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Renormalization flows in complex networks
Filippo Radicchi∗,1 Alain Barrat,2, 1 Santo Fortunato,1 and Jose´ J. Ramasco1
1Complex Systems and Networks Lagrange Laboratory (CNLL), ISI Foundation, Turin, Italy
2CPT (CNRS UMR 6207), Luminy Case 907, F-13288 Marseille Cedex 9, France
Complex networks have acquired a great popularity in recent years, since the graph representation
of many natural, social and technological systems is often very helpful to characterize and model
their phenomenology. Additionally, the mathematical tools of statistical physics have proven to be
particularly suitable for studying and understanding complex networks. Nevertheless, an important
obstacle to this theoretical approach is still represented by the difficulties to draw parallelisms
between network science and more traditional aspects of statistical physics. In this paper, we explore
the relation between complex networks and a well known topic of statistical physics: renormalization.
A general method to analyze renormalization flows of complex networks is introduced. The method
can be applied to study any suitable renormalization transformation. Finite-size scaling can be
performed on computer-generated networks in order to classify them in universality classes. We also
present applications of the method on real networks.
PACS numbers: 89.75.Hc, 05.45.Df
Keywords: Networks, renormalization, fixed-points
I. INTRODUCTION
Many real systems in nature, society and technology
can be represented as complex networks [1, 2, 3, 4, 5, 6].
Independently of their natural, social or technological ori-
gin, most networks share common topological features,
like the “small-world” property [7] and a strong topolog-
ical heterogeneity. The small-world property expresses
the fact that the average distance between nodes, as de-
fined in the graph-theoretical sense, is small with respect
to the number of nodes, and typically grows only loga-
rithmically with it. Networks are topologically heteroge-
neous in that the distributions of the number of neigh-
bors (degree) of a node are broad, typically spanning
several orders of magnitude, with tails that can often be
described by power laws (hence the name “scale-free net-
works” [8]).
While “scale-free-ness” implies the absence of a char-
acteristic scale for the degree of a node, it is not a
priori clear how this can be related to the notion of
self-similarity, often studied in statistical physics, and
also typically related to the occurrence of power laws.
In this context, several recent works have focused on
defining and studying the concept of self-similarity for
networks. The notion of self-similarity is related to
a renormalization transformation, properly adapted to
graphs, introduced by Song et al. [9]. The renormal-
ization procedure is analogous to standard length-scale
transformations, used in classical systems [10, 11], and
can be simply performed by using a box covering tech-
nique interpreted in a graph-theoretical sense. The anal-
ysis of this transformation in real networks [9] has re-
vealed that some of them, such as the WWW, social,
∗Correspondence should be addressed to FR. Electronic address:
f.radicchi@gmail.com
metabolic and protein-protein interaction networks, ap-
pear to be self-similar while others, like the Internet, do
not. Self-similarity here means that the statistical fea-
tures of a network remain unchanged after the applica-
tion of the renormalization transformation. Many succes-
sive papers have focused on this subject, performing the
same analysis on several networks, introducing new box-
covering techniques and trying to explain the topological
differences between self-similar and non-self-similar net-
works [12, 13, 14, 15, 16, 17, 18] (for a recent review on
this topic see [19]).
In this context, the analysis of renormalization flows
of complex networks [20] represents a new perspective
to study block transformations in graphs. Differently
from all former studies, the study of Ref. [20] is not de-
voted to observe the effect of a single transformation,
but to analyze the renormalization flow produced by re-
peated iterations of the transformation. Starting from
an initial network, the iteration of the renormalization
procedure allows to explore the space of network con-
figurations just as standard renormalization is used to
explore the phase space of classical systems in statistical
physics [10, 11]. For these reasons, the analysis of renor-
malization flows of complex networks represents not only
an important theoretical step towards the understanding
of block-transformations in graphs, but also a further at-
tempt to link traditional statistical physics and network
science.
In this paper, we substantially extend the analysis pre-
sented in [20]. We perform a numerical study of renor-
malization flows for several computer-generated and real
networks. The numerical method is applied to different
renormalization transformations. For a particular class
of transformations, we find that the renormalization flow
leads non-self-similar networks to a condensation tran-
sition, where a few nodes attract a large fraction of all
links. The main result of the paper lies in the robustness
of the scaling rules governing the renormalization flow

2of a network: independently of the transformation, the
renormalization flow of non-self-similar networks is char-
acterized by the same set of scaling exponents, which
identify a unique universality class. In contrast, the
renormalization flow of self-similar networks allows to
classify these networks in different universality classes,
characterized by a set of different scaling exponents.
The paper is organized in the following way. In sec-
tion II, we describe the standard technique used in order
to renormalize a network and define the renormalization
flow of a graph. We then start with the analysis of renor-
malization flows of different networks. In the case of
computer-generated graphs, we distinguish the behavior
of non-self-similar (section IIIA) and self-similar (sec-
tion III B) networks. Section IV is devoted to the analy-
sis of the renormalization flows of real complex networks.
Finally, in section V we summarize and comment the re-
sults.
II. RENORMALIZING COMPLEX NETWORKS
Differently from classical systems, graphs are not em-
bedded in Euclidean space. As a consequence, standard
length-scale transformations cannot be performed on net-
works since measures of length have a meaning only in
a graph-theoretical sense: the length of a path is given
by the number of edges which compose the path; the dis-
tance between two nodes is given by the length of the (or
one of the) shortest path(s) connecting the two nodes.
Based on this metrics, Song et al. [9] proposed an orig-
inal technique for renormalizing networks (see Fig. 1).
Given the length of the transformation ℓB, their method
is given by the following steps:
• Tile the network with the minimal number of boxes
NB; each box should satisfy the condition that all
pairs of nodes within the box have distance less
than ℓB.
• Replace each box with all nodes and mutual edges
inside with a supernode.
• Construct the renormalized network composed of
all supernodes: two supernodes are connected if
in the original network there is at least one link
connecting nodes of their corresponding boxes.
The former recipe represents a transformation RℓB ap-
plicable to any unweighted and undirected network lead-
ing to the generation of a new unweighted and undirected
network, the “renormalized” version of the original one.
In principle, there are many ways to tile a network and
therefore the transformation RℓB is not invertible. More-
over, finding the best coverage of a network (i.e., the one
which minimizes the number of boxes NB) is computa-
tionally expensive: up to now, the best algorithm intro-
duced in this context is the greedy coloring algorithm [15]
(GCA), a greedy technique inspired by the mapping of
the problem of tiling a network to node-coloring, a well
FIG. 1: Renormalization procedure applied to a simple graph.
(left) The original network is divided into boxes and the renor-
malized graph (right) is generated according to this tiling.
known problem in graph theory [21]. An analogous tech-
nique, leading to a qualitatively and quantitatively sim-
ilar transformation RrB , is random burning (RB) [14].
In RB boxes are spheres of radius rB centered at some
seed nodes, so that the maximal distance between any
two nodes within a box does not exceed 2rB . Nodes in
boxes defined through the transformationRrB satisfy the
condition defining boxes of the transformation RℓB , for
ℓB = 2rB + 1. However, the search for minimal box cov-
erage is much more effective for the GCA than for RB,
and this may occasionally yield different results, as we
shall see.
The strict meaning of self-similarity is that any part of
an object, however small, looks like the whole [22]. Sim-
ilarly, complex networks are self-similar if their statisti-
cal properties are invariant under a proper renormaliza-
tion transformation. Song et al. [9] have shown that the
degree distribution of several real networks remains un-
changed if a few iterations of the renormalization trans-
formation are performed. Moreover, when this feature
is verified, the number of boxes NB needed to tile the
network for a given value of the length parameter ℓB de-
creases as a power law as ℓB increases:
NB (ℓB) ∼ ℓ−dBB . (1)
The exponent dB is called, in analogy with classical sys-
tems, the fractal exponent of the network [22]. This prop-
erty has been verified for several real networks in various
studies [9, 12, 14]. On the other hand, not all real net-
works are self-similar, i.e. Eq.(1) and the invariance of
the degree distribution do not hold for them. For consis-
tency, these networks are called non-self-similar.
In contrast to former studies which mostly dealt with
a single step of renormalization, we are interested here in
analyzing renormalization flows of complex networks, i.e.
the outcome of repeated iterations of the renormalization
procedure described above. Starting from a graph G0,

3with N0 nodes and E0 edges, we indicate as Gt (with
Nt nodes and Et edges) the network obtained after t
iterations of the transformation R:
G1 = R (G0) , G2 = R (G1) = R2 (G0) , . . .
. . . , Gt = R (Gt−1) = . . . = Rt (G0) . (2)
Note that in Eq.(2) we have suppressed the subscript ℓB
(or rB) for clarity of notation. In our analysis, we follow
the flow by considering several observables. We mainly
focus on the variables
κt = Kt/ (Nt − 1) , (3)
where Kt is the largest degree of the graph Gt, and
ηt = Et/ (Nt − 1) , (4)
which is basically the average degree of the graph Gt
divided by two. κt and ηt can assume non-trivial values
in any graph, excluding trees (for which ηt = 1, ∀t). We
monitor also the fluctuations of the variable κt along the
flow by measuring the susceptibility
χt = N0
(
〈κ2t 〉 − 〈κt〉2
)
, (5)
where 〈·〉 denotes averages taken over different realiza-
tions of the covering algorithm. Moreover, we consider
other quantities like the average clustering coefficient
Ct [7]. All these observables are monitored as a func-
tion of the relative network size xt = Nt/N0, which is a
natural way of following the renormalization flow of the
variables under study.
III. COMPUTER-GENERATED NETWORKS
We first consider artificial networks. In the case of
computer-generated networks, it is in fact possible to con-
trol the size N0 of the initial graphG0 and to perform the
well known finite-size scaling analysis for the renormal-
ization flow. For every computer-generated graph and
every transformation R, we find that the observable κt
obeys a relation of the type
κt = F
[
(xt − x∗)N1/ν0
]
, (6)
where F (·) is a suitable function depending on the
starting network and the particular transformation used.
Analogous scaling relations hold for the other observ-
ables (ηt and Ct) we considered. The susceptibility χt
needs an additional exponent γ since it obeys a relation
of the type: χt = Nγ/ν0 G
[
(xt − x∗)N1/ν0
]
, with G (·) a
suitable scaling function. In general, the scaling expo-
nent ν does not depend on the particular transformation
R used to renormalize the network, but depends on the
starting network G0: we always obtain ν = 2 for any
non-self-similar network (Sec. III A) and values of ν de-
pending on the initial network in the case of self-similar
graphs (Sec. III B). On the other hand, we obtain x∗ = 0
in all cases, except for the particular transformations ob-
tained for rB = 1 and ℓB = 2 on non-self-similar networks
(Sec. III A 1). In the next sections, we show our numeri-
cal results, obtained from the analysis of renormalization
flows of computer-generated networks, distinguishing be-
tween the various cases. All values of ν and x∗ are listed
in Table I. We emphasize the importance of the fact
that the exponent ν is able to classify artificial networks
in different universality classes.
A. Non-self-similar networks
We consider several computer-generated networks for
which Eq.(1) does not hold. Eq.(6) is able to describe
the renormalization flows of any of these network models.
The scaling exponent ν = 2 identifies a single universal-
ity class for all these models. The only difference is given
by the finite value of x∗ > 0 obtained when renormaliza-
tion is performed with ℓB = 2 or rB = 1 (Sec. III A 1).
Instead, for ℓB > 2 and rB > 1 we always obtain x∗ = 0
(Sec. III A 2).
1. rB = 1, ℓB = 2.
For rB = 1 or ℓB = 2, the transformation R has a par-
ticular behavior. In the case of GCA and ℓB = 2, at each
stage of the renormalization flow, the boxes in which the
network is tiled have the peculiarity to be fully connected
subgraphs or cliques [23]. In the case of renormalization
performed with RB and rB = 1, spheres are compact sub-
graphs composed only of neighbors of the selected seed
nodes. In both cases, at each stage of the renormaliza-
tion flow, the contraction of the network is much slower if
compared with the same transformations run for higher
values of ℓB or rB.
In Fig. 2, we show some numerical results obtained
following the renormalization flow of the Erdo¨s-Re´nyi
(ER) [24] model with average degree 〈k〉 = 2. For both al-
gorithms used for renormalizing the networks, we clearly
see a point of intersection between all the curves occur-
ring at a particular value x∗ > 0. Interestingly, the same
values of ν and x∗ hold also for the susceptibility χt and
the average clustering coefficient Ct. Numerical results
for both quantities and their relative scaling are reported
in Fig. 3.
The same behavior is observed for all the non-self-
similar networks that we have studied. To mention a
few, we performed numerical simulations also on the
Baraba´si-Albert (BA) model [8] and its generalization
given by scale-free networks generated with linear prefer-
ential attachment [25]. We report in Fig. 4 the numerical
results obtained for the BA model: the quantities κt and
ηt are shown as a function of the renormalization flow’s
variable xt. Again, a clear crossing point x∗ > 0 can be

50.03 0.04 0.05 0.06 0.07 0.08 0.09
xt
0
200
400
600
800
χ t
N0 = 1000
N0 = 2000
N0 = 5000
N0 = 10000
N0 = 20000
-2 -1 0 1 2
(xt - x*) N01/ν
0
0.005
0.01
0.015
0.02
χ t


N 0
-
γ/ν
a
ER <k> = 2
0.03 0.04 0.05 0.06 0.07 0.08 0.09
xt
0
0.05
0.1
0.15
0.2
C t
N0 = 1000
N0 = 2000
N0 = 5000
N0 = 10000
N0 = 20000
-2 -1 0 1 2
(xt - x*) N01/ν
0
0.05
0.1
0.15
0.2b
ER <k> = 2
0.11 0.12 0.13 0.14 0.15 0.16 0.17
xt
0
50
100
150
200
χ t
N0 = 500
N0 = 1000
N0 = 2000
-1 0 1 2
(xt - x*) N01/ν
0
0.01
0.02
χ t


N 0
-
γ/ν
c
ER <k> = 2
0.11 0.12 0.13 0.14 0.15 0.16 0.17
xt
0
0.02
0.04
0.06
0.08
0.1
0.12
C t
N0 = 500
N0 = 1000
N0 = 2000
-1 0 1
(xt - x*) N01/ν
0
0.05
0.1
d
ER <k> = 2
FIG. 3: Study of renormalization flows on ER model with 〈k〉 = 2. The box covering has been performed by using RB with
rB = 1 (a,b) and GCA with ℓB = 2 (c,d). The figures display the susceptibility χt (a,c) and the average clustering coefficient
Ct (b,d) as a function of the relative network size xt. The insets display the scaling function of the variable (xt − x∗)N1/ν0 for
χt and Ct. Here the scaling exponent ν = 2 and the susceptibility exponent γ = ν in all cases.
As a prototype of computer-generated self-similar net-
work, we consider the Fractal Model (FM) introduced by
Song et al. [13]. The FM is self-similar by design, as it is
obtained by inverting the renormalization procedure. At
each step, a node turns into a star, with a central hub and
several nodes with degree one. Nodes of different stars
can then be connected in two ways: with probability e
one connects the hubs with each other, with probability
1 − e a non-hub of a star is connected to a non-hub of
the other. The resulting network is a tree with power
law degree distribution, the exponent of which depends
on the probability e.
In the case of the FM network it is possible to de-
rive the scaling exponent ν, by inverting the construc-
tion procedure of the graph. In this way one recovers
graphs with identical structure at each renormalization
step and one can predict how κt, for instance, varies as
the flow progresses. Since we are interested in renormal-
izing the graph, our process is the time-reverse of the
growth described in [13], and is characterized by the fol-
lowing relations
Nt−1 = nNt,
kt−1 = s kt,
β = 1 + log n
log s ,
(7)
where n and s are time-independent constants determin-
ing the value of the degree distribution exponent β of
the network. Here Nt and kt are the number of nodes

60.98
0.99
1
κ
t
N0 = 1000
N0 = 2000
N0 = 5000
N0 = 10000
0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13
xt
1
1.1
1.2
1.3
1.4
η t
-1 0 1
(xt - x*) N01/ν
0.9
0.95
1
-1 0 1
(xt - x*) N01/ν
1
1.2
1.4
1.6
a BA m = 3
0.96
0.97
0.98
0.99
1
κ
t
N0 = 500
N0 = 1000
N0 = 2000
0.22 0.23 0.24 0.25 0.26
xt
1
1.2
1.4
1.6
η t
-1 0 1 2
(xt - x*) N01/ν
0.96
0.98
1
-2 -1 0 1 2
(xt - x*) N01/ν
1
1.2
1.4
1.6
b
BA m = 3
FIG. 4: Study of renormalization flows on the BA model with 2m = 〈k〉 = 6 (m indicates the number of connections introduced
by each node during the construction of the BA model). The box covering has been performed by using RB with rB = 1 (a)
and GCA with ℓB = 2 (b). The figures display the variables κt (a,b top) and ηt (a,b bottom) as a function of the relative
network size xt. The insets display the scaling function of the variable (xt − x∗)N1/ν0 for κt and ηt. Here the scaling exponent
ν = 2 in both cases.
0.03 0.04 0.05 0.06 0.07 0.08 0.09
xt
0
50
100
150
200
t
N0 = 1000
N0 = 2000
N0 = 5000
N0 = 10000
103 104N0
101
102
103
~N0
103 104N0
0
10
20
30
40
~log(N0)
a
ER <k> = 2
xt = 0.03
xt = 0.059
0.11 0.12 0.13 0.14 0.15 0.16 0.17
xt
40
80
120
160
t
N0 = 500
N0 = 1000
N0 = 2000
N0 = 5000
103 104N0
101
102
~N0
0.8
103 104N0
10
20
30
40
~log(N0)
b
ER <k> = 2
xt = 0.12
xt = 0.15
FIG. 5: Study of renormalization flows on the ER model with 〈k〉 = 2. The box covering has been performed by using RB
with rB = 1 (a) and GCA with ℓB = 2 (b). The figures display the number of iterations t as a function of the relative network
size xt. The fixed point [i.e., xt = x∗ = 0.059 (RB), 0.015 (GCA)] is reached in a number of renormalization steps growing
logarithmically with the initial system size N0 (see the insets on the right in each figure). In contrast, the number of stages
needed to go out from the fixed-point (i.e., to reach a given xt < x∗) grows as a power of N0 (see the insets on the left in each
figure).
and a characteristic degree of the network at step t of
the renormalization; we choose the maximum degree Kt.
The initial network has size N0 and shrinks due to box-
covering transformations. In this case, for the variable
κt one obtains
κt ∼
Kt
Nt
=
K0
N0
( s
n
)−t
=
K0
N0
(Nt
N0
)− β−2β−1
=
K0
N0
x−
β−2
β−1
t ∼ (N0 xt)−
β−2
β−1 ,
(8)

7100 101 102 103 104 105
k
10-12
10-10
10-8
10-6
10-4
10-2
100
P(
k)
ER
BA
perturbed WS
perturbed FM
a
100 101 102 103 104 105
k
10-6
10-5
10-4
10-3
10-2
10-1
100
C(
k)
b
100 101 102 103 104 105
k
10-5
10-4
10-3
10-2
10-1
100
k n
n
(k)
<
k>
/<k
2 >
c
FIG. 6: Statistical properties of the fixed point in the case of computer-generated networks. Renormalization has been performed
by using GCA with ℓB = 2. Initial network sizes are: N0 = 106 for the ER model, N0 = 106 for the BA model, N0 = 106 for the
Watts-Strogatz (WS) model (with ratio of rewired links p = 0.01), and N0 = 156251 for the fractal model (FM), with ratio of
added connections p = 0.05. Dashed lines have slopes −1.5 in (a), −1.2 in (b) and −1 in (c). (d) The graphical representation
of the fixed point has been obtained by starting from an ER model with N0 = 30000 and 〈k〉 = 2.
where we used s = n1/(β−1), Nt/N0 = n−t and K0 ∼
N1/(β−1)0 , derived from Eqs. (7). We see that the scaling
exponent ν = 1 is obtained for any value of the expo-
nent β. From Eq.(8) we actually get the full shape of
the scaling function, that is a power law: our numerical
calculations confirm this prediction (see Fig. 8b). We re-
mark that this holds only because one has used precisely
the type of transformation that inverts the growth pro-
cess of the fractal network. This amounts to applying
the GCA with ℓB = 3, as we did Fig. 8b.
If we consider instead the renormalization procedure
defined by RB with rB = 1 (or by GCA with ℓB = 2),
the centers of the boxes will be mostly low degree nodes,
as discussed above. Hubs are thus included in boxes only
as neighbors of low degree nodes and, as a consequence,
the supernode corresponding to a box with a large hub
inside will have a degree which is essentially the same
as the degree of the hub before renormalization. It is
therefore reasonable to assume that Kt ∼ K0, and we
get
κt ∼
Kt
Nt
∼ K0Nt
∼ N
1/(β−1)
0
Nt
= (N
β−2
β−1
0 xt)−1 (9)
which is again a scaling function of the variable N1/ν0 xt,
with ν = (β − 1)/(β − 2), as we found numerically (see
Fig. 8a).
Qualitatively similar numerical results can be shown

810-1
100
κ
t
N0 = 10000
N0 = 20000
N0 = 50000
N0 = 100000
10-3 10-2
xt
1
2
3
4
5
6
η t
100 101
xt N0
1/ν
10-1
100
100 101
xt N0
1/ν
2
4
6
a ER <k> = 2
10-2
10-1
100
κ
t
N0 = 2000
N0 = 10000
N0 = 10000
N0 = 20000
10-4 10-3 10-2 10-1 100
xt
0
5
10
η t
10-2 100 102
xt N0
1/ν
10-2
10-1
100
10-2 100 102
xt N0
1/ν
0
5
10
b BA m = 3
FIG. 7: Study of renormalization flows on non-self-similar artificial graphs. Renormalization has been performed by using GCA
with ℓB = 3. The figures display the variables κt (a,b top) and ηt (a,b bottom) as a function of the relative network size xt,
for the renormalization flow of an ER model with 〈k〉 = 2 (a) and a BA model with 2m = 〈k〉 = 6 (b). The insets display the
scaling function of the variable (xt − x∗)N1/ν0 for κt and ηt. Here the scaling exponent is ν = 2 in both cases.
10-2
10-1
100
κ
t
N0 = 1251
N0 = 6251
N0 = 31251
10-3 10-2 10-1 100
xt
100
101
102
103
χ t
10-2 100 102
xt N0
1/ν
10-2
10-1
100
10-2 100 102
xt N0
1/ν
10-3
10-2
χ t

N 0
-
γ/ν
a FM e = 0.5
10-2
10-1
100
κ
t
N0 = 251
N0 = 1251
N0 = 6251
N0 = 31251
10-5 10-4 10-3 10-2 10-1 100
xt
10-1
100
101
102
103
χ t
100 102 104
xt N0
1/ν
10-2
10-1
100
100 102 104
xt N0
1/ν
10-4
10-2
χ t

N 0
-
γ/ν
b
FM e = 0.5
FIG. 8: Study of renormalization flows on self-similar artificial graphs. The box covering has been performed by using RB
with rB = 1 (a) and GCA with ℓB = 3 (b). The graph is an FM network with e = 0.5, where e is the probability for hub-hub
attraction [13]. The figures display κt (a, b, top), and χt (a, b, bottom) as a function of the relative network size xt. The scaling
function of the variable (xt − x∗)N1/ν0 for κt and χt is displayed in the insets. We find that the two box covering techniques
yield different exponent values: ν = 2.2 (RB) and ν = 1 (GCA). The dashed lines indicate the predicted behavior of the scaling
function. In (a) the exponent of the power law decay for the scaling function is −1, independently of the exponent β of the
degree distribution of the initial graph; in (b) instead the scaling function decays with an exponent −(β − 2)/(β − 1) = −0.45.
We still find γ = ν for both transformations.
also for other self-similar models of networks: un-
perturbed Watts-Strogatz (WS) model [7] (i.e., one-
dimensional lattice), hierarchical model [27] and the
Apollonian (AP) network model [28] (see Table I).
C. Effect of small perturbations on self-similar
networks
Self-similar objects correspond by definition to fixed
points of the transformation. To study the nature of

9Type Network R ν x∗
Non-self-similar
ER 〈k〉 = 2
rB = 1 2.0(1) 0.059(1)
ℓB = 2 2.0(1) 0.15(1)
rB = 2 2.0(1) 0
ℓB = 3 2.0(1) 0
BA m = 3
rB = 1 2.0(1) 0.098(2)
ℓB = 2 2.0(1) 0.245(5)
rB = 2 2.0(1) 0
ℓB = 3 2.0(1) 0
Self-similar
WS 〈k〉 = 4
rB = 1 1.0(1) 0
ℓB = 2 1.0(1) 0
ℓB = 3 1.0(1) 0
FM e = 0.5
rB = 1 2.2(1) 0
ℓB = 2 2.2(1) 0
ℓB = 3 1.0(1) 0
AP
ℓB = 2 4.8(2) 0
ℓB = 3 1.0(1) 0
Perturbed self-similar
WS 〈k〉 = 4
rB = 1 2.0(1) 0.004(2)
ℓB = 3 2.0(1) 0
FM e = 0.5 rB = 1 2.1(1) 0.118(2)ℓB = 3 2.0(1) 0
AP
rB = 1 2.0(1) 0.045(2)
ℓB = 2 2.0(1) 0.05(1)
ℓB = 3 2.0(1) 0
TABLE I: We list the values of the scaling exponent ν and
of the fixed point threshold x∗ (fourth and fifth column, re-
spectively) for all networks we consider in our numerical anal-
ysis. Computer-generated networks (specified in the second
column) are divided in non-self-similar, self-similar and per-
turbed self-similar (first column). The perturbation is made
by rewiring a fraction p = 0.01 of all links in the WS model
and by adding a fraction p = 0.05 or p = 0.01 of all connec-
tions in the FM or AP networks, respectively. The third col-
umn specifies the type of transformation used to analyze the
renormalization flow. We associate to each numerical value
of ν and x∗ its error.
these fixed points, we have repeated the analysis of the
renormalization flows for the self-similar networks con-
sidered, but perturbed by a small amount of random-
ness, through the addition or rewiring of a small frac-
tion p of links. The results are shown in Fig. 9 for
WS small-world networks, which are simply linear chains
(trivially self-similar) perturbed by a certain amount of
rewiring [7], and FM networks with randomly added
links. In both cases we recover the behavior observed
for non-self-similar graphs, with a scaling exponent ν = 2
(this holds for all values of rB or ℓB investigated, see also
[20]). This clearly implies that the original self-similar
fixed points are unstable with respect to disorder in the
connections, and highlights once again the robustness of
the exponent value ν = 2. Furthermore, the statisti-
cal properties of the fixed point reached at x∗, when it
exists (i.e., for rB = 1 or ℓB = 2) are again the same
as those obtained starting from non-self-similar networks
(see Fig. 6). For these particular renormalization flows
(for rB = 1 or ℓB = 2), the picture obtained is therefore
a global flow towards the structure depicted in Fig. 6,
with isolated unstable fixed points given by the artificial
self-similar graphs.
IV. REAL NETWORKS
For real-world networks, a finite-size scaling analysis is
not available because of the uniqueness of each sample.
On the other hand, it is still possible to apply repeat-
edly the renormalization transformation and to study the
evolution of the network properties (a similar numerical
study has been performed also in [29]). In Fig. 10, we
measure some basic statistical properties of two real net-
works along the renormalization flow. We consider the
Actor Network [8], a graph constructed from the Internet
Movie Database [32] where nodes are connected if the
corresponding actors were cast together in at least one
movie, and the link graph of the Web pages of the domain
of the University of Notre Dame (Indiana, USA) [26].
Both networks have been claimed to be self-similar, since
Eq.(1) holds for both of them [9]. On the one hand,
the degree distributions P (k) are only slightly affected
by the renormalization transformation, and retain their
main characteristics even after several stages of renormal-
ization (in particular for the Web graph, see Fig. 10b).
This first result points towards an effective self-similarity
of P (k) under the action of the renormalization flow.
The degree distribution by itself is however not enough
to characterize complex networks, since many different
topologies can correspond to the same P (k). Important
information is in particular encoded in the clustering co-
efficient spectrum C(k), defined as the average clustering
coefficient of nodes of degree k, and in the average de-
gree of the neighbors of nodes of degree k, knn(k), which
is a measure of the degree correlations between nearest
neighbors in the graph. In this context, Fig. 10c, d, e,
and f show that even a single renormalization transfor-
mation induces large changes in these quantities. In this
respect, the apparent self-similarity exhibited by the de-
gree distribution does not extend to higher order corre-
lation patterns.
Interestingly, in the case rB = 1 or ℓB = 2, the it-
eration of the renormalization transformation leads all
real networks investigated [either self-similar or not, as
defined by Eq.(1)] towards the same kind of structure (il-
lustrated in Fig. 6) which is reached by non-self-similar
artificial networks (see Fig. 11). Note that, for real net-
works, no change in the initial size can be performed, so
we simply show the structure obtained after a few steps
of renormalization, which remains stable for many steps
due to the peculiarity of the case rB = 1 or ℓB = 2, as
explained above.
All these results allow us to discuss the exact self-
similarity of real-world networks: as we have seen in
the case of computer-generated self-similar networks,
all fixed-points correspond to strongly regular topolo-
gies and minimal perturbations are enough to break the
picture of self-similarity. Since randomness is an un-
avoidable element in real complex networks, exact self-

11
100 101 102 103 104
k
10-8
10-6
10-4
10-2
100
P(
k)
t = 0
t = 1
t = 2
t = 4
t = 10
a
Actor Network
100 101 102 103 104 105
k
10-10
10-8
10-6
10-4
10-2
100
P(
k)
t = 0
t = 1
t = 4
t = 8
b
WWW
100 101 102 103 104
k
10-4
10-3
10-2
10-1
100
C(
k)
c
Actor Network
100 101 102 103 104 105
k
10-5
10-4
10-3
10-2
10-1
100
C(
k)
d
WWW
100 101 102 103 104
k
10-3
10-2
10-1
100
k n
n
(k)
<
k>
/<k
2 >
e
Actor Network
100 101 102 103 104 105
k
10-3
10-2
10-1
100
k n
n
(k)
<
k>
/<k
2 >
f
WWW
FIG. 10: Statistical properties of real networks after t steps of renormalization. We consider two examples: a network of
392, 340 actors, where nodes are connected if the corresponding actors were cast together in at least one movie [8] (a, c and e);
the link graph of the WWW, consisting of 325, 729 Web pages of the domain of the University of Notre Dame (Indiana, USA)
and of their mutual hyperlinks [26] (b, d and f). The box covering was performed with the GCA (ℓB = 2), but the results
hold as well for other transformations. The clustering spectrum and the degree correlation pattern change drastically already
after a single transformation. In particular, the actor network displays assortativity, but after two transformations it becomes
disassortative. The solid line in (b) has slope −2.1.

12
100 101 102 103 104 105
k
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
P(
k)
Actor Network
Scientific Collaboration Network
WWW
Protein Interaction Network Yeast
a
100 101 102 103 104 105
k
10-6
10-5
10-4
10-3
10-2
10-1
100
C(
k)
b
100 101 102 103 104 105
k
10-5
10-4
10-3
10-2
10-1
100
k n
n
(k)
<
k>
/<k
2 >
c
FIG. 11: Statistical properties of the ’fixed point’ in the case of real networks. Renormalization has been performed by using
GCA with ℓB = 2. The properties of the networks are measured after a certain number (less than ten) of renormalization
steps. Dashed lines have the same slopes as those appearing in Fig. 6. The real networks considered in this figure are: the
actor network [8], the scientific collaboration network [30], the network of Web pages of the domain of the University of
Notre Dame [26] and the protein-protein interaction network of the yeast Saccharomyces cerevisae [31]. (d) The graphical
representation of the fixed point has been obtained by starting from the protein-protein interaction network of the yeast.
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