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Towards a better understanding of the characteristics of fractal networks
Applied Network Science volume 8, Article number: 17 (2023)
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 socalled boxcovering method. A network is called fractal if the minimum number of boxes needed to cover the entire network follows a powerlaw 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, longrangerepulsive correlation, and small edge betweenness centralities. However, these assertions are usually based on tailormade 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 realworld networks originating from six domains. We clarify which characteristics are universally present in fractal networks and which features are just artifacts or coincidences.
Introduction
Network science has received a great deal of research interest in the past two decades since networks can efficiently model numerous realworld structures and phenomena, including the Internet, the WWW, cellular networks, and social networks (Newman 2018). 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 (Forster et al. 2020), the mechanisms underlying the disease (Prasad et al. 2020) and the social interactions of communities (Yum 2020; Karaivanov 2020).
The breakthrough in network science dates back around the millennium since the rapid and largescale development of computer science made it possible to store and efficiently analyze complex networks (Molontay and Nagy 2021). 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 scalefree property (Barabási and Albert 1999) and the smallworld property (Watts and Strogatz 1998).
The fractality of networks is another wellstudied characteristic. While the notion of fractal scaling was originally introduced in geometry, it has been extended to complex networks as well (Song et al. 2005). The book of Rosenberg (2020) and the survey of Wen and Cheong (2021) give an extensive overview of fractal networks. Fractal scaling was verified in various realworld networks (Song et al. 2005; Yook et al. 2005; Gallos et al. 2007) and has been associated with numerous important properties, such as robustness against intentional attacks (Song et al. 2006) and accelerated flow (Gallos et al. 2008). 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 (Song et al. 2006; Fujiki et al. 2017; Kitsak et al. 2007; Wei and Wang 2016) 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 realworld and modelgenerated networks.
The investigated characteristics that have been connected to the fractality of networks are the following:

1
Yook et al. (2005) and Song et al. (2006) argued that fractality originates from the disassortativity of the network and the repulsion (disconnectedness) of the hubs.

2
Fujiki et al. (2017) and Rybski et al. (2010) demonstrated, using different approaches, that there is a connection between longrange anticorrelation and fractality.

3
Wei and Wang (2016) 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
Csányi and Szendrői (2004) were the first to draw attention to the opposing relationship between fractality and smallworldness, however, among many others, they also mention that the transition between fractal and smallworld is smooth and these two properties can also be present simultaneously.

5
Finally, Kitsak et al. (2007) argued that in fractal networks there is a weaker correlation between the degree and the betweenness centrality of the nodes than in nonfractal 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 socalled boxcovering method, using the length of the shortest path between two nodes as the distance metric. The method can be summarized as follows (Song et al. 2005): 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.:
The \(d_B\) exponent is called the box dimension or fractal dimension of the network.
Boxcovering is proved to be an NPhard problem (Song et al. 2007), 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 boxcovering algorithms, we refer to the work of Kovács et al. (2021). 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 (Song et al. 2007):

1
Let \(C\) be the set of uncovered nodes.

2
Randomly choose a \(c\in C\) node, and remove it from \(C\).

3
Remove every node from \(C\), which is at distance at least \(l_B\) from \(C\).

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
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 boxcovering 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. (2009), 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 NPhard nature of the boxcovering 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 \(l_B\) (Clauset et al. 2009; Rozenfeld et al. 2009). 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 loglog plot, fit a straight line, and decide about the goodnessoffit by the mean squared error, the coefficient of determination or by simply looking at the plots (Fujiki et al. 2017; Zheng et al. 2014). 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. (2021). 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 (2014), which takes advantage of the observation that while in fractal networks the \(N_B(l_B) \sim l_B^{d_B}\) relation holds, for nonfractal networks \(N_B(l_B) \sim e^{d_e l_B}\) is true (Rozenfeld et al. 2009). Here, we apply a modified version presented by Akiba et al. (2016). 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 rootmeansquare errors of the two curves:
The idea of normalization of data points and the use of RMSE allows us to compare the goodnessoffit of different networks.
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 nonfractal. 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 nonfractal networks, we still need to create a cutoff point. We observed in both real and modelgenerated 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 nonfractal 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 modelgenerated 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.
SongHavlinMakse model
One of the most wellknown fractal network models is the SongHavlinMakse model (SHM) (Song et al. 2006). 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
The initial graph is a simple structure, e.g. two nodes connected via a link.

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
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 nonfractal for \(p=0\) (Song et al. 2006; Molontay 2015). 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 nonfractal, which is illustrated in Fig. 3. It can also be seen that the transition from fractal to nonfractal is smooth, hence in the \(0.4\le p\le 0.6\) 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 cutoff point at \(p=0.5\), because we do not want to exclude the intermediate networks.
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 SongHavlinMakse model (SHM) in such a way that the new mechanism can generate fractal networks with strong hub attraction (Kuang et al. 2015). 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 withinbox linkgrowth 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 (Kuang et al. 2015):

1
The initial condition and the growth of the model are the same as in step 1 and 2 of the SongHavlinMakse model.

2
We rewire the (x, y) edge at time \(t+1\) with probability
$$\begin{aligned} p_{xy}&= {\left\{ \begin{array}{ll} a, &{}\text { if } \frac{deg_t(x)}{deg_t^{max}}>T \text { and } \frac{deg_t(y)}{deg_t^{max}}>T\\ b, &{}\text { otherwise, } \end{array}\right. } \end{aligned}$$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 nonhubs.

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 (Kuang et al. 2015), consequently, as a slight abuse of notation, the probabilities we use here are equivalent to \(1a\) and \(1b\) with regard to the original article.
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 (Kuang et al. 2015). 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 \(b\le 0.1\) we can generate nonfractal networks, independently of parameter a given that \(a<b\). This is wellillustrated in Fig. 5.
(u, v)flower
The family of (u, v)flowers was introduced by Rozenfeld et al. (2007). 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
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 \(u\le v\).

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.
Two networks generated with different parameter settings are shown in Fig. 6. Rozenfeld et al. (2007) showed that the model generates fractal networks when \(u>1\), and nonfractal ones when \(u=1\). This statement is illustrated in Fig. 7. Furthermore, it was also shown by Rozenfeld et al. (2007) that in the \(u=1\) case the resulting networks are smallworld, which supports the idea that fractal and smallworld are conflicting properties. This statement is investigated in the Smallworld property section.
Repulsion based fractal model
In our earlier work (ZakarPolyák et al. 2022a), we introduced the repulsionbased fractal (RBF) model, which is also based on the SHM model (Song et al. 2006) and adapts some concepts of the HADG model as well (Kuang et al. 2015). The model evolves through time and rewires edges with probability based on the degree of the endpoints to create repulsion among nodes. The withinbox linkgrowth method of Kuang et al. (2015) is also adapted by the model to increase the clustering coefficient, hence creating more realistic networks. The growing mechanism of the repulsionbased fractal model is as follows:

1
The initial condition and the growth of the model are the same as in step 1 and 2 of the SongHavlinMakse model.

2
In iteration step \(t+1\) we remove every edge (x, y) with probability
$$\begin{aligned} p_{xy}^Y = 1  \left Y  \frac{\deg _t(x)+\deg _t(y)}{2\cdot \deg _{t, \max }}\right , \end{aligned}$$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
We add \(\deg _t(y)\) edges among the newly generated offspring of every old node y. In order to avoid creating selfloops, this step is only executed, when \(m>1\).
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 Fig. 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 (ZakarPolyák et al. 2022a).
Lattice smallworld transition model
The lattice smallworld transition model (LSwTM) was also introduced by ZakarPolyák et al. (2022a). It utilizes the fractal nature of gridlike structures and also adapts the preferential attachment mechanism to work against fractal scaling. The model is defined as follows:

1
We start with a ddimensional (practically \(d=2\)) grid graph with \(n_1 \times n_2 \times \dots \times n_d\) vertices.

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 \(p_{z}\):
$$\begin{aligned} p_{z} = \frac{1}{1 + \exp \left( a \cdot \left( \frac{\deg (z)}{\deg _{\max }}  \frac{1}{2}\right) \right) }, \end{aligned}$$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.
Even a small probability of rewiring results in a network that differs greatly from the initial grid graph, as illustrated in Fig. 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 nonfractal networks (ZakarPolyák et al. 2022a). As Fig. 11 demonstrates, it is reasonable to choose \(p=0.01\) as a cutpoint. It was also shown by ZakarPolyák et al. (2022a) that this model demonstrates a fractal—smallworld transition.
Data
To gain a complete understanding of the relationship between fractality and other network properties, it is essential to consider a diverse and largescale collection of realworld and modelgenerated 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 networkrelated calculations, including the network generation process, were done using the NetworkX package (Hagberg et al. 2008).
Modelgenerated 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 Tables 1 and 2.
For those analyses, where the evaluation is done on a networkbynetwork 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 nonfractal networks (except for the RBFM, where only fractal networks can be generated).
Real networks
Real networks were collected from various online repositories (Stark et al. 2006; Clauset et al. 2016; Vázquez and Naik 2003; Kunegis 2013; Rossi and Ahmed 2015; Barabási 2016; Kasthuri et al. 2008, ^{Footnote 1}^{,} ^{Footnote 2} Cho et al. 2014; Bu et al. 2006; Jeong et al. 2001; Nagy and Molontay 2022). 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 realworld 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 nonfractal networks from all size categories presented in the domain.
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 boxcovering algorithm, we repeated the procedure 15 times and averaged their outcomes. The resulting class distribution of modelgenerated networks, real networks, and all combined networks is shown in Fig. 12. It can be seen that there are much more fractal networks amongst both the modelgenerated and the real networks, but the number of nonfractal networks is also significant.
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 (Yook et al. 2005; Song et al. 2006). It has been much disputed whether the fractal nature of networks originates from these characteristics, there are papers that support this assertion (Zhang et al. 2007), but there are more works that confute it (Fujiki et al. 2017; Wei and Wang 2016; Kuang et al. 2015; Nagy 2018).
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 disassortativity
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 nonfractal networks are assortative, as Fig. 13b shows.
While the fractal networks that the SongHavlinMakse and the Hub attraction dynamical growth models generate are disassortative, the nonfractal networks generated by these two models are also disassortative. Although in the case of the SHM model, the figures in the supplementary material (ZakarPolyák et al. 2022b) suggest that for a specific parameter setting (\(m=2\)) the fractal networks are more disassortative than the nonfractal ones, for \(m=1\) and \(m>2\), fractal networks typically have a higher assortativity coefficient than the nonfractal 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 nonfractal 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 Fig. 13a). In this sense, the LSwT model behaves in the opposite way to the (u, v)flower.
In the case of realworld networks, we can say that fractal networks are often disassortative, but there are numerous examples of assortative cases too, which is well illustrated in Fig. 14. Moreover, if we consider not only the binary fractal/nonfractal 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 (ZakarPolyák et al. 2022b).
Overall, our findings partially support the conclusion of Kuang et al., namely, that fractality is independent of the assortative mixing (Kuang et al. 2015), 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., nonfractal) 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 nonfractal networks can be clearly separated according to their HCS. Fractal (u, v)flowers have HCS close to 0, while for nonfractal (u, v)flowers, this measure is at least 1, which can be seen in Fig. 15a.
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 nonfractal networks (i.e. when \(b\le 0.1\)) 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 Kuang et al. (2015) 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 nonfractal networks generated by the HADG model.
Similar behavior is demonstrated by the other models as well, that is fractal networks show stronger hub repulsion than nonfractals. 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 Repulsionbased fractal model is able to create fractal networks with HCS being close to or even above 1, while in the case of the SongHavlinMakse model, only the nonfractal networks possess such high scores. Illustrations of the results for the SHM, RBF, and LSwT models can be found in the supplementary material (ZakarPolyák et al. 2022b).
Similarly, as Fig. 15b 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 nonfractals, although there are also many examples of nonfractal networks with lower scores. Similar observations can be made if we consider the R values of the networks (see: ZakarPolyák et al. 2022b). 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 nonfractal networks, hence it cannot be considered as the reason behind fractality.
Longrange correlation
Besides direct degree correlation, the longrange correlation has also been associated with fractal scaling (Fujiki et al. 2017; Rybski et al. 2010). Both studies suggest, based on different approaches, that there is a connection between longrange anticorrelation and fractality. Here, we apply both of the methods (Fujiki et al. 2017; Rybski et al. 2010) in addition to a more immediate extension of neighborlevel degree correlation measures, introduced by Mayo et al. (2015).
In the work of Rybski et al. (2010), a fluctuation analysis approach was proposed to measure longrange correlations. The steps of this method can be summarized as follows:

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

3
Repeat steps 1 and 2 for all possible d.

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 longrange correlations.
An extension of the concept of hub repulsion to longrange scales was proposed by Fujiki et al. (2017). The authors examined how the distribution of hub distances looks in fractal and nonfractal networks. The procedure can be summarized by the following steps:

1
Calculate the distance of all pairs of hubs.

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
Calculate \(\tilde{P}(l)\) by dividing \(\hat{P}(l)\) by the number of possible edges among hubs, i.e. \(\tilde{P}(l) = \hat{P}(l)/\left( {\begin{array}{c}N_{hub}\\ 2\end{array}}\right)\).
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 Fujiki et al. (2017), for this analysis, we cut off the hubs at the 98th percentile of the degree distribution.
The third approach to capture longrange correlations was introduced by Mayo et al. (2015) and has not been used before to study the relationship between fractality and longrange 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
Fix m, and for every node x, take the average degree of the nodes that are at distance m from x.

2
Calculate \(\langle k_m\rangle (k)\) by taking the average of the outputs of step 1 over nodes with degree k.

3
Examine the relation of k and \(\langle k_m\rangle (k)\).
Following the line of Mayo et al. (2015), 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 realworld and modelgenerated networks in Fig. 17. Generally, it can be said that for the (u, v)flower and the SongHavlinMakse model, F(d) scales as a power of d with exponent less than \(\frac{1}{2}\) in the fractal cases, while in the nonfractal cases the relation is rather exponential, which supports the observations of Rybski et al. (2010).
However, in the case of the Repulsionbased fractal model, F(d) does not follow a power law. It may not be immediately visible from Fig. 17b, but the exponential curve provides a better fit than the power law. For the fitting, we use the powerlaw Python package (Alstott et al. 2014).
For the Hub attraction dynamical growth model, as b increases, the powerlaw 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 smallworld transition model, none of the previously mentioned relations seem to hold on F(d) for the fractal networks.
Among realworld networks, there are some cases, where the expected behavior of F(d) can be observed, as Fig. 17c shows. However, there are examples, where power law relation cannot be detected, thus longrange correlations cannot be concluded, as it is illustrated in Fig. 17d. In conclusion, we can say that longrange 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 Fujiki et al. (2017), that in fractal networks hub distances have a wide range, while in nonfractal networks hubs cannot get far from each other.
However, a surprising observation can be made based on the SongHavlinMakse model. As Fig. 18a 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.
In the case of the LSwTM, for \(p\le 0.1\), 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 Fig. 18b show two networks of the same size where the fractal network clearly possesses smaller hub distances than the nonfractal one. Moreover, the third and fourth subplots of Fig. 18b show two fractal networks of the same size with completely different hub distances.
Results with longrange neighbor connectivity
Finally, the results obtained by the third approach, i.e., the neighbor connectivity (Mayo et al. 2015), suggest that there is no apparent connection between the fractality and the longrange correlation profile of networks.
The Hub attraction dynamical growth model seems to be the only exception, because the nonfractal 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 \(m\ge 3\).
In the case of the Repulsionbased fractal model and the SongHavlinMakse model, usually, no correlation can be detected for \(m\ge 3\) 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.
Networks generated by the Lattice smallworld transition model preserve their correlation profile for all m distances, i.e., fractal networks have positive degree correlations, while nonfractal networks have negative correlations, even in the longrange scale (see Fig. 19).
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 (ZakarPolyák et al. 2022b).
Overall, we can conclude from the results of all three approaches that fractality and longrange correlation profiles do not have a clear ubiquitous connection.
Edge betweenness centrality
Wei and Wang (2016) 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.
The SongHavlinMakse, 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 nonfractal networks can be observed in the aforementioned three models: fractal networks tend to obtain larger values than nonfractal networks of the same size (see the supplementary material (ZakarPolyák et al. 2022b)). Similar observations can be made on the Lattice smallworld transition model (see Fig. 22a–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 nonfractal 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 nonfractals. 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 Fig. 22d–f.
Figure 22g–i suggests that fractal real networks often have larger values concerning these edge betweenness centralityrelated measures than nonfractals. However, there are also examples of nonfractal networks with high edge BC measures, and when the values are low the fractal and nonfractal 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 ZakarPolyák et al. 2022b).
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 nonfractal 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. (2007) studied the betweenness centrality of fractal and nonfractal networks. The authors analyzed seven SHM modelgenerated networks and four relatively large realworld 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 nonfractal networks. The authors argue that the Pearson correlation coefficient is not a suitable metric to characterize the difference between fractal and nonfractal networks because the average betweenness centrality for a given degree does not change much (Kitsak et al. 2007). 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. (2007), 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 (ZakarPolyák et al. 2022b). Figure 23 shows the mean CV of the betweenness centralities for three network models and the realworld networks. In the case of network models, for each parameter setting, we took the average of the results of 15 graphs.
Figure 23 suggests that the SongHavlinMakse model (Fig. 23b) indeed supports the conjecture of Kitsak et al. (for networks with at least 100 nodes), but the other network models and realworld networks do not seem to be in alignment with this conjecture.
As Fig. 23a illustrates, in the case of the LSwT model, the dispersion (CV) of betweenness centrality is low for both pure fractal and pure nonfractal networks. Surprisingly, those networks have the highest dispersion (i.e., lowest correlation) that possess a mixture of the fractal and nonfractal properties (\(0<p<1\)).
In the case of the HADG model (Fig. 23c) while it is true that the purely nonfractal 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 nonfractal, 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 (ZakarPolyák et al. 2022b)), and we found that while on average the nonfractal 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. (2007).
Finally, Fig. 23d demonstrates that in real networks the distribution of the coefficient of variation of the betweenness centrality in fractal and nonfractal networks does not differ.
Smallworld property
Another widely studied topic is the relationship between the smallworld 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 smallworld, but fractal on a macroscopic scale (Csányi and Szendrői 2004). Several other papers also reported that there is a connection between the lack of the smallworld property and the emergence of fractality (Zheng et al. 2014; Rozenfeld et al. 2007; Tian and Shi 2008; Mokhlissi et al. 2020; Zhang et al. 2008). On the other hand, plenty of models have been introduced, that exhibit transition from fractal to smallworld networks (Song et al. 2006; Li et al. 2017; Watanabe et al. 2015; Rozenfeld et al. 2010; Zhang et al. 2008). Moreover, modifications of existing models have been proposed to demonstrate the simultaneous presence of the two properties in the generated networks (Song et al. 2006; Zhang et al. 2007; Barrière et al. 2006; Kim et al. 2007; Ikeda 2020).
The possible connection between the fractal and smallworld property of networks has received a great deal of research interest, however, it has to be mentioned that the term “smallworld network” is often used nonrigorously. By definition, a network can be considered smallworld, if \(l \sim \log N\) holds (Watts and Strogatz 1998), 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 smallworld networks (Porter 2012).
Here, we rather distinguish between the two approaches and examine the relation of fractality to the smallworld 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 smallworld 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 (Nagy and Molontay 2022).
In the case of the (u, v)flower and the Lattice smallworld 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 Fig. 24a, which shows the change in the normalized diameter and average path length for the (u, v)flower.
In the case of the SongHavlinMakse 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 comparingbased 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 Fig. 24b illustrates, similar observations can be made on the real networks, too. The distances are mostly quite small both in nonfractal 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 nonfractal networks do not tend to have large distances, and based on our dataset, a cutoff 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 cutoff 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 smallworld to the nonsmallworld property driven by the main parameter that also drives the fractality of the network: as fractality weakens, smallworld property arises. This transition is not sharp, and there are intermediate states, where both properties are significant. The smallworld 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 smallworld transition model. It can be seen that there are a few states for both models, where the smallworld property is met, while the networks are also fractal.
Except for the Lattice smallworld 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 nonfractal, indeed power law holds, while in the nonfractal 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 (ZakarPolyák et al. 2022b).
Overall, if we take into consideration both of the approaches concerning the smallworld property we can say that there is a significant relation between fractality and nonsmallworld property, however, they do not necessarily exclude each other since there are examples for networks, which are fractal and smallworld 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 nonfractal 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 treebased classification algorithms to distinguish between fractal and nonfractal 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 (Nagy and Molontay 2022, 2019) with features that have been associated with fractality. The list of our explanatory variables together with their description can be found in Table 5.
We consider three datasets to perform the task on, one consisting of the modelgenerated 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 scikitlearn (Pedregosa et al. 2011) and XGBoost (Chen et al. 2016) 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 nonfractal networks indeed differ in the considered network characteristics.
Table 7 shows the three most important features with the corresponding permutation importance scores for every algorithm and dataset. We can observe that for modelgenerated 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 nonfractal 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.
From the results detailed above, we can conclude that the magnitude of distances in a network certainly has a connection 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 nonfractal 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 realworld and modelgenerated networks, in order to prevent making conclusions based on coincidences. Our most important findings are summarized in Table 8.
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 smallworld 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 nonfractal networks (smaller hub connectivity score). The real networks also suggest that a large hub connectivity score (hub attraction) is a property of nonfractal networks, but counterexamples on both sides make hub repulsion a nonuniversal characteristic of fractal networks.
The possible connection of longrange 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 modelgenerated and real networks, which support the suggestion of anticorrelation in fractal networks, even on the longrange 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 nonfractals. We have found that the SongHavlinMakse 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 smallworld property. We have found that those network models which are able to generate both fractal and nonfractal 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 nonfractality. Moreover, a transition from “smallworldness” to large distances can be observed in the Repulsion based fractal model as well, which generates only fractal networks. When we examined the smallworld property on growing network models we experienced that the vast majority of the considered modelgenerated networks support the conflicting relation of fractality and smallworld 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 nonfractal networks based on other network properties. We solved the problem with stateoftheart 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 nonfractal 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 nonfractal networks, such as network embedding techniques. If the networks could be embedded into a vector space, where the two classes are wellseparated, then the properties of this space could reveal what the difference lies in.
Availability of data and materials
The networks and the data analyzed in this study are publicly available at the following GitHub repository : https://github.com/marcessz/fractalnetworks
Change history
22 May 2023
A Correction to this paper has been published: https://doi.org/10.1007/s41109023005458
Notes
Transportation Networks for Research Core Team: Transportation Networks for Research. https://github.com/bstabler/TransportationNetworks.
The Centre for Water Systems (CWS) at the University of Exeter: Centre for Water Systems. http://emps.exeter.ac.uk/engineering/research/cws/resources/benchmarks/.
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Acknowledgements
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).
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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.
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ZakarPolyák, E., Nagy, M. & Molontay, R. Towards a better understanding of the characteristics of fractal networks. Appl Netw Sci 8, 17 (2023). https://doi.org/10.1007/s41109023005378
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DOI: https://doi.org/10.1007/s41109023005378