Complex networks renormalisation: flows and fixed points Filippo Radicchi,1 Jos´e Javier Ramasco,1 Alain Barrat,1, 2 and Santo Fortunato∗1
arXiv:0803.3637v1 [physics.soc-ph] 25 Mar 2008
2
1 Complex Networks Lagrange Laboratory (CNLL), ISI Foundation, Torino, Italy Laboratoire de Physique Th´eorique (CNRS UMR 8627), Universit´e de Paris-Sud, France
Complex networks in nature, society and technology share a set of topological features, reflecting some common organisational principles. Recently, it has been claimed that some complex networks are self-similar under a convenient renormalisation procedure. Here we present a general method to systematically study renormalisation flows in graphs. We find that the behaviour of some variables under renormalisation, such as the maximum and the average number of connections of a node, is described by simple scaling laws, characterised by critical exponents. This result holds for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalisation flows for graphs display features similar to those found in the well-known renormalisation of spin systems. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graph topologies that are inaccessible to a standard analysis. PACS numbers: 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 feature of fractals [1] and it expresses the invariance of a geometrical set under a length-scale transformation. Many complex systems such as the World-Wide-Web (WWW), the Internet, social and biological systems, have a natural 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 scalefree: they remain invariant under a rescaling of the degree variable, suggesting that suitable transformations of the networks’ structure may leave their statistical properties 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 includes 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 mutual 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 renormalisation transformation RℓB . For some real networks, such as the WWW, social, metabolic and protein-protein interaction networks, a few iterated applications 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,
∗ Correspondence should be addressed to SF. Electronic address:
[email protected]
are instead found not to be self-similar under RℓB . Iterated applications of RℓB generate renormalisation flows in the space of all possible graphs. Studying the behaviour 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 uncover unknown similarities. In this paper we perform a systematic study of the renormalisation transformation RℓB , its flows and fixed points. We denote a generic graph of N0 nodes and E0 links by G0 and the renormalisation transformation by R, for simplicity. 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 heterogeneous networks, where boxes built around large degree nodes (hubs) determine a considerable contraction of the system at each step. Such effects may perturb the analysis of the renormalisation flow, which therefore has not been investigated so far. We have devised a general procedure that overcomes this difficulty and allows to study the renormalisation flows. Tiling a network means covering it with the minimum number of boxes. We adopted two popular techniques for box covering: the greedy colouring algorithm [9] (GCA) and random burning [10] (RB). GCA is a greedy technique inspired by the mapping of the problem of tiling a network to node-colouring, a well known problem in graph theory [11]. In RB, boxes are spheres of radius rB centred 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. Important characteristics of a network are its average degree and its largest degree. We therefore focus on the variables κt = Kt /(Nt −1), where Kt is the largest degree of graph Gt , and ηt = Et /(Nt − 1) (ηt is basically the
2 average degree of the graph, divided by two). As the number of renormalisation steps t increases, we study the flows of κt and ηt 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, expressed by the susceptibility χt = N0 hκt 2 i − hκt i2 ; here the averages, denoted by h·i are taken over various realisations of the covering algorithm. In Figs. 1 and 2, we see how the variables evolve for an Erd¨ os-R´enyi [12] (ER) graph with average degree hki = 2, which thus contains a giant component and has loops, and a scale-free network ` a la Barab´ asiAlbert [13] (BA). Such networks are not self-similar according to box-covering renormalisation [7]. The box covering was carried out with both RB and GCA, with several choices for the box size. We find that the functions κt (xt ), ηt (xt ) and χt (xt ) are all scaling functions of the 1/ν variable (xt − x∗ )N0 , as indicated by the remarkable data collapse of the insets. The threshold x∗ is non-zero when there is a crossing of the curves for κt (ηt ), as we see e.g. in Fig. 1a, c. Such a non zero x∗ occurs for the RB transformation with rB = 1 as well as for the GCA with ℓB = 2 on all non-self-similar networks analysed. It implies the existence of a special stable fixed point for these transformations, which holds in the limit of infinite network size. This particular fixed point is a graph where a few nodes attract a large fraction of all links (i.e., κt (x∗ ) ≈ 1), but is not found if different transformations, such as ℓB = 3 or rB = 2, are used for the renormalisation procedure (Fig. 2d). Our analysis gives however the deep result that, independently of the particular fixed point, the scaling relations hold on a very general ground, 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 relation ν = 2 seems to be generally valid. The scaling of the susceptibility curves requires another exponent γ, which controls the divergence of the peaks (see insets of Figs.1b and 1d). We obtain γ = ν for all graphs and transformations. In Fig. 3 we study the flows for graphs which are self-similar under box-covering renormalisation: the fractal model (FM) introduced by Song et al. [14] (see Appendix A) (Figs. 3a and 3c), which is a prototype of selfsimilar network obtained by inverting a particular renormalisation procedure; the Watts-Strogatz (WS) smallworld network with no rewiring, i.e. a one-dimensional chain [15] (Fig. 3b and 3d). In these cases, renormalisation affects the graphs appreciably only after a transient in which they maintain their statistical features. Nevertheless, the scaling behaviour of the resulting curves is the same we have observed for non-self-similar graphs, but with different exponents. For the FM network, it can be shown analytically that the exponent ν found with RB depends on the exponent β of the power law degree distribution according to the simple relation ν = (β − 1)/(β − 2), whereas the GCA for ℓB = 3 yields ν = 1, for any value of β (see
Appendix B). 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 renormalisation flows for the self-similar networks considered, but perturbed by a small amount of randomness, through the addition (for FM) or rewiring (for WS) of a small fraction p of links. The results are shown in Fig. 4 for WS networks (4a and 4c) and FM networks (4b and 4d). In both cases we recover the behaviour observed for non-self-similar graphs, with a scaling exponent ν = 2, which implies that the original fixed points are unstable with respect to disorder in the connections. To complete our analysis, we have studied the renormalisation flows for many other artificial networks, either self-similar or not, such as scale-free networks generated with linear preferential attachment [16], ER graphs at the threshold for the formation of the giant component (hki = 1), hierarchical and Apollonian networks [17, 18]. In all cases we have found the same scaling behaviour for κt , ηt 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 renormalisation for lattice models [8]. Still, we find a coherent picture: non-self-similar graphs are characterised by the exponent ν = 2; self-similar graphs yield different values for ν. In conclusion, our results show that renormalisation flows in graphs, as defined by the box-covering method, display a clear scaling behaviour, opening a new promising research avenue in the field of complex networks, with close contacts to real space renormalisation in lattice models. Our analysis uses the well-established finitesize scaling and real space renormalisation techniques and could be easily generalised to other possible renormalisation schemes. For a full classification of networks in universality classes it seems necessary to explore further the robustness of the critical exponents under renormalisation, and to study the flow of other variables, which may deliver new interesting scaling functions and exponents. 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.
ACKNOWLEDGMENTS
We thank A. Flammini, S. Havlin, V. Loreto, H. A. Makse and A. Vespignani for discussions and feedback on the manuscript.
3 APPENDIX A: THE FRACTAL MODEL OF SONG ET AL.
The Fractal Model is self-similar by design. It is obtained by inverting the renormalisation 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 (Mode I), with probability 1 − e a 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 depends on the probability e. APPENDIX B: PREDICTIONS OF SCALING 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 renormalisation step and one can predict how κt , for instance, varies as the flow progresses. Since we are interested in renormalising the graph, our process is the time-reverse of the growth described in [14], and is characterised by the following relations Nt−1 = n Nt , kt−1 = s kt , log n β =1+ , log s
(B1)
where n and s are time-independent constants determining 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
[1] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, 1982. [2] A.-L. Barab´ asi & R. Albert, Rev. Mod. Phys. 74, 47 (2002). [3] S. N. Dorogovtsev & J. F. F. Mendes, Evolution of Networks: from biological nets to the Internet and WWW, Oxford Univ. Press, Oxford, 2003. [4] M. E. J. Newman, SIAM Rev. 45, 167 (2004). [5] R. Pastor-Satorras & A. Vespignani, Evolution and structure of the Internet: A statistical physics approach Cambridge Univ. Press, Cambridge, 2004. [6] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez & D. U. Hwang, Phys. Rep. 424, 175 (2006). [7] C. Song, S. Havlin & H. A. Makse, Nature 433, 392 (2005). [8] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1971.
the renormalisation; we choose the maximum degree Kt . The initial network has size N0 and shrinks due to boxcovering transformations. In this case, for the variable κt one obtains β−2 − β−1 Kt K 0 Nt K0 s −t κt ∼ = = Nt N0 n N0 N0 β−2 β−2 K0 − β−1 ∼ (N0 xt )− β−1 , = x N0 t
(B2)
where we used s = n1/(β−1) , Nt /N0 = n−t and K0 ∼ 1/(β−1) N0 , derived from Eqs. B1. We see that the scaling exponent ν = 1 is obtained for any value of the exponent β. From Eq. B2 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. For the specific graphs of Figs. 3a and 3c, this amounts to applying the GCA with ℓB = 3, as we did in Fig. 3c. If we consider instead the renormalisation procedure defined by RB with rB = 1 (or by GCA with ℓB = 2), the centres of the boxes will be mostly low degree nodes. Hubs would be included in boxes only as neighbours of low degree nodes. As a consequence, the supernode corresponding to a box with a big hub inside will have a degree which is essentially the same as the degree of the hub before renormalisation. If we assume that Kt ∼ K0 we get 1/(β−1)
κt ∼
K0 N Kt ∼ ∼ 0 Nt Nt Nt
β−2
= (N0β−1 xt )−1
(B3) 1/ν
which is again a scaling function of the variable N0 with ν = (β − 1)/(β − 2), as we found numerically.
xt ,
[9] C. Song, L. K. Gallos, S. Havlin & H. A. Makse, J. Stat. Mech. P03006 (2007). [10] K.-I. Goh, G. Salvi, B. Kahng & D. Kim, Phys. Rev. Lett. 96, 018701 (2006). [11] B. Bollob´ as, Modern Graph Theory, Springer Verlag, New York, 1998. [12] P. Erd¨ os & A. R´enyi, Publ. Math. Debrecen 6, 290 (1959). [13] A.-L. Barab´ asi & R. Albert, Science 286, 509 (1999). [14] C. Song, S. Havlin & H. A. Makse, Nature Phys. 2, 275 (2006). [15] D. J. Watts & S. H. Strogatz, Nature 393, 440 (1998). [16] S. N. Dorogovtsev, J. F. F. Mendes & A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000). [17] E. Ravasz & A.-L. Barab´ asi, Phys. Rev. E 67, 026112 (2003). [18] J. S. Andrade, H. J. Herrmann, R. F. S. Andrade & L. R. da Silva, Phys. Rev. Lett. 94, 018702 (2005).
4 1
a
ER
= 2
BA m = 3
1
1
0.8
κt
κt
0.8
1
c
0.6 0.6 -1
-2
0.4
1
0
0.99
0.9 -1
2
1/ν (xt - x*) N0
1.3 N0 = 1000
0
(xt - x*) N0
1.1
N0 = 2000
1
1/ν
ηt
ηt
1
1.4 1.6
1.3 1.2 1.1 1 -1
1.2
0
1/ν (xt - x*) N0
0.98
1.4 1.4 1.3
0.95
1.4 1.2
1.2
N0 = 1000
1 -1
0
(xt - x*) N0
N0 = 5000
1.1
N0 = 10000
N0 = 2000
1
1/ν
N0 = 5000 N0 = 10000
N0 = 20000
11
0.03
0.04
0.05
0.06
xt
0.07
0.08
1
0.09
b
0.06
0.08
0.1
xt
0.12
0.14
d 250
0.01
N0 = 5000 N0 = 10000
0.005 0 -2
-1
0
χt
1
-γ/ν
BA m = 3
0.002 0 -1
150
2
1/ν
400
0
0.004
N0 = 20000
(xt - x*) N0
200
200
χt N 0
N0 = 2000
0
1
(xt - x*) N0
χt
600
N0 = 1000
0.015
χt N 0
800
-γ/ν
0.02
2 1/ν
100 ER = 2
0.03
0.04
50
0.05
0.06
xt
0.07
0.08
0.09
0
0.06
0.08
xt
0.1
0.12
0.14
FIG. 1: Study of renormalisation flows on non-self-similar artificial graphs. The box covering was performed with random burning (RB), with rB = 1. After t iterations of the renormalisation procedure, the graph Gt has Nt nodes and Et links and its maximal degree is Kt . a, b) ER network with average degree hki = 2. c, d) BA network with number of links per new added node m` = 3 (hki = 2m ´ = 6). The figures display κt = Kt /(Nt − 1) (a and c, top), ηt = Et /(Nt − 1) (a and c, bottom) and χt = N0 hκt 2 i − hκt i2 (b and d) as a function of the relative network size xt = Nt /N0 . The insets display the scaling 1/ν function of the variable (xt − x∗ )N0 for κt , ηt and χt . Here ν = 2 and the susceptibility exponent γ = ν in both cases.
5 0
1
a
ER = 2
0.8
ER = 2 0
1
10
κt
κt
0.9
d 10
0.8
N0 = 10000 N0 = 20000
-1
0.6
0.7
-1
-1
0
(xt - x*) N0
0.6 1.4
10
1/ν N0
6
5
N0 = 1000
4
N0 = 2000
1.2
4
ηt
1
-1
(xt - x*) N01/ν
2
3
1
0
0
0.12
0.14
xt
0.16
10
1/ν
xt N0
2 1
1 0.1
1
10
-3
0.18
-2
10
10
xt
e
b 200
2 N0 = 20000 N0 = 50000 N0 = 100000
0
10
0.01
-3
1
0
(xt - x*) N0
100
10
-γ/ν
0 -1
2
χt
150
N0 = 10000
0.02
χt N 0
-γ/ν
10
1/ν
-2
10
χt N 0
ηt
xt
6
1.4
1.2
N0 = 100000
1
10
N0 = 500
1.1
χt
0
1.6
1.3
N0 = 50000
10
1
1/ν
10
-6
10
-9
10
ER = 2
N0 = 500 N0 = 1000 N0 = 2000
50
1
0
2
10
10
-4
10
xt N0
10 1/ν
ER = 2 -6
10
0 0.1
0.14
xt
0.16
0.99
-3
0.18
1
c
f
xt
0
10
10
0
BA m = 3 0
10
0.98 0.98
-1
10
-1
10
N0 = 2000
BA m = 3
0.97 0.96 0.96
10
10
1
-1
-1
-2
10
κt
κt
0.12
N0 = 10000 -2
1
0
(xt - x*) N01/ν
10
2
-2
0
10
10
2
10
1/ν
x t N0
-2
N0 = 10000 N0 = 20000
10
1.6
1.6
1.4
1 -2
1.2
10
1.2
10 5
ηt
ηt
1.4
-1
0
1
(xt - x*) N01/ν
2
N0 = 500
5
0 -2 10
0
10
2
1/ν
10
x t N0
N0 = 1000 N0 = 2000
1
0.22
0.24
xt
0.26
0 -5 10
-4
10
-3
10
-2
xt
10
-1
10
0
10
FIG. 2: Study of renormalisation flows on non-self-similar artificial graphs. The box covering was performed with the greedy colouring algorithm (GCA), with ℓB = 2 (a, b, c) and ℓB = 3 (d, e, f). a, b, d, e) ER network with average degree hki = 2. c, f) BA network with number of links per new added node m`= 3 (hki = 2m ´ = 6). The figures display κt = Kt /(Nt − 1) (a, c, d, f, top), ηt = Et /(Nt − 1) (a, c, d, f, bottom) and χt = N0 hκt 2 i − hκt i2 (b and e) as a function of the relative network size 1/ν xt = Nt /N0 . As in Fig. 1, the insets display the scaling function of the variable (xt − x∗ )N0 for κt , ηt and χt . Again ν = 2 and the susceptibility exponent γ = ν.
6 0 N0 = 1251
FM e = 0.5
FM e = 0.5 -1
N0 = 31251
0
10
10
-1
κ t ~ xt
-1
10
-1
-2
10
-2
10
-2
0
10
10
-2
1/ν 10
0
4
2
10
10 1/ν
10
x t N0
-3
2
10
2
-2
10
10
-3
10
-2
0
10
10
χt
χ t N0
10
-γ/ν
-γ/ν
10 3
χt
10 10
2
10
x t N0
-2
0
10
κt
κt
10
N0 = 6251
-1
10
0
c
10
χ t N0
a
2
1/ν10
1
x t N0
1
N0 = 251
10
-4
10
0
10
N0 = 1251
4
2
1/ν 10
10
x t N0
N0 = 6251
0
10
-2
10
N0 = 31251
10
-0.45
κ t ~ xt
-1
0
10 -3 10
-1
-2
10
10
xt
10
1
0
10
10
b
-4
-3
10
-2
10
10
xt
-1
10
0
10
1
10
d 0
10
10
0
N0 = 5000
-1
N0 = 5000
N0 = 10000
WS
N0 = 20000 N0 = 50000
10
N0 = 10000
WS
-1
N0 = 20000 N0 = 50000
10
0
10
10
-2
10
-3
10
χt N 0
-1
-2
10
-4
10
-3
10
1
0
10
10
2
-5
10
3
10
xt N0
-4
10 -6 10
-2
10
-γ/ν
10
χt N 0
-3
10
χt
0
-2
10
-γ/ν
χt
N0 = 100000
10
4
0
10
-3
10
xt
-2
10
-1
10
0
10
10 -5 10
10
10
xt N0
-4
-4
10
4
2
10
1/ν
-4
10
1/ν
-3
10
-2
xt
10
-1
10
0
10
FIG. 3: Study of renormalisation flows on self-similar artificial graphs. The box covering was performed with RB, with rB = 1 (a, b) and GCA with ℓB = 3 (c, d). a and c) FM network with e = 0.5, where e is the probability for hub-hub attraction [14]. b and d) Unperturbed WS network (i.e., one-dimensional chain) with hki = 4. The figures display κt = Kt /(Nt − 1) (a, b, top), ´ ` and χt = N0 hκt 2 i − hκt i2 (a, b, bottom and c, d) as a function of the relative network size xt = Nt /N0 . The scaling function 1/ν of the variable (xt − x∗ )N0 for κt and χt is displayed in the insets. We find ν = 1 for the WS network, both with RB and with GCA. For the FM, instead, the two box covering techniques yield different exponent values: ν = 2.2 (RB) and ν = 1 (GCA), see Appendix B. The dashed lines in Figs. 3a and 3c indicate the predicted behaviour of the scaling function (see Appendix B). In Fig. 3a 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 Fig. 3c instead the scaling function decays with an exponent −(β − 2)/(β − 1) = −0.45. We still find γ = ν for both graphs and transformations.
7 1
a
N0 = 5000
WS
N0 = 10000
1
0
c
10
0.9 -0.25
0
0.9
N0 = 20000
10
N0 = 50000
-2
10
-3
10
0.25
1/ν (xt - x*) N0
N0 = 10000
10
-2
κt
κt
0.95
N0 = 100000
N0 = 5000
0
10
N0 = 50000
0.95
WS
-1
N0 = 20000
-4
10
-2
-4
0
10
10
2
10
10
1/ν
xt N0
-5
10
0 -0.3
0
0.3
0.6
-4
10
0.9
(xt - x*) N01/ν
-4
10
-γ/ν
0.001
χt N 0
1000
0
10
0.002
χt
χt
2000
χt N 0
-γ/ν
0.003
-8
10
-12
10
-16
-8
10
10
-2
0
10
2
10
10 1/ν
FM e = 0.5
N0 = 6251
-1
-2
0
10
10
xt
10
FM e = 0.5 0
10
-5
5
0
1/ν
10
-1
-1 10
10
N0 = 1251
-2
10
N0 = 156251
N0 = 6251
-1
1
0
10
N0 = 31251
2
10
10
10
1/ν
x t N0
-2
0 10000
0.06
-5
5
0
(xt - x*)
1
10
10
1/ν N0
-γ/ν
0.02 0 -10
5000
2
10
0.04
χt N 0
-γ/ν
10
χt N 0
χt
-3
10
d 10 (xt - x*) N0
N0 = 31251
0.2
-4
10
0
1 0.8 0.6 0.4 0.2 0 -10
0.6 0.4
10
0.006
xt
1 0.8
κt
0.004
κt
b
-12
0.002
χt
0
xt N0
-2
10
-3
10
-4
10
0
10
-1
10
1
0
10
10
1/ν
0 0
x t N0
-1
0.1
0.2
xt
0.3
0.4
10 -5 10
-4
10
-3
10
-2
xt
10
-1
10
0
10
FIG. 4: Effect of a small random perturbation on renormalisation flows. The box covering was performed with RB, with rB = 1 (a, b) and GCA with ℓB = 3 (c, d). a and c) WS network with hki = 4 and a fraction p = 0.01 of randomly rewired links. b and d) FM network with e ´= 0.5 and a fraction p = 0.05 of added links. The figures display κt = Kt /(Nt − 1) (a, b, c, d, top), ` and χt = N0 hκt 2 i − hκt i2 (a, b, c, d, bottom) as a function of the relative network size xt = Nt /N0 . We see that now RB with rB = 1 yields a crossing of the κt -curves for both the WS and the FM networks. The exponents are now very different from the unperturbed case: we recover ν = 2. The relation γ = ν seems to hold here as well.
