Emergence of scaling in random networks

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Emergence of scaling in random networks

arXiv:cond-mat/9910332v1 [cond-mat.dis-nn] 21 Oct 1999 Emergence of Scaling in Random Networks Albert-L´aszl´o Barab´asi∗ and R´eka Albert Department of Physics, University of Notre-Dame, Notre-Dame, IN 46556 Systems as diverse as genetic networks or the world wide w... [收起]
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Emergence of scaling in random networks
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arXiv:cond-mat/9910332v1 [cond-mat.dis-nn] 21 Oct 1999 Emergence of Scaling in Random Networks

Albert-L´aszl´o Barab´asi∗ and R´eka Albert

Department of Physics, University of Notre-Dame, Notre-Dame, IN 46556

Systems as diverse as genetic networks or the world wide web are best described as
networks with complex topology. A common property of many large networks is that the
vertex connectivities follow a scale-free power-law distribution. This feature is found to be
a consequence of the two generic mechanisms that networks expand continuously by the
addition of new vertices, and new vertices attach preferentially to already well connected
sites. A model based on these two ingredients reproduces the observed stationary scale-
free distributions, indicating that the development of large networks is governed by robust
self-organizing phenomena that go beyond the particulars of the individual systems.

∗alb@nd.edu

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The inability of contemporary science to describe systems composed of non-identical
elements that have diverse and nonlocal interactions currently limits advances in many dis-
ciplines, ranging from molecular biology to computer science (1). The difficulty in describing
these systems lies partly in their topology: many of them form rather complex networks,
whose vertices are the elements of the system and edges represent the interactions between
them. For example, living systems form a huge genetic network, whose vertices are proteins
and genes, the edges representing the chemical interactions between them (2). At a different
organizational level, a large network is formed by the nervous system, whose vertices are
the nerve cells, connected by axons (3). But equally complex networks occur in social sci-
ence, where vertices are individuals or organizations, and the edges characterize the social
interaction between them (4), or describe the world wide web (www), whose vertices are
HTML documents connected by links pointing from one page to another (5, 6). Due to their
large size and the complexity of the interactions, the topology of these networks is largely
unknown.

Traditionally, networks of complex topology have been described using the random graph
theory of Erd˝os and R´enyi (ER) (7), but in the absence of data on large networks the
predictions of the ER theory were rarely tested in the real world. However, driven by the
computerization of data acquisition, such topological information is increasingly available,
raising the possibility of understanding the dynamical and topological stability of large
networks.

In this paper we report on the existence of a high degree of self-organization characterizing
the large scale properties of complex networks. Exploring several large databases describing
the topology of large networks that span as diverse fields as the www or the citation patterns
in science we show that, independent of the system and the identity of its constituents, the
probability P (k) that a vertex in the network interacts with k other vertices decays as a
power-law, following P (k) ∼ k−γ. This result indicates that large networks self-organize
into a scale-free state, a feature unexpected by all existing random network models. To
understand the origin of this scale invariance, we show that existing network models fail to

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incorporate growth and preferential attachment, two key features of real networks. Using a
model incorporating these two ingredients, we show that they are responsible for the power-
law scaling observed in real networks. Finally, we argue that these ingredients play an easily
identifiable and important role in the formation of many complex systems, implying that
our results are relevant to a large class of networks observed in Nature.

While there are a large number of systems that form complex networks, detailed topo-
logical data is available only for a few. The collaboration graph of movie actors represents
a well documented example of a social network. Each actor is represented by a vertex, two
actors being connected if they were cast together in the same movie. The probability that
an actor has k links (characterizing his or her popularity) has a power-law tail for large
k, following P (k) ∼ k ,−γactor where γactor = 2.3 ± 0.1 (Fig. 1A). A more complex network
with over 800 million vertices (8) is the www, where a vertex is a document and the edges
are the links pointing from one document to another. The topology of this graph deter-
mines the web’s connectivity and, consequently, our effectiveness in locating information on
the www (5). Information about P (k) can be obtained using robots (6), indicating that
the probability that k documents point to a certain webpage follows a power-law, with
γwww = 2.1 ± 0.1 ( Fig. 1B) (9). A network whose topology reflects the historical patterns
of urban and industrial development is the electrical powergrid of western US, the vertices
representing generators, transformers and substations, the edges corresponding to the high
voltage transmission lines between them (10). Due to the relatively modest size of the net-
work, containing only 4941 vertices, the scaling region is less prominent, but is nevertheless
approximated with a power-law with an exponent γpower ≃ 4 (Fig. 1C). Finally, a rather
large, complex network is formed by the citation patterns of the scientific publications, the
vertices standing for papers published in refereed journals, the edges representing links to
the articles cited in a paper. Recently Redner (11) has shown that the probability that a
paper is cited k times (representing the connectivity of a paper within the network) follows
a power-law with exponent γcite = 3.

The above examples (12) demonstrate that many large random networks share the com-

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mon feature that the distribution of their local connectivity is free of scale, following a

power-law for large k, with an exponent γ between 2.1 and 4 which is unexpected within the

framework of the existing network models. The random graph model of ER (7) assumes that

we start with N vertices, and connect each pair of vertices with probability p. In the model

the probability that a vertex has k edges follows a Poisson distribution P (k) = e−λλk/k!,



where λ = N  N −1  pk(1 − p)N −1−k . In the small world model recently introduced by
 

k

Watts and Strogatz (WS) (10), N vertices form a one-dimensional lattice, each vertex being

connected to its two nearest and next-nearest neighbors. With probability p each edge is

reconnected to a vertex chosen at random. The long-range connections generated by this

process decrease the distance between the vertices, leading to a small-world phenomenon

(13), often referred to as six degrees of separation (14). For p = 0 the probability distri-

bution of the connectivities is P (k) = δ(k − z), where z is the coordination number in the

lattice, while for finite p, P (k) is still peaked around z, but it gets broader (15). A common

feature of the ER and WS models is that the probability of finding a highly connected ver-

tex (that is, a large k) decreases exponentially with k, thus vertices with large connectivity

are practically absent. In contrast, the power-law tail characterizing P (k) for the studied

networks indicates that highly connected (large k) vertices have a large chance of occurring,

dominating the connectivity.

There are two generic aspects of real networks that are not incorporated in these models.

First, both models assume that we start with a fixed number (N) of vertices, that are then

randomly connected (ER model), or reconnected (WS model), without modifying N. In

contrast, most real world networks are open, they form by the continuous addition of new

vertices to the system, thus the number of vertices, N, increases throughout the lifetime

of the network. For example, the actor network grows by the addition of new actors to

the system, the www grows exponentially in time by the addition of new web pages (8),

the research literature constantly grows by the publication of new papers. Consequently, a

common feature of these systems is that the network continuously expands by the addition

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of new vertices that are connected to the vertices already present in the system.
Second, the random network models assume that the probability that two vertices are

connected is random and uniform. In contrast, most real networks exhibit preferential
connectivity. For example, a new actor is cast most likely in a supporting role, with more
established, well known actors. Consequently, the probability that a new actor is cast with
an established one is much higher than casting with other less known actors. Similarly, a
newly created webpage will more likely include links to well known, popular documents with
already high connectivity, or a new manuscript is more likely to cite a well known and thus
much cited paper than its less cited and consequently less known peer. These examples
indicate that the probability with which a new vertex connects to the existing vertices is not
uniform, but there is a higher probability to be linked to a vertex that already has a large
number of connections.

We next show that a model based on these two ingredients naturally leads to the observed
scale invariant distribution. To incorporate the growing character of the network, starting
with a small number (m0) of vertices, at every timestep we add a new vertex with m(≤ m0)
edges that link the new vertex to m different vertices already present in the system. To
incorporate preferential attachment, we assume that the probability Π that a new vertex
will be connected to vertex i depends on the connectivity ki of that vertex, such that
Π(ki) = ki/ j kj. After t timesteps the model leads to a random network with t + m0
vertices and mt edges. This network evolves into a scale-invariant state with the probability
that a vertex has k edges following a power-law with an exponent γmodel = 2.9±0.1 (Fig. 2A).
As the power-law observed for real networks describes systems of rather different sizes at
different stages of their development, it is expected that a correct model should provide a
distribution whose main features are independent of time. Indeed, as Fig. 2A demonstrates,
P (k) is independent of time (and, subsequently, independent of the system size m0 + t),
indicating that despite its continuous growth, the system organizes itself into a scale-free
stationary state.

The development of the power-law scaling in the model indicates that growth and pref-

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erential attachment play an important role in network development. To verify that both
ingredients are necessary, we investigated two variants of the model. Model A keeps the
growing character of the network, but preferential attachment is eliminated by assuming
that a new vertex is connected with equal probability to any vertex in the system (that
is, Π(k) = const = 1/(m0 + t − 1)). Such a model (Fig. 2B) leads to P (k) ∼ exp(−βk),
indicating that the absence of preferential attachment eliminates the scale-free feature of
the distribution. In model B we start with N vertices and no edges. At each time step we
randomly select a vertex and connect it with probability Π(ki) = ki/ j kj to vertex i in the
system. While at early times the model exhibits power-law scaling, P (k) is not stationary:
since N is constant, and the number of edges increases with time, after T ≃ N2 timesteps
the system reaches a state in which all vertices are connected. The failure of models A and
B indicates that both ingredients, namely growth and preferential attachment, are needed
for the development of the stationary power-law distribution observed in Fig. 1.

Due to the preferential attachment, a vertex that acquired more connections than another
one will increase its connectivity at a higher rate, thus an initial difference in the connectivity
between two vertices will increase further as the network grows. The rate at which a vertex
acquires edges is ∂ki/∂t = ki/2t, which gives ki(t) = m(t/ti)0.5, where ti is the time at which
vertex i was added to the system (see Fig. 2C), a scaling property that could be directly tested
once time-resolved data on network connectivity becomes available. Thus older (smaller ti)
vertices increase their connectivity at the expense of the younger (larger ti) ones, leading
with time to some vertices that are highly connected, a ”rich-gets-richer” phenomenon that
can be easily detected in real networks. Furthermore, this property can be used to calculate γ
analytically. The probability that a vertex i has a connectivity smaller than k, P (ki(t) < k),
can be written as P (ti > m2t/k2). Assuming that we add the vertices at equal time intervals
to the system, we obtain that P (ti > m2t/k2) = 1 − P (ti ≤ m2t/k2) = 1 − m2t/k2(t + m0).
The probability density P (k) can be obtained from P (k) = ∂P (ki(t) < k)/∂k, which, at

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long times, leads to the stationary solution

P (k) = 2m2 ,
k3

giving γ = 3, independent of m. While it reproduces the observed scale-free distribution, the
proposed model cannot be expected to account for all aspects of the studied networks. For
this we need to model these systems in more detail. For example, in the model we assumed
linear preferential attachment, that is Π(k) ∼ k. However, while in general Π(k) could have
an arbitrary nonlinear form Π(k) ∼ kα, simulations indicate that scaling is present only
for α = 1. Furthermore, the exponents obtained for the different networks are scattered
between 2.1 and 4. However, it is easy to modify our model to account for exponents
different from γ = 3. For example, if we assume that a fraction p of the links is directed,
we obtain γ(p) = 3 − p, which is supported by numerical simulations (16). Finally, some
networks evolve not only by adding new vertices, but by adding (and sometimes removing)
connections between established vertices. While these and other system-specific features
could modify the exponent γ, our model offers the first successful mechanism accounting for
the scale-invariant nature of real networks.

Growth and preferential attachment are mechanisms common to a number of complex
systems, including business networks (17, 18), social networks (describing individuals or
organizations), transportation networks (19), etc. Consequently, we expect that the scale-
invariant state, observed in all systems for which detailed data has been available to us,
is a generic property of many complex networks, its applicability reaching far beyond the
quoted examples. A better description of these systems would help in understanding other
complex systems as well, for which so far less topological information is available, including
such important examples as genetic or signaling networks in biological systems. We often
do not think of biological systems as open or growing, since their features are genetically
coded. However, possible scale-free features of genetic and signaling networks could reflect
the evolutionary history dominated by growth and aggregation of different constituents,
leading from simple molecules to complex organisms. With the fast advances in mapping

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out genetic networks, answers to these questions might not be too far. Similar mechanisms
could explain the origin of the social and economic disparities governing competitive systems,
since the scale-free inhomogeneities are the inevitable consequence of self-organization due
to the local decisions made by the individual vertices, based on information that is biased
towards the more visible (richer) vertices, irrespective of the nature and the origin of this
visibility.

References and Notes
1. R. Gallagher and T. Appenzeller, Science 284, 79 (1999); R. F. Service, Science 284,
80 (1999).
2. G. Weng, U. S. Bhalla, R. Iyengar, Science 284, 92 (1999).
3. C. Koch and G. Laurent, Science 284, 96 (1999).
4. S. Wasserman and K. Faust, Social Network Analysis, (Cambridge University Press,
Cambridge, 1994).
5. Members of the Clever project, Sci. Am 280, 54 (June 1999).
6. R. Albert, H. Jeong and A.-L. Barab´asi, Nature 401, 130 (1999), see also
http://www.nd.edu∼networks.
7. P. Erd˝os, and A. R´enyi, Publ. Math. Inst. Hung. Acad. Sci 5, 17 (1960); B. Bollob´as,
Random Graphs (Academic Press, London, 1985).
8. S. Lawrence and C. L. Giles, Science 280, 98 (1998); Nature 400, 107 (1999).
9. Note that in addition to the distribution of incoming links, the www displays a number
of other scale-free features, characterizing the organization of the webpages within a domain
[B. A. Huberman and L. A. Adamic, Nature 401, 131 (1999)], the distribution of searches
[B. A. Huberman, P. L. T. Pirolli, J. E. Pitkow and R. J. Lukose, Science 280, 95 (1998)],
or the number of links per webpage (6).
10. D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).
11. S. Redner, European Physical Journal B 4, 131 (1998).
12. We also studied the neural network of the worm Caenorhabditis elegans (3, 10) and
the benchmark diagram of a computer chip

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(http://vlsicad.cs.ucla.edu/∼cheese/ispd98.html). We find that P (k) for both is consistent
with power-law tails, despite the fact that for C. elegans the relatively small size of the
system (306 vertices) limits severely the data quality, while for the wiring diagram of the
chips vertices with over 200 edges have been eliminated from the database.

13. S. Milgram, Psychol. Today 2, 60 (1967); M. Kochen (ed.) The Small World (Ablex,
Norwood, NJ, 1989).

14. J. Guare, Six Degrees of Separation: A play (Vintage Books, New York, 1990).
15. M. Barth´el´emy and L. A. N. Amaral, Phys. Rev. Lett. 82, 15 (1999).
16. Note that for most networks the connectivity m of the newly added vertices is not
constant. However, choosing m randomly will not change the exponent γ [Y. Tu, private
communication].
17. W. B. Arthur, Science 284, 107 (1999).
18. Note that preferential attachment was also used to model correlations between stock
prices [L. A. N. Amaral and M. Barth´el´emy, private communication].
19. J. R. Banavar, A. Maritan and A. Rinaldo, Nature 399, 130 (1999).
20. We thank D. J. Watts for providing the C. elegans and the power grid data, B. C.
Tjaden for supplying the actor data, H. Jeong for collecting the data on the www and L. A.
N. Amaral for helpful discussions. This work was partially supported by NSF Career Award
DMR-9710998.

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FIGURES

10-1 100 100 C

AB 101

10-2
10-2 10-1

P(k) 10-3

10-4 10-4 10-2

10-5 10-6 10-3

10-6 100 101 102 10-8 10-4
103 100 101 102 103 104 100

k

FIG. 1. The distribution function of connectivities for various large networks. (A) Actor col-
laboration graph with N = 212, 250 vertices and average connectivity k = 28.78; (B) World wide
web, N = 325, 729, k = 5.46 (6); (C) Powergrid data, N = 4, 941, k = 2.67. The dashed lines
have slopes (A) γactor = 2.3, (B) γwww = 2.1 and (C) γpower = 4.

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A B C100 100 103

10-2
10-2 102

10-4

10-4 101
10-6
P(k)
P(k)
ki(t)

10-8 10-6 100
100 101 102 103 0 20 40 60 80 100 101 102 103 104 105

k kt

FIG. 2. (A) The power-law connectivity distribution at t = 150, 000 (o) and t = 200, 000 (2) as
obtained from the model (see text), using m0 = m = 5. The slope of the dashed line is γ = 2.9. (B)
The exponential connectivity distribution for model A, in the case of m0 = m = 1 (o), m0 = m = 3
(2), m0 = m = 5 (3) and m0 = m = 7 (△). (C) Time evolution of the connectivity for two
vertices added to the system at t1 = 5 and t2 = 95. The dashed line has slope 0.5.

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