# Detecting different topologies immanent in scale-free networks with the same degree distribution

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Edited by Terrence J. Sejnowski, Salk Institute for Biological Studies, La Jolla, CA, and approved February 22, 2019 (received for review September 29, 2018)

## Significance

This paper highlights that not all scale-free (SF) networks arise through a Barabási−Albert (BA) preferential attachment process. Although evident from the literature, this fact is often overlooked by many researchers. For this purpose, it is demonstrated, with simulations, that established network measures cannot distinguish between BA networks and other SF networks (random-like and lattice-like) with the same degree distribution. Additionally, it is examined whether an existing self-similarity metric is also capable of distinguishing different SF topologies with the same degree distribution. To contribute to this discrimination, this paper introduces a spectral metric, which is shown to be more capable of distinguishing between different SF topologies with the same degree distribution, in comparison with the existing metrics.

## Abstract

The scale-free (SF) property is a major concept in complex networks, and it is based on the definition that an SF network has a degree distribution that follows a power-law (PL) pattern. This paper highlights that not all networks with a PL degree distribution arise through a Barabási−Albert (BA) preferential attachment growth process, a fact that, although evident from the literature, is often overlooked by many researchers. For this purpose, it is demonstrated, with simulations, that established measures of network topology do not suffice to distinguish between BA networks and other (random-like and lattice-like) SF networks with the same degree distribution. Additionally, it is examined whether an existing self-similarity metric proposed for the definition of the SF property is also capable of distinguishing different SF topologies with the same degree distribution. To contribute to this discrimination, this paper introduces a spectral metric, which is shown to be more capable of distinguishing between different SF topologies with the same degree distribution, in comparison with the existing metrics.

- network science
- Barabási−Albert networks
- preferential attachment
- pattern recognition
- power-law degree distribution

The scale-free (SF) property is a fundamental concept in the study of complex networks (1, 2), which describes networks where their degree distribution *p*(*k*) follows asymptotically a power-law (PL) pattern, according to*k* is the node degree and *γ* is the PL exponent that should be *γ* > 1 so that the Riemann zeta function will be finite (3). It has been found that many real-world networks have the SF property, such as biological, citation, spatial, economic, and social networks (4⇓–6), along with the Internet and the World Wide Web (2). For such networks, the PL exponent usually ranges within the interval 2 < *γ* < 3 (1), although sometimes it may exceed these bounds (7).

The SF networks have been named so because PLs have their functional form *f*(*x*) = α·*x*^{−β} unaltered at all scales. This implies that a rescale (*x*: = *cx*) of the independent variable *x* changes the form of *f*(*x*) only through a multiplicative factor that depends on the PL exponent, according to the equation *f*(*cx*) = (*c*^{–β})·*f*(*x*). However, this property, with respect to rescaling, is, by definition, being satisfied only for the degree, and therefore it cannot be considered as a universal property that describes all of the measurable structural attributes in the SF networks (4). For instance, it is not certain that, in an SF network, the distribution of the local clustering coefficient or of the betweenness centrality also follows a PL pattern similarly to the degree (4, 8), and thus the SF property does not, by default, describe these measures too. Moreover, the definition of the SF networks is based on a statistical property (i.e., on fitting PLs to the degree distribution of real-world networks) (1⇓⇓–4, 6), which renders more an empirical (or approximate) and less a structural nature to this definition. Therefore, the definition of the SF property is very broad, and it is not linked directly to a characteristic type of network topology.

A step toward linking the SF property with a characteristic network topology was made when the authors of ref. 2 proposed a procedure generating SF networks, which is commonly known as the Barabási−Albert (BA) model. This procedure is based on growth and on the so-called mechanism of preferential attachment (1, 9), according to which an SF network is produced over time when the probability for a node to gain a new connection is proportional to the node’s degree. This implies that new nodes entering the network “prefer” to connect with the already highly connected ones and thus that an SF network forms hierarchies. In this hierarchical structure, a few nodes (the hubs) undertake the major load of connectivity, a fact that is reflected in the PL shape of the degree distribution (1). The BA networks abound in the scientific literature, and thus they have become the standard SF reference model (1, 2, 4⇓–6). This is obviously because the BA was the first model that successfully described a procedure generating SF networks, but also because the hub-and-spoke hierarchical structure of BA networks is quite important in many disciplines, such as in applied, biological, and socioeconomic research (4⇓–6, 10), as well as in other real-world applications (1, 4, 11).

Although the BA model is capable of producing SF networks, it is not the only model with this capability (4). Indicatively, an alternative to the BA model is the so-called fitting model (or DMS model, which was named so from the initials of its authors) (12), and it is based on a linear preferential attachment procedure, where an additional parameter of the nodes’ initial attractiveness is considered in the model’s algorithm (4). Recently, the authors of ref. 13 proposed a model generating star-like SF networks (called “superstar networks”). This model has a stronger bias toward high-degree nodes than exhibited by standard preferential attachment.

The use of different null models (i.e., reference models, which are generated by a random process and describe a set of features of certain network topology) to generate SF networks does not by default result in the same network topology (13, 14). This happens even in cases where the degree distribution of these models is the same. For instance, the authors of ref. 14 studied, with simulations, some structural properties of SF networks with the same degree distribution (the BA model, the Molloy−Reed model, the Kalisky model, and two SF models proposed by the authors, named “MA” and “MB”), and they observed that these networks have different structural properties in terms of their number of components, their components’ size, and global efficiency (i.e., the harmonic mean of the geodesic edge lengths; see ref. 4). According to this approach, the structures of the examined models ranged between a decentralized pattern with a larger number of components and a centralized BA pattern with all vertices included in a single component and with a medium to high global efficiency.

Within this context, the definition of the SF property was submitted to criticism for being abstract. Indicatively, the authors of ref. 15 noted that SF networks inherit from the literature “an imprecision as to what exactly SF means.” In particular, based on the relevant literature, the authors summarized that the SF networks have a scaling (PL) degree distribution, they are generated by certain random processes (one of which is preferential attachment), they have highly connected hubs, they preserve their property under random degree-preserving rewiring, they are self-similar (i.e., their total is similar to one or more of its parts; see ref. 8), and they are independent of specific domain attributes. Toward an attempt to specialize this broad definition, for the detection of the SF property, the authors of ref. 15 proposed a self-similarity metric defined by the formula*s*_{max} = max{*s*(*H*)}, and *H* is the set of graphs with degree distribution identical to that of *G*.

The *S*(*G*) metric indicates the existence of the SF property when it is (maximum and) close to 1 [*S*(*G*) ≈ 1], which denotes that the hubs in the network are connected to each other. However, with the introduction of the superstar SF networks, which are dominated by a single hub, the authors of ref. 13 have shown that connectivity between hubs is not a defining condition for the SF networks. Thereupon, they proposed an approach for distinguishing between the BA and the superstar SF topology. This approach was based on the counting of the number (and size) of hubs and on the definition of the minimum degree of hubs in terms of the theoretic exponent of the degree distribution.

As is evident from the previous short review, the SF property is very broad to describe a singular topology in networks. Substantially, this property defines a family of networks (the SF networks), where the BA model is only one member of this family. Since the BA networks prevail in the scientific literature (1, 2, 4⇓–6), many researchers seem to overlook their difference from the SF networks. One reason for this is probably that the detection of the SF property is an easy task, based on the PL definition (1, 2, 4), and thus no special tools have been developed for discrimination among the SF networks. A bright exception to this observation may be the work of ref. 13, which proposed a discrimination method between BA and superstar SF networks (although it suggested as optimal the structure of the superstar SF networks against the BA’s), but this approach still depends on a degree distribution consideration, and it is based on a (superstar) SF null model where its programming code is not broadly available; thus it is not easy to implement this approach in order to evaluate its ability to discriminate among other members of the SF networks’ family. Further, when taking into account the recent work of ref. 16, the authors of which claim that “SF networks are rare” in nature, it can be argued that the BA model is the most important of the SF networks because it suggests the common choice for empirical studies in network science. Therefore, the development of tools discriminating between the BA model and other SF networks suggests an up-to-date and important issue for network science.

Within this context, this paper highlights that not all networks with a PL degree distribution arise through a BA preferential attachment growth process, a fact which, although evident from the literature, is often overlooked by many researchers. Toward this direction, it is demonstrated, with simulations, that many established measures of network topology do not suffice to distinguish between BA networks and other SF networks (i.e., belonging to the family of SF networks) with the same degree distribution. Additionally, it is examined whether the *S*(*G*) structural metric proposed by the authors of ref. 15, which provides a self-similar definition of the SF property, is also capable to distinguish between different SF topologies with the same degree distribution. Finally, to contribute to this discrimination, this paper introduces a spectral metric, which is defined with reference to the main diagonal of the adjacency matrix. The proposed measure is shown to be more capable of distinguishing between different SF topologies with the same degree distribution, in comparison with the existing metrics.

The remainder of this paper is organized as follows: Section 1 describes the SF null models used in the analysis, and section 2 shows the simulation results, performs a statistical inference analysis on some fundamental network measures, and examines the ability of the *S*(*G*) metric to distinguish among some members of the SF networks family with the same degree distribution. Section 3 proposes a spectral metric and examines the ability of this metric to distinguish among the available SF networks, and, finally, in section 4, conclusions are given.

## 1. Null Models

The undirected (source) null models with the SF property were generated using the algorithm of the Generalized BA model (2), which is available in the open-source software of ref. 17 (version 0.8.2). The initial parameters of the generator were set each time to the default values (*SI Appendix*), where their modifications may provide addresses for further research. The number of the algorithm’s steps ranged between 10 and 15,000 (*SI Appendix*, Table S1), which was submitted to a customized systematic sampling with a gradually increasing lag, aiming to produce networks with an increasing number of nodes. Null models for more than 15,000 steps were not generated, due to computational constraints. Due to the probabilistic architecture in the generator’s algorithm, the produced SF null models include isolated nodes. However, these isolated nodes were ignored to apply PL fittings to the degree distributions of the null models.

Further, associated random-like (RL) and lattice-like (LL) null models, with the same degree distribution as the source BA network (see *SI Appendix*, Fig. S1), were generated using the “randomization” (18) and “latticization” (18, 19) iterative algorithms, which are available in *m*-file format from ref. 20. According to the randomization algorithm, network nodes are randomly being chosen in quadruplets (*u*, *v*, *w*, and *z*), so that the edges *e*_{uv} and *e*_{wz} belong to the network [*e*_{uv},*e*_{wz} ∈*E*(*G*), where *E*(*G*) ≡*E* is the edge set of the network *G*], whereas the edges *e*_{uz} and *e*_{vw} do not (*e*_{uv},*e*_{wz}∉*E*). These edges are then rewired so that *e*_{uv},*e*_{wz} ∉*E* and *e*_{uv},*e*_{wz} ∈*E*, provided that none of the new edges already exist in the network; if new edges already exist, the rewiring step is aborted and a new quadruplet is selected. This procedure preserves the degree distribution even in cases of directed networks. The latticization algorithm follows the same procedure with the randomization algorithm, under the constraint that “swaps are only carried out if the resulting matrix has nonzero entries that are located closer to the main diagonal (thus approximating a lattice or ring topology). This algorithm is implemented as a probabilistic optimization using a weighted cost function” (19).

## 2. Simulations and Analysis

Simulations were conducted on 44 undirected BA (SF) networks (see *SI Appendix*, Table S1), and on 44 RL (*G*_{RL}) and 44 LL (*G*_{LL}) associated null models, all of which were produced from the BA models, and they have the same number of nodes (*n*) and edges (*m*) and the same degree distribution *p*(*k*) as the source (BA) networks. The topologies of the BA, RL, and LL null models were embedded (visualized) in the 2D Euclidean space using the “Force-Atlas” layout which is available in the open-source software of ref. 17. This layout is produced by a force-directed algorithm (see ref. 21), which is developed by the software’s developers and is based on applying repulsion strengths between the network hubs while arranging the hubs’ connections into surrounding clusters. The Force-Atlas algorithm (17) is used on the software's default parameters (see *SI Appendix*).

An indicative picture of the topologies produced for the available network types is shaped in Fig. 1, which displays the topological layouts and the sparsity (spy) plots (i.e., matrix plots displaying nonzero elements with dots; see ref. 10) of their adjacency matrices of the *G*_{(i)}(4,981, 7,469) null models, where *G*(*n*,*m*) is a network with *n* nodes and *m* edges and *i* = BA, RL, LL indicates the network type (where appropriate, the *i* index referring to the model type will not be written, due to simplicity). Since all graphs are subjected to the same embedding (and thus to the same transformation rules) (6, 22), comparisons among the layouts shown in Fig. 1 are possible. As can be observed from Fig. 1, the topological layouts and spy plots of the BA, RL, and LL null models appear considerably different. In particular, the topological layout of the LL model is obviously different from the others because it configures a dense torus of nodes with a small central core, whereas the BA and the RL models configure greater cores surrounded by node rings (expressing isolated nodes) of negligible thickness (number of nodes). The topological layouts of the BA and RL models appear similar, where a slight difference in the density of their cores can be observed, which, for the BA model, seems bigger. A similar picture is also shaped by the examination of the spy plots of these three null models, where the LL pattern configures a dense strap along the main diagonal with small concentrations in the top right and bottom left corners of the matrix. Conversely, the BA and RL models configure a scattered pattern throughout the area of the adjacency matrix with a dense concentration in the top left corner. In general, in their vast majority, except the case of *G*(*n* = 4, *m* = 3), the null models illustrate a similar topological picture to that shown in Fig. 1.

In an attempt to quantify the previous observations based on the topological maps, a statistical inference analysis is applied to a set of measures describing fundamental topological aspects in networks. In particular, the measures participating in the analysis are the network diameter (dG), which is the length of the longest path describing the scale of network (23), the modularity (*Q*), which is an objective function expressing the network’s ability to be divided into communities (24), the number of connected components (NC), which expresses the level of network connectivity (6, 10), the average (*C*) clustering coefficient, which express (in global and local level, respectively) the degree to which nodes in a graph tend to cluster together (25), the average path length (*r*), which measures the preference of network nodes to attach to other similar (13). These measures were selected in the analysis from a broader set of network measures available in the literature (1, 6, 13), because they were the only ones having distinct statistical properties for any pair of the three network types BA, RL, and LL, and thus their overall view may give insights about what network topology is, in total (see ref. 10).

Within this context, Fig. 2 shows 95% confidence intervals (CIs) (26) for these topological measures, which are computed for each network type (BA, RL, and LL) of the available null models. As can be observed, in all cases, the CIs of the BA and RL models are different but overlaid (i.e., their mean values can be considered statistically equal), whereas the LL null models are distinct and do not overlay the other network types (i.e., their mean values can be considered statistically different), except the measures dG and NC. This analysis shows that classic network topological measures do not succeed in discriminating among these three types of null models, where the cases of BA and RL are of quite similar topologies and therefore are difficult to discriminate.

Next, to evaluate the capability of the SF metric proposed by the authors of ref. 15 to discriminate among these three types of null models, 95% and 99% CIs are computed for the *s*(*G*) and *S*(*G*) metrics, as is shown in Fig. 3. Computations are conducted on the set *H* = {BA, RL, LL}, according to Eq. **2**. As can be observed, the *s*(*G*) metric does not succeed in discriminating among these available types (BA, RL, and LL), whereas the *S*(*G*) shows quite distinct results. In particular, *S*(*G*) scores within the interval [0.988, 0.998] correspond (with 95% certainty) to the BA topology, and, within the interval [0.941, 0.964], they correspond to the RL topology, whereas, within the interval [0.871, 0.905], they correspond to the LL topology.

As a further analysis, 99% CIs—produced based on the Student’s distribution—are computed on “dynamic” sample size consisting of samples with a sequentially decreasing number of cases (null models). In particular, let’s consider as *X*_{1} = {*G*_{1}, *G*_{2}, *G*_{3},…, *G*_{44}} the total set of the available null models (it can be either *G*: = BA, or *G*: = RL, or *G*: = LL; see *SI Appendix*, Table S1). Then, we define the set *X*_{i} = *X*_{i–1} – {*G*_{i}} = {*G*_{i}, *G*_{i+1},…, *G*_{44}}, where *i* = 2,3,…41 (samples with less than four null models were not considered). The number of the null models in the set *X*_{i} is 45 – *i*. In this analysis, the lengths of the CIs differ due to the different number of cases included in each set *X*_{i}. Additionally, the Student’s distribution which produces broader intervals is chosen for the computation of the 99% CIs, to counterbalance the uncertainty due to the probabilistic nature of the simulation and due to sampling constraints. Within this context, the calculation of the CIs on the dynamic sample size shows that, for almost 24% of the available samples (10 out of 41), for which the *S*(*G*) metric was computed, it is not possible to discriminate between the null models RL and LL. This is especially observed for samples including bigger networks (*X*_{32}, *X*_{33},…, *X*_{41}), where the number of nodes is *n* ≥ 1,384, although this threshold can be considered as an approximate rather than a legitimate cutoff. Increasing the precision in this threshold may suggest an avenue for further research.

## 3. Proposing an SF Detection Spectral Metric

The previous analysis has shown, first, that it is not possible to discriminate the BA topology among the null models BA, RL, and LL (which have the same degree distribution) by using classic measures of network topology and, second, that the *S*(*G*) metric proposed by the authors of ref. 15 is a good measure to succeed this discrimination, but it is difficult to discriminate between the BA and RL topologies when the network size gets bigger. Within this context, this paper proposes a spectral metric for discrimination of the BA, RL, and LL topologies, which was inspired by the spatial constraint used in the latticization algorithm (18, 19) and by the spy plots’ layouts of the adjacency matrices that are shown in Fig. 1. In particular, the proposed metric measures the average distance from the main diagonal of the nonzero elements in the adjacency matrix of a graph (see *SI Appendix*, Fig. S2), and it is defined by the following math formula:*dd*_{ij} is the distance of the element (*i*,*j*) from the main diagonal of an adjacency *A*, *x*_{ij} = |(*i*,*i*)–(*i*,*j*)|, *y*_{ij} = |(*j*,*j*)–(*i*,*j*)|, *n* is the number of network nodes, and

The proposed metric is given the name “diagonal distance” (*DD*) of the adjacency *A*, and it expresses the average of distances (heights) *dd*_{ij} that intersect the right angle of the triangles (*α*_{ii}*α*_{ij}*α*_{jj}) shown in *SI Appendix*, Fig. S2. The *DD* may also suggest a measure useful to recurrence quantification analysis (RQA), especially to the family of the diagonal-referenced RQA measures and metrics (see ref. 27), but its evaluation in this field suggests a topic for further research. However, the *DD* is sensitive under node reordering (or relabeling). For instance, let’s consider a network *G*(10,1), with *n* = 10 nodes and a single edge (*m* = 1) connecting the first (*n*_{1}) and the second (*n*_{2}) nodes (*e*_{1,2}∈*E*). For this network, the edge’s distance to the main diagonal equals *n*_{2} ↔ *n*_{10}), the distance to the main diagonal of the new edge will become *DD* metric for pattern recognition.

A first answer can be given from the study of the programming codes available from ref. 20, where it can be observed that the node labeling in null models expresses the node age, namely, the step at which each node was created by the generator algorithm during the stepwise procedure of the model’s construction (obviously, in this procedure, all nodes have different ages). Therefore, pattern recognition using the *DD* metric is possible, when the node ages in the network are known and thus when the nodes are labeled according to their age, provided that all ages are different (if not, their ordering suggests a topic of further research). Further, in Fig. 1, we can observe that node labeling according to node age can produce configurations in the adjacency matrices that are representative of network topologies. This is evident by the correspondences that can be made between the topological layouts (Fig. 1*A*) and the spy plots (Fig. 1*B*), where we can observe that cores in the layouts correspond to dot concentrations in the matrix corners. Especially for the LL topology, we can observe that the node torus in the layout corresponds to the diagonal strap in the adjacency matrix. It should be noted that applying a layout in the open-source software used in the analysis does not affect the node placement in the adjacency matrix.

However, in most of the cases, the node age is not known for real-world networks. Therefore, to repair the sensitivity of *DD* to node reordering, this measure has to be calculated after defining a standard relabeling of the network nodes. Such a relabeling can be achieved under the control of a node attribute (degree, local clustering, betweenness centrality, closeness centrality, etc.), namely, by using as new labels the rank (either ascending or descending) of the nodes according to a standard attribute. Through a trial and error approach, a successful relabeling can be achieved under the control (a descending order was chosen) of the eigenvector centrality (CE) (see ref. 11), which is a spectral measure computed on the eigenvectors of the adjacency matrix. The results of the relabeling under the control of CE are shown in Fig. 4, where distinct topologies emerge among the three network types and, also, nonoverlaid CIs are produced for their respective *DD*s. Some avenues for further research in this topic can be the examination of other ordering choices (i.e., to compute the *DD* under the control of other node attributes) and also to seek consistently for minimum or maximum possible values of the *DD*.

The *DD* is subjected to the same testing as the metric of ref. 15, to comparatively evaluate its capability to discriminate among the three network types BA, RL, and LL. The results of the analysis are shown in Fig. 5, where scores of the *DD* within the interval [0.484, 0.510] correspond (with 95% certainty) to the BA topology, scores within the interval [0.437, 0.476] correspond to the RL topology, and scores within the interval [0.291, 0.352] correspond to the LL topology. The analysis of *DD* on the dynamic sample size shows that, for almost 15% of the available samples (6 out of 41), it is not possible to discriminate between the null models BA and RL. This is especially observed for samples including the smallest networks (*X*_{1}, *X*_{2}, and *X*_{3}), and for the least samples (*X*_{39}, *X*_{40}, and *X*_{41}), although the latter case is subjected to uncertainty due to small sample sizes.

According to the previous analysis, the proposed metric can discriminate for more samples between the available SF network types than the metric of ref. 15. Within this framework, the second answer that can be given about the ability of the metric *DD* to be used for pattern recognition is also positive, and it is based on the node reordering under the control of the CE. Overall, the proposed metric *DD* is shown to be useful for the detection of the SF topology produced by the BA preferential attachment growth model. The *DD* provides a comparable to the *S*(*G*) metric performance alongside providing advantages in its spectral definition (it measures the concentration of nonzero elements to the main diagonal of the adjacency matrix), in its comparison-free definition [since the *S*(*G*) metric is defined by the maximum value extracted from a set of SF topologies], in its degree-free configuration (it is not subjected to the constraint of being defined by one network measure), and in the ease of implementing into code.

## 4. Conclusions

This paper highlighted that not all networks with a PL degree distribution arise through a BA preferential attachment growth process, a fact that, although evident from the literature, is often overlooked by many researchers. The analysis showed, with simulations, that classic measures of network topology do not succeed in discriminating between BA networks and other (RL and LL) SF networks with the same degree distribution.

Toward this direction, an existing self-similarity metric *S*(*G*) was examined, which was proposed for the detection of the SF property, to evaluate the capability of this metric to discriminate among the three available topologies (BA, RL, and LL) with the same degree distribution. The analysis showed that *S*(*G*) is capable of producing distinct results at 95% confidence level, but, when submitted to a dynamic 99% CI (based on the Student’s distribution) analysis, an inconsistency was observed for samples including bigger networks.

Within this context, this paper proposed a spectral metric (diagonal distance, *DD*) defined as the average distance of the nonzero elements from the main diagonal of a network’s adjacency matrix. The proposed measure was submitted to the same evaluation as the existing SF metric, and it was found to be also capable of discriminating among the available SF topologies but was more consistent than the *S*(*G*) in terms of the network size. Also, the analysis pointed to some avenues for further research. Some indicative directions are to examine the effects on diagonal distance by changing the algorithms’ defaults, to increase precision by including bigger sample sizes, to compute *DD* under other ordering choices or by considering its minimum or maximum values, to examine changes in *DD* on LL models generated by reordered arrangements of the same adjacency, and to detect differences in *DD* between the BA and the SF superstar topologies.

Overall, this paper highlighted the difference between BA and SF networks, provided insights about differences in the topology of the SF networks, examined the potential of established measures to distinguish among different topologies of SF networks with the same distribution, and introduced a metric for pattern recognition among the members of SF networks family.

## Acknowledgments

I thank the two anonymous reviewers for their valuable comments that significantly improved the quality of the paper.

## Footnotes

- ↵
^{1}Email: tsiotas{at}uth.gr.

Author contributions: D.T. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1816842116/-/DCSupplemental.

Published under the PNAS license.

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