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Scaling properties of d-dimensional complex networks

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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 r −α A 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 q constant 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 q monotonically decreasing with αA/d increasing from its critical value to a characteristic value αA/d ≃ 5. Finally, the third regime is Boltzmannian (with q = 1), and corresponds to short-range interactions.
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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 Torica 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
rαA
ij (αA0). 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 q1), and
corresponds to short-range interactions.
I. INTRODUCTION
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 [1], cosmology [2], quan-
tum information theory [3] and social networks [4]. 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 αd, 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: samuraigab@gmail.com.br
E-mail address: thiago.cris@yahoo.com.br
E-mail address: luciano@fisica.ufrn.br
§E-mail address: tsallis@cbpf.br
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 αA0. The results
that were obtained reinforced the connection between
nonextensive statistical mechanics and the networks the-
ory [13]. 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 [21], is Boltzmannian-like (q1)
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 q1, 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
2
number of the sites increases at time and the connections
between the sites follow a preferential attachment rule,
given by:
ΠikiriαA.(1)
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
p(r)1
rd+αG(αG>0; d= 1,2,3,4),(2)
where r1 is the Euclidean distance from the newly ar-
rived site to the center of mass of the pre-existing system.
For more details see [13]. 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)
eqk/κ (αA, αG, d), with ez
q[1 + (1 q)z]1
1q. The re-
lation between qand γ(the exponent of the asymptotic
power law) is given by γ1/(q1) (see [5] for more
details). When αA= 0 we recover the Barab´asi-Albert
(BA) model [25] 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 [13], 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 [13], in
order to analyse these properties, we choose the typical
value αG= 2 and vary the parameters (αA, d).
II. RESULTS
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
ki(t)t
tiβ
where tit. (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 q1). It cannot be excluded that the terminal
value of βis achieved only at the limit αA/d → ∞ (see
Fig. 2b).
100
101
102
103
hk10i
(a) (b)
d= 1 d= 2 d= 3 d= 4
(c)
101102103104
t/t10
100
101
102
103
hk10i
(d)
101102103104
t/t10
(e)
101102103104
t/t10
(f)
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
3
0246810
αA
0.0
0.1
0.2
0.3
0.4
0.5
0.6
β
(a) d= 1
d= 2
d= 3
d= 4
01/2 12345678910
αA/d
0.0
0.1
0.2
0.3
0.4
0.5
0.6
β
β(αA/d) =
1
2if 0 αA/d 1/2
0.41e0.45αA
d+ 0.11 if αA/d > 1/2
(b) d= 1
d= 2
d= 3
d= 4
FIG. 2. (a) βdecreases with αAand increases with d.(b)
As we can see, from the rescaling αAαA/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
β0.11, 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
hli=2
N(N1) X
i<j
dij ,(4)
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
αAαA/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
αA
8
10
12
14
16
hli
(a)
d= 1
d= 2
d= 3
d= 4
012345678910
αA/d
0
2
4
6
8
10
hli(1 + αA/d)1
(b)
d= 1 d= 2 d= 3 d= 4
0 1 2 4 6 8 10
αA/d
8
6
4
2
0
∆(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 αAαA/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 [26]. 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 (q1) the network is not
scale-free anymore since it presents an exponential degree
distribution, in agreement with some results available in
the literature [27].
We have computed the degree distribution entropy S
for each value of αAand d= 1,2,3,4. We computed the
4
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 [13] (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)eqk/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 SqPkpklnq(1/pk) for the same data,
where lnqzz1q1
1qis the inverse of the q-exponential
function. When αA→ ∞ (q1), 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
achieved.
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:
hCi=1
NX
i
2ni
ki(ki1),(5)
where kiis the degree of the site i,niis the number
of connections between the neighbours of the site iand
ki(k11)/2 is the total number of possible links between
them.
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 ∼ N0.75 (in later works, Baraasi analyt-
ically claimed that hCi ∼ (log N)2/N; for further details
0 1 2 3 4 5 6 7 8 9
0.8
1.0
1.2
1.4
entropy
(a)
SBG
Sq
0 5 10 15
(b)
SBG
Sq
0 10 20
αA
0.8
1.0
1.2
1.4
entropy
(c)
SBG
Sq
0 10 20 30
αA
(d)
SBG
Sq
FIG. 4. Measure of entropy in complex networks. Com-
parison between qentropy (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→ ∞ (q1) 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 [28]). 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 ∼ Nǫ(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 αAαA/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 [29].
CONCLUSION
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-
5
0123456789
αA
0.8
1.0
1.2
1.4
Sq
(a)
d= 1
d= 2
d= 3
d= 4
0123456789
αA/d
0.8
1.0
1.2
1.4
Sq
Sq(αA/d) =
0.82 if 0 αA/d 1
0.56e1αA
d+ 1.38 if αA/d > 1
(b) d= 1
d= 2
d= 3
d= 4
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 αAαA/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 αA5.
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 [18] 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 [32]. 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 [33]. 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
αA
0.0
0.1
0.2
0.3
0.4
0.5
0.6
hCi
(a)
d= 1
d= 2
d= 3
d= 4
012345678910
αA/d
0.0
0.5
1.0
1.5
2.0
[hCi(1 + αA)(d+ 2)]A
(b)
d= 1
d= 2
d= 3
d= 4
0.025 0.050 0.075
(1/N)0.55
0.00
0.05
0.10
0.15
hCi
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)]Aand αAαA/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
Boltzmannian-like regime.
103
102
101
hCi
(a)
αA= 0.0
αA= 0.2
αA= 0.4
αA= 0.6
αA= 0.8
αA= 1.0
αA= 1.5
αA= 2.0
αA= 4.0(b)
αA= 0.0
αA= 0.5
αA= 1.0
αA= 1.5
αA= 2.0
αA= 2.5
αA= 3.0
αA= 4.0
αA= 6.0
102103104105106
N
103
102
101
hCi
(c)
αA= 0.0
αA= 1.0
αA= 1.5
αA= 2.0
αA= 2.5
αA= 3.0
αA= 4.0
αA= 6.0
αA= 8.0
102103104105106
N
(d)
αA= 0.0
αA= 1.0
αA= 2.0
αA= 3.0
αA= 4.0
αA= 5.0
αA= 6.0
αA= 8.0
αA= 10.0
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 αA2don, 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
6
0246810
αA
0.0
0.2
0.4
0.6
0.8
1.0
ǫ(αA, d)
(a) d= 1
d= 2
d= 3
d= 4
01234
αA/d
0.0
0.2
0.4
0.6
0.8
1.0
ǫ(αA, d)
(b) d= 1
d= 2
d= 3
d= 4
FIG. 8. Analysis of the exponent ǫ(αA, d). (a) This graph
reminds us the behavior of the maximum Lyapunov exponent
[29]. 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 αAαA/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 ǫ0 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-
ature.
ACKNOWLEDGMENTS
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-
tions.
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... The relationship between asymptotically scale-free d-dimensional geographic networks and nonextensive statistical mechanics started being explored in 2005 [28][29][30][31][32][33], where a preferential attachment index α A and a growth index α G were included. These studies showed that geographic networks exhibit three regimes: (a) 0 ≤ α A /d ≤ 1, corresponding to strongly long-range interactions, ...
... We verify that the energy distribution is completely independent from α G in all situations, as already discussed in earlier publications [28][29][30][31]33]. Consequently, we fix it to be α G = 1 in all our simulations. ...
... Also, we observe that p(ε) remains invariant when we fix α A /d and we modify the values of d = 1, 2, 3, 4, as shown in Fig. 2. This implies an universality property, namely that the energy distribution depends on the ratio α A /d and does not depend independently on α A and on d [29][30][31][32][33]. In consequence, we present our results by simply running d = 2. ...
Article
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Systems that consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of d -dimensional geographic networks (characterized by the index α G ⩾ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely P ( w ) ∝ e − | w − w c | / τ , w being the weight ( w c ⩾ 0; τ > 0). In this model, each site has an evolving degree k i and a local energy ε i ≡ ∑ j = 1 k i w i j / 2 ( i = 1, 2, …, N ) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability Π i j ∝ ε i / d i j α A ( α A ⩾ 0 ) , where d ij is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to α A / d > 1 and 0 ⩽ α A / d ⩽ 1; α A / d → ∞ corresponds to interactions between close neighbors, and α A / d → 0 corresponds to infinitely-ranged interactions. The site energy distribution p ( ɛ ) corresponds to the usual degree distribution p ( k ) as the particular instance ( w c , τ ) = (1, 0). We numerically verify that the corresponding connectivity distribution p ( ɛ ) converges, when α A / d → ∞, to the weight distribution P ( w ) for infinitely wide distributions (i.e. τ → ∞, ∀ w c ) as well as for w c → 0, ∀ τ . Finally, we show that p ( ɛ ) is well approached by the q -exponential distribution e q − β q | ε − w c ′ | [ 0 ⩽ w c ′ ( w c , α A / d ) ⩽ w c ] , which optimizes the nonadditive entropy S q under simple constraints; q depends only on α A / d , thus exhibiting universality.
... The relationship between asymptotically scale-free d-dimensional geographic networks and nonextensive statistical mechanics started being explored in 2005 [27][28][29][30][31][32], where a preferential attachment index α A and a growth index α G were included. These studies showed that geographic networks exhibit three regimes: (a) 0 ≤ α A /d ≤ 1, corresponding to strongly long-range interactions, (b) 1 ≤ α A /d < ∼ 5, corresponding to moderately long-range interactions, and (c) α A /d > ∼ 5, corresponding to the BG-like regime, i.e., q 1 (short-range interactions). ...
... We verify that the energy distribution is completely independent from α G in all situations, as already discussed in earlier publications [27][28][29][30]32]. Consequently, we fix it to be α G = 1 in all our simulations. ...
... Also, we observe that p(ε) remains invariant when we fix α A /d and we modify the values of d = 1, 2, 3, 4, as shown in Fig. 2. This implies an universality property, namely that the energy distribution depends on the ratio α A /d and does not depend independently on α A and on d [28][29][30][31][32]. In consequence, we present our results by simply running d = 2. ...
Preprint
Full-text available
Systems which consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of $d$-dimensional geographic networks (characterized by the index $\alpha_G\geq0$; $d = 1, 2, 3, 4$) whose links are weighted through a predefined random probability distribution, namely $P(w) \propto e^{-|w - w_c|/\tau}$, $w$ being the weight $ (w_c \geq 0; \; \tau > 0)$. In this model, each site has an evolving degree $k_i$ and a local energy $\varepsilon_i \equiv \sum_{j=1}^{k_i} w_{ij}/2$ ($i = 1, 2, ..., N$) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability $\Pi_{ij}\propto \varepsilon_{i}/d^{\,\alpha_A}_{ij} \;\;(\alpha_A \ge 0)$, where $d_{ij}$ is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to $\alpha_A/d>1$ and $0\leq \alpha_A/d \leq 1$; $\alpha_A/d \to \infty$ corresponds to interactions between close neighbors, and $\alpha_A/d \to 0$ corresponds to infinitely-ranged interactions. The site energy distribution $p(\varepsilon)$ corresponds to the usual degree distribution $p(k)$ as the particular instance $(w_c,\tau)=(2,0)$. We numerically verify that the corresponding connectivity distribution $p(\varepsilon)$ converges, when $\alpha_A/d\to\infty$, to the weight distribution $P(w)$ for infinitely narrow distributions (i.e., $\tau \to \infty, \,\forall w_c$) as well as for $w_c\to0, \, \forall\tau$.
... During the initial years, network science and nonextensive statistical mechanics were seen as completely different areas. But meaningful connections started in 2005 [16][17][18][19][20][21][22] . It is nowadays known that the degree distribution of asymptotically scale-free networks at the thermodynamic limit is of the form P(k) ∝ e −k/κ q , where the q-exponential function is defined as e z q ≡ [1 + (1 − q)z] ...
... We have in fact analyzed a large amount of typical cases in the space (d, α A , α G , w 0 , η) , and have systematically found the same results for d = 1, 2, 3 within the intervals (α A /d ∈ [0, 10]; α G ∈ [1, 10]; w 0 ∈ [0.5, 10]; η ∈ [0.5, 3]) . Similarly to previous works 16,[19][20][21] , p(ε) does not depend on α G ; also, it does not depend independently on d and α A , but only, remarkably, on the ratio α A /d ; p(ε) also depends on w 0 and η (see Fig. 2a-d). Because of these features, and without loss of generality, we have once for ever fixed α G = 1 , and d = 2 . ...
... In Fig. 4a,d we show the numerical results for the index q as function of α A /d . This result is consistent with 19,20 , where the behaviour of q characterises the existence of three regimes. As can be seen, q is constant and equal to 4/3 in the range 0 ≤ α A /d ≤ 1 . ...
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Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space–time entanglement. Its generalization based on nonadditive q -entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d -dimensional geographically located network with weighted links and exhibit its ‘energy’ distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann–Gibbs exponential factor is generically substituted by its q -generalisation, and is recovered in the $$q=1$$ q = 1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.
... During the initial years, network science and nonextensive statistical mechanics were seen as completely different areas. But meaningful connections started in 2005 [16][17][18][19][20][21][22] . It is nowadays known that the degree distribution of asymptotically scale-free networks at the thermodynamic limit is of the form P(k) ∝ e −k/κ q , where the q-exponential function is defined as ...
... . Similarly to previous works 16,[19][20][21] , p(ε) does not depend on α G ; also, it does not depend independently on d and α A , but only, remarkably, on the ratio α A /d; p(ε) also depends on w 0 and η (see Fig. 2(a)-(d)). Because of these features, and without loss of generality, we have once for ever fixed α G = 1, and d = 2. ...
... In Fig. 4(a,d) we show the numerical results for the index q as function of α A /d. This result is consistent with 19,20 , where the behaviour of q characterises the existence of three regimes. As can be seen, q is constant and equal to 4/3 in the range 0 ≤ α A /d ≤ 1. ...
Preprint
Full-text available
Boltzmann-Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving strong space-time entanglement. Its generalization based on nonadditive $q$-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a $d$-dimensional geographically located network with weighted links and exhibit its 'energy' distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann-Gibbs exponential factor is generically substituted by its $q$-generalisation, and is recovered in the $q=1$ limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.
... The unprecedented use of a nonadditive entropy (a conceptual possibility which, in one way or another, had already been considered historically [11][12][13]) in order to generalize the BG statistical mechanics, opened a door that has been being widely explored since 1988 [14,15]. It is for a wide class of anomalous situations, including analogous geometrical random systems such as asymptotically scale-free networks (see, for instance, [16][17][18] and references therein involving distance-dependent couplings of the type r −α a ; for the particular case of α A = 0, see [19][20][21][22]) that non-Boltzmannian entropies and related formalisms become useful. A neat explanation of the difference between entropic additivity and entropic extensivity will be provided below. ...
... Also worth mentioning are selected entropic applications beyond BG in other areas of knowledge: complex networks [16][17][18]; economics [149][150][151][152][153][154][155][156]; geophysics (earthquakes, atmosphere) [157][158][159][160][161][162][163][164][165][166]; general and quantum chemistry [139,[167][168][169][170][171]; hydrology and engineering (water engineering [172] and materials engineering [173,174]); power grids [175]; the environment [176]; medicine [177][178][179]; biology [180,181]; computational processing of medical images (microcalcifications in mammograms [182] and magnetic resonance for multiple sclerosis [183]) and time series (e.g., ECG in coronary disease [184] and EEG in epilepsy [185,186]); train delays [187]; citations of scientific publications and scientometrics [188,189]; global optimization techniques [190][191][192]; ecology [193][194][195]; cognitive science [196][197][198]; mathematics (functions [199], uniqueness theorems and related axiomatic approaches [200][201][202][203][204][205], central limit theorems, and generalized Fourier transform [206][207][208][209][210][211][212][213][214]); probabilistic models [215][216][217]; information geometry [218,219]. ...
... We see here that both the thermal and the geometrical model exhibit the interesting α/d scaling. The same happens for the α-Heisenberg inertial ferromagnet [100], the α-Fermi-Pasta-Ulam model [101][102][103][104], and other asymptotically scale-free networks [17,18], thus exhibiting the ubiquity of this grounding scaling law. Before proceeding, let us clarify why the statistical mechanical description of scale-free networks appears as a particular instance of q-statistics. ...
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The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy S B G started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.
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