A dandelion structure of eigenvector preferential attachment networks

Abstract

In this paper we introduce a new type of preferential attachment network, the growth of which is based on the eigenvector centrality. In this network, the agents attach most probably to the nodes with larger eigenvector centrality which represents that the agent has stronger connections. A new network is presented, namely a dandelion network, which shares some properties of star-like structure and also a hierarchical network. We show that this network, having hub-and-spoke topology is not generally scale free, and shows essential differences with respect to the Barabási–Albert preferential attachment model. Most importantly, there is a super hub agent in the system (identified by a pronounced peak in the spectrum), and the other agents are classified in terms of the distance to this super-hub. We explore a plenty of statistical centralities like the nodes degree, the betweenness and the eigenvector centrality, along with various measures of structure like the community and hierarchical structures, and the clustering coefficient. Global measures like the shortest path statistics and the self-similarity are also examined.

Introduction

Understanding the intricate characteristics exhibited by contemporary real networks constitutes the primary focus of Complex Networks research. These distinctive attributes, such as the clustering coefficient and betweenness, defy explanation through the conventional Erdős-Rényi random graph model¹ and regular lattice structures. In response to this challenge, Watts-Strogatz’s small-world model² and Barabási–Albert’s (BA) scale-free (SF) model³ emerged as efforts to elucidate the non-trivial properties observed in diverse networks such as computer networks, the internet, social networks, biological networks, and brain networks^4–10.

Among these models, BA model has been put in an intense focus as the first microscopic model for SF real networks with a variety of behaviors like power-law and scaling behaviors, which identifies an important universality class in complex networks. The mechanism of preferential attachment in this model is used to describe the process of agent attraction within various networks^11–19. In growing complex networks, the preferential attachment mechanism refers to the preference of the newly added agents to be attached to an agent with a higher centrality, which is chosen to be the degree centrality in BA model. The likelihood that a newly added node links to a node i with degree $k_i$ is proportional to a power of its degree according to^16–19

$$\begin{array}{c}\hfill {\mathrm{\Pi }}^{(\text{BA})}(i)=\frac{k_i^{\zeta }}{{\sum }_j{k}_j^{\zeta }},\end{array}$$ 1

where $\zeta$ is a nonlinearity exponent, which is one for the BA model with power law degree distribution^11–15.

The graph spectrum provides a way to analyze the diffusion rate on a network^20,21. The diffusion rate, essentially how quickly information or influence spreads across the network, is dictated by the eigenvalues of the graph’s adjacency matrix²¹. These eigenvalues determine the rate of convergence of the diffusion process. Eigenvector centrality, correspondent to the largest eigenvalue, identifies influential nodes that can significantly impact the spread of information or disease²². The graph spectrum plays a crucial role in analyzing random walks on graphs²³, which simulate how a walker navigates the network by randomly moving between connected nodes. The second smallest eigenvalue, ${\lambda }_2$, of the adjacency matrix (or Laplacian matrix for some applications) is closely related to the mixing time of a random walk²³. The spectral properties of a graph can also offer valuable information about assortative or disassortative mixing patterns, shedding light on the underlying structural characteristics of the network^21,24.

The preferential attachment algorithm, which might change from system to system, has found many applications in real networks like protein network evolution¹³, online networks¹⁶, internet infrastructure²⁵, internet encyclopedia Wikipedia¹⁴, see^12,26 for a good reference. To explain various real networks, a lot of variants were proposed, each of which modeling the attachment algorithm. For example, in a model termed Bianconi-Barabási Network, the stochastisity comes to play via a fitness factor, or attractivity²⁷. Homophillic model²⁸, and Euclidean Distance Model²⁹ (resulting to q-Gaussian distribution for the degree) are the other examples. The Fitness Model and Homophilic Model with euclidean distance³⁰ are mixed other models, see³¹ for more details. Re-wiring the connections is another strategy to modify the BA universality class³². In some growing complex networks, particularly those composed of traders (trade complex networks)^33–35 one may expect that the preferential attachment scheme should be very effective^36,37. The same is expected in Socio-political complex networks³⁸. In such networks, the addition of a new member to the system may be driven by their pursuit of more influential individuals capable of connecting it to a center of power for expediting their objectives. The sought-after centrality of the candidate does not exclusively correspond to the highest degree within the network, but rather, it can manifest through various forms of centrality. One such example is the level of communication the person maintains with effective individuals. This particular centrality can be quantitatively captured through the concept of eigenvector centrality, defined as the summation of the centralities of the neighbors of a given node³⁹. A high eigenvector centrality does not necessarily translate to a high degree centrality, but it means that the node under focus has very good connections.

In this study, we present a growth model in which preferential attachment is a linear function of the nodes’ eigenvector centrality (rather than their degree centrality). Our model, which we call as a dandelion network, demonstrates similarities with a winner-takes-all scenario which is identified by the fact that one node catches a significant proportion of all links. Such networks have a hub-and-spoke structure, which are often identified in air transportation networks, cargo delivery networks, telecommunication networks, and healthcare organization structures^40–44, to mention a few.

The paper organizes as follows: In the next section we introduce our model. The network structure section is devoted to the results that our model leads to. It contains degree and eigenvector statistics, central point dominance, degree dynamics, finite size scaling, shortest path and closeness statistics, clustering coefficient, assortativity and community structure, and self-similarity. We close the paper with Concluding remarks and Methods sections.

The model and motivation

In this section we describe our preferential attachment network model, in which the nodes with greater eigenvector centralities are more likely to be picked as target nodes by newly added nodes at each time step. There is a huge literature on the eigenvector centrality structure of the complex networks^45–52. It is shown that the eigenvector centrality is more important than the degree, closeness and betweenness features for finding prominent or key author in research professionals’ relationship network⁵¹. Eigenvector centrality can handle networks with signed graphs and different edge weights, while degree, betweenness, and closeness centrality are normally studied for binary connections⁴⁵. The effect of (extended) eigenvector centrality in temporal networks was considered in⁵⁰, where it was shown that stronger connections between layers in a network significantly impact how influence (centrality) is distributed and how it changes over time.

Eigenvector centralities are also used to study geometrical and local properties of geographical and social networks^47,48,51,52. For a good review on the geographical and social structures as well as capital flow over the spread of disease; see⁴⁷. It is shown that the eigenvector centrality is very affected by the graph structure and weights, as examined for a twitter data which shows that there is a significant difference among 10 most influential users⁵². Eigenvector centrality has already been employed for urban street networks which incorporates information from both topology and data residing on the nodes⁴⁸.

Despite this huge literature, a little attention has been paid to the networks where the eigenvector centrality has a role in constructing the network, shaping its geometry and topology, see⁵² for an instance. While the reviewed studies are related to such a network constructed based on the eigenvector centrality, there is not a known map to relate them. Here we propose a model where the eigenvector centrality directly influences the network growth through the preferential attachment setup. We believe that such a setup is logically possible in practice.

We define an external integer m in our model which is the number of links added upon adding one node to the system. The attachment probability in our model is defined via

$$\begin{array}{c}\hfill \mathrm{\Pi }(i)=\frac{v_i^{\text{max}}}{{\sum }_j{v}_j^{\text{max}}},\end{array}$$ 2

which should be compared with Eq. (1). In this equation $v_i^{\text{max}}$, as the eigenvector centrality of node i, is defined as the eigenvector of the adjacency matrix corresponding to maximum eigenvalue. More precisely, if we represent A as the adjacency matrix of an unweighted, undirected network with the entries $a_{\mathit{ij}}=1$ if i and j are connected, and zero otherwise, then

$$\begin{array}{c}\hfill (A-I\lambda )\mathbf{v}=0.\end{array}$$ 3

If we show the maximum eigenvalue as ${\lambda }_{\text{max}}>0$ (the positivity is guaranteed by the Perron-Frobenius theorem), then this relation can be written as

$$\begin{array}{c}\hfill v_i^{\text{max}}=\frac{1}{{\lambda }_{\text{max}}}\sum _{j\in K_i}{v}_j^{\text{max}}.\end{array}$$ 4

where $K_i$ is the set of neighbors of i, i.e. $a_{\mathit{ij}}=1$. For simplicity, from now on we show the eigenvectors corresponding to the maximum eigenvalues as $v_i\equiv v_i^{\text{max}}$. This relation explains why the eigenvector centrality of node i is about the strength of connections that it has.

The model is defined as follows: We start with a fully connected network with $m+1$ nodes, and add nodes one by one to the system. In each step, the newly added node to the system establishes m links to different nodes according the probability given in Eq. (2). Note that no node can receive more than one link from a given node, so that the graph is a simple graph rather than a multi-graph. Therefore, in each times step t, we find the eigenvalues and eigenvectors of the $t\times t$ adjacency matrix A. The centrality of the nodes that we concern are the one corresponding to the maximum eigenvalue. It is important to have in mind that the eigenvector centrality of all nodes are updated upon adding a new node even if no connection is established to it. Note that to have a unique centarlity for each site, we need normalization in each time step as follows

$$\begin{array}{c}\hfill \sum _i^N{v}_i^2=1.\end{array}$$ 5

In connected networks with the number of nodes greater than two the maximum possible value of $v_i$ is $\frac{\sqrt{2}}{2}$, which corresponds to the highest central node. The system size N is defined as the final time of the simulations ($N\equiv t_{\text{final}}+m+1$).

The network structure

This section is devoted to the simulations results. In the following subsections we analyze the network measures one by one, and compare them with the same quantities in the BA model.

Figure 1 depicts a sample network for $m=1$, representing a dandelion structure with the so-called hub-and-spoke topology. There is a super-hub at the network’s core with the highest eigenvector centrality value of $\approx \frac{\sqrt{2}}{2}$. In the remainder of this paper, we introduce this structure and statistically investigate its properties.

Figure 1.

A dandelion network with $N=4000$ nodes and $m=1$.

Degree and eigenvector centralities

The node degree k ($\#$ links that end up in a given node) and the eigenvector v, and the corresponding distribution functions (p(k) and p(v) respectively) are analyzed in this section. Fig. 2a,d show the degree distribution of different network sizes for $m=1$ and $m=2$, respectively. First, observe that these functions do not show power-law behavior, they show two separate peaks, i.e. our network is not SF. Additionally they comprise of two parts which move away as N increases. The discontinuities and the observed gap are caused by the degree difference between the super-hubs (the right part of the plot) and the remaining nodes. This super-hub is the central point in Fig. 1, and the smooth (left) peak that is observed in intermediate k values is due to the accumulation of nodes that are directly connected to the super-hub. As the network grows in size, the degree difference between the super-hubs and the rest of the nodes becomes more pronounced. This implies that super-hubs attract new entering nodes at a faster pace than regular nodes.

Figure 2.

Here, we explore how network size affects super-hub formation and eigenvector centrality distribution in our model compared to the Barabási–Albert (BA) model ($m=1$). Panels (a) and (d) show the degree distribution for different network sizes ($m=1$ and $m=2$, respectively). As networks grow larger, super-hubs (nodes with significantly higher degrees than others) emerge, and their degree increases at a much faster rate compared to other nodes. Panels (b) and (e) depict the eigenvector centrality distribution for our model (m$=1$ and $m=2$). The main figures only display results for one network size for clarity, with insets providing zoomed-in views highlighting a peak in eigenvector centrality. Green, orange, and blue data points represent different network sizes of 4000 (32000), 8000 (64000) and 16000 (128000) for the case of $m=1$ ($m=2$). As the network grows in size, the distribution width of the zoomed region narrows. Panels (c) and (f) show the eigenvector centrality distribution for the BA model.

The distribution function of the eigenvector centrality is shown in Fig. 2b,e (correspondent to $m=1$ and $m=2$, respectively), according to which the separated parts are observed: the most important nodes or super-hubs have the highest eigenvector centrality value of almost $\simeq \frac{\sqrt{2}}{2}$.

The first neighbors of the super-hub (called the second-level hubs) form the second peak which occur in the intermediate k and v values as one can see in the Fig. 2a,b (or equivalently in the Fig. 2d,e). As the network grows in size, the distribution width of this group of nodes narrows as it is shown in the insets of the Fig. 2b,e. The peak in p(v) is explained in terms of the plateau in the $v-k$ plot, Fig. 3a,b, leading to an accumulation of points in the peak point. The plateau forms due to the direct connection to a super-hub node. One may use Eq. (4) to explain this flatness, according to which the eigenvector centrality of a second-level hub (i) in the case of $m=1$ is found to be

$$\begin{array}{c}\hfill v_{i\in \text{second level}}=\frac{1}{{\lambda }_{\text{max}}}\left[v_{\text{sh}},+,\sum _{j\in \text{third level}},a_{\mathit{ij}},v_j,+,.,.,.\right],\end{array}$$ 6

where $v_{\text{sh}}$ is the eigenvector centrality of the super hub, and the summation is over all nodes with distance 2 from the super hub. The number of links that are connected to the super hub (=$\#$ second level nodes) is roughly $N^{\frac{1}{2}}$, and the number of third level nodes is $\frac{1}{2}N$, so that the average degree of the second level nodes is $\frac{1}{2}{N}^{\frac{1}{2}}$. Given that $v_{j\in \text{third level}}\approx {10}^{-3}$ for $N=10000$, we conclude that the second term in the bracket is of order $5\times {10}^{-2}$ for $N=10000$, which is pretty smaller than the first term that is of order $\sqrt{2}/2$. We calculate the amount of the plateau in the Supplementary for a star graph, and show that it coincide with our results.

Figure 3.

(a) Here, we show the relationship between eigenvector centrality and node degree in our model for various network sizes. The panels (a) and (b) depict this relationship for parameter values $m=1$ and $m=2$, respectively. As network size increases, a discontinuity emerges in the eigenvector centrality versus node degree plot. Panel (c) serves as a comparison point, showcasing the correlation function for the Barabási–Albert (BA) model ($m=1$) where no discontinuity is observed, even for larger network sizes.

In Figs. 2c,f and 3c, we show the same quantities for the Barabási–Albert networks. While it turns out that the eigenvector centrality first saturates, and then undergoes a boost to a new value (with some corresponding peaks in the distribution function), there is no discontinuity in the graph. Such discontinuities are observed e.g. in Figs. 2a,b and 3a which shows that increasing the system size leads to disparities in the degree and eigenvector centralities of the super-hubs on one hand and the remainder of the nodes on the other. Such discontinuities will appear in networks with central point dominance (CPD) values near to one (see the following subsection).

Central point dominance

As a global feature, CPD is defined⁵³ as

$$\begin{array}{c}\hfill \text{CPD}=\frac{1}{N-1}\sum _i(B_{\text{max}}-B_i),\end{array}$$ 7

where $B_{\text{max}}$ is the network’s greatest value of betweenness centrality, and $B_i$ represents the betweenness centrality of node i.

This feature is used to provide a perspective on network topology in such a way that networks with a CPD value of zero (the lowest possible value) have a complete graph structure, whilst networks with a CPD value of one (the greatest possible value) have a star graph structure. The greater the CPD value of a network, the higher the number of hubs in that network. Networks with a CPD value of one have only one hub $B_{\text{max}}=B_{\text{hub}}=1$, and $B_{i\ne \text{hub}}=0$ which is a star graph. We call such a node as the super-hub. Figure 4a shows betweenness in terms of the node degree for our model. For finite N values there are two distinct regions with two different slopes (exponents), while a super hub is distinguishable as a separate region which is separated from the others by a gap. While the network is not scale free (SF) in a general sense, in the mentioned two regimes we see different power-law behaviors like $b^{(i)}\propto k^{(i)}$ where $i=1,2$ counts the number of the intervals, and ${\eta }^{(1)}=1.374\pm 0.008$ and ${\eta }^{(2)}=2.04\pm 0.02$ for $N=128000$ and $m=2$. This should be compared with the Barabási–Albert model in which $b^{(\text{BA})}\propto k^{{\eta }_{\text{BA}}}$, where ${\eta }_{\text{BA}}=1.68\pm 0.03$ in our calculations with the same parameters. The dependence of CPD on N is shown in Fig. 4b, from which we observe how CPD approaches 1 as $N\to \infty$ for $m=1$. This should be compared with the BA model, Fig. 4c, where CPD shows a different trend. Note that for $m>1$ the graphs tend to increase and approach 1 for larger N values, but the faith of the system needs runs for larger N values which was not accessible for us. This finding indicates that the network’s topology changes over time and eventually resembles a star graph in terms of CPD, particularly when $m=1$. It is important to highlight that this similarity to star-like graphs is limited to certain characteristics like CPD and only becomes apparent over extended periods. While the dandelion network’s structure differs from hub-and-spoke networks, it eventually displays properties similar to them, with a super-hub acting as the central entity^54–58.

Figure 4.

(a) Betweenness centrality as a function of the degree of the nodes for three network sizes of 32000, 64000 and 128000 considering $m=2$ for all of them. Super-hubs have the highest betweenness centrality. There is also a meaningful difference between the betweenness centrality of the super-hubs and that of the other nodes. The global CPD feature is presented for the different system sizes for the (b) our model and (c) BA model. The BA model keeps its initial structre; i.e. its CPD value doesn’t change while this isn’t true for our model. By increasing the size of the system, the structure of the networks is gradually converted to a star-like structure that is the CPD value tends to one.

Assortativity

The tendency of nodes to connect to other nodes of the network with similar degrees is called assortativity. In contrast, disassortativity refers to a state in which nodes with low degrees prefer to link to high degree nodes, and vice versa. There can also be intermediate states in which there are no correlations between low degree and high degree nodes. Pearson correlation coefficient of the degrees²⁴ is used in order to identify this quantity. The assortativity is defined as

$$\begin{array}{c}\hfill r=\frac{M^{-1}{\sum }_{j>i}{k}_i{k}_j{a}_{\mathit{ij}}-{\left[M^{-1},{\sum }_{j>i},(1/2),(k_i+k_j),a_{\mathit{ij}}\right]}^2}{M^{-1}{\sum }_{j>i}(1/2)\left(k_i^2,+,k_j^2\right)a_{\mathit{ij}}-{\left[M^{-1},{\sum }_{j>i},(1/2),(k_i+k_j),a_{\mathit{ij}}\right]}^2},\end{array}$$ 8

with M as the total number of the edges in the network. Assortativity varies in the range of $r=[-1,1]$. The network is assortative if $r>0$ and disassortative if $r<0$; there is no correlation between degrees of nodes when $r=0$. It is not surprising that the hub-and-spoke networks are dissortative⁵⁹.

We have calculated this feature for both the BA and our model for two values of $m=1$ and 2. Figure 5 uncovers the fact that the assortativity is negative for finite N values for both models (BA is more disassortative than ours meaning that the low k nodes are more likely to connect to higher k nodes and vice versa, (see Ref.⁶⁰ for more details). The assortativity tend asymptotically ($N\to \infty$) to zero for these two models in large enough networks regardless of the value of m. This means that the node degrees are uncorrelated in this limit.

Figure 5.

Assortativity as a function of the system size.

Community and hierarchical structure

In this analysis, we examine community and the hierarchical structures of both the BA model and our model. This comparison allows us to gain insights into the community structure perspectives of dandelion and BA networks.

A qualitative definition defines communities as network sub-graphs in which the number of links inside them exceeds the number of edges connecting them. Taking the preceding requirements into account, the modularity parameter, Q, was proposed by Newman and Girvan⁶¹ as

$$\begin{array}{c}\hfill Q=\sum _{i=1}^N\left[e_{\mathit{ii}},-,{\left(\sum _j,e_{\mathit{ij}}\right)}^2\right].\end{array}$$ 9

In this equation, if the index i counts the number of communities found in the network, $e_{\mathit{ii}}$ is the fraction of links within the community i, $e_{\mathit{ij}}$ is the fraction of links connecting communities i and j. Our model does not show community structure, i.e. there is one community with $Q=1$ for all m values.

The hierarchical structure shows a more pronounced structure in our model. It refers to how nodes are organized into groups or clusters with varying levels of granularity or resolution. A network with a hierarchical structure can be broken down into smaller subnetworks, and these subnetworks can further be subdivided into even smaller ones. This hierarchical arrangement can provide valuable information about the modularity, functionality, and evolution of the network ²⁶. For the special case of $m=1$, the method by which we obtain the hierarchy and the corresponding communities is to remove hubs from the network in a descending arrangement, i.e. we first remove super-hub ($n=1$), which results to $n_1$ separate connected graphs. Then we proceed by removing the nodes with distance 1 from the super hub ($n=2$) resulting to $n_2$ separate connected graphs, etc. The number of communities in the mth level ($n_m$) shows the number of communities in that level of hierarchy. At each level, the degree of the hub determines the number of communities in that branch. We continue this process until there are no communities left in the network. A single node is not considered as a community, i.e. we define subgraphs to be communities with at least one link between their nodes, so that the minimum size of the communities is two.

In the Fig. 6, the process of removing the main hubs from the system at two levels for the BA model and our model has been shown. Figure 7 shows the number of the hierarchical communities identified at the first level as a function of the size of the networks. This number reveals a power-law behaviour for both models

$$\begin{array}{c}\hfill N_{\text{comm}}\propto N^{\beta },\end{array}$$ 10

with the exponents ${\beta }_{\text{BA}}=0.51\pm 0.09$ and ${\beta }_{\text{dandelion}}=0.72\pm 0.04$. The result for the BA model is rather expected given that the node degree growth exponent of this model is $\beta =\frac{1}{2}$ ⁶², while this exponent is expected to vary depending on the level in our model. Figure 8 displays the number of hierarchical communities at different levels where we have fixed the size of the network for both models. It is worth noting that the decreasing behavior (higher n values) is due to removing the communities with just one member. While this number has a maximum in level 2 for our model, it is size dependent for the BA model. Larger networks show the maximum number of the communities in higher levels.

Figure 6.

This figure presents a three-column visualization process. The leftmost column displays the original graphs at level-0. The center column shows the level-1 graphs obtained by removing the super-hub and its connecting links. Finally, the rightmost column depicts level-2 graphs after eliminating nodes with a distance of one from the super-hub. The top row showcases graphs generated by our model, while the bottom row illustrates the same trimming process applied to BA model graphs. All graphs at level-0 were generated with parameters $N=2000$ and $m=1$. See the text for more details.

Figure 7.

Number of the communities as a function of the size of the network calculated only for the first level for (a) our model and (b) BA model. This number, as a result of fitting shown by the red line, reveals a power-law behaviour in both models with the exponents $0.72\pm 0.04$ and $0.51\pm 0.09$ respectively.

Figure 8.

Number of the hierarchical communities as a function of the different levels of obtaining communities for different network sizes; (a) our model and, (b) BA model. While this number has a maximum in level 2 for our model, it is size dependent for the BA model. Larger networks show the maximum number of the communities in higher levels.

Clustering coefficient and hierarchical organization

Clustering coefficient in the graph theory is a measurement of how closely connected nodes in a graph tend to be (for $m>1$). Most real-world networks, especially social networks, have nodes that naturally form close-knit clusters with a high density of ties. There are two versions of this measurement: local and global. The local version shows how embedded single nodes are, while the global version is intended to show how the network is clustered overall. The local clustering coefficient is defined² as

$$\begin{array}{c}\hfill c_i=\frac{2n_i}{k_i(k_i-1)},\end{array}$$ 11

where $n_i$ is the number of edges between the neighbors of node i. The global clustering coefficient¹¹ is defined as

$$\begin{array}{c}\hfill C=\frac{3N_{▵}}{N_3},\end{array}$$ 12

where $N_{▵}$ and $N_3$ are the total number of loops of length three and the number triplets in the network, respectively. Ravasz and Barabási⁶³ discuss that many natural and social networks exhibit two generic properties: they are SF and have a high degree of clustering. They show that these two characteristics are the result of hierarchical structure, suggesting that small groups of nodes aggregate into increasingly large groups while maintaining a SF topology.

Figure 9a depicts the result of the local clustering coefficients as a function of the nodes’ degree. In contrast to the BA model in which these two features are uncorrelated, they are correlated in our dandelion structure according to which

$$\begin{array}{c}\hfill c(k)\sim k^{-{\gamma }_{\mathit{ck}}},\end{array}$$ 13

where ${\gamma }_{\mathit{ck}}=1.3\pm 0.2$ for $10\lesssim k\lesssim 100$, and other values in the other intervals, especially it is positive for $k\lesssim 10$ and $k\gtrsim 100$. The reason of formation of these intervals is the different structures for various hierarchical communities. For example the intermediate k values correspond to the nodes which are immediately connected to the super-hub node. The dynamics and structure of these nodes are different from the other communities like the super-hub nodes, or the high n-level communities. Again, we observe that while the dandelion network is not a SF networks, it shows power-law behavior in distinct intervals. Figure 9b,c suggests that our model complies with the hierarchical structure requirements outlined by Ravasz and Barabási⁶³ for these node degree ranges.

Figure 9.

(a) Local clustering coefficient as a function of the degree of nodes for the network sizes of 64000 and 128000 given $m=2$; there are some minimums and maximums. These two features are correlated in our model and reveal a power-law behaviour with constant exponents in large enough networks. Super-hubs have the lowest value of the local clustering coefficient. (b) We have isolated the largest network ($N=128000$) of the previous subfigure (a) and specified two values of its degrees by two vertical yellow lines. The local clustering coefficient reveals one maximum and one minimum in terms of the degrees identified by the left and the right yellow lines, respectively. The local clustering coefficient has a scaling power-law relation in terms of the degrees of the interval made by those yellow lines with the slope of $-1.28\pm 0.19$. (c) The degree distribution of the network presented in (b). Those two specific degrees have been identified here again by two vertical yellow lines. These two special degrees specify values in terms of them, the degree distribution reveal two peaks. The degree distribution is not power-law in general but one can find the degree distribution to be scale-free within the specified interval with the slope of $0.81\pm 0.02$. (d) Global clustering coefficient as a function of the size of the networks for different values of m; the black ($m=2$), green ($m=4$) and blue ($m=8$) lines indicate the slope of $-0.69\pm 0.04$, $-0.722\pm 0.004$ and $0.701\pm 0.007$ respectively.

The global clustering coefficients analysis demonstrate the following power-law behavior

$$\begin{array}{c}\hfill C\propto N^{-{\gamma }_{\mathit{CN}}},\end{array}$$ 14

where ${\gamma }_{\mathit{CN}}=0.69\pm 0.04$ (for $m=2$ as in the Fig. 9d), which should be compared with the exponents of the BA (${\gamma }_{\mathit{CN}}^{(\text{BA})}=\frac{3}{4}$⁶⁴) and Erdős-Rényi (${\gamma }_{\mathit{CN}}^{(\text{ER})}=1$^64,65) models.

Global measurements

Understanding the global properties of complex networks is crucial as they unveil the fundamental patterns, mechanisms, and functionalities that the networks represent. By global properties we mean the properties that are related to non-local observables and deal with the walks over the network’s edges. This includes path length statistics, closeness, and the re-scaling self-similarity.

Average shortest path length and closeness

The average number of links along the shortest paths for all possible pairs of network nodes is known as average shortest-path length, which is an important measure in network topology. For undirected graphs, the average shortest path length is defined as

$$\begin{array}{c}\hfill l=\frac{2}{N(N-1)}\sum _{j>i}{d}_{\mathit{ij}},\end{array}$$ 15

where $d_{\mathit{ij}}$ is the shortest path length between nodes i and j. The average shortest path length of most well-known networks, such as the BA SF network for which $l\sim \frac{lnN}{ln(lnN)}$⁶⁶, ER network for which $l\sim lnN$^64,67, and Watts-Strogatz small-world network for which $l\sim N$ if $N<N^{\ast }$ and $l\sim lnN$ if $N>N^{\ast }$ with $N^{\ast }$ as p-dependant crossover size separating the small and large world regimes⁶⁸ increases by the size of network. Indeed, the behavior of the average shortest path length is a gauge for the universality class of the complex networks⁶⁹.

The closeness of a node i is defined as the reciprocal of its farness relative to the other nodes multiplied by $N-1$,

$$\begin{array}{c}\hfill C_i=\frac{N-1}{{\sum }_j{d}_{\mathit{ij}}},\end{array}$$ 16

so that

$$\begin{array}{c}\hfill l=\frac{1}{N}\sum _i\frac{1}{C_i}.\end{array}$$ 17

A node with higher closeness is more central in the sense that it is more accessible by the others, like super-hub in dandelion networks. The average shortest path of the hub-and-spoke networks follows the statistics of star graph, which is trivially constant for large enough networks. More precisely, for a star graph one expects that $C_{\text{SH}}=1$ and $C_{\text{nSH}}=\frac{N-1}{2N-3}$ (SH/nSH stands for super-hub/non-super-hub), so that $l=2\frac{N-1}{N}$, which goes to $C_{\text{SH}}=1$, $C_{\text{nSH}}=\frac{1}{2}$ and $l=2$ as $N\to \infty$ (see the Supplementary for details). Therefore, based on the findings of the previous sections we expect that the average shortest path for the dandelion network saturates to a constant as the network size becomes sufficiently large. Figure 10 shows the average shortest path length as a function of the system size for the dandelion and BA networks. Fig. 10b confirms the expected behavior for the BA model ⁶⁶, i.e. l is linear in terms of $\frac{lnN}{ln(lnN)}$. In the dandelion network for $m=1$, the average shortest path length tends to a constant, which is $l\to 4.78\pm 0.04$, showing that in spite of similarities, it is topologically different from star graph. This observation can be explained by the fact that the majority of newly added nodes in our model tend to connect to the super-hub node, so that the average shortest path length of the network will not surpass a certain value providing the network is sufficiently large. For $m>1$ it turns out that the average shortest path length saturates to some m-dependent values ($l\to 4.30\pm 0.03$ for $m=2$), while for a better characterization larger N values are needed. The greater the value of m in a network, the shorter its asymptotic average shortest path length.

Figure 10.

Average shortest path length in terms of network size for eight values of N including 125, 250, 500, 1000, 2000, 4000, 8000, and 16000, from left to right, with different values of m for (a) our model and (b) BA model.

Figure 11a depicts a $m=1$ dandelion network based on the nodes’ closeness centrality with the size of points proportional to their closeness centrality. The network’s super-hub has been identified by largest size and white color showing its highest closeness. A very different behavior between dandelion and BA for $m=1$ networks is evident in Fig. 11b,c, the former showing a hierarchy based on their closeness centralities. The communities arisen from this closeness hierarchy is actually related to the hierarchical communities found based on the deletion of nodes, discussed in Sec. Community and hierarchical structure. These peaks correspond to the branches of the dandelion; the nth peak corresponds to a nth level branch with distance n form the super hub. Consider a node with distance n from super-hub. The closeness centrality of the node j distinct from the super-hub is given by

$$\begin{array}{c}\hfill C_j^{(n)}\approx \frac{N-1}{{\sum }_k\left(d_{\text{SH}}^{(k)},+,n\right)}=\frac{C_{\text{SH}}}{1+n{C}_{\text{SH}}},\end{array}$$ 18

where $d_{\text{SH}}^{(k)}$ is the distance of node k from the super-hub, and $C_{\text{SH}}=\frac{N-1}{{\sum }_j{d}_{\text{SH}}^{(j)}}$ is the closeness centrality of the super-hub. On the other hand the Eq. (17) tells us that

$$\begin{array}{c}\hfill Nl=\frac{1}{C_{\text{SH}}}+\frac{n_1}{C_1}+\frac{n_2}{C_2}+...=\frac{1}{C_{\text{SH}}}\sum _{m=0}^M{n}_m\left(1,+,m,C_{\text{SH}}\right),\end{array}$$ 19

where $n_m$ and $C_m$ and the number of nodes and the centrality of the nodes in the mth level, and M is the maximum m value available in a dandelion network with size N (the total number of the communities). Noting that ${\sum }_{m=0}^M{n}_m=N$, and that $n_m\propto N^{\frac{3}{4}m}$ (see Fig. 7a) we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & Nl{C}_{\text{SH}}=N\left(1,+,C_{\text{SH}},f,(M)\right)\hfill \\ \hfill & \to C_{\text{SH}}=\frac{1}{l-f_N(M)},\hfill \end{array}\end{array}$$ 20

where

$$\begin{array}{c}\hfill f_N(M)\equiv \frac{{\sum }_{m=0}^{M}m{N}^{\frac{3}{4}m}}{{\sum }_{m=0}^M{N}^{\frac{3}{4}m}}=\frac{M{N}^{\frac{3}{4}(M+2)}-(M+1)N^{\frac{3}{4}(M+1)}+N^{\frac{3}{4}}}{N^{\frac{3}{4}}(N^{\frac{3}{4}}-1)(N^{\frac{3}{4}M}-1)}.\end{array}$$ 21

Noting that ${lim}_{N\to \infty }{f}_N(M)\to M$, we finally find

$$\begin{array}{c}\hfill C_{\text{SH}}=\frac{1}{l-M},M<l.\end{array}$$ 22

This equation, when combined with Eq. (18) gives ($M<l$)

$$\begin{array}{c}\hfill C_{\text{SH}}^{(n)}=\frac{1}{l-M+n},n=0,1,2,...,M.\end{array}$$ 23

This turns out to be the discrete spectrum (the peak points) of the closeness centrality. For $l<M$ this argument fails since short connections prevent Eq. (18) to be valid.

Figure 11.

(a) A representation of a network generated by our model with $N=2000$ and $m=1$. This visualization is based on different values of the nodes’ closeness feature. Different sizes and colors are used to identify the lowest and highest values of closeness. The nodes with the smallest size and darkest color have the lowest value of closeness, whereas the brightest and largest nodes have the highest degree of closeness. The network’s super-hub has the highest value of closeness. Closeness distribution of (b) our model and (c) BA model for different system sizes given $m=1$. While the BA model indicates a nearly continuous distribution, our model reveals a discrete one with some peaks at specified closeness amounts.

Re-scaling self-similarity

The fractal nature of shapes is often assessed by using the ideas of fractality, fractal dimensions, and the box-counting technique. Fractal networks are based on the rules of fractal geometry, and their fractal dimension is calculated in a similar way as for shapes. The box-covering technique, also called box-counting, uses the shortest path between the points of a graph. This method works as follows for networks for a given characteristic box length $l_b$:

• A set of nodes is said to fit within a box of size $l_b$ if the shortest distance between any two of them is less than $l_b$.

• The network is covered by a bunch of boxes of size $l_b$ if the nodes are divided so that each group fits into them.

• The minimum number of boxes required to cover the network is identified as $N_b(l_b)$.

• If the minimal number of boxes scales as a power of the box size, i.e.,

  $$\begin{array}{c}\hfill N_b(l_b)\propto l_b^{-d_b},\end{array}$$ 24

  then the network is fractal with a finite fractal dimension or box dimension $d_b$.

We examine the fractal properties of both the BA and dandelion networks in this section. We use the random sequential algorithm^70,71 to calculate the fractal dimension of the networks, among other methods. The random sequential algorithm uses the burning idea (breadth-first search). The boxes are created by burning (expanding) them from a randomly chosen center (or seed) node to its adjacent nodes. Moreover, nodes are “burned out” once they are allocated to a box. At each level, a random unburned center node s is selected, and then a sphere of radius $r_b$ (this radius is connected to $l_b$ as $l_b=2r_b+1$) is built around s; more specifically, the algorithm picks those unburned nodes that are no more than $r_b$ away from the center node s. These freshly burned nodes make up a new box.

The outcomes for the dandelion and BA networks are displayed in Fig. 12. In a log-log plot, Eq. (24) implies that a linear graph should be obtained, whose slope is the fractal dimension. No linear pattern is discerned at any scale, which leads us to infer that the networks we examined are not fractal.

Figure 12.

None of (a) our model and (b) the BA model are self-similar. Here, we have used the random sequential algorithm ⁷⁰. Evaluations are the result of 10 realizations for each model given $N=16000$ and $m=1$.

Degree dynamics, mean field theory

In this section we focus on the dynamical aspects of the nodes on a mean field level. The average value of the degree $k_i$ is represented by a continuous real variable, the growth rate of which is expressed according to Eq. (2)

$$\begin{array}{c}\hfill \frac{d{k}_i}{\mathit{dt}}\propto \frac{v_i}{{\sum }_j{v}_j}.\end{array}$$ 25

First we estimate ${\sum }_j{v}_j$ using the simulation results. Figure 13a, shows that there is a power-law relation between ${\sum }_j{v}_j$ and N as

$$\begin{array}{c}\hfill \sum _j{v}_j\sim N^{\alpha },\end{array}$$ 26

with $\alpha =0.333\pm 0.006$. This exponent, $\alpha \simeq \frac{1}{3}$, is m-independent and plays a crucial role in the network. Given that $v_{\text{SH}}\simeq \frac{\sqrt{2}}{2}$ one can solve Eq. (25) as (knowing that $N\simeq t$ for large Ns)

$$\begin{array}{c}\hfill k_{\text{max}}(t)\sim t^{\frac{2}{3}}\end{array}$$ 27

This result is consistent with the numerical result for the time evolution of the super-hub’s degree, as shown in Fig. 13b. For the nodes in the second level (distance 1 from the super-hub) the typical eigenvector centrality (corresponding to the first peak point in the distribution function of the eigenvector centrality) runs with time approximately as (Fig. 13c)

$$\begin{array}{c}\hfill v_{\text{peak}}(t)\propto t^{-\frac{1}{3}}.\end{array}$$ 28

We then are able to estimate the corresponding degree centrality ($k_{\text{peak}}$) as

$$\begin{array}{c}\hfill k_{\text{peak}}(t)\propto t^{\frac{1}{3}}.\end{array}$$ 29

This equation is valid for all non-super-hub nodes. We now understand the reason of the discontinuities seen for instance in the Figs. 2a,b and 3a. In large enough networks (long enough time steps), the aforesaid discontinuities will increase as the degree of the super-hub grows faster than the degree of the other nodes. Figure 13d displays the rate of growth in the degree of the nodes inserted into the network at various time steps. One can easily spot the super-hub (node with index zero) here as its degree growth rate differs significantly from that of the other nodes. In the BA model, the rate at which the degree of the nodes changes is $\beta =0.5$ for all nodes but it is not the case here. Note that, for $m>1$, more steps are required to see the discontinuities observed in $m=1$.

Figure 13.

(a) Variation of ${\sum }_{j=1}^N{v}_j$ as function of N given $m=2$. There is a scaling realtion with the exponent $0.333\pm 0.006$. (b) The growth of the hub’s degree with time steps given $m=2$. The solid red line has the slope of $0.73\pm 0.03$. (c) The scaling relation of the $v_{\mathit{peak}}$ in terms of the size of the system given $m=2$. This scaling shown by solid red line reveals an exponent of $0.3159\pm 0.0005$. (d) Degree evolution of some nodes a network with $N=64000$ and $m=2$. The nodes are added at time steps (indicating their indexes) 0, 2, 4 and 6. The node with index two (added at time step two) has had the chance to be chosen as the hub node. Here, one can easily spot the super-hub (node with index two) as its degree growth rate, $0.710\pm 0.001$, differs significantly from that of the other three nodes with an average exponent of $0.296\pm 0.001$.

To confirm the results found in the mean field arguments, we use data collapse analysis. Fig. 14a,b are analogous to the Figs. 2a and 3a in which we have scaled data representing non-hub nodes with the scaling factor of $N^{-\alpha }$. We find the scaling relation as

$$\begin{array}{c}\hfill f(k,N)\sim N^{-\alpha }\varphi (k/N^{\alpha })\end{array}$$ 30

with $\alpha =1/3$. f(k, N) can represent either the probability distribution function of nodes’ degree, p(k, N), or eigenvector centrality of the nodes, v(k, N). That is, if we plot $f(k,N)N^{\alpha }$ against $k/N^{\alpha }$ for different network sizes, we will get a unite curve all data collapsed on it. The data representing the super-hubs will also collapse into a unite curve if we replace $N^{-\alpha }$ with $N^{\alpha -1}$ in the former equation.

Figure 14.

At large enough networks, one can collapse the data representing the non-hub nodes in either (a) the degree distribution or (b) eigenvector-degree dependencies in one unite curve by plotting $f(k,N)N^{\alpha }$ against $k/N^{\alpha }$ for different network sizes with f(k, N) as degree distribution or eigenvector centrality of the nodes.

Concluding remarks

In the Barabási–Albert model, the nodes with high degree centralities have a good chance of being connected by new nodes, the paradigm known as the rich get richer. There are however a lot of non-scale-free networks showing winner-takes-all phenomenon, where only one super-hub appears in the network. The latter, called hub-and-spoke complex network, appears everywhere in various systems; air transportation networks, cargo delivery networks, telecommunication networks, and healthcare organization structures^40–44 are some instances of these networks. In this paper we propose an important class of preferential attachment complex networks showing winner-takes-all property based on the eigenvector centrality which is reminiscent of a dandelion for $m=1$. One node becomes super-hub for long periods, which explains the opening of a gap in the degree centrality distribution.

As a non-scale-free network, our model displays unique characteristics such as a star-like pattern (where the CPD value approaches one in sufficiently large networks), a correlation between the degree of nodes and their clustering coefficient, and a distinct distribution of closeness unlike the continuous distribution seen in the BA model. Additionally, certain areas of degree centrality exhibit a hierarchical structure. Globally, the model maintains a steady average shortest path length over extended periods, qualifying it as a small-world network. We also evaluated the model for fractal characteristics and determined it does not possess this quality. Numerous statistical aspects of the model were analyzed, including betweenness, closeness, and h-index, from different statistical viewpoints; these are detailed in the Tables 1 and 2. This model offers deeper understanding of the structure and behavior of real-world networks that display a hub-and-spoke structure.

Table 1.

In this table we summarize the properties of the networks we found in our model and the BA model.

	Scale-free	Fractality	Star graph	Small-world	l	HS$_{}^{\ast }$	Assortativity	CS$_{}^{+}$
Our model	Some regions	$\times$	$✓$ ($N\to \infty$)	$✓$	Const ($N\to \infty$)	Some regions	$\times$	$\times$
BA model	$✓$	$\times$	$\times$	$✓$	$\sim lnN/ln(lnN)$	$\times$	$\times$	$\times$

*Hierarchical structure.

+ community structure.

Table 2.

Correlation coefficient of different features in our model given $N=128000$ and $m=2$.

	Degree	h-index$_{}^{\ast }$	Eigenvector	betweenness	Closeness	Clustering coefficient
degree	1.00	–	–	–	–	–
h-index	0.99	1.00	–	–	–	–
eigenvector	1.00	0.95	1.00	–	–	–
betweenness	1.00	0.96	1.00	1.00	–	–
closeness	0.97	0.96	0.77	0.99	1.00	–
clustering coefficient	$-0.33$	$-0.84$	$-0.23$	$-0.85$	0.05	1.00

*In the academic context, the H-index⁷² also known as the Hirsch index is determined as the highest number h for which there are h publications that have been cited at least h times each. On the other hand, within the realm of complex networks, a node’s H-index is the highest number h where the node has h or more connections, each with a connection degree of at least h.

Methods

We implemented our numerical calculations using Python programming language. This work leverages the Python libraries graph-tool⁷³ and NetworkX⁷⁴ to compute both the eigenvector centrality as well as betweenness centrality and the other relevant statistical features. The graph-tool particularly well-suited for the demands of large network analysis tasks since its core algorithms and data structures are implemented in C++. Additionally, it leverages OpenMP for efficient parallelization on multi-core architectures.

In our preferential attachment model, an incoming node attach to those with higher eigenvector centrality, so that in each time step we have to calculate the eigenvector centrality of all nodes, the process which leads the simulation to be much slower than other preferential attachment complex networks like BA model. To decrease the run time, we needed optimization of the already-existing codes in both graph-tool and NetworkX. The network starts with a fully connected seed network of $m+1$ nodes. For the case of $m=1$, we utilized the NetworkX library to efficiently construct the desired network, while for $m\ge 2$, the graph-tool library provides more efficient ways for simulation.

The algorithm used to compute eigenvector centrality of the (weighted) adjacency matrix A of a network is the so-called power method. This method’s complexity depends on the network structure and is expressed as $O\left(N,\times ,\frac{-log\epsilon }{log|{\lambda }_1/{\lambda }_2|}\right)$. Here, N represents the number of nodes in the network up to time step t, $\epsilon$ is a small tolerance parameter, and ${\lambda }_1$ and ${\lambda }_2$ are the largest and second-largest eigenvalues of the adjacency matrix, respectively.

We explored the influence of the parameter m on network behavior by simulating four m values: 1, 2, 4, and 8. For each m, we generated graphs of varying sizes 125, 250, 500, 1000, 2000, 4000, 8000, 16000 accompanied by a corresponding number of samples 64000, 32000, 16000, 8000, 4000, 2000, 1000 and 500, respectively. These network sizes were sufficient to capture the qualitative behaviors for $m=1$. However, our model restricts nodes to forming only one connection per existing node (simple graph). As a result, for $m>1$, we required larger network sizes to observe the same qualitative behaviors seen with $m=1$. This is because with higher m, each node forms more connections, requiring a larger network to exhibit similar behavior. For $m=2$, we achieved similar results with network sizes up to 128000 (networks with $N=32000,64000,128000$ realized by 250, 125 and 64 samples, respectively). We chose not to explore sizes beyond 16000 for m = 4 and 8 because of the computational limitations. For the simulation we used Intel(R) Xeon(R) CPU E5-2680 v3 @ 2.50GHz. The time it took to complete computations on a network of size $N=16000$ with a single CPU and $m=1$ was $2.16\times {10}^6$ seconds across 500 samples. In contrast, when using 48 cores to parallelize computations on a network of size $N=128000$ with $m\ge 2$, the completion time was $1.3\times {10}^6$ seconds over 64 samples.

Author contributions

V.A. and Z.E. analyzed the data, V.A. wrote The first version of the paper, M.N.N. designed and supervised the problem. All authors reviewed the manuscript.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-67896-9.