# Geometric explanation of the rich-club phenomenon in complex networks

Máté Csigi  MTA-BME Information Systems Research Group, Budapest University of
Technology and Economics, H-1117 Budapest, Magyar tudósok krt. 2, Hungary
Attila Kőrösi  MTA-BME Information Systems Research Group, Budapest University
of Technology and Economics, H-1117 Budapest, Magyar tudósok krt. 2, Hungary
József Bíró  MTA-BME Information Systems Research Group, Budapest University
of Technology and Economics, H-1117 Budapest, Magyar tudósok krt. 2, Hungary
Zalán Heszberger  MTA-BME Information Systems Research Group, Budapest
University of Technology and Economics, H-1117 Budapest, Magyar tudósok krt.
2, Hungary  Yury Malkov  Federal state budgetary institution of science,
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul’yanov
Street, 603950 Nizhny Novgorod, Russia  András Gulyás  MTA-BME Information
Systems Research Group, Budapest University of Technology and Economics,
H-1117 Budapest, Magyar tudósok krt. 2, Hungary  A. Gulyás was supported by
the Janos Bolyai Fellowship of the Hungarian Academy of Sciences
Corresponding author: András Gulyás<EMAIL_ADDRESS>
###### Abstract

The rich club organization (the presence of highly connected hub core in a
network) influences many structural and functional characteristics of networks
including topology, the efficiency of paths and distribution of load. Despite
its major role, the literature contains only a very limited set of models
capable of generating networks with realistic rich club structure. One
possible reason is that the rich club organization is a divisive property
among complex networks which exhibit great diversity, in contrast to other
metrics (e.g. diameter, clustering or degree distribution) which seem to
behave very similarly across many networks. Here we propose a simple yet
powerful geometry-based growing model which can generate realistic complex
networks with high rich club diversity by controlling a single geometric
parameter. The growing model is validated against the Internet, protein-
protein interaction, airport and power grid networks.

###### keywords:

complex networks, rich club coefficient, metric space, geometry

## Introduction

The rich club organization plays a central role in the structure and function
of networks [1, 2, 3, 4, 5, 6, 7, 8]. Some networks (e.g. the human brain [7],
airport networks, social networks [1] and the Internet [8]) have a strong rich
club meaning that their hubs are densely connected to each other. Others (e.g.
protein-protein interaction networks [1], the power grid [9]) behave quite the
contrary as the subgraphs made out of their hubs are very sparse. This high
variation across networks is illustrated in Figure 1, which shows the
normalized rich club coefficient $\rho(k)$ [1] as the function of degree $k$
for the airport network, the Internet and the protein-protein interaction
network.

Figure 1: Illustration of the diverse rich club organization in real networks.
The plot shows the normalized rich club coefficient $\rho(k)$ as the function
of degree $k$ as: $\rho(k)=\frac{\phi(k)}{\phi_{\text{unc}}(k)}$, where
$\phi(k)$ is the density of the subgraph $G_{k}$ of the network containing
only the nodes with $\text{degree}>k$ . The $\phi_{\text{unc}}$ is the same
for the maximally randomized version of the network conserving the degree
distribution. One can see that the subgraph of hubs in the airport network
possesses about $1.5$ times more links than in the randomized version. On the
contrary, the hubs in the PPI network have less than the half of the
connections experienced in its randomized counterpart.

The explanation and reproduction of this great rich club diversity is highly
non trivial. The state-of-the-art models targeting the rich club organization
are based on heavy randomization techniques [10, 11, 12, 13], which shuffle
network connections until a given organization structure is artificially
imitated. Although these randomization-based models are fairly usable, they do
not give deeper insight into the mechanisms causing this diversity during the
evolution of the networks. Consequently, _growing_ models capable of
incorporating various rich club networks in a simple and intuitive manner
would be useful towards deeper understanding the underlying evolutionary
reasons of this diversity.

Here we propose a simple geometry-based _growing_ model which can explain the
emergence of the rich club variability in real networks by adjusting a single
spatial parameter. Our model is built upon the real-world observation that in
some networks the establishment of very long connections is not feasible. For
example in power grid networks, the electric current cannot be transferred
efficiently (i.e. without huge losses of energy) over large distances without
intermediate transformations at middle stations [14, 15]. Similarly, optical
networks apply signal re-generators for the transmission of light signals over
large distances to be able to sustain the signal-to-noise ratio [16]. Also in
certain social networks, middlemen as intermediate nodes may play crucial role
in enhancing cooperation between the individuals or groups [17]. Such networks
seem to implement an “artificial” threshold above which no direct connections
are allowed. Other networks do not have such inherent thresholds and the
length of the edges is only limited by the “natural” geometric boundary of the
network. For example in airport networks we can find very long links, because
transferring passengers over large distances is not an issue with the current
aviation technologies.

In this paper we confine these observations into a simple geometric growing
model, in which we introduce a length threshold for creating edges. We show
that such a growing model can naturally reproduce and account for the
experienced diversity in the rich-club organization of networks, while keeping
other network statistics (diameter, degree distribution and clustering)
intact. The applied geometric representation of networks is an active and
quickly advancing research direction in network science[18]. There are
numerous studies describing networks as random geometric graphs, performing
some functions [19, 20] (e.g. navigation, information transmission) or
structural properties (e.g. small-world, clustering, modularity) [21, 22] of
networks in a geometric context, and disclosing some fundamental relations
between topology and hidden metric spaces[23]. A the proper choice of geometry
(e.g. Euclidean, Bolyai-Lobachevskian hyperbolic geometry or other metric
space) can also promote the interpretation of numerous network processes [24,
25, 26].

## Results

In our model $N$ nodes are randomly generated one after another on an
Euclidean 2D $R$-disk with uniformly distributed coordinates. When adding a
new node, it selects the $m$ closest nodes already residing on the disk (if
there are less than $m$ nodes on the disk then it selects all of them). The
distances between the new node and the old ones are calculated by the
Euclidean distances normalized by a function of the old node degrees (as in
the Growing Homophilic model[22]). If this “effective” distance between the
new node and a selected one is smaller than the threshold $T$ then they are
directly connected, otherwise a so-called "bridge" node to the midpoint of the
two nodes is established and connects to both nodes. The formal description of
the network generation process is performed in panel (a) of Figure 2, while
panel (b) shows a small network generated with the model.

Parameters:$N$, $R$, $m$, $T$

$V=\\{\\}$, $E=\\{\\}$

for i=0, i<N do

$V=V\bigcup i$

$i_{r}=\sqrt{UR^{2}}$, where $U=\text{random}(0,1)$

$i_{a}=\text{random}(0,2\pi)$

for j in closest $\max(m,|V|)$ nodes to i in V do

$\text{d}_{\text{eff}}=\text{EuclDist}(i,j)/\sqrt{\text{k}_{j}}$

if $\text{d}_{\text{eff}}<T$ then

$E=E\bigcup(i,j)$

else

$V=V\bigcup b^{ij}$,

$[b^{ij}_{r},b^{ij}_{a}]=\text{midpoint}(\overline{ij})$

$E=E\bigcup\\{(i,b^{ij}),(j,b^{ij})\\}$

(a)(b)(c)(d)

Figure 2: Details of the model. Panel (a) shows the exact pseudocode of the
generation process. Panel (b) plots a sample network generated by the model.
Panels (c-d) shows time evolution of the network for $m=2$. In panel (c) a
green node is added to a network consisting the two blue nodes only. The red
circle represents the threshold around the new node. As neither of the two
blue points reside within the red circle two bridge nodes (red) are inserted
to the midpoints of the edges. Panel (d) shows the same network in a later
time instant. In this later phase the new green node can connect directly
(bridge nodes are not needed) to the two nearest blue nodes as they are closer
than $T$.

Time evolution of the model, as new nodes are inserted into the network at
different stages is shown in Figure 2. For the sake of simplicity, in this
illustration the distance normalization by node degrees is omitted. At the
beginning of the generation process, many bridge nodes are inserted as the
distance between the nodes is typically larger than $T$ (see panel (c) in
Figure 2). As the network grows, the average node density and degrees
increases, so the typical normalized distance between the nodes will fall
below $T$ and no more bridge nodes are added (panel (d) in Figure 2). From
this stage the model falls back to the growing homophilic model analyzed in
[22]. Setting $T$ to a very large value (e.g. $T>2R$) completely recovers the
model in [22] because bridge nodes are never inserted to the graph. We show,
that by varying $T$, the model generates complex networks with diverse rich-
club organization, while having scale-free degree distribution, small diameter
and large clustering. In the remaining of the paper we will use the settings
summarized in Table 1 in our analytical and simulation results.

Network | $N$ | $m$ | $R$ | $T$
---|---|---|---|---
Generated $T=12$ | 5000 | 3 | 50 | 12
Generated $T=30$ | 5000 | 3 | 50 | 30
Generated $T=100$ | 5000 | 3 | 50 | 100
Table 1: Simulation settings.

### Number of bridge nodes

First, we show that the total number of bridge nodes quickly converges to a
relatively small value compared to the reasonable network size ($N$) during
the generation of the graph, and this value is independent of the graph size.
To support this observation we give a recursive estimation of the expected
number of new bridge nodes generated at each step of the model, and based on
this recursion a mathematical expression is given to the limit of the expected
total number of bridge nodes (see methods for more details Methods). By
analyzing the recursion one can show that the expected number of bridge nodes
at step $N$ denoted by $b_{N}$ can approximately be expressed in the form

$b_{N}\approx\exp(-f_{1}N+f_{2}\log N+f_{3})\ .$ (1)

where the functions $f_{1}$, $f_{2}$ and $f_{3}$ may depend on $R,T,m$ but are
independent from $N$. From this it immediately follows, that for the total
number of bridge nodes $B_{N}$

$B_{N}\approx\int_{x=1}^{N}\exp(-f_{1}x+f_{2}\log x+f_{3}){\rm
d}x\rightarrow\exp(f_{3}){E}_{-f_{2}}(f_{1})\ {\rm as}\ N\rightarrow\infty$
(2)

where ${E}$ is the exponential integral function. The vanishing term during
the convergence in $B_{N}$ is $N^{1+f_{2}}E_{-f_{2}}(f_{1}N)$ and also
approximately exponential.

In Figure 4 the expected total number of bridge nodes ($B_{N}$) calculated by
recursion (6) is plotted in each iteration together with the simulation result
for the same parameters. The two plots readily justify that $B_{N}$ has a
characteristic flat after certain iterations, which means that $B_{N}$
converges to a finite fixed value during the graph generation process. This
also illustrates that for sufficiently large network the total number of
bridge nodes is negligible comparing to the network size. Furthermore,
according to statistical tests the overall distribution of the nodes on the
$R$-disk is apparently not affected by the bridge nodes, and still can be
treated as uniform.

Figure 3: The recursively estimated number of bridge nodes in each iteration
is plotted with the simulated values. The $T=12$ line shows the average of
$15$ simulation runs.

Figure 4: Small world property of $T=12$ networks. The diameter of the network
grows logarithmically with the number of nodes.

### Diameter, clustering and degree distribution

The diameter of all three generated networks (see Table 1) is around $9$-$10$,
similar to the real networks (Table 2). Figure 4 shows that the diameter of
the $T=12$ networks is an approximate logarithmic function of the network
size, which confirms the small-world property. Also the generated networks
have high clustering coefficients with values very close to that of real
networks. Finally, Table 2 confirms that the clustering coefficient is
insensitive to the threshold parameter.

Network | N | Edges | Diameter | Avg. dist. | Clustering coefficient
---|---|---|---|---|---
Generated $T=12$ | 5030 | 15024 | 10 | 4.54 | 0.66
Generated $T=30$ | 5004 | 14998 | 9 | 4.51 | 0.67
Generated $T=100$ | 5000 | 14994 | 10 | 4.46 | 0.69
PPI network | 5084 | 22148 | 10 | 3.98 | 0.12
Airport network | 2845 | 10409 | 10 | 3.75 | 0.59
Internet | 23748 | 58414 | 10 | 3.52 | 0.61
Table 2: Basic topological properties of real and generated networks.

Now we show that the generated networks has scale-free degree distribution
independently of $T$.

###### Theorem 1

The networks produced by the model have scale-free degree distribution with
$\gamma=3$ when $N\to\infty$.

Proof: Suppose we compute the effective distance as
${{d}_{\text{eff}}}=\frac{{{d}_{Euc}}}{\sqrt{k}}$. At each insertion step the
algorithm connects a new element to exactly $m$ neighbors that globally
minimize the normalized distance. To infer the degree distribution of the
neighbor elements, we temporary fix the distance to the $m+1$-th nearest
neighbor $d_{\text{eff}}^{m+1}$ and randomly shuffle positions of the $m$
neighbor nodes under the condition that they all remain the $m$ nearest
neighbors with respect to the new element (i.e. having effective distance to
the new element less than $d_{\text{eff}}^{m+1}$).
For every possible value of the neighbor degree $k$, possible element
positions are bounded in the initial Euclidean space by a radius
${{r}_{\text{Euc}}}=d_{\text{eff}}^{m+1}\sqrt{k}$. Since the nodes are
distributed uniformly in the Euclidean space, the probability of having an
element with degree k proportional to the $r_{\text{Euc}}$-ball volume. Thus
under fixed $d_{\text{eff}}^{m+1}$ the overall probability of connecting to an
element with degree $k$ is proportional to $(k)$.
The probability inferred for a fixed value of $d_{\text{eff}}^{m+1}$ does not
depend on either the value of $d_{\text{eff}}^{m+1}$, or the positions of the
nodes that are not the closest neighbors of the inserted elements, so that is
true for every possible positions of the elements in the Euclidean space and
overall probability of connection to a node is proportional to its degree
$(k)$. This means that new nodes connect to the old ones with probability
proportional to $k$, which is equivalent to the Barabasi-Albert model [27],
proved to produce scale-free networks with $\gamma=3$.

The Figure 6 shows the degree distributions of three networks generated with
our model with various values of $T$. The plot readily confirms that the
degree distributions are indeed scale-free with $\gamma=3$ independently of
$T$.

Figure 5: The degree distributions of our simulated networks. The plot
confirms that the choice of $T$ does not effect the degree distribution, which
is a power-law with $\gamma=3$.

Figure 6: Rich-club organization in networks generated by the model with
different settings of $T$. The plot readily confirms that our model is able to
generate rich-club diversity by simply adjusting $T$. We note for the
remarkable similarity with the same plot for real networks in Figure 1.

### Rich-club coefficient

Although the insertion of bridge nodes keeps degree distribution, clustering
and diameter intact, the simulation results plotted in Figure 6 clearly show
that the graphs generated by the model differ greatly in their rich-club
organization depending on $T$. Setting $T$ to the diameter of the $R$-disk
($T=100$, red triangles in Figure 6), the model does not limit the lengths of
the edges artificially, so the only limiting factor is the natural geometry of
the disk itself. In this case we obtain a network with a strong rich-club,
similarly to the airport network. Conversely, adjusting $T$ to $12$, the model
will create only edges having $d_{\text{eff}}<12$. This is a strong
“artificial” limitation for the edge lengths imposed by the generation
process. As a result, the model yields a network with no rich-club ($T=12$,
blue squares in Figure 6), likewise the PPI network. We note the appealing
similarity between Figure 6 and Figure 1, showing the rich-club diversity in
real networks.

## Discussion

An intriguing question could be whether our model captures something
fundamental from the growth processes of real networks, or exhibit similar
rich-club diversity simply by chance. For answering this question we have
performed the CCDF’s (Complementary Cumulative Distribution Function) of the
normalized edge length distribution in a rich-club (airports with flights in
the US) and a non rich-club network (the North American Power Grid) together
with the networks generated with our model in Figure 7. Panel (a) shows
continuously significant (on all length scale) decrease of edge length
distributions before the final “natural” cutoff for the airport and the
$T=100$ networks caused by the geometry of the continent and the $R$-disk
respectively. On panel (b) however we can observe a clearly visible plateau
before the cutoff of the edge lengths in the power grid network. This means
that edge lengths are much denser near the cutoff, which in this case is
rather “artificial” and caused by the growth process of the network and not
the underlying geometry. Our model produces a very similar edge length
distribution for the setting $T=12$. These results hint that networks having
no rich-clubs use a very similar limiting for the length of the connections as
our model do. As a consequence, this length-limiting phenomenon can also
account for the emergence of the observed diverse rich-club organization in
real networks.

These two examples also underline that our method is parsimonious in a sense
that the rich club organization can be tuned by only a single geometric
threshold parameter in a growing homophilic model. We think the results
presented in this paper are strong indications that the rich club diversity
can be placed at all on a growing/evolutionary perspective, and provide deeper
insight into the mechanisms resulting certain rich club behavior during the
growth of networks.

(a)(b)

No plateau

Plateau

Figure 7: Edge length distribution in networks with different rich club
organization. While panel (a) shows a continuously decreasing length
distribution in case of rich-club networks, there is a clearly visible plateau
before the cutoff of the CCDF function in networks having no rich-clubs (panel
(b)).

## Methods

### Recursive estimation of the number of bridge nodes

Let $A(r,T,R)$ be the area of the intersection of an $r-$centered disk with
radius $T$ and the $R-$disk, and let $p(r,T,R)$ be the fraction of $A(r,T,R)$
and the area of the $R-$disk, i.e. $p(r,T,R)=\frac{A(r,T,R)}{R^{2}\pi}$.
Further, let us assume that there are already $j$ nodes in the network. The
$(j+1)^{\rm th}$ randomly generated node will connect to the $m$ nearest
neighbors. For calculating the necessary bridge nodes in this step, the task
is to determine what are the nodes among the $m$ nearest neighbors which are
farther than $T$. To ease the computation, the degrees of the neighbors are
substituted by their expectation values (denoted by $\bar{k}_{j}$ and to be
determined later) subject to the whole network at this stage. Since the
effective distance is computed as the Euclidean distance divided by
$\sqrt{\bar{k}_{j}}$, it is approximately equivalent to investigate the
expected number of points among the $m$ nearest ones being outside of the
$(j+1)^{\rm th}$ node $T\sqrt{\bar{k}_{j}}$ \- radius vicinity. This will be
equal to the expected number of newly inserted bridge nodes at this step.
Denote the radial coordinate of the $(j+1)^{\rm th}$ node by $r$ and assume
that the previously generated random points and established bridge nodes are
still evenly distributed on the $R-$disk. With this, the probability that
$i,0\leq i\leq j$ nodes among the $j$ ones are closer to the $(j+1)^{\rm th}$
node than $T\sqrt{\bar{k}_{j}}$ is

$\binom{j}{i}p(r,T\sqrt{\bar{k}_{j}},R)^{i}(1-p(r,T\sqrt{\bar{k}_{j}},R))^{j-i}$

and hence the expected number of necessary bridge nodes at this step is

$\sum_{i=0}^{\min(m-1,j-1)}\min(m-i,j-i)\binom{j}{i}p(r,T\sqrt{\bar{k}_{j}},R)^{i}(1-p(r,T\sqrt{\bar{k}_{j}},R))^{j-i}:=\beta_{j}(r,T\sqrt{\bar{k}_{j}},R,m)$

Note, that this is still a conditional expectation value which is to be de-
conditioned by the density of the radial coordinate $r$. Towards the de-
conditioning, first the function $A(r,T,R)$ is to be determined. Clearly,
$A(r,T,R)=T^{2}\pi$ if $r\leq R-T$, i.e. there is no intersection of the two
disks. Otherwise, if $r\geq R-T$ then by using straightforward geometrical
calculations

$A(r,T,R)=\alpha R^{2}+\gamma T^{2}-2\sqrt{s(s-R)(s-T)(s-r)}$ (3)

where

$\alpha=\arccos\frac{R^{2}+r^{2}-T^{2}}{2rR}\ ,\
\gamma=\arccos\frac{r^{2}+T^{2}-R^{2}}{2rT}\ ,\ s=\frac{R+T+r}{2}\ .$ (4)

Now, the de-conditioning is possible with $p(r,T,R)=A(r,T,R)/(R^{2}\pi)$ and
the density of the radial coordinate $r$, which is $\frac{2r}{R^{2}}$.
Further, let $j(N)=N+b_{1}+b_{2}+\ldots+b_{N}$ where $N$ is the randomly
generated points and $b_{l}\ l=1,\ldots,N$ is the expected number of bridge
nodes established after the $l^{\rm th}$ random node. For completing the
recursive estimation, the expected degree should also be expressed upon the
$l^{\rm th}$ random node generation. This is

$\bar{k}_{l}=\frac{2(lm-\frac{m(m+1)}{2}+2\sum_{i=1}^{l}b_{i})}{l+\sum_{i=1}^{l}b_{i}}\
,\ {\rm for}\ l>m\ ,\ {\rm else}\
\bar{k}_{l}=\frac{2(\frac{l(l-1)}{2}+2\sum_{i=1}^{l}b_{i})}{l+\sum_{i=1}^{l}b_{i}}\
.$ (5)

For $l=1$ let $b_{1}=0$, $\bar{k}_{1}=0$ and let $B_{N}=\sum_{i=1}^{N}b_{i}$.
The main recursion can now be expressed as

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!b_{N+1}=\int_{r=0}^{R}\beta_{j(N)}(r,T\sqrt{\bar{k}_{j(N)}},R,m)\frac{2r}{R^{2}}{\rm
d}r\ .$ (6)

### Data Availability

The data that support the findings of this study are available from public
data repositories. In particular, the topology of the AS level Internet has
been downloaded from CAIDA (Center for Applied Internet Data Analysis,
www.caida.org). We have downloaded the airport network from the OpenFlights
database (www.openflights.org). We used the DIP [28] database as a source for
the protein-protein interaction network of the S. cerevisiae. Finally, the map
of the north american power grid has been downloaded from [29].

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## Acknowledgements

The research work leading to these results was partially supported by HSNLab
and Ericsson. Project no. 108947 and 123957 has been implemented with the
support provided from the National Research, Development and Innovation Fund
of Hungary, financed under the OTKA-K (2013/1) funding scheme. Yury Malkov is
grateful for the support from RFBR, according to the research project No.
16-31-60104 mol_a_dk and by the Government of Russian Federation (agreement
#14.Z50.31.0033 with the Institute of Applied Physics of RAS).

## Author contributions statement

M.Cs. and A.G. have developed the model and contributed to the experiments.
A.K., J.B., Z.H. and Y.M. contributed to the analysis and the numerical
results. All authors reviewed the manuscript.

## Additional information

Competing financial interests

The authors declare no competing financial interests.