arXiv:1810.01686v2 [cond-mat.stat-mech] 3 Jan 2019
Scaling properties of d-dimensional complex networks
Samura´ı Brito1,∗Thiago C. Nunes2,†Luciano R. da Silva2,3,‡and Constantino Tsallis3,4,5,6§
1International Institute of Physics, Universidade Federal do Rio Grande do Norte,
Campus Universit´ario, Lagoa Nova, Natal-RN 59078-970, Brazil
2Departamento de F´ısica Te´orica e Experimental,
Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-900, Brazil
3National Institute of Science and Technology of Complex Systems, Brazil
4Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
5Santa Fe Institute, 1399 Hyde Park Road, New Mexico 87501, USA and
6Complexity Science Hub Vienna, Josefstaedter Strasse 39, A 1080 Vienna, Austria.
The area of networks is very interdisciplinary and exhibits many applications in several ﬁelds
of science. Nevertheless, there are few studies focusing on geographically located d-dimensional
networks. In this paper, we study scaling properties of a wide class of d-dimensional geographically
located networks which grow with preferential attachment involving Euclidean distances through
ij (αA≥0). We have numerically analyzed the time evolution of the connectivity of sites, the
average shortest path, the degree distribution entropy, and the average clustering coeﬃcient, for
d= 1,2,3,4, and typical values of αA. Remarkably enough, virtually all the curves can be made
to collapse as functions of the scaled variable αA/d. These observations conﬁrm the existence of
three regimes. The ﬁrst one occurs in the interval αA/d ∈[0,1]; it is non-Boltzmannian with very-
long-range interactions in the sense that the degree distribution is a q-exponential with qconstant
and above unity. The critical value αA/d = 1 that emerges in many of these properties is replaced
by αA/d = 1/2 for the β-exponent which characterizes the time evolution of the connectivity of
sites. The second regime is still non-Boltzmannian, now with moderately long-range interactions,
and reﬂects in an index qmonotonically decreasing with αA/d increasing from its critical value to
a characteristic value αA/d ≃5. Finally, the third regime is Boltzmannian-like (with q≃1), and
corresponds to short-range interactions.
Networks are everywhere, from the Internet to social
networks. We are living in the network age and the emer-
gence of more and more related researches is natural. The
theory of networks has applications in a diversity of sci-
entiﬁc ﬁelds, such as medicine , cosmology , quan-
tum information theory  and social networks . For a
long time, it was diﬀusely believed that the statistics gov-
erning complex networks was only the Boltzmann-Gibbs
(BG) one. However, in 2005 the connection between net-
works and nonextensive statistical mechanics started to
be explored [5–7], and is presently very active [8–14].
In the literature, systems with long-range interactions
are characterized by paired potentials that decay slowly
with the distance. A potential of the form 1/rαis typi-
cally said to be long-range if 0 ≤α≤d, where dis the
spatial dimension of the system. Some examples of such
potentials are gravitational systems, two-dimensional hy-
drodynamic systems, two-dimensional elastic systems,
charged systems and dipole systems. Unlike the case
of classical systems with short-range interactions (usu-
ally described within BG statistics), where many results
∗E-mail address: email@example.com
†E-mail address: firstname.lastname@example.org
‡E-mail address: luciano@ﬁsica.ufrn.br
§E-mail address: email@example.com
are well understood, there is a lack of complete knowl-
edge about the dynamic and statistical properties of sys-
tems with long-range interactions (for which BG statis-
tics fails). In this sense, many theories have been pro-
posed to understand the systems that interact at long-
range, and q-statistics has shown satisfactory results for
this regime [15–20].
In 2016, we studied a d-dimensional network model
where the interactions are short- or long-ranged depend-
ing on the choice of the parameter αA≥0. The results
that were obtained reinforced the connection between
nonextensive statistical mechanics and the networks the-
ory . In that work, we found some quantities which
present a universal behaviour with respect to the partic-
ular variable αA/d and observed the existence of three
regimes. In the ﬁrst one, namely 0 ≤αA/d ≤1, q
is constant and larger than unity, characterizing a non-
Boltzmannian regime with very-long-range interactions.
In the second one, qmonotonically decreases as αA/d
increased from its critical value αA/d = 1 to a charac-
teristic value αA/d ≃5. The third regime, above this
characteristic value , is Boltzmannian-like (q≃1)
and corresponds to short-range interactions. For the β
exponent (deﬁned here below) the behaviour is somewhat
diﬀerent: a ﬁrst regime is exhibited for 0 ≤αA/d ≤1/2,
a second regime appears between αA/d = 1/2 and a char-
acteristic value once again close to 5, and a third regime,
Boltzmannian-like with q≃1, between this value and
inﬁnity; it cannot be excluded that the purely Boltzman-
nian behaviour only occurs for αA/d → ∞.
Our model was constructed through two stages: the
number of the sites increases at time and the connections
between the sites follow a preferential attachment rule,
Each newly created site can connect to mothers. In
the present work, all results were obtained for m= 1.
The growth of the network starts with one site at the
origin, and then, we stochastically locate a second site
(and then a third, a fourth, and so on up to N) through
the d-dimensional isotropic distribution
rd+αG(αG>0; d= 1,2,3,4),(2)
where r≥1 is the Euclidean distance from the newly ar-
rived site to the center of mass of the pre-existing system.
For more details see . This network is characterized
by three parameters αA,αGand d, where αAcontrols the
importance of the distance in the preferential attachment
rule, αGis associated with the geographical distribution
of the sites, and dis the dimension of the system.
The connectivity distribution was the only property
studied in the previous work. Our results showed that the
degree distribution of this model is very well described by
the q-exponential functions that emerges from nonexten-
sive statistical mechanics [22–24], more precisely P(k)∼
eq−k/κ ∀(αA, αG, d), with ez
q≡[1 + (1 −q)z]1
1−q. The re-
lation between qand γ(the exponent of the asymptotic
power law) is given by γ≡1/(q−1) (see  for more
details). When αA= 0 we recover the Barab´asi-Albert
(BA) model  with q= 4/3 (γ= 3). Remarkably
enough, our previous results showed that the index qand
κexhibit universal behaviours with respect to the scaled
variable αA/d (∀d).
Motivated by the results in , in the present work
we are interested in investigating, for the same network
model, other possible universal behaviors with respect
to the same scaled variable αA/d. Besides that, we also
are interested in verifying the existence of the same three
regimes that we have previously observed. We have ana-
lyzed the exponent β, which is associated with the time
evolution of the connectivity of sites, the average shortest
path hli, the degree distribution entropy Sqand the aver-
age clustering coeﬃcient hCi. Along the lines of , in
order to analyse these properties, we choose the typical
value αG= 2 and vary the parameters (αA, d).
A. Time evolution of the connectivity of sites
One of the most common analyses that are done in net-
works theory is to verify how the degree of a site changes
at time. This property is usually referred to as connec-
tivity time evolution and it usually follows the behaviour
where ti≤t. (3)
We analyzed the time evolution of the connectivity of
sites for typical values of αAand d= 1,2,3,4 (see Fig. 1).
In order to do that, we choose the site i= 10 (the result
is independent of i), and then we compute the time evo-
lution of its connectivity. All simulations were made for
105sites and 103samples.
We observe that the dynamic exponent βis not con-
stant, in discrepancy with its value for the BA model:
βdecreases with αAand increases with d(see Fig. 2a).
Moreover, we notice that βexhibits universal curves with
respect to the scaled variable αA/d. When αA/d ≥0 up
to the critical value αA/d = 1/2, the system is in the
same universality class of the BA model with β= 1/2
and it is in the non-Boltzmannian very-long-range in-
teractions regime (qis constant above unity). From
αA/d > 1/2 on, β < 1/2 and decreases nearly exponen-
tially with αA/d down to the value 0.11 for αA/d ≈5.
For αA/d above this value up to inﬁnity, βremains prac-
tically constant, indicating a Boltzmannian-like regime
(with q≃1). It cannot be excluded that the terminal
value of βis achieved only at the limit αA/d → ∞ (see
d= 1 d= 2 d= 3 d= 4
FIG. 1. Time evolution of the connectivity of the site i= 10
in log −log plot for diﬀerent values of αAand d= 1,2,3,4.
The ﬁgure sublabels refer to (a) αA= 0, (b) αA= 2, (c)
αA= 3, (d) αA= 5, (e) αA= 6 and (f ) αA= 8. We can see
that ki∝(t/ti)β(αA,d)where β(αA, d) is the asymptotic slope
of the curves. For αA= 0, independent of the dimension, we
recover the BA model with β= 1/2, and when αA→ ∞ the
dimension does not matter either.
B. Average shortest path length
The average shortest path length is a concept, in net-
work theory, deﬁned as the average number of steps along
the shortest paths for all possible pairs of sites of the net-
work. In real networks, a short path makes it easier to
(a) d= 1
2if 0 ≤αA/d ≤1/2
d+ 0.11 if αA/d > 1/2
(b) d= 1
FIG. 2. (a) βdecreases with αAand increases with d.(b)
As we can see, from the rescaling αA→αA/d, all the curves
of βcollapse, and three regions clearly emerge. The ﬁrst one
is from αA/d = 0 up to the critical value αA/d = 1/2, β=
1/2. The second regime, from the critical value αA/d = 1/2
up to the characteristic value αA/d ≃5, βdecreases nearly
exponentiallly. When αA/d &5, βreaches a terminal value
β≃0.11, and the Bolzmannian-like regime is achieved. The
simulations have been run for 103samples and N= 105.
transfer information and can reduce costs. Mathemati-
cally, the average shortest path length is deﬁned by
where dij is the shortest path (smaller number of edges)
between the sites iand j. We have computed the aver-
age shortest path length hlifor typical values of αAand
∀d. When αA= 0 the results are the same as the BA
model where hli ∼ ln N(for m= 1), independent of the
dimension of the system. We have numerically veriﬁed
that hlidepends on (αA,d), increasing with αAand de-
creasing with d(see Fig. 3a). Remarkably enough, all
the curves can be made to collapse through the scalings
αA→αA/d and hli → hli(1 + αA/d)−1(see Fig. 3b).
Again, we can see the existence of three regimes. The
non-Boltzmannian very-long-range interactions go up to
the critical value αA/d = 1, as we can see in the inset
plot (Fig. 3b). The non-Boltzmannian moderate long-
range interactions go to up to the characteristic value
αA/d ≃5, as can be seen from the derivative of the
collapse curve. And ﬁnally, from αA/d &5 on, the
Boltzmannian-like limit is reached.
0 1 2 3 4 5 6 7 8 9 10
hli(1 + αA/d)−1
d= 1 d= 2 d= 3 d= 4
0 1 2 4 6 8 10
∆(hli(1 + αA/d)−1)/∆(αA/d)
FIG. 3. Shortest path length. (a) We can see that the
chemical distance hliincreases with αAand decreases with
d.(b) We can observe that the curves exhibit universal-
ity when we re-scale the axis replacing αA→αA/d and
hli → hli(1 + αA/d)−1. In the inset plot we show the deriva-
tive collapsed curve in order to see more precisely the exis-
tence of the three regimes. The critical value αA/d = 1, show
us the end of the non-Boltzmanninan very-long-range inter-
actions. The second regime, the non-Boltzmannian moder-
ate long-range interactions, go to up the characteristic value
αA/d ≃5. Finally, the third regime, from the characteristic
value up to αA/d → ∞, we see the Boltzmannian-like behav-
ior characterizing short-range interactions. This results are
for N= 104and 103samples.
C. Degree distribution entropy
The computation of the entropy in complex networks
is important to verify the heterogeneity and structure of
the network . The degree distribution entropy mea-
sures the quantity of randomness present in the connec-
tivity distribution. In our simulations, it is possible to
realize the change of the topology of the network. When
αA= 0 the network is a scale-free with an asymptotically
power law connectivity distribution. As we increase the
value of αAthe randomness of the degree distribution
also increases. For αA→ ∞ (q→1) the network is not
scale-free anymore since it presents an exponential degree
distribution, in agreement with some results available in
the literature .
We have computed the degree distribution entropy S
for each value of αAand d= 1,2,3,4. We computed the
q-entropy (Sq), from nonextensive statistical mechanics,
and the Boltzmann-Gibbs (BG) (SBG) entropy (alterna-
tively referred to as Shannon entropy) for the same con-
nectivity distributions studied in  (see Fig. 4). The
BG entropy was calculated from SBG =Pkpkln(1/pk),
where pkis the probability to ﬁnd sites with kdegree
and the sum is over k= 1 up to kmax under the con-
straint Pkpk= 1. Since we have P(k)∼eq−k/κ, to
each value αA/d, a pair of parameters (q, κ) is associ-
ated. This enables, in particular, the computation of
the q-entropy Sq≡Pkpklnq(1/pk) for the same data,
1−qis the inverse of the q-exponential
function. When αA→ ∞ (q→1), both entropies con-
verge to the same asymptotic limit. This result was of
course expected since, for q= 1, the q-entropy recovers
the standard entropy SBG .
Our results show that there is a region where the two
entropies are diﬀerent. It is known that the BG entropy
is not appropriate for systems where long-range interac-
tions are allowed. So, this result provides evidence that
Sqis adequate to describe the interactions in this nonex-
tensive domain. Besides this result, we also studied the
dependence of Sqwith both (αA, d) and αA/d. We ver-
iﬁed that, although Sqdepends on αAand dseparately
(see Fig. 5a), the curves exhibit universal behavior with
regard to the scaled variable αA/d (see Fig. 5b). Once
again, we clearly see the existence of three regimes. In
the ﬁrst one, Sqhas a constant value up to the critical
value αA/d = 1. From that value on, the characteristic
value αA/d ≃5, Sqincreases nearly exponentially and
then, from αA/d &5 on, the Boltzmannian-like limit is
D. Average clustering coeﬃcient
The average clustering coeﬃcient is an important mea-
sure in the theory of networks and it is associated with
how the neighbours of a given node are connected to each
other. This coeﬃcient is deﬁned as follows:
where kiis the degree of the site i,niis the number
of connections between the neighbours of the site iand
ki(k1−1)/2 is the total number of possible links between
In order to compute it, we run our network model for
m= 2 (because hCi= 0 when m= 1) and analyzed
how hCichanges with both (αA, d) and αA/d. We see
that hCiincreases with αAand decreases with d(see
Fig. 6a). The larger αAthe more aggregated the network
is. In the standard Barab´asi-Albert model (αA= 0),
the clustering coeﬃcient is inﬂuenced by the size Nof
the network, such that hCican be numerically approxi-
mated by hCi ∼ N−0.75 (in later works, Barab´asi analyt-
ically claimed that hCi ∼ (log N)2/N; for further details
0 1 2 3 4 5 6 7 8 9
0 5 10 15
0 10 20
0 10 20 30
FIG. 4. Measure of entropy in complex networks. Com-
parison between q−entropy (Sq) and the standard entropy
(SBG ). The BG entropy was calculated in the network us-
ing SBG =−kPkpkln pk(we used k= 1), whereas the q-
entropy was calculated using Sq=−Pkpq
klnqpk, where pk
is the probability of ﬁnding a site with connectivity kand
Pkpk= 1. In the region of long-range interactions we can
see that Sqis very diﬀerent from SBG, exhibiting that Sq
is more sensitive for describing this model in this domain.
When αA→ ∞ (q→1) both entropies converges to the same
asymptotic behaviour. The sublabels refer to (a) d= 1, (b)
d= 2, (c) d= 3, and (d) d= 4.
see ). From this behaviour we can see that, in the
thermodynamical limit (N→ ∞), hCi → 0. So, we have
numerically veriﬁed that when N→ ∞ hCi → 0 not only
for αA/d = 0, but for 0 ≤αA/d ≤1 (see the inset plot in
Fig. 6b). We also analyzed how the clustering coeﬃcient
changes with Nand we found that hCi ∼ N−ǫ(αA,d).
This power-law form was in fact expected since it agrees
with the numerical result previously found for the par-
ticular case αA= 0. However, surprisingly enough, when
αA&2dthe clustering coeﬃcient does not change with
Nanymore (see Fig. 7). Analyzing how ǫ(αA, d) changes
with both (αA, d) and αA/d, we see that this exponent
decreases with αA, but increases with d(see Fig. 8a).
Although we did not get collapse for these curves, by
rescaling αA→αA/d we clearly can see that all curves
perfectly intersect in αA/d = 1, strongly indicating a
change of regime (see Fig. 8b). The results found for
ǫ(αA, d) are somewhat reminiscent of the κ(αA, d) expo-
nent associated with the maximal Lyapunov exponent for
the generalized Fermi-Pasta-Ulam (FPU) model .
Our present results reveal an intriguing ubiquity of the
variable αA/d for the class of networks focused on here,
where both topological and metric aspects exist. Sur-
prisingly, the use of this variable indeed provides col-
0.82 if 0 ≤αA/d ≤1
d+ 1.38 if αA/d > 1
(b) d= 1
FIG. 5. Entropy dependence of αAand d= 1,2,3,4. (a)
Sqincreases with αAand decreases with d.(b) Once again,
we obtain the collapse of Sqwhen rescaling αA→αA/d.
The entropy Sqhas a constant value up to the critical value
αA= 1 and a nearly exponential behavior emerges up to the
characteristic value αA≃5.
lapses or quasi collapses for all the properties studied
here. Another interesting point is the existence of three,
and not only two, regimes. The ﬁrst one is a non-
Boltzmannian regime characterized by very-long-range
interactions and it goes from αA/d = 0 up to the critical
value αA/d = 1, except for βwhose critical value turns
out to be αA/d = 1/2, curiously enough. The second
one, from the critical value up to the characteristic value
αA/d ≃5, is still non-Boltzmannian and corresponds
to moderate long-range interactions. The third and last
regime, from αA/d ≃5 on, is Boltzmannian-like and is
characterized by short-range interactions. The existence
of the intermediate regime has also been observed in clas-
sical many-body Hamiltonians, namely the α-generalized
XY [16, 17] and Heisenberg  rotator models as well
as the Fermi-Pasta-Ulam model [19, 20, 29–31].
The present work neatly illustrates a fact which is not
always obvious to the community working with com-
plex networks, more precisely those exhibiting asymp-
totic scale-free behavior. Such networks frequently be-
long to the realm of applicability of nonextensive statis-
tical mechanics based on nonadditive entropies, and to
its superstatistical extensions . Such connections be-
tween thermal and geometrical systems are by no means
rare in statistical mechanics since the pioneering and en-
lightening Kasteleyn and Fortuin theorem . In the
present scenario, such connection can be naturally un-
derstood if we associate half of each two-body interac-
tion energy between any two sites of the Hamiltonian
0 1 2 3 4 5 6 7 8 9 10
[hCi(1 + αA)(d+ 2)]/αA
0.025 0.050 0.075
d= 1 αA= 0.75
FIG. 6. Clustering coeﬃcient for typical values of αAand
d= 1,2,3,4. (a) hCiincreases with αAbut it decreases with
d.(b) All curves collapse with the rescaling hCi → hCi[(αA+
1)(d+2)]/αAand αA→αA/d. In the thermodynamical limit
hCi → 0 from αA= 0 up to the critical value αA/d = 1.
In the inset plot we show an example for d= 1 and αA=
0.75. From the characteristic value αA/d ≃5 on we reach the
FIG. 7. Clustering coeﬃcient as a function of Nfor (a) d= 1,
(b) d= 2, (c) d= 3, and (d) d= 4. hCidecreases with N,
but increases with αA. Interestingly, from αA∼2don, hCi
does not change any more with N.
to each of the connected sites, thus generating, for each
node, a degree (number of links) in the sense of networks.
Through this perspective, it is no surprise that the degree
distribution corresponds to the q-exponential function
which generalizes the Boltzmann-Gibbs weight within
thermal statistics. The fact that in both of these geo-
metrical and thermal systems, the scaled variable αA/d
(a) d= 1
(b) d= 1
FIG. 8. Analysis of the exponent ǫ(αA, d). (a) This graph
reminds us the behavior of the maximum Lyapunov exponent
. The exponent ǫ(αA, d) decreases with αAand does so
faster for smaller d.(b) Although we did not get collapse for
these curves through the rescaling αA→αA/d, we can clearly
see that all curves appear to perfectly intersect at αA/d =
1. This result strongly indicates a change of regime, with
something special occurring at the value αA/d = 1. When
αA/d &2 we observe that ǫ→0 agrees with the result showed
in Fig. 7.
plays a preponderant role becomes essentially one and the
same feature. Mathematically-based contributions along
such lines would be more than welcome. Last but not
least, it would surely be interesting to understand how
come the critical point of the βexponent diﬀers from
the all the critical points that we studied here. This is
somewhat reminiscent of the two-dimensional XY ferro-
magnetic model with short-range interactions for which
nearly all properties exhibit a singularity at the positive
temperature of Kosterlitz and Thouless [34, 35], whereas
the order parameter critical point occurs at zero temper-
We gratefully acknowledge partial ﬁnancial support
from CAPES, CNPq, Funpec, Faperj (Brazilian agencies)
and the Brazilian ministries MEC and MCTIC. We thank
the High Performance Computing Center at UFRN for
providing the computational facilities to run the simula-
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