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Classes of smallworld networks

Communicated by Herman Z. Cummins, City College of the City University of New York, New York, NY (received for review April 20, 2000)
Abstract
We study the statistical properties of a variety of diverse realworld networks. We present evidence of the occurrence of three classes of smallworld networks: (a) scalefree networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broadscale networks, characterized by a connectivity distribution that has a power law regime followed by a sharp cutoff; and (c) singlescale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broadscale and singlescale 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 smallworld 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 “realworld” complex systems (8) such as chemicalreaction networks (9), neuronal networks (2), food webs (10–12), social networks (13, 14), scientificcollaboration networks (15), and computer networks (4, 16–19).
Smallworld 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 smallworld 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 smallworld 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 smallworld 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 movieactor collaborations, and (c) the neuronal network of the worm Caenorhabditis elegans seem to be smallworld networks (2). Further, it was proposed (5) that these three networks (a–c) as well as the worldwide web (4) and the network of citations of scientific papers (22, 23) are scalefree—that is, they have a distribution of connectivities that decays with a power law tail.
Scalefree 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). Scalefree networks are also smallworld 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 scalefree 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 scalefree 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 highvoltage 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. 1 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 movieactor 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 movieactor 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. 2 shows the connectivity distribution for these social networks. The scalefree (power law) behavior of the movieactor 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 smallworld networks of ref. 2. Fig. 3 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. 3c).
Discussion
Thus far, we presented empirical evidence for the occurrence of three structural classes of smallworld networks: (a) scalefree networks, characterized by a connectivity distribution with a tail that decays as a power law (4, 22, 23); (b) broadscale or truncated scalefree 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) singlescale 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 smallworld 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 scalefree 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 scalefree model of ref. 5 but introduce aging and cost constraints of varying strength. In the original scalefree 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 wellknown 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. 4 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 smallworld networks have an analogy in the theory of critical phenomena (34). At the gasliquid critical point, the distribution of sizes of the droplets of the gas (or of the liquid) is scalefree, as there is no freeenergy cost in their formation (34). As for the case of a scalefree 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 nonnegligible surface tension introduces a freeenergy cost for droplets that limits their sizes such that their distribution becomes broadscale: 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 singlescale 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 whom reprint requests should be addressed. Email: amaral{at}buphy.bu.edu.

↵† Present address: CEA, Service de Physique de la Matière Condensée, 91680 BruyeresleChatel, France.

↵‡ 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.
 Received April 20, 2000.
 Accepted July 13, 2000.
 Copyright © 2000, The National Academy of Sciences
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