Scale-Free Networks with the Same Degree Distribution: Different Structural Properties

We have analysed some structural properties of scale-free networks with the same degree distribution. Departing from a degree distribution obtained from the Barab\'asi-Albert (BA) algorithm, networks were generated using four additional different algorithms a (Molloy-Reed, Kalisky, and two new models named A and B) besides the BA algorithm itself. For each network, we have calculated the following structural measures: average degree of the nearest neighbours, central point dominance, clustering coefficient, the Pearson correlation coefficient, and global efficiency. We found that different networks with the same degree distribution may have distinct structural properties. In particular, model B generates decentralized networks with a larger number of components, a smaller giant component size, and a low global efficiency when compared to the other algorithms, especially compared to the centralized BA networks that have all vertices in a single component, with a medium to high global efficiency. The other three models generate networks with intermediate characteristics between B and BA models. A consequence of this finding is that the dynamics of different phenomena on these networks may differ considerably.


Introduction
he degree distribution (), deined as the fraction of vertices in the network with degree ,isanimportantproperty of a complex network.In particular, the degree distribution of many real world networks [1-] was accurately itted by a scale-free (power-law) degree distribution where is a scaling parameter.
A power-law degree distribution was observed, for instance, in networks of animal movements [5].Such networks are examples of networks whose degree distribution may be either estimated using a questionnaire in which the number of contacting farm holdings is assessed or through the analysis of animal movement records.When there is a large number of farm holdings in the network and a data bank of animal movements is not available, we might assess the degree distribution using a questionnaire.From the estimated degree distribution, one may be interested in recovering approximately the real network to simulate, for instance, the potential spread of infectious diseases such as foot-andmouth disease and bovine brucellosis, for which the network of animal movements is an important means of dissemination [6][7][8].Nevertheless, the process of recovering a possible real network from the estimated degree distribution may lead to a misleading inference.he presence of a scale-free degree distribution does not guarantee that the recovered network will show the same topology as the original one.M o r eth a no n em e th odm a yg e n e r a t ean e tw o r kth a ts h o w s a scale-free degree distribution, and, from these diferent methods, networks can emerge with diferent structural properties, which may impact the outcomes of the simulation of dynamical phenomena on the network.
In this paper, we depart from a given degree distribution and we show how to generate networks using diferent algorithms and the implications in the network topology of choosing one of these algorithms to generate networks when all you have is the network's degree distribution.
A well-known method to generate a scale-free network is the preferential attachment [9,10], in which links are added to vertices based on their degree.In this approach, a network is generated, and then the resulting power-law distribution is evaluated.We use the preferential attachment approach to generate a network with a scale-free degree distribution.Basedonthisdistribution,networksaregeneratedusingfour diferent algorithms (two of them proposed for the irst time).Toqualitativelycomparethesenetworks,wecalculatesomeof their structural properties [11].
hispaperisorganizedasfollows.InSection 2,wediscuss the calculation and properties of the chosen parameters to compare the networks.In Section 3,w ed e s c r i b et h ei v e algorithms used to generate the networks.In Section ,w e show the results of the calculations of the structural measures for the scale-free networks obtained.Finally, in Section 5,we discuss the implications of our indings.

Structural Properties
It is worth to mention that our objective is not to perform an extensive review of all possible metrics but just highlight some global features of diferent networks instead.Also, there is an unlimited set of topological measurements, and they are oten correlated, implying redundancy in most of the cases [11].We calculated the following structural properties [11]: average degree of the nearest neighbors, central point dominance [12], clustering coeicient [13], the Pearson correlation coeicient [1 ], and global eiciency [15].Some of these parameters are related to local properties of the networks (average degree of the nearest neighbor a n dc l u s t e r i n gc o e i c i e n t ) ,a n do t h e r sa r er e l a t e dt og l o b a l properties (central point dominance, global eiciency, and the Pearson correlation coeicient).All chosen parameters relect global networks' trends and also provide a meaningful interpretation regarding the networks' dynamical properties.
2.1.Average Degree of the Nearest Neighbors.he average degree of the nearest neighbors of a vertex may be calculated as where is the element of the adjacency matrix, deined as =1if there is an edge between vertices and and =0, otherwise.he average degree of the nearest neighbors checks for correlations between the degrees of diferent vertices.If there are no correlations, nn () is independent of .W hen nn () is an increasing function of , vertices of high degree tend to connect with vertices of high degree, and the network is classiied as assortative, whereas whenever nn () is a decreasing function of , vertices of high degree tend to connect with vertices of low degree, and the network is called disassortative [11].
2.2.Clustering Coeicient.he clustering coeicient (CC) for undirected networks may be calculated using the following deinition [1,13]: where CC is deined as: and is the total number of vertices in the network.CC relects the network's tendency to group together nodes with common links, thus raising the number of triangles found inside the network.
2.3.Central Point Dominance.he central point dominance (CPD) [12]isameasurerelatedtothebetweennesscentrality of the most central vertex in a network.Its value is 0 for networks in which the betweenness centralities of all vertices a r eeq uala nd1f o rthewheelo rs ta rnetw o rk.heeq ua tio n for the CPD is [12] where max and are, respectively, the largest values of the relative betweenness centrality in the network and the relative betweenness centrality of vertex .he relative betweenness centrality is the ratio between the betweenness centrality of a vertex and its maximum possible value, ( 2 −3+2)/2, which corresponds to the betweenness of the central vertex in a star network.CPD relects an important network characteristic, which is the network's dependence on speciic vertices to maintain its information low.Networks with higher values of CPD rely on fewer vertices to pass their information to other vertices, while networks with lower values of CPD have their low and pathways distributed in a more decentralized way, thus being more resilient to random vertices removal.
2. .Global Eiciency.he global eiciency (GE) is a measurement that quantiies the eiciency of the network in sending information between vertices, deined as [15] where is the shortest path length between vertices and .Networks with high GE can send information much faster and to a larger number of vertices than networks with low GE.

Correlation Coeicient.
A detailed deinition for the correlation coeicient ()m a yb ef o u n di n [ 1 ].Basically, it is simply the Pearson correlation coeicient between the degrees at either ends of an edge, consisting of another way to determine the degree correlation, besides the average degree of the nearest neighbors.

Algorithms
To guarantee that all the networks generated follow the same degree distribution, allowing comparisons between them, we have irstly generated a network following the Barabási-A l b e r t( B A )a l g o r i t h m ,a n dt h e nw eh a v ea p p l i e dt h eo t h e r algorithms to generate networks based on the degree distribution of the BA network.Due to the growth process inherent in the BA algorithm, it would be diicult or even impossible to generate a BA network from a given () distribution.
For the sake of completeness, we describe below all the algorithms used.
(1) We start with a disconnected set of 0 vertices.(2) At each time step, a new vertex with (< 0 ) edges is added, linking the new vertex to diferent vertices already in the system.(3) When choosing the vertices to which the new vertex connects, we assume that the probability that a new vertex will be connected to vertex depends on the degree of vertex (preferential attachment), such that We have used the BA algorithm implemented in the igraph package of the Statistical Sotware [16].

Molloy-Reed Model.
To generate networks using the Molloy-Reed (MR) model, we have used the following algorithm.
(1) For each vertex, we choose a degree from the distribution.(2) At each time step, we connect randomly a pair of vertices, taking into account that the probability of selecting a vertex is directly proportional to the number of its open connections, deined as the number of remaining links [17].(3) he previous step is repeated until there are no more open connections.
In this version of the MR algorithm, multiple edges are ignored, self-edges are not allowed, and open connections may be discarded if there is only one vertex remaining.

Kalisky Model
. he algorithm proposed by Kalisky et al. [17] is based on the MR model.he aim of the Kalisky algorithm is to force a hierarchy on the MR model, deining layers in the graph, as follows.
(1) A degree is assigned to each vertex.

. Model A.
In this algorithm, called hereater as Model A (MA), the vertices are randomly sampled from a list of vertices with available links. he algorithm is as follows.
(1) Wechoosethevertexwiththehighestavailabledegree (ℎ) in the network (in the irst step, this is the vertex with the maximum degree).
(2) We connect that vertex with ℎ other vertices, randomlyselectedfromalistwithavailablevertices,thus exhausting the links of the chosen vertex.
(3) Steps (i) and (ii) are repeated until there are no more vertices with open connections.

Model B.
In this algorithm, called hereater as Model B (MB), a vector, whose elements are the degrees of all vertices obtained from the BA degree distribution, is randomly generated.hen, the vertices are selected in sequence, following the order of the vector elements.he algorithm is as follows.
(1) Wechoosethevertexwiththehighestavailabledegree (ℎ) in the network (in the irst step, this is the vertex with the maximum degree).
(2) We connect that vertex with the irst ℎ other vertices of the generated vector, thus exhausting the links of the chosen vertex.
(3) Steps (i) and (ii) are repeated until there are no more vertices with open connections.
We stress that the last two models (MA and MB) automatically avoid the generation of multiple edges and self-edges.MA generates networks with vertices connected randomly, starting the connection process with the hubs, while an interesting feature of MB is that it generates networks in which every hub is connected to the other hubs.As far as we know, these two algorithms have not been proposed before.
he computer codes used to generate the networks are availableuponrequest.Formakingthecodesfreelyavailable, we implemented the algorithms using the Statistical Sotware [18], along with the Matrix package [19].

Results
Figure 1 shows the scale-free networks generated using the algorithms by Barabási-Albert (Figure 1 2 shows the generated networks and the degree distribution for =2 .W eh a v eu s e dt h e Kamada-Kawai visualization algorithm implemented in the "network" package [20]. We not ice in Figure 1 that the BA network has only one component, while the other models generate networks with several components.For =1 , this behavior may be observed in Figure 3.F o r=2 , however, all models tend to generate only one giant component, with the exception of MB, which generates a larger number of components (Figure 3).To assess the assortativity of the diferent networks, we analyzed the average degree of the nearest neighbors of vertices with degree , ⟨ nn ⟩, as a function of for networks with 10 3 vertices and =1or =2(Figure ).We have also analyzed networks with 10 2 , 10 3 ,a n d10 4 vertices with =2and =3 , but the qualitative results were similar.As a general behavior, the algorithms used provide disassortative mixing.he exception is the network generated using MB (Figures (e) and (j)), for which an assortative mixing is observed for degrees up to a critical value (between 10 and 15), followed by a disassortative mixing onwards.P r o ba b l yd u et oah i ghl ev e lo fr ed u n da n cyi nth egia n t component, for =1 , the median clustering coeicient for the MB network (0.08) is higher than the values In Figure 5(c), we notice that the network generated using MB is clearly less eicient than the other networks due to its higher number of small components (Figure 3).On the other hand, for =1 , the BA network has the highest median GE (0.08), probably because in this network there is always only one component.However, the number of components is not the only factor inluencing GE, since BA network has a higher GE for =2than for =1 , showing that the number of links also has a major impact in GE, as expected.Also, for =2 ,a sw eca nseeinFigure 3, with the exception of the MB network, all the networks have only one component and a similar GE (median of 0.15).
Estimates of the correlation coeicient for the diferent types of networks are shown in Figure 5(d  may notice that positive values were mainly observed in the MB network.his inding is consistent with the analysis of the ⟨ nn ⟩, since negative correlation coeicients were found for the networks with a disassortative mixing pattern.For =2 , negative values for the correlation coeicient were also observed for the MB network.Table 1 summarizes the results of the average number of components and the average size of the giant component ( i np e r c e n t a g eo ft h ee n t i r en e t w o r k )f o rt h ei v em o d e l s .Comparing the models, the extreme cases are the MB and the BA networks: the MB networks show a larger number of components, a smaller giant component size, and a very low (for =1 )tolow(for=2 )GEandCPD;whiletheB A networks have only one component, medium (for =1 ) to high (for =2 )GE,a ndveryhigh(f o r=1 )tovery high (for =2 ) CPD. he other three models analyzed generate networks with intermediate characteristics between MB and BA models but approaching the BA model when =2 .I np a r t i c u l a r ,f o r=2 , regarding the average number of components, the MA networks are closer to the BA networks.MR and Kalisky networks show similar number of components and giant component size.
No. components 1.00 using the algorithms and then simulate the dynamics of an infectious disease on these networks.An important inding of [21] is that the simulations for the susceptible-infectedsusceptible (SIS) infectious diseases models show that the disease prevalence in MB networks is lower than in the other networks, which may be related to the MB network structure, in which a large set of vertices are not connected to the main component of the network.Regarding the results observed, an aspect that calls attention is that the network generated using algorithm MB difers (by visual inspection) from the networks generated using the other models.In fact, the MB algorithm generates giant component size, if compared to the other algorithms, as shown in Table 1 and Figure 3.
he MB networks show lower CPD and global eiciency values, and assortative mixing for low degree values when compared to the other networks.hese properties are probably a consequence of the distribution of components in the MB network, with one giant component and a large number of small components.On the other hand, for =1 ,t h e BA networks show the higher CPD and global eiciency median values, possibly relecting the existence of only one component in these networks.For =2,asimilarcomment applies to all models with the exception of MB.
Based on the indings presented in this paper, we may hypothesize that, based only on the observed degree distribution (),itmaynotbepossibletomakeanaccurateinference about some structural properties of the network.A consequence of this remark is that diferent scale-free networks (and possibly other types of networks, except lattice and similar networks) with the same degree distribution may have distinct structural properties so that the dynamics of diferent phenomena on these networks may difer considerably.
Diferent algorithms may be invented to generate networks from a given degree distribution.Provided that a network is generated, sets of vertices may be rearranged to increase or decrease the components' sizes.In this paper, we analyzed ive speciic algorithms, ranging from the BA model, which always generates a network with a single component, to the MB algorithm, which can generate a network with several components, and with three other intermediate cases.he efects of our indings are clearly evident, with one model (MB) giving decentralized and low eicient networks and another one (BA) giving networks much more eicient and centralized, with three cases in the middle, all of which with exactly the same degree distribution.
A word of caution is in order: when generating a scale-free network from a given degree distribution, researchers should state and, if necessary, describe clearly which algorithm was used.Otherwise, from the same (),t h es i m u l a t i o n of dynamical phenomena can result in diferent outcomes depending on the algorithm used to generate the network.
hus, for those interested in applying questionnaires to infer the network structure, based only on the degree distribution, it is possible to estimate the average degree, the degree variance and other moments of the statistical distribution, that is, properties that derive directly from the degree distribution, but it is not possible to infer the dynamical properties.If the interest is to analyse dynamical processes on the network, the degree distribution is not enough, it is necessary to have the adjacency matrix.In other words, it is necessary to know the links within the network.

( 2 )
Westartfromthemaximaldegree() vertex, which is connected to open connections.he set composed by this vertex and its neighbors is the irst layer of vertices.(3) hesecondlayerisilledoutinthesameway:weconnect all open connections emerging from vertices in the irst layer to randomly chosen open connections.( ) his process continues until the set of open connections is empty.
Figure1shows the scale-free networks generated using the algorithms by Barabási-Albert (Figure1(b)), Molloy-Reed (Figure 1(c)), Kalisky et al. (Figure 1(d)), Model A (Figure 1(e)), Model B (Figure 1(f)), and the corresponding degree distribution () (Figure 1(a)) based on an original network generated using the BA model with 0 = 1000 vertices and adding, at each time step, a new edge between two vertices (=1).Figure2shows the generated networks and the degree distribution for =2 .W eh a v eu s e dt h e Kamada-Kawai visualization algorithm implemented in the "network" package[20].We not ice in Figure1that the BA network has only one component, while the other models generate networks with several components.For =1 , this behavior may be observed in Figure3.F o r=2 , however, all models tend to generate only one giant component, with the exception

Figure
Figure1shows the scale-free networks generated using the algorithms by Barabási-Albert (Figure1(b)), Molloy-Reed (Figure 1(c)), Kalisky et al. (Figure 1(d)), Model A (Figure 1(e)), Model B (Figure 1(f)), and the corresponding degree distribution () (Figure 1(a)) based on an original network generated using the BA model with 0 = 1000 vertices and adding, at each time step, a new edge between two vertices (=1).Figure2shows the generated networks and the degree distribution for =2 .W eh a v eu s e dt h e Kamada-Kawai visualization algorithm implemented in the "network" package[20].We not ice in Figure1that the BA network has only one component, while the other models generate networks with several components.For =1 , this behavior may be observed in Figure3.F o r=2 , however, all models tend to generate only one giant component, with the exception

Figure 1 :
Figure 1: Network generated using the BA model with 0 = 1000 vertices and =1(b) and corresponding degree distribution (a), from which the other scale-free networks were derived using the following algorithms: (c) MR model, (d) Kalisky model, (e) model A, and (f) model B.

Figure 2 :Figure 3 :Figure :
Figure 2: Network generated using the BA model with 0 = 1000 vertices and =2(b) and corresponding degree distribution (a), from which the other scale-free networks were derived using the following algorithms: (c) MR model, (d) Kalisky model, (e) model A, and (f) model B.