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Complex networks renormalization: flows and fixed points
Filippo Radicchi,1 Jose´ Javier Ramasco,1 Alain Barrat,1, 2 and Santo Fortunato1
1Complex Systems Lagrange Laboratory (CNLL), ISI Foundation, Torino, Italy
2Laboratoire de Physique The´orique (CNRS UMR 8627), Universite´ de Paris-Sud, France
Recently, it has been claimed that some complex networks are self-similar under a convenient
renormalization procedure. We present a general method to study renormalization flows in graphs.
We find that the behavior of some variables under renormalization, such as the maximum number
of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true
for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs.
Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems.
An analysis of classic renormalization for percolation and the Ising model on the lattice confirms
this analogy. Critical exponents and scaling functions can be used to classify graphs in universality
classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.
PACS numbers: 89.75.Hc, 05.45.Df
Keywords: Networks, renormalization, fixed points
Generally speaking, an object is self-similar if any part
of it, however small, maintains the general properties of
the whole object. Self-similarity is a characteristic fea-
ture of fractals [1] and it expresses the invariance of a ge-
ometrical set under a length-scale transformation. Many
complex systems such as the World-Wide-Web (WWW),
the Internet, social and biological systems, have a natu-
ral representation in terms of graphs, which often display
heterogeneous distributions of the number of links per
node (the degree k) [2, 3, 4, 5, 6]. These distributions
can be described by a power law decay, i.e. are scale-
free: they remain invariant under a rescaling of the de-
gree variable, suggesting that suitable transformations of
the networks’ structure may leave their statistical proper-
ties invariant. Since graphs however are not embedded in
Euclidean space, a standard length-scale transformation
cannot be performed. The concept of length can only be
defined in the graph-theoretical sense of the number of
links along any shortest path between two nodes. In this
context, Song et al. [7] proposed to transform a network
by means of a box-covering technique, in which a box in-
cludes nodes such that the distance between each pair of
nodes within a box is smaller than a threshold ℓB. After
tiling the network, the nodes of each box and their mu-
tual links are replaced by a supernode: supernodes are
connected if in the original network there is at least one
link between nodes of their corresponding boxes. This
defines a renormalization transformation RℓB . For some
real networks, such as the WWW, social, metabolic and
protein-protein interaction networks, a few iterated ap-
plications of this procedure seem to leave their degree
distribution invariant, which led to the claim that they
are self-similar [7]. Other networks, such as the Internet,
are instead found not to be self-similar under RℓB .
Iterated applications of RℓB generate renormalization
flows in the space of all possible graphs. Studying the be-
havior of such flows is crucial: the existence of possible
fixed points of the transformation would allow to identify
universality classes of networks, much like it happens for
second-order phase transitions in statistical physics [8].
This could offer a natural way to classify graphs and un-
cover unknown similarities. In this paper we perform a
systematic study of the renormalization transformation
RℓB , its flows and fixed points.
We denote a generic graph of N0 nodes and E0 links by
G0 and the renormalization transformation by R, for sim-
plicity. A series of t successive transformations R on G0
leads to the graph Gt = Rt(G0), with Nt nodes and Et
links. Finite size effects are strong especially in hetero-
geneous networks, where boxes built around large degree
nodes (hubs) determine a considerable contraction of the
system at each step. Such effects may perturb the anal-
ysis of the renormalization flow, which therefore has not
been investigated so far. We have devised a general pro-
cedure that overcomes this difficulty and allows to study
the renormalization flows.
Tiling a network means covering it with the minimum
number of boxes. We adopted two popular techniques for
box covering: the greedy coloring algorithm [9] (GCA)
and random burning [10] (RB). GCA is a greedy tech-
nique inspired by the mapping of the problem of tiling a
network to node-coloring, a well known problem in graph
theory [11]. In RB, boxes are spheres of radius rB cen-
tered at some seed nodes, so that the maximal distance
between any two nodes within a box does not exceed
2rB. The correspondence between the two methods is
obtained for ℓB = 2rB +1. The main results of our anal-
ysis appear robust with respect to the particular adopted
box covering technique.
An important characteristic of a network is its largest
degree. We therefore focus on the variable κt = Kt/(Nt−
1), where Kt is the largest degree of graph Gt. As the
number of renormalization steps t increases, we study
the flows of κt as a function of the relative network size
xt = Nt/N0 (N0 is the initial network size). We also
study the fluctuations of the variable κt along the flow,

2expressed by the susceptibility χt = N0
(
〈κt2〉 − 〈κt〉2
)
;
here the averages, denoted by 〈·〉 are taken over various
realizations of the covering algorithm.
In Fig. 1 we see how the variables evolve for an Erdo¨s-
Re´nyi [12] (ER) graph with average degree 〈k〉 = 2, which
thus contains a giant component and has loops. Such net-
work is not self-similar according to box-covering renor-
malization [7]. The box covering was carried out with
GCA, with ℓB = 3. We find that the functions κt(xt)
and χt(xt) are scaling functions of the variable xtN1/ν0 ,
as indicated by the remarkable data collapse of the in-
sets. The scaling relations hold on a very general ground,
10-2
10-1
100
κ t N0 = 1000
N0 = 2000
N0 = 5000
N0 = 10000
10-4 10-2 100
xt
10-4
10-2
100
χ t
10-2 100 102
xt N0
1/ν
10-2
100
10-2 100 102
xt N0
1/ν
10-8
10-4
χ t

N 0
-
γ/ν
ER <k> = 2
FIG. 1: Study of renormalization flows on non-self-similar ar-
tificial graphs. The box covering was performed with GCA
for ℓB = 3. After t iterations of the renormalization proce-
dure, the graph Gt has Nt nodes and Et links and its maximal
degree is Kt. The graph is an ER network with average de-
gree 〈k〉 = 2. The figures display κt = Kt/(Nt − 1) (top) and
χt = N0
`
〈κt2〉 − 〈κt〉2
´
(bottom) as a function of the relative
network size xt = Nt/N0. The insets display the scaling func-
tion of the variable xtN1/ν0 for κt and χt. Here ν = 2.0(1)
and the susceptibility exponent γ = ν (within errors).
namely for all the box covering procedures investigated,
with exponents identifying a narrow set of universality
classes. In the case of non-self-similar objects the esti-
mates for the exponent ν are consistent with the value 2.
The scaling of the susceptibility curves requires another
exponent γ, which controls the divergence of the peaks
(see inset of Fig.1, bottom). We obtain γ = ν for all
graphs and transformations.
In Fig. 2 we study the flows for a class of graphs which
are self-similar under box-covering renormalization: the
fractal model (FM) introduced by Song et al. [13]. The
Fractal Model 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 con-
nects the hubs with each other (Mode I), with probabil-
ity 1− e a non-hub of a star is connected to a non-hub of
the other (Mode II). The resulting network is a tree with
power law degree distribution, the exponent of which de-
10-3
10-2
10-1
100
κ t
N0 = 251
N0 = 1251
N0 = 6251
N0 = 31251
κt ~ xt
-0.45
10-4 10-3 10-2 10-1 100 101
xt
10-1
100
101
102
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
FM e = 0.5
FIG. 2: Study of renormalization flows on self-similar artificial
graphs. The box covering was performed with GCA for ℓB =
3. The graph is an FM network with e = 0.5, where e is the
probability for hub-hub attraction [13]. The figure displays
κt = Kt/(Nt − 1) (top), and χt = N0
`
〈κt2〉 − 〈κt〉2
´
(bot-
tom) as a function of the relative network size xt = Nt/N0.
The scaling function of the variable xtN1/ν0 for κt and χt
is displayed in the insets. The exponent is ν = 1.05(5).
The dashed lines indicate the predicted behavior of the scal-
ing function. The scaling function decays with an exponent
−(β−2)/(β−1) = −0.45. We still find γ = ν (within errors).
pends on the probability e.
This type of graphs maintain their statistical features
under renormalization. Nevertheless, the scaling behav-
ior is the same we have observed for non-self-similar
graphs, but with different exponents. In the case of the
FM network it is possible to derive the scaling exponent
ν, by inverting the construction 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 renormalizing the graph, our process is
the time-reverse of the growth described in [13], and is
characterized by the following relations
Nt−1 = nNt, kt−1 = s kt, β = 1 +
log n
log s , (1)
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
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−2
β−1
=
K0
N0
x−
β−2
β−1
t ∼ (N0 xt)−
β−2
β−1 ,
(2)
where we used s = n1/(β−1), Nt/N0 = n−t and K0 ∼
N1/(β−1)0 , derived from Eqs. 1. We see that the scaling

310-4
10-3
10-2
10-1
100
κ t N0 = 5000
N0 = 10000
N0 = 20000
N0 = 50000
10-4 10-2 100
xt
10-4
10-2
100
χ t
10-2 100 102
xt N0
1/ν
10-4
10-2
100
10-2 100 102
xt N0
1/ν
10-8
10-4
χ t

N 0
-
γ/ν
a
WS
10-2
10-1
100
κ t N0 = 251
N0 = 1251
N0 = 6251
N0 = 31251
N0 = 156251
10-4 10-2 100
xt
10-4
10-2
100
102
χ t
10-2 100 102
xt N0
1/ν
10-2
100
10-2 100 102
xt N0
1/ν
10-6
10-4
10-2
100
χ t

N 0
-
γ/ν
b
FM e = 0.5
FIG. 3: Effect of a small random perturbation on renormal-
ization flows. The box covering was performed with GCA,
with ℓB = 3. a) WS network with 〈k〉 = 4 and a fraction
p = 0.01 of randomly rewired links. b) FM network with
e = 0.5 and a fraction p = 0.05 of added links. The figures dis-
play κt = Kt/(Nt−1) (a, b, top), and χt = N0
`
〈κt2〉 − 〈κt〉2
´
(a, b, bottom) as a function of the relative network size
xt = Nt/N0. We see that the exponents are now very dif-
ferent from the unperturbed case: we recover ν = 2.0(1), as
in the case of non-self similar graphs. The relation γ = ν is
still satisfied within errors.
exponent ν = 1 is obtained for any value of the exponent
β. From Eq. 2 we actually get the full shape of the scaling
function, that is a power law: our numerical calculations
confirm this prediction. We remark that this holds only
because one has used precisely the type of transformation
that inverts the growth process of the fractal network.
This amounts to applying the GCA with ℓB = 3.
Self-similar objects correspond by definition to fixed
points of the transformation. To study the nature of
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 randomness,
through the addition or rewiring of a small fraction p of
links. The results are shown in Fig. 3 for Watts-Strogatz
(WS) small-world networks [14] and FM networks. In
both cases we recover the behavior observed for non-self-
similar graphs, with scaling exponents ν = 2.0(1) and
2.0(1), which implies that the original fixed points are
unstable with respect to disorder in the connections. To
complete our analysis, we have studied the renormaliza-
100 102
xt L0
100
P t


,
|m
| t
perc. L0 = 81
L0 = 243
L0 = 729
Pt ~ (xtL0)
-5/48
Ising L0 = 128
L0 = 256
L0 = 512
|m|t ~ (xtL0)-1/8
FIG. 4: Analogues of our scaling plots for real space renormal-
ization in percolation and the Ising model in two dimensions.
For percolation we use a triangular lattice, and the renormal-
ization reduces the volume by 1/9 at each step. For Ising we
use a square lattice, with a volume contraction factor of 1/4.
The relative system size xt now refers to the linear dimension
L of the lattices. So, the values of xt are multiples of 1/3
for percolation, of 1/2 for Ising. The plot illustrates the flows
obtained starting from the critical value of the control pa-
rameter, corresponding to the occupation density p = 0.5 for
percolation and the temperature kT = 2.269 for Ising. The
two order parameters, the percolation strength (relative size
of the percolating cluster) and the magnetization, scale with
xt. We recover the well known exponents of percolation and
Ising (βp/νp = 5/48, βI/νI = 1/8).
tion flows for many other artificial networks, either self-
similar or not, such as scale-free networks a´ la Baraba´si-
Albert [15] or generated with linear preferential attach-
ment [16], ER graphs at the threshold for the formation
of the giant component (〈k〉 = 1), hierarchical and Apol-
lonian networks [17, 18]. In all cases we have found the
same scaling behavior for κt and χt. We warn that the
values of the exponents may a priori also be affected
by the specific transformation adopted, as it happens in
real space renormalization for lattice models [8]. Still, we
find a coherent picture: non-self-similar graphs are char-
acterized by exponents consistent with ν = 2; self-similar
graphs yield different values for ν.
The scaling relations we have found are somewhat un-
usual, as the scaling variable entails the relative system
volume xt and not a control parameter. To disclose the
meaning of our scaling, we repeated our analysis for two
traditional systems of statistical physics: percolation and
the Ising model in two dimensions. We have applied real
space renormalization to percolation configurations on a
triangular lattice, and to Ising configurations on a square
lattice. For percolation, a triangular cell is replaced by
a supernode, which is occupied if the majority of sites of
the cell are occupied, empty otherwise (the procedure is
described in [19]). For Ising we have applied a classical

4majority rule scheme to square cells with four spins. In
Fig. 4 we show the relation between the order parameter
and xt (which here indicates the contraction in the linear
dimension L0), for different initial lattice sizes, starting
from configurations at the critical point. We observe a
clean scaling, just as in network renormalization.
The plots are analogues of the standard finite size scal-
ing plots. The order parameter scales as L−β/ν (L being
the linear dimension of the lattice) at the critical den-
sity, which in our case reads (xtL0)−β/ν and matches the
trend observed in the figure. At variance with finite size
scaling, where one always considers configurations of the
same system, here the renormalization may bring the sys-
tem to configurations corresponding to the critical state
of other systems in the same universality class, but the
scaling holds. We have also repeated the analysis starting
from system configurations in the subcritical and super-
critical phases, in which cases no scaling is observed.
The scaling of Fig. 4 does not give new insight about
percolation and Ising, as it just reproduces well known
exponents. Standard finite size scaling does the same
job, but there one has a control parameter (occupation
density, temperature) that allows to identify the state of
the system. In the case of networks, the state is repre-
sented by the topology of the system and there is no ob-
vious control parameter, so our approach seems the only
possibility to extract information about possible critical
properties. The scaling for self-similar graphs in Fig. 2
corresponds to the critical scaling of Fig. 4.
In conclusion, our results show that renormalization
flows in graphs, as defined by the box-covering method,
display a clear scaling behavior, opening a new promising
research avenue in the field of complex networks, with
close contacts to real space renormalization in lattice
models. Our analysis uses the well-established finite-size
scaling and real space renormalization techniques and
could be easily generalized to other possible renormaliza-
tion schemes. For a full classification of networks in uni-
versality classes it seems necessary to explore further the
robustness of the critical exponents under renormaliza-
tion, and to study the flow of other variables, which may
deliver new interesting scaling functions and exponents.
The analogies we have found with the classic renormaliza-
tion of percolation and the Ising model on the lattice are
intriguing and give more insight to our picture. Finally,
an interesting open question concerns the possibility to
assign real-world networks to specific universality classes.
This is a challenging issue, as for real graphs a finite-size
scaling analysis is not available because of the uniqueness
of each sample. A possibility could be to estimate their
”distance” from the self-similar (unstable) fixed points of
the transformation.
We thank A. Flammini, S. Havlin, V. Loreto, H. A.
Makse and A. Vespignani for discussions and feedback on
the manuscript. We also thank an anonymous referee for
suggesting a closer analysis of the relation between our
findings and the standard renormalization of statistical
mechanics models.
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