real networks, but the number of non-fractal networks is also significant.

Domain | Number of nodes | Number of edges | Diameter
---|---|---|---
AVG | (MIN, MAX) | AVG | (MIN, MAX) | AVG | (MIN, MAX)
Brain | 1073 | (65, 2989) | 7485 | (730, 31548) | 14 | (2, 43)
Metabolic | 815 | (11, 2831) | 2099 | (10, 20448) | 9 | (1, 22)
Cheminformatics | 58 | (44, 125) | 100 | (77, 149) | 16 | (9, 37)
Infrastructural | 1268 | (8, 6474) | 2396 | (7, 15645) | 37 | (3, 122)
Foodweb | 115 | (19, 765) | 655 | (38, 6613) | 5 | (2, 9)
Social | 4465 | (86, 9763) | 7300 | (117, 24806) | 14 | (6, 28)

Table 4: Some of the main features of the collected real networks. The
average, minimum, and maximum values of the number of nodes, the number of
edges, and the diameter of the networks by domains.

[width=0.9]figures/class_distribution.pdf

Figure 12: Class distribution concerning fractality for (a) model generated,
(b) real-world, (c) all of the examined networks together.

## Analysis of network characteristics

In this chapter, we intend to give a comprehensive analysis of the relation of
fractality with other network properties by revisiting some assertions from
the literature.

### Disassortativity and hub repulsion

The first network properties, which were associated with the origin of
fractality are disassortativity and repulsion between large degree nodes,
i.e., hubs [12, 14]. It has been much disputed whether the fractal nature of
networks originates from these characteristics, there are papers that support
this assertion [46], but there are more works that confute it [16, 18, 29,
47].

The concepts of disassortativity and hub repulsion are often used
interchangeably, although the latter can be considered only as the practical
interpretation of the former. For this reason, we rather separate the two
notions: First, we measure the assortativity of a network by the classic
assortativity coefficient. Second, we define a novel hub connectivity score
($HCS$) as the number of edges among hubs divided by the number of hubs, thus
it shows how many hub neighbors a hub has on average. Formally:
$HCS=\frac{E_{hub}}{N_{hub}},$ where $N_{hub}$ denote the number of hubs in
the network and $E_{hub}$ is the number of edges among these hubs. In this
way, $HCS$ is large for those networks, in which hubs tend to connect to each
other (strong attraction), and small, when there are only a few or no edges
among them (strong repulsion).

Here, we define hubs as nodes whose degrees are at least two times the average
degree of the network. If there is no such node, we set its hub connectivity
score to $-1$. Furthermore, both the assortativity coefficient and the hub
connectivity score are averaged over 15 realizations of the network models for
each parameter setting.

#### Results for dissasortativity

For mathematical network models, we study how the assortativity coefficient
depends on the parameter of the model which influences the fractality of the
network. Except for the LSwT model, all models generate disassortative fractal
networks, however, the $(u,v)$-flower is the only model where the fractal
networks are disassortative, and the non-fractal networks are assortative, as
Figure 13 (b) shows.

[width=0.95]figures/assortativity_model.pdf

Figure 13: Assortativity as a function of the parameter, which influences
fractality for the (a) Lattice small-world transition model, (b)
$(u,v)$-flower. The grey dotted reference line shows Assortativity = 0. The
results for the other models can be found in the supplementary material [48].

While the fractal networks that the Song-Havlin-Makse and the Hub attraction
dynamical growth models generate are disassortative, the non-fractal networks
generated by these two models are also disassortative. Although in the case of
the SHM model, the figures in the supplementary material [48] suggest that for
a specific parameter setting ($m=2$) the fractal networks are more
disassortative than the non-fractal ones, for $m=1$ and $m>2$, fractal
networks typically have a higher assortativity coefficient than the non-
fractal ones. Hence, in general, based on the (dis)assortativity of the
network generated by the SHM or the HADG models, no conclusions can be drawn
about whether the network is fractal or not.

Similarly, the RBF model also generates disassortative fractal networks, but
since the RBF model can only generate fractal networks, based on this model,
no conclusions can be drawn about the assortativity of the non-fractal
networks.

The LSwTM serves as a counterexample to the aforementioned assertion because
it not only generates fractal networks with assortative mixing, but we can
observe a positive correlation between fractality and assortativity, i.e., the
more fractal the model is, the higher the assortativity is (see Figure 13
(a)). In this sense, the LSwT model behaves in the opposite way to the
$(u,v)$-flower.

[width=0.95]figures/assortativity_real.pdf

Figure 14: Assortativity of real networks plotted against the number of nodes
on a semi-log scale. Subfigure (a) shows the non-fractal, subfigure (b) the
fractal networks. The grey dotted reference line shows Assortativity = 0. The
difference in the distribution of the assortativity of the fractal and non-
fractal networks is statistically significant (Mann-Whitney U rank test’s
$p$-value: $2.17\cdot 10^{-7}$).

In the case of real-world networks, we can say that fractal networks are often
disassortative, but there are numerous examples of assortative cases too,
which is well illustrated in Figure 14. Moreover, if we consider not only the
binary fractal/non-fractal categories but the continuous $R$ coefficient of
the networks (see the Network models section), we cannot recognize any
remarkable pattern in the assortativity. An illustration of this result can be
found in the supplementary material [48].

Overall, our findings partially support the conclusion of Kuang et al.,
namely, that fractality is independent of the assortative mixing [29], because
there are numerous counterexamples on both sides for the conjecture that
fractality originates from disassortativity. However, disassortativity is
still common amongst fractal networks.

#### Results for hub repulsion

Regarding the hub repulsion, we can observe that most models support the
conjecture that this property may lie behind fractality. For instance, for the
$(u,v)$-flower, in the $u=1$ (i.e., non-fractal) case, the HCS scores are much
higher than in the fractal cases. Disregarding small networks (i.e. if the
number of nodes is less than 100) due to the lack of hubs, we can say that
fractal and non-fractal networks can be clearly separated according to their
$HCS$. Fractal $(u,v)$-flowers have $HCS$ close to 0, while for non-fractal
$(u,v)$-flowers, this measure is at least 1, which can be seen in Figure 15
(a).

[width=0.95]figures/hub_rep_uv_real.pdf

Figure 15: Plot of hub connectivity score for the (a) $(u,v)$-flower, (b) real
networks. The $HCS$ is plotted as a function of the number of nodes on a semi-
log scale. The difference of the distribution of the $HCS$ of the fractal and
non-fractal networks is statistically significant in both cases (Mann-Whitney
U rank test’s $p$-value is $0.016$ for the $(u,v)$-flower, and $3.06\cdot
10^{-9}$ for the real networks).

Besides the $(u,v)$-flower, the Hub attraction dynamical growth model also
seems to support the conjecture. Figure 16 shows how the hub connectivity
score characterizes the different cases of the HADG model. For the non-fractal
networks (i.e. when $b\leq 0.1$) the $HCS$ is higher than for the fractal
networks. Furthermore, we can claim that the extent of the hub connection (or
repulsion) depends on the parameter $b$, which influences the fractality of
the network, and not on the parameter $a$, which creates the repulsion. Thus,
as Kiang et al. [29] showed, the hubs in fractal networks can be directly
connected, but our results show that the hub connectivity score is still
capable of distinguishing the fractal and non-fractal networks generated by
the HADG model.

[width=0.95]figures/hub_rep_hadgm.pdf

Figure 16: Plots of the hub connectivity score for the HADG model. $HCS$ is
plotted against the logarithm of the number of nodes. The grouping of points
is based on (a) parameter $b$, which influences fractality, (b) parameter $a$,
which influences hub connectivity.

Similar behavior is demonstrated by the other models as well, that is fractal
networks show stronger hub repulsion than non-fractals. However, it must be
mentioned that $HCS$ usually stays between 0 and 1 for these models, and the
difference concerning fractality can only be observed for each model
separately. For example, the Repulsion-based fractal model is able to create
fractal networks with $HCS$ being close to or even above 1, while in the case
of the Song-Havlin-Makse model, only the non-fractal networks possess such
high scores. Illustrations of the results for the SHM, RBF, and LSwT models
can be found in the supplementary material [48].

Similarly, as Figure 15 (b) shows, no clear consensus can be drawn on the
conjecture based on real networks. There are examples of fractal networks with
large $HCS$ (i.e., strong hub attraction), however, it can be said that
networks with high $HCS$ are typically non-fractals, although there are also
many examples of non-fractal networks with lower scores. Similar observations
can be made if we consider the $R$ values of the networks (see: [48]). It is
important to note that the scores are generally higher for real networks than
for those that are generated by models, regardless of fractality.

In conclusion, we can say that similarly to disassortativity, strong hub
repulsion is also common amongst fractal networks, but this property still
cannot distinguish perfectly fractal and non-fractal networks, hence it cannot
be considered as the reason behind fractality.

### Long-range correlation

Besides direct degree correlation, the long-range correlation has also been
associated with fractal scaling [16, 19]. Both studies suggest, based on
different approaches, that there is a connection between long-range anti-
correlation and fractality. Here, we apply both of the methods [16, 19] in
addition to a more immediate extension of neighbor-level degree correlation
measures, introduced in [49].

In [19] a fluctuation analysis approach was proposed to measure long-range
correlations. The steps of this method can be summarized as follows:

  1. 1.

Consider all shortest paths in the network of length $d$. For all of these
paths, calculate the average degree of the nodes on the path.

  2. 2.

Calculate $F(d)$, which is the standard deviation of the mean degrees
calculated in step 1.

  3. 3.

Repeat steps 1 and 2 for all possible $d$.

  4. 4.

Examine if $F(d)$ scales as a power of $d$ with exponent $\alpha$. If so,
$-\frac{1}{2}<\alpha<0$ suggests positive, $-1<\alpha<-\frac{1}{2}$ negative
long-range correlations.

An extension of the concept of hub repulsion to long-range scales was proposed
in [16]. The authors examined how the distribution of hub distances looks in
fractal and non-fractal networks. The procedure can be summarized by the
following steps:

  1. 1.

Calculate the distance of all pairs of hubs.

  2. 2.

For all distance $l$ calculate $\hat{P}(l)$, which is the number of hub pairs
separated by the shortest path of length $l$.

  3. 3.

Calculate $\tilde{P}(l)$ by dividing $\hat{P}(l)$ by the number of possible
edges among hubs, i.e. $\tilde{P}(l)=\hat{P}(l)/\binom{N_{hub}}{2}$.

In this way, $\tilde{P}(l)$ is the probability that a randomly selected pair
of hubs is at distance $l$ from each other. In order to be consistent with the
results of [16], for this analysis, we cut off the hubs at the 98th percentile
of the degree distribution.

The third approach to capture long-range correlations was introduced in [49]
and has not been used before to study the relationship between fractality and
long-range correlation. This method extends the notion of neighbor
connectivity to nodes at a distance larger than one. The main idea of the
method can be summarized as follows.

  1. 1.

Fix $m$, and for every node $x$, take the average degree of the nodes that are
at distance $m$ from $x$.

  2. 2.

Calculate $\langle k_{m}\rangle(k)$ by taking the average of the outputs of
step 1 over nodes with degree $k$.

  3. 3.

Examine the relation of $k$ and $\langle k_{m}\rangle(k)$.

Following the line of [49], we consider the values of $m$ up to 5, and assume
power law relation between $k$ and $\langle k_{m}\rangle(k)$.

#### Results with fluctuation analysis

The results of the fluctuation analysis are illustrated for some real-world
and model-generated networks in Figure 17. Generally, it can be said that for
the $(u,v)$-flower and the Song-Havlin-Makse model, $F(d)$ scales as a power
of $d$ with exponent less than $-\frac{1}{2}$, while in the non-fractal cases
the relation is rather exponential, which supports the observations of [19].

[width=0.95]figures/fluctuation.pdf

Figure 17: Results of the fluctuation analysis on a log-log scale for some
examples of the (a) $(u,v)$-flower, (b) Repulsion-based fractal model. (c)
illustrates the cases of some metabolic, (d) infrastructural real networks.
The dotted grey lines on subfigures (a) and (c) are guides for the eye.
Results for all of the examined networks can be found in the supplementary
material [48].

However, in the case of the Repulsion-based fractal model, $F(d)$ does not
follow a power law. It may not be immediately visible from Figure 17(b), but
the exponential curve provides a better fit than the power law. For the
fitting, we use the powerlaw Python package [50].

For the Hub attraction dynamical growth model, as $b$ increases, the power-law
relation indeed appears, but the transition is smooth and there are fractal
networks that do not show the desired relation. In the case of the Lattice
small-world transition model, none of the previously mentioned relations seem
to hold on $F(d)$ for the fractal networks.

Among real-world networks, there are some cases, where the expected behavior
of $F(d)$ can be observed, as Figure 17(c) shows. However, there are examples,
where power law relation cannot be detected, thus long-range correlations
cannot be concluded, as it is illustrated in Figure 17(d). In conclusion, we
can say that long-range anticorrelation captured by fluctuation analysis is
not a universal property of fractal networks.

#### Results with hub distances

Concerning the distribution of hub distances, we can say that the HADGM, RBFM,
and the $(u,v)$-flower support the suggestion of [16], that in fractal
networks hub distances have a wide range, while in non-fractal networks hubs
cannot get far from each other.

However, a surprising observation can be made based on the Song-Havlin-Makse
model. As Figure 18(a) illustrates, the range of hub distances expands as $p$
grows, but in the case of the pure fractal case $p=1$, it falls back to the
level of intermediate networks. Consequently, this model does not seem to
support that stronger fractal property always comes with larger hub distances.

[width=0.9]figures/long_range_hub.pdf

Figure 18: Distribution of hub distances for (a) some cases of the Song-
Havlin-Makse model, (b) some real networks. On subfigure (b), for the first
two subplots, two networks with similar sizes ($\sim 140$ number of nodes) are
considered, and similarly to the third and fourth subplots ($\sim 2300$ number
of nodes). The results for the other examined networks can be found in the
supplementary material [48].

In the case of the LSwTM, for $p\leq 0.1$, i.e., when the model is fractal, no
hubs are formed, hence this analysis cannot be carried out for this model.

Investigating real networks suggests that the examined property is independent
of fractality. The first two subplots of Figure 18(b) show two networks of the
same size where the fractal network clearly possesses smaller hub distances
than the non-fractal one. Moreover, the third and fourth subplots of Figure
18(b) show two fractal networks of the same size with completely different hub
distances.

#### Results with long-range neighbor connectivity

Finally, the results obtained by the third approach, i.e., the neighbor
connectivity [49], suggest that there is no apparent connection between the
fractality and the long-range correlation profile of networks.

The Hub attraction dynamical growth model seems to be the only exception,
because the non-fractal networks generated by this model usually preserve
their disassortative structure for large distances as well, while fractal
networks mostly do not show any correlation for distance $m\geq 3$.

In the case of the Repulsion-based fractal model and the Song-Havlin-Makse
model, usually, no correlation can be detected for $m\geq 3$ and until that,
the correlation profile does not change.

For the $(u,v)$-flower, at distance $m=3$ or $m=4$ the reverse of the
correlation profile of the $m=1$ case can be observed for all networks,
independently of fractality.

[width=0.95]figures/mchain_mixture.pdf

Figure 19: Degree correlations of the two extreme cases of the LSwT model at
distances from 1 to 5. The first row corresponds to the $p=0$, and the second
to the $p=1$ case. $\langle k_{m}\rangle(k)$ is plotted against $k$ on a log-
log scale, and the line fitted to the log-transformed data is also provided.

Networks generated by the Lattice small-world transition model preserve their
correlation profile for all $m$ distances, i.e., fractal networks have
positive degree correlations, while non-fractal networks have negative
correlations, even in the long-range scale (see Figure 19).

[width=0.95]figures/mchain_brain.pdf

Figure 20: Degree correlations of a fractal (top row) and a non-fractal
(bottom row) brain network at distances from 1 to 5. $\langle k_{m}\rangle(k)$
is plotted against $k$ on a log-log scale, and the line fitted to the log-
transformed data is also provided.

[width=0.95]figures/mchain_social.pdf

Figure 21: Degree correlations of two fractal social networks at distances
from 1 to 5. $\langle k_{m}\rangle(k)$ is plotted against $k$ on a log-log
scale, and the line fitted to the log-transformed data is also provided.

Degree correlations of the real networks – independently of their fractality –
are usually preserved or reversed for larger distances but do not seem to
disappear. Figure 20 shows two brain networks, one of them is fractal, the
other one is not, and their correlation profile is very similar for all $m$
distances. Figure 21 shows two fractal social networks with a negative
correlation on the direct neighbor level and a positive correlation at $m=2$.
Illustrations of the results for all of the examined networks can be found in
the supplementary material [48].

Overall, we can conclude from the results of all three approaches that
fractality and long-range correlation profiles do not have a clear ubiquitous
connection.

### Edge betweenness centrality

In [18], the authors reported that even a small number of edges with high
betweenness centrality ($BC$) can destroy the fractal scaling of a network.
Although, they investigated this conjecture from the perspective of minimum
spanning trees, here we rather study the suggestion explicitly on the
networks. In other words, we examine the question of whether fractal networks
can have edges with high betweenness centrality. To this end, we calculate
multiple measures concerning the edge betweenness centralities: the average
and maximum $BC$ and the average of the top 5% of edge betweenness
centralities. We examine whether fractal networks tend to possess smaller
values of the aforementioned measures.

#### Results

Some of the investigated models can only generate networks with edges having a
small betweenness centrality, other models can also create networks with edges
with quite large $BC$ as well, therefore a comparison between models cannot be
made.

[width=0.93]figures/ebc_lswtm_uv_real.pdf

Figure 22: Measures concerning the edge betweenness centralities of (a)-(c)
the Lattice small-world transition model, (d)-(f) the $(u,v)$-flower, (g)-(i)
the real networks. (a),(d),(g) Average edge betweenness centrality;
(b),(e),(h) maximum edge betweenness centrality; (c),(f),(i) average of the
top 5% of edge betweenness centralities plotted against the logarithm of the
number of nodes. The $p$-values in the title of each subfigure correspond to
the Mann-Whitney U rank test performed on the corresponding metric values to
see the statistical significance of the difference in the distribution of
fractal and non-fractal networks.

The Song-Havlin-Makse, the Hub attraction dynamical growth, and the Repulsion
based fractal models generate networks for which the examined measures range
from 0 to $0.6$, and they decrease as the number of nodes increases. A
difference in fractal and non-fractal networks can be observed in the
aforementioned three models: fractal networks tend to obtain larger values
than non-fractal networks of the same size (see the supplementary material
[48]). Similar observations can be made on the Lattice small-world transition
model (see Figure 22 (a)-(c)). However, here the edge betweenness centralities
are low in general for all parameter settings, regardless of fractality.
Furthermore, we can observe that as the value of parameter $p$ grows, i.e., as
the network becomes less fractal, the betweenness centrality of its edges
decreases which contradicts the conjecture.

Contrary to the previous models, for $(u,v)$-flowers, for any given network
size, the maximal edge $BC$ is typically larger for non-fractal than for
fractal networks. The same observation can be made for the average of the top
5% betweenness centralities for small networks (fewer than 100 nodes),
however, for larger networks, this property disappears, and the values are
higher for fractal networks than for non-fractals. Moreover, the average of
all betweenness centralities also shows that fractal networks have a higher
average edge $BC$. The aforementioned results are well illustrated in Figure
22 (d)-(f).

Figure 22 (g)-(i) suggests that fractal real networks often have larger values
concerning these edge betweenness centrality-related measures than non-
fractals. However, there are also examples of non-fractal networks with high
edge $BC$ measures, and when the values are low the fractal and non-fractal
categories cannot really be distinguished. Moreover, similar observations can
be made, even if we investigate the continuous measure of fractality, the $R$
coefficient (see the supplementary material [48]).

To sum up, we can conclude that fractal networks can have edges with high
betweenness centrality as well, furthermore, the related metric values seem to
be higher on average for the fractal than non-fractal networks, which
contradicts the suggested connection between edge betweenness and fractality.

### Correlation of degree and betweenness centrality

In one of the earliest works on fractal networks, Kitsak et al. [17] studied
the betweenness centrality of fractal and non-fractal networks. The authors
analyzed seven SHM model-generated networks and four relatively large real-
world networks. They have found that there is a smaller correlation between
the betweenness centrality and the degree of a node in fractal networks than
in non-fractal networks. The authors argue that the Pearson correlation
coefficient is not a suitable metric to characterize the difference between
fractal and non-fractal networks because the average betweenness centrality
for a given degree does not change much [17]. Hence, the authors measure the
standard deviation of the betweenness centralities for a given degree and they
compare it to that of the counterpart networks made by the configuration
model. Due to the large computing complexity, here we apply a slightly
different approach.

In this work, we measure the coefficient of variation (ratio of standard
deviation to the mean) of the betweenness centralities for given degrees and
then take the average along the degrees. A low mean coefficient of variation
(CV) means that the correlation between the betweenness centrality and the
degree is high, and similarly, a high CV means that the correlation is low. We
also computed the Pearson correlation, and a weighted mean of the coefficient
of variation, where similarly to Kitsak et al. [17], we weighted by the degree
distribution.

#### Results

We found that the different metrics for measuring the correlation between
betweenness centrality and degree gives consistent results. Here we discuss
our findings concerning the mean CV in detail, for results about the other
metrics see the supplementary material [48]. Figure 23 shows the mean CV of
the betweenness centralities for three network models and the real-world
networks. In the case of network models, for each parameter setting, we took
the average of the results of 15 graphs.

[width=.45]figures/degree_btw/deg_bwc_cv_lswtm.pdf
[width=.45]figures/degree_btw/deg_bwc_cv_shm.pdf
[width=.45]figures/degree_btw/deg_bwc_cv_hadgm2.pdf
[width=0.45]figures/degree_btw/deg_bwc_cv_real.pdf

Figure 23: The mean coefficient of variation (CV) of the betweenness
centrality per degree. The fractal networks are colored red and the non-
fractal networks are blue. The difference in the distribution of the mean CV
of the fractal and non-fractal real networks is not statistically significant
(Mann-Whitney U rank test’s $p$-value: 0.125).

Figure 23 suggests that the Song-Havlin-Makse model (Figure 23 (b)) indeed
supports the conjecture of Kitsak et al. (for networks with at least 100
nodes), but the other network models and real-world networks do not seem to be
in alignment with this conjecture.

As Figure 23 (a) illustrates, in the case of the LSwT model, the dispersion
(CV) of betweenness centrality is low for both pure fractal and pure non-
fractal networks. Surprisingly, those networks have the highest dispersion
(i.e., lowest correlation) that possess a mixture of the fractal and non-
fractal properties ($0<p<1$).

In the case of the HADG model (Figure 23 (c)) while it is true that the purely
non-fractal network has lower variance (i.e., the higher correlation between
degree and betweenness centrality) and that the purely fractal network has
high dispersion. It is also apparent that when $b=0.1$ the model is still non-
fractal, but the deviation of the betweenness centralities is as high as in
the case of the fractal networks ($b=0.5$ and $b=1.0$).

We also studied the $(u,v)$-flower (the results are included in the
supplementary material [48]), and we found that while on average the non-
fractal networks ($u=1$) have lower CV values, it is also possible to generate
fractal networks that have nearly zero variance regarding the betweenness
centralities for given degrees. Hence, the $(u,v)$-flower model also does not
really support the observation of Kitsak et al. [17].

Finally, Figure 23 (d) demonstrates that in real networks the distribution of
the coefficient of variation of the betweenness centrality in fractal and non-
fractal networks does not differ.

### Small-world property

Another widely studied topic is the relationship between the small-world and
fractal properties of networks. Csányi and Szendrői suggest that these two are
conflicting properties of networks, however, they also mention that mixed
property could also be possible in such a way that a network is
microscopically small-world, but fractal on a macroscopic scale [20]. Several
other papers also reported that there is a connection between the lack of the
small-world property and the emergence of fractality [25, 30, 51, 52, 53]. On
the other hand, plenty of models have been introduced, that exhibit transition
from fractal to small-world networks [14, 54, 55, 56, 57]. Moreover,
modifications of existing models have been proposed to demonstrate the
simultaneous presence of the two properties in the generated networks [14, 46,
58, 59, 60].

The possible connection between the fractal and small-world property of
networks has received a great deal of research interest, however, it has to be
mentioned that the term small-world network is often used non-rigorously. By
definition, a network can be considered small-world, if $l\sim\log N$ holds
[8], i.e., when the average distance in the network ($l$) grows proportionally
to the logarithm of the number of nodes ($N$). However, this definition can
only be applied to networks evolving with time or where different states can
be compared, so typically to network models. Consequently, networks having a
relatively small diameter or average path length compared to their size are
also often referred to as small-world networks [61].

Here, we rather distinguish between the two approaches and examine the
relation of fractality to the small-world property using two slightly
different approaches. First, we consider the length of the diameter and
average path length for all networks to see whether fractal networks have
larger distances. In the second approach, we consider growing network models
to study whether the original concept of small-world property and the fractal
property can simultaneously be present in a network or whether these are
exclusive characteristics.

#### Results with normalized average path length and diameter

For the first approach, to be able to compare the distances of networks of
different sizes, both the average path length and the diameter are normalized
by the logarithm of the number of nodes [45].

In the case of the $(u,v)$-flower and the Lattice small-world transition
model, apart from networks with very few nodes, the different cases determined
by the main parameter ($u$ for the $(u,v)$-flower and $p$ for the LSwTM)
clearly separate, and the distances are growing as the networks become
fractal. This phenomenon is well illustrated by Figure 24 (a), which shows the
change in the normalized diameter and average path length for the
$(u,v)$-flower.

[width=0.95]figures/avgpath_diam.pdf

Figure 24: Normalized diameter (i.e. diameter divided by the logarithm of the
size) plotted against the size on a semi-log scale for the cases of the (a)
$(u,v)$-flower, (b) real networks. On subplot (a) the coloring is based on the
$u$ parameter of the model. The insets are showing the semi-log plots of the
normalized average path length plotted against the size of the network. The
difference between the distribution of normalized diameter and the normalized
average path length of the fractal and non-fractal real networks are
statistically significant (Mann-Whitney U rank test’s $p$-value: $7.66\cdot
10^{-6}$ and $1.19\cdot 10^{-5}$). The results for the other models, as well
as an $R$-value-based illustration of the real networks, can be found in the
supplementary material [48].

In the case of the Song-Havlin-Makse and the HADG model, this clear separation
can only be observed for fixed values of the $(n,m)$ parameter pair. Namely,
the parameter, which influences fractality, also affects the distances
similarly to LSwTM and $(u,v)$-flower.

Since the Repulsion based fractal model always generates fractal networks, the
previous comparing-based examination cannot be applied to this model, however,
it can be said that the average path length and the diameter of this model are
similar to those of the fractal networks generated by the SHM and HADG models.

As Figure 24 (b) illustrates, similar observations can be made on the real
networks, too. The distances are mostly quite small both in non-fractal and
fractal networks and for small values the two classes can hardly be separated
based on the normalized diameter and average path length. However, we can
observe that non-fractal networks do not tend to have large distances, and
based on our dataset, a cut-off point at 2.5 for the normalized average path
length, and at 5 for the normalized diameter can be created. Above these
values, there seem to be only fractal networks. However, we have to emphasize
that there are numerous fractal networks below these cut-off points as well,
which means that fractality does not originate just from large distances.

#### Results with growing network models

All the investigated network models undergo a transition from the small-world
to the non-small-world property driven by the main parameter that also drives
the fractality of the network: as fractality weakens, small-world property
arises. This transition is not sharp, and there are intermediate states, where
both properties are significant. The small-world transition can also be
observed in the Repulsion based fractal model, which generates only fractal
networks. Figure 25 illustrates some cases of the Repulsion based fractal
model and the Lattice small-world transition model. It can be seen that there
are a few states for both models, where the small-world property is met, while
the networks are also fractal.

[width=0.95]figures/smallworld.pdf

Figure 25: Average path length as a function of the logarithm of the size for
some cases of the (a) Repulsion-based fractal model, (b) Lattice small-world
transition model. On subplot (a) the networks are generated by setting
parameter $Y$ to 0, and the coloring is based on parameter $n$. On subplot (b)
the coloring is based on parameter $p$. The insets are showing the log-log
plots of the same cases.

Except for the Lattice small-world transition model, the relation of the
average path length to the size can also be examined considering the
iterations, through which the network evolves. That is we fix all the
parameters, except the iteration number ($n$), and see if the average path
length grows proportionally to the logarithm of the size or power law holds
instead. It can be said that in this approach, for the cases where the
resulting networks are rather fractal than non-fractal, indeed power law
holds, while in the non-fractal cases logarithmic relation can be observed.
The only exception is the Repulsion based fractal model because it has a few
(fractal) cases, where the connection is rather logarithmic, for which an
illustration can be found in the supplementary material [48].

Overall, if we take into consideration both of the approaches concerning the
small-world property we can say that there is a significant relation between
fractality and non-small-world property, however, they do not necessarily
exclude each other since there are examples for networks, which are fractal
and small-world at the same time.

## A machine learning approach

In the previous sections, we investigated the network characteristics that
have been associated with fractality one by one. Here, we address the problem
as a binary classification task and distinguish between fractal and non-
fractal networks based on a few selected network characteristics. The benefit
of this approach is that we can investigate numerous network metrics at the
same time, identify the most important features and also recognize how the
combination of metrics affects the fractal scaling of networks.

Here, we use three decision tree-based classification algorithms to
distinguish between fractal and non-fractal networks: simple decision tree,
random forest, and XGBoost. We select the explanatory variables to get a
collection of characteristics, which represents the structure of the networks
well, but they are not too correlated. Moreover, we aim to make these metrics
as independent of the network size as possible, hence where it is reasonable,
normalization is also performed. We extend the set of metrics that we used in
our earlier studies [45, 62] with features that have been associated with
fractality. The list of our explanatory variables together with their
description can be found in Table 5.

Name | Description
---|---
avg_deg | Average degree
assortativity | Assortativity coefficient
avg_clust | Average of the local clustering coefficients
max_eigen | Maximum of the eigenvector centralities
skew_deg_dist | Skewness of the degree distribution
diam_logsize | Diameter, divided by the logarithm of the number of nodes
max_deg_n | Maximum degree, divided by the number of nodes minus 1
hub_connectivity | Hub connectivity score
ebc_avg_95 |  | Average of the edge betweenness centralities
---
over the top 5% of edges

Table 5: Name and description of the chosen explanatory variables, for the
classification task.

We consider three datasets to perform the task on, one consisting of the
model-generated networks, one of the real networks, and one which combines the
two sets, thus including all examined networks. Moreover, we drop the small
networks from all datasets, i.e. the ones whose number of nodes is less than
100, because in most of these cases fractality can hardly be defined, as was
also mentioned earlier. We use 2/3 of the datasets for training and the
remaining 1/3 for testing to avoid overfitting. Two evaluation metrics are
used to measure the performance of the algorithms, accuracy and the Area Under
the ROC Curve (AUC). The hyperparameter optimization for the algorithms is
carried out based on the latter one because, in the case of an unbalanced
class distribution, the accuracy score can often be misleading. For the data
preparation, training, and evaluation of the algorithms, we use the scikit-
learn [63] and XGBoost [64] Python packages.

To identify the most important variables, we calculate the permutation
importance score of the features. This score shows how much the performance of
the model decreases if the values of a given attribute are randomly permuted.

#### Results

The performance of the models measured on the test sets is summarized in Table
6. It can be said that all of the algorithms can solve the problem with high
accuracy and AUC score, thus we can conclude that fractal and non-fractal
networks indeed differ in the considered network characteristics.

| Decision tree | Random forest | XGBoost
---|---|---|---
model | real | all | model | real | all | model | real | all
Accuracy | 0.97 | 0.86 | 0.90 | 0.95 | 0.91 | 0.95 | 0.95 | 0.86 | 0.95
AUC | 0.98 | 0.90 | 0.97 | 1.00 | 0.95 | 0.98 | 0.99 | 0.82 | 0.98

Table 6: Accuracy and AUC scores of the different algorithms on the
fractal/non-fractal classification task, on the three datasets.

|  | First feature | Second feature | Third feature
---|---|---|---|---
model | DT | diam_logsize (0.285) | hub_connectivity (0.198) | assortativity (0.109)
RF | hub_connectivity (0.136) | diam_logsize (0.071) | avg_clust (0.020)
XGB | diam_logsize (0.250) | hub_connectivity (0.159) | assortativity (0.071)
real | DT | avg_deg (0.107) | hub_connectivity (0.104) | diam_logsize (0.067)
RF | avg_clust (0.040) | assortativity (0.038) | max_deg_n (0.029)
XGB | assortativity (0.112) | hub_connectivity (0.044) | diam_logsize (0.022)
all | DT | diam_logsize (0.290) | assortativity (0.118) | hub_connectivity (0.098)
RF | hub_connectivity (0.086) | diam_logsize (0.082) | avg_clust (0.058)
XGB | diam_logsize (0.272) | avg_deg (0.089) | hub_connectivity (0.088)

Table 7: The most important features for every algorithm and dataset. For all
the cases the three features with the highest permutation importance score are
listed, showing the exact scores as well.

Table 7 shows the three most important features with the corresponding
permutation importance scores for every algorithm and dataset. We can observe
that for model-generated networks, the normalized diameter and the hub
connectivity score both have significant importance for all algorithms. In
addition, the assortativity coefficient also seems to be important for most of
the methods. In the case of real networks, the set of important features
varies for the different machine learning (ML) models. The assortativity
coefficient, the hub connectivity score, and the normalized diameter are among
the most important attributes for two of the three algorithms, but the average
clustering coefficient and the average or maximum degree can also be
considered important features for some ML models. In the combined dataset, the
normalized diameter and the hub connectivity score turned out to be the most
important characteristics of all algorithms. The assortativity coefficient,
average clustering coefficient, and average degree also seem to have notable
importance for some of the methods.

Figure 26 shows two scatterplots of the combined dataset with respect to
different network characteristics. It can be seen that while a large
(normalized) diameter is a characteristic of only the fractal networks, an
additional feature is still not enough to clearly separate fractal and non-
fractal networks when the diameter is small. Similarly, most of the
investigated fractal networks possess a small hub connectivity score, but
there are a significant number of them with large $HCS$, and considering the
normalized maximum degree as well, we still cannot separate the two classes
clearly.

[width=0.9]figures/ML_05.pdf

Figure 26: Scatter plots of the dataset consisting of both real and model
generated networks, with respect to two selected important features. (a)
Assortativity coefficient plotted against the normalized diameter. (b)
Normalized maximum degree plotted against the hub connectivity score.

From the results detailed above, we can conclude that the magnitude of
distances in a network is certainly connected to fractality. It may not be the
distances between hubs, which influence fractality, but average distances
generally. However, the connectivity of hubs, as well as assortativity indeed
seem to have a distinguishing ability. Although alone they are not enough to
separate fractal and non-fractal networks, together with other properties,
they could contribute to the distinction. It seems that, although a single
network characteristic does not clearly determine the fractal property, the
combination of several metrics can achieve excellent distinguishing power.

## Summary

In this work, we investigated which characteristics could cause the emergence
of fractal scaling in complex networks. Our analyses relied on a large dataset
of both real-world and model-generated networks, in order to prevent making
conclusions based on coincidences. Our most important findings are summarized
in Table 8.

| SHM | HADGM |  $(u,v)$-flower | RBFM | LSwTM | Real networks
---|---|---|---|---|---|---
Disassortativity |  frac.$\Longrightarrow$disass. disass.$\centernot\Longrightarrow$frac. |  frac.$\Longrightarrow$disass. disass.$\centernot\Longrightarrow$frac. | ✓ | ✓ | ✗ | ✓
Hub repulsion | ✓ | ✓ | ✓ | $\bigcirc$ | ✓ | ✓
Long range anticorrelation |  | Fluctuation
---
analysis
✓ | $\bigcirc$ | ✓ | ✗ | ✗ | $\bigcirc$
Hub distances | $\bigcirc$ | ✓ |  ✓($n>3$) | ✓ | – | $\bigcirc$
| Neighbor
---
connectivity
$\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | ✗ | $\bigcirc$
Small edge BC | Mean EBC | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
Max EBC | ✗ | ✗ | ✓ | ✗ | ✗ | ✗
Mean of top 5% | ✗ | ✗ |  | ✓($N\leq 100$)
---
✗($N>100$)
✗ | ✗ | ✗
| Weak correlation between
---
degree and node BC
✓($N>100$) | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$
Non-small-world property |  | Average path length
---
and diameter
✓ | ✓ | ✓ | $\bigcirc$ | ✓ |  | large dist. $\Longrightarrow$ frac.
---
small dist. $\centernot\Longrightarrow$ non-frac.
| Growing network
---
models
$\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | $\bigcirc$ | –

Table 8: The relation of network models and real networks to the examined
conjectures concerning fractal characteristics. The check mark (✓) denotes
that the model/real networks support the statement (for real networks it means
that the distribution of the metric is statistically significantly different
in fractal and non-fractal networks), the cross mark (✗) means that they
contradict it, i.e. they behave in the opposite way, the circle ($\bigcirc$)
refers to mixed results and when no clear conclusion can be drawn, moreover,
the hyphen (–) denotes when the analysis could not be carried out due to the
nature of the model. Finally, $N$ denotes the number of nodes and $n$ denotes
the number of iterations.

Concerning the disassortativity of fractal networks, we have found that
although most of the considered mathematical models suggest that fractality
correlates with disassortativity, there is also one model, the Lattice small-
world transition model, which completely contradicts the statement.
Consequently, we can conclude that although disassortativity is common amongst
fractal networks, just based on the disassortativity, we cannot clearly tell
whether a network is fractal or not, which is suggested by real networks as
well. We conclude that disassortativity cannot be considered the reason behind
fractality, Somewhat similar observations can be made in the case where hub
repulsion was measured directly. All of the considered network models show
that in fractal networks hubs are less connected than in non-fractal networks
(smaller hub connectivity score). The real networks also suggest that a large
hub connectivity score (hub attraction) is a property of non-fractal networks,
but counterexamples on both sides make hub repulsion a non-universal
characteristic of fractal networks.

The possible connection of long-range anticorrelation to fractality was
reviewed using three different methods. Here we could not find a clear
connection between the correlation of node degrees and fractal scaling.
Although for all three methods, we could find examples of both model-generated
and real networks, which support the suggestion of anticorrelation in fractal
networks, even on the long-range scale, there are numerous counterexamples as
well.

The suggestion of the connection of edge betweenness centrality with
fractality was also reviewed. We examined whether fractal networks can possess
edges with large betweenness centrality. We have come to the conclusion that
fractal networks show no tendency to have edges mostly with small betweenness
centrality. Almost all of the examined network models show the opposite of the
statement, while real networks suggest that small edge betweenness centrality
does not depend on fractality.

In addition to the connection of fractality and edge betweenness centrality, a
suggestion regarding node betweenness centrality was also revised. Namely, we
revisited the conjecture that the correlation between degree and node BC is
weaker in fractal networks than in non-fractals. We have found that the Song-
Havlin-Makse model supports this statement, but all the other mathematical
models and the real networks rather contradict it.

We investigated thoroughly the suggested conflicting relation of fractality
and the small-world property. We have found that those network models which
are able to generate both fractal and non-fractal instances support the
observation that the distances (average path length and diameter) are larger
in fractal networks. In the case of real networks, we have also found that
large distances are present only in fractal networks, however, small distances
do not imply non-fractality. Moreover, a transition from small-worldness to
large distances can be observed in the Repulsion based fractal model as well,
which generates only fractal networks. When we examined the small-world
property on growing network models we experienced that the vast majority of
the considered model-generated networks support the conflicting relation of
fractality and small-world property.

Finally, we introduced a novel approach to analyze the origin of fractal
networks. We formulated a binary classification problem with the goal to
distinguish fractal and non-fractal networks based on other network
properties. We solved the problem with state-of-the-art machine learning
algorithms and identified the characteristics with high distinguishing
ability. The results suggest that although a single characteristic is not
enough, a combination of several metrics can distinguish between fractal and
non-fractal networks efficiently. The normalized average distance is possibly
one of the most essential properties in recognizing fractal scaling, moreover,
hub connectivity and assortativity can also contribute to the characterization
of fractal networks together with other properties.

An important direction of further studies could be to directly examine the
possible connection of the proposed joint properties to fractality. For these
analyses, the extension of the dataset with additional models and real
networks may be necessary. Furthermore, other network characteristics should
be involved in such studies, which have not been considered in previous works.
Different approaches could also be used to distinguish fractal and non-fractal
networks, such as network embedding techniques. If the networks could be
embedded into a vector space, where the two classes are well-separated, then
the properties of this space could reveal what the difference lies in.

## Competing interests

The authors declare that they have no known competing financial interests.

## Author’s contributions

RM conceived the study. EZP reviewed the literature. EZP and MN implemented
and carried out the analyses. EZP prepared the original draft. MN and RM
reviewed and edited the manuscript. All authors read and approved the final
manuscript.

## Acknowledgment

This work has been partially supported by the National Research, Development,
and Innovation Office (NKFIH, Project K142169) and by the ”Fractal geometry
and applications” Research Group (NKFIH, Project KKP144059).

## Data availability

The networks and the data analyzed in this study are publicly available at the
following GitHub repository [48]: https://github.com/marcessz/fractal-networks

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