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# Classes of small-world networks

^{*}To whom reprint requests should be addressed. E-mail: ude.ub.yhpub@larama.

^{†}Present address: CEA, Service de Physique de la Matière Condensée, 91680 Bruyeres-le-Chatel, France.

## Abstract

We study the statistical properties of a variety of diverse
real-world networks. We present evidence of the occurrence of three
classes of small-world networks: (*a*) scale-free
networks, characterized by a vertex connectivity distribution that
decays as a power law; (*b*) broad-scale networks,
characterized by a connectivity distribution that has a power law
regime followed by a sharp cutoff; and (*c*) single-scale
networks, characterized by a connectivity distribution with a fast
decaying tail. Moreover, we note for the classes of broad-scale and
single-scale networks that there are constraints limiting the addition
of new links. Our results suggest that the nature of such constraints
may be the controlling factor for the emergence of different classes of
networks.

Disordered networks, such as small-world networks are the focus of recent interest because of their potential as models for the interaction networks of complex systems (1–7). Specifically, neither random networks nor regular lattices seem to be an adequate framework within which to study “real-world” complex systems (8) such as chemical-reaction networks (9), neuronal networks (2), food webs (10–12), social networks (13, 14), scientific-collaboration networks (15), and computer networks (4, 16–19).

Small-world networks (2), which emerge as the result of randomly
replacing a fraction *P* of the links of a *d*
dimensional lattice with new random links, interpolate
between the two limiting cases of a regular lattice
(*P* = 0) and a random graph
(*P* = 1). A small-world network is characterized by the
following properties: (*i*) the local neighborhood is
preserved (as for regular lattices; ref. 2); and (*ii*) the
diameter of the network, quantified by average shortest distance
between two vertices (20), increases logarithmically with the number of
vertices *n* (as for random graphs; ref. 21). The latter
property gives the name small-world to these networks, because it
is possible to connect any two vertices in the network through
just a few links, and the local connectivity would suggest the network
to be of finite dimensionality.

The structure of small-world networks and of real networks has been
probed through the calculation of their diameter as a function of
network size (2). In particular, networks such as (*a*) the
electric power grid for Southern California, (*b*) the network
of movie-actor collaborations, and (*c*) the neuronal network
of the worm *Caenorhabditis elegans* seem to
be small-world networks (2). Further, it was proposed (5) that these
three networks (*a–c*) as well as the world-wide web (4) and
the network of citations of scientific papers (22, 23) are
scale-free—that is, they have a distribution of connectivities that
decays with a power law tail.

Scale-free networks emerge in the context of a growing network in which
new vertices connect preferentially to the more highly connected
vertices in the network (5). Scale-free networks are also small-world
networks, because (*i*) they have clustering
coefficients much larger than random networks (2) and
(*ii*) their diameter increases logarithmically with the
number of vertices *n* (5).

Herein, we address the question of the conditions under which disordered networks are scale-free through the analysis of several networks in social, economic, technological, biological, and physical systems. We identify a number of systems for which there is a single scale for the connectivity of the vertices. For all these networks, there are constraints limiting the addition of new links. Our results suggest that such constraints may be the controlling factor for the emergence of scale-free networks.

## Empirical Results

First, we consider two examples of technological and economic
networks: (*i*) the electric power grid of Southern California
(2), the vertices being generators, transformers, and substations and
the links being high-voltage transmission lines; and (*ii*)
the network of world airports (24), the vertices being the airports and
the links being nonstop connections. For the case of the airport
network, we have access to data on number of passengers in transit and
of cargo leaving or arriving at the airport, instead of data on the
number of distinct connections. Working under some reasonable
assumptions,^{‡} one can expect that the number of
distinct connections from a major airports is proportional to the
number of passengers in transit through that airport, making the two
examples, *i* and *ii*, comparable. Fig.
Fig.11 shows the connectivity distribution for
these two examples. It is visually apparent that neither case has a
power law regime and that both have exponentially decaying tails,
implying that there is a single scale for the connectivity
*k*.

Second, we consider three examples of “social” networks:
(*iii*) the movie-actor network (2), the links in this network
indicating that the two actors were cast at least once in the same
movie; (*iv*) the acquaintance network of Mormons (25), the
vertices being 43 Utah Mormons and the number of links the number of
other Mormons they know; and (*v*) the friendship network of
417 Madison Junior High School students (26). These three examples
describe apparently distinct types of social networks with very
different sample sizes. In fact it can be argued that the network of
movie-actor collaborations is not really a social network but is
instead an economic network. However, because it was considered in
other publications (1, 2, 5) as a social network, we classify it
similarly here. We feel that the acquaintance and friendship networks
may be better proxies of real social networks and, as such, expect
similar results from the analysis of both networks. Fig.
Fig.22 shows the connectivity distribution for
these social networks. The scale-free (power law) behavior of the
movie-actor network (5) is truncated by an exponential tail. In
contrast, the network of acquaintances of the Utah Mormons and the
friendship network of the high school students display no power law
regime, but instead we find results consistent with a Gaussian
distribution of connectivities, indicating the existence of a single
scale for
*k*.^{§}

Third, we consider two examples of networks from the natural sciences:
(*vi*) the neuronal network of the worm *C. elegans*
(2, 27, 28), the vertices being the individual neurons and the links
being connections between neurons; and (*vii*) the
conformation space of a lattice polymer chain (29), the vertices being
the possible conformations of the polymer chain and the links being the
possibility of connecting two conformations through local movements of
the chain (29). The conformation space of a linear polymer chain seems
to be well described (29) by the small-world networks of ref. 2. Fig.
Fig.33 *a* and *b* shows for
*C. elegans* the cumulative distribution of *k* for
both incoming and outgoing neuronal links. The tails of both
distributions are well approximated by exponential decays, consistent
with a single scale for the connectivities. For the network of
conformations of a polymer chain, the connectivity follows a binomial
distribution, which converges to the Gaussian (29); thus, we also find
a single scale for the connectivity of the vertices (Fig.
(Fig.33*c*).

## Discussion

Thus far, we presented empirical evidence for the occurrence of
three structural classes of small-world networks: (*a*)
scale-free networks, characterized by a connectivity distribution with
a tail that decays as a power law (4, 22, 23); (*b*)
broad-scale or truncated scale-free networks, characterized by a
connectivity distribution that has a power law regime followed by a
sharp cutoff, like an exponential or Gaussian decay of the tail (see
example *iii*); and (*c*) single-scale networks,
characterized by a connectivity distribution with a fast decaying tail,
such as exponential or Gaussian (see examples *i*,
*ii*, and *iv–vii*).

A natural question is “what are the reasons for such a rich range of possible structures for small-world networks?” To answer this question, let us recall that preferential attachment in growing networks gives rise to a power law distribution of connectivities (5). However, preferential attachment can be hindered by two classes of factors.

#### Aging of the vertices.

This effect can be pictured for the network of actors; in time, every actor will stop acting. For the network, this fact implies that even a very highly connected vertex will, eventually, stop receiving new links. The vertex is still part of the network and contributes to network statistics, but it no longer receives links. The aging of the vertices thus limits the preferential attachment preventing a scale-free distribution of connectivities.

#### Cost of adding links to the vertices or the limited capacity of a vertex.

This effect is exemplified by the network of world airports: for reasons of efficiency, commercial airlines prefer to have a small number of hubs where all routes connect. In fact, this situation is, to a first approximation, indeed what happens for individual airlines; however, when we consider all airlines together, it becomes physically impossible for an airport to become a hub to all airlines. Because of space and time constraints, each airport will limit the number of landings/departures per hour and the number of passengers in transit. Hence, physical costs of adding links and limited capacity of a vertex (30, 31) will limit the number of possible links attaching to a given vertex.

### Modeling.

To test numerically the effect of aging and cost constraints on the
local structure of networks with preferential attachment, we simulate
the scale-free model of ref. 5 but introduce aging and cost constraints
of varying strength. In the original scale-free model, a network
grows over time by the addition of new vertices and links. A vertex
newly added to the network randomly selects *m* other
vertices to establish new links, with a selection probability that
increases with the number of links of the selected vertex. This
mechanism generates faster growth of the most connected vertices—in a
process identical to the city growth model of Simon and Bonini
(32)—and it is well-known that the mechanism leads to a steady state
with a power law distribution of connectivities (33).

We generalize this model by classifying vertices into one of two
groups: active or inactive. Inactive vertices cannot receive new links.
All new vertices are created active but in time may become inactive. We
consider two types of constraints that are responsible for the
transition from active to inactive. In the first, which we call
“aging,” vertices may become inactive each time step with a
constant probability *P*_{i}. This fact implies that
the time a vertex may remain active decays exponentially. In the
second, which we call “cost,” a vertex becomes inactive when it
reaches a maximum number of links
*k*_{max}. Fig.
Fig.44 shows our results for both types of
constraint. It is clear that both lead to cutoffs on the power law
decay of the tail of connectivity distribution and that, for strong
enough constraints, no power law region is visible.

### Analogy with Critical Phenomena.

We note that the possible distributions of connectivity of the
small-world networks have an analogy in the theory of critical
phenomena (34). At the gas-liquid critical point, the distribution of
sizes of the droplets of the gas (or of the liquid) is scale-free, as
there is no free-energy cost in their formation (34). As for the case
of a scale-free network, the size *s* of a droplet is power
law distributed: *P*(*s*) ≈
*s*^{−}^{α}. As we
move away from the critical point, the appearance of a non-negligible
surface tension introduces a free-energy cost for droplets that limits
their sizes such that their distribution becomes broad-scale:
*P*(*s*) ≈
*s*^{−}^{α}*f*(*s*/ξ),
where ξ is the typical size for which surface tension starts to be
significant, and the function *f*(*s*/ξ)
introduces a sharp cutoff for droplet sizes *s* > ξ.
Far from the critical point, the scale ξ becomes so small that no
power law regime is observed and the droplets become single-scale
distributed: *P*(*s*) ≈
*f*(*s*/ξ). Often, the distribution of sizes in
this regime is exponential or Gaussian.

## Acknowledgments

We thank J. S. Andrade, Jr., R. Cuerno, N. Dokholyan, P. Gopikrishnan, C. Hartley, E. LaNave, K. B. Lauritsen, F. Liljeros, H. Orland, F. Starr, and S. Zapperi for stimulating discussions and helpful suggestions. The Center for Polymer Studies is funded by the National Science Foundation and National Institutes of Health (NCRR P41 RP13622).

## Footnotes

^{‡}To be able to compare the two types of
distributions, one must make two assumptions. The first assumption is
that there is a typical number of passengers per flight. This
assumption is reasonable, because the number of seats in airplanes does
not follow a power law distribution. The second assumption is that
there is a typical number of flights per day between two cities. This
assumption is also reasonable, because at most there will be about 20
flights per day and per airline between any two cities; thus, the
distribution of number of flights per day between two cities is
bounded.

Article published online before print: *Proc. Natl. Acad. Sci. USA*,
10.1073/pnas.200327197.

Article and publication date are at www.pnas.org/cgi/doi/10.1073/pnas.200327197

^{§}Note that even though the sample sizes of these two
networks is rather small, the agreement with the Gaussian distribution
is very good, suggesting that our results are reliable. Moreover, a
power law distribution would curve the opposite way in the semilog
plot.

## References

*J. Stat. Phys.*, in press.

**National Academy of Sciences**

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