VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

12

Emergence of a Small World from Local Interactions: Modeling Acquaintance Networks

Jörn Davidsen, Holger Ebel, and Stefan Bornholdt*Institut für Theoretische Physik, Universität Kiel, Leibnizstrasse 15, D-24098 Kiel, Germany

(Received 20 August 2001; revised manuscript received 6 December 2001; published 8 March 2002)

How do we make acquaintances? A simple observation from everyday experience is that often oneof our acquaintances introduces us to one of his or her acquaintances. Such a simple triangle interactionmay be viewed as the basis of the evolution of many social networks. Here, it is demonstrated that thisassumption is sufficient to reproduce major nontrivial features of social networks: short path length, highclustering, and scale-free or exponential link distributions.

DOI: 10.1103/PhysRevLett.88.128701 PACS numbers: 89.75.Hc, 87.23.Ge, 89.75.Da

A remarkable feature of many complex systems is theoccurrence of large and stable network structures as, forexample, networks on the protein or gene level, ecologi-cal webs, communication networks, and social networks[1–3]. Simple models based on disordered networks arequite successful in describing basic properties of such sys-tems. When addressing topological properties, however,neither random networks nor regular lattices provide anadequate framework. A helpful concept along this line isthe idea of “small-world networks” introduced by Wattsand Strogatz [4,5] which initiated an avalanche of scien-tific activity in this field [6–11]. Small-world networksinterpolate between the two limiting cases of regular lat-tices with high local clustering and of random graphs withshort distances between nodes. High clustering impliesthat, if node A is linked to node B, and B is linked tonode C, there is an increased probability that A will alsobe linked to C. Another useful measure is the distancebetween two nodes, defined as the number of edges alongthe shortest path connecting them. A network is calleda “small-world network” if it exhibits the following twocharacteristic properties [4,12]: (i) high clustering and(ii) a small average shortest path between two randomnodes (sometimes called the diameter of the network), scal-ing logarithmically with the number of nodes. Thus, tworandomly chosen nodes in the network are likely to be con-nected through only a small number of links. The mostpopular manifestation of a small world is known as “sixdegrees of separation,” a postulate by the social psycholo-gist Stanley Milgram [13] stating that most pairs of peoplein the United States can be connected through a path ofonly about six acquaintances [14].

Let us here focus on social networks, and acquaintancenetworks in particular, which are typical examples for thesmall-world property [1,2,12]. What does the concept ofsmall-world networks tell us about such real world sys-tems? In its original definition [4,5] it serves as an eleganttoy model demonstrating the consequences of high cluster-ing and short path length. However, since these networksare derived from regular graphs, their applicability to realworld systems is very limited. In particular, how a net-work in a natural system dynamically forms a small-world

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topology, often starting from a completely random struc-ture, remains unexplained. The main goal of this Letter isto provide one possible answer to this problem.

A similar problem of dynamical origin is faced (andmuch progress has been made) in a different, but notcompletely unrelated field: the dynamics of scale-freenetworks. Scale-free properties are commonly studiedin diverse contexts from the stability of the internet [15]to the spreading of epidemics [16] and are observed insome social networks [1–3,17,18]. The origin of scale-freeproperties is well understood in terms of interactions thatgenerate this topology dynamically, e.g., on the basis ofnetwork growth and preferential linking [2,3,19]. Whilethese models generate scale-free structures they do not, ingeneral, lead to clustering and are therefore of limited usewhen modeling social networks.

In this Letter, an attempt is made to unify ideas fromthe two fields of “small-world networks” and “scale-freenetworks” in order to address the dynamics of social net-works and the dynamical emergence of the small-worldstructure. In particular, a simple dynamical model for theevolution of acquaintance networks is studied. It generateshighly clustered networks with small average path lengthsthat scale logarithmically with network size. Furthermore,for small death-and-birth rates of nodes this model con-verges towards scale-free degree distributions, in additionto its small-world behavior. Basic ingredients are a localconnection rule based on “transitive linking,” and a finiteage of nodes.

To be specific, let us formulate a model of an acquain-tance network with a fixed number N of nodes (as per-sons) and undirected links between those pairs of nodesthat represent people who know each other. Acquaintancenetworks evolve, with new acquaintances forming betweenindividuals, and old ones dying. Let us assume that peopleare introduced to each other by a common acquaintanceand that the network is formed only by people who arestill alive. The dynamics is defined as follows:

(i) One randomly chosen person picks two random ac-quaintances of his and introduces them to each another.If they have not met before, a new link between themis formed. In case the person chosen has less than two

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VOLUME 88, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 25 MARCH 2002

acquaintances, he introduces himself to one other randomperson.

(ii) With probability p, one randomly chosen person isremoved from the network, including all links connected tothis node, and replaced by a new person with one randomlychosen acquaintance.

These steps are then iterated. Note that the number ofnodes remains constant, neglecting fluctuations in the num-ber of individuals in acquaintance networks. The finite ageimplies that the network reaches a stationary state which isan approximation of the behavior of many social networks,and is in contrast to most models based on network growth[2,18,20]. The probability p determines the separation ofthe two time scales in the model. In general, the rate atwhich people make social contacts can be as short as min-utes or hours, while the time scale on which people join orleave the network may be as long as years or decades. Inthe following, we therefore focus on the regime p ø 1.

Once the network reaches a statistically stationary state,one of its characteristic quantities is the degree distributionP�k� of the network. In Fig. 1, the degree distribution isshown for different values of p. Because of the limitedlifetime of persons in the network, the observed numbersof acquaintances of different persons do not grow forever,but rather fall into some finite range. This can be seenin the degree distribution where the cutoff at high k re-sults from the finite age of nodes. In the regime p ø 1,the degree distribution is dominated by the transitive link-ing process (i), resulting in a power-law range which in-creases in size with decreasing p. For larger values of p,the Poissonian death process (ii) competes with the transi-tive linking process (i), resulting in a stretched exponentialrange in the degree distribution until, in the case p � 1,the random linking of (ii) dominates with its Poissonian

100

101

102

103

k

10-6

10-5

10-4

10-3

10-2

10-1

100

P(k)

p = 0.0025

p = 0.01

p = 0.04

.

FIG. 1. The degree distribution P�k� of the model in the sta-tistically stationary state. The distribution exhibits a power-lawregime for small p, with an exponent of 1.35 for p � 0.0025.Note that the distribution is largely insensitive to system size N ,which here is N � 7000. The exponential cutoff is a result ofthe finite age of nodes.

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dynamics. Therefore, the above model generates degreedistributions spanning scale-free and exponential regimes,both of which are observed in the statistics of social net-works. For large enough network size N , the specific dis-tribution P�k� only depends on the single free parameterof the model p. From experimental data, one can esti-mate p as well, which is typically very small p ø 1 suchthat the two time scales of the network dynamics are wellseparated.

As noted above, small-world networks are characterizedby a high degree of clustering C and a small average short-est path length � which scales logarithmically with thenumber of nodes. The degree of clustering is measuredby the clustering coefficient defined as follows: For a dis-tinct node i, the clustering coefficient Ci is given by theratio of existing links Ei between its ki neighbors to thepossible number of such connections 1

2ki�ki 2 1�. Thenthe clustering coefficient C of the network is defined asthe average over all nodes,

C � �Ci�i �

ø2Ei

ki�ki 2 1�

¿i. (1)

For the above model, C can be related to the mean de-gree �k� and p because in the stationary state the numberof added links on average equals the number of removedlinks:

1 2 C � p��k� 2 1� . (2)

In Table I, the clustering coefficient of the above model inthe stationary state—calculated according to Eq. (1)— isshown for different values of p. Within numerical preci-sion, the same values are obtained via Eq. (2) as one caneasily deduce from Table I. In comparison to the clus-tering of a random network with the same size and samemean degree Crand, the model coefficient C is consistentlyof a much larger size. The clustering coefficient of a ran-dom network, i.e., a network with constant probability oflinking each pair of nodes plink � �k���N 2 1� and, there-fore, a Poissonian distribution of the node degree, is justthis probability Crand � plink. Obviously, Crand is propor-tional to the mean degree �k� for constant network size.For further comparison, let us derive an estimate C0 foran upper bound of the average clustering coefficient of anetwork with the same degree distribution, but randomly

TABLE I. Clustering coefficient for different values of p anda network size of N � 7000. C 0 is an upper bound for theaverage clustering coefficient of a network with the same degreedistribution without transitive linking. Crand is the clusteringcoefficient of a random network of the same size and with thesame average degree �k�.

p �k� �k2� C C0 Crand

0.04 14.9 912 0.45 0.036 0.00210.01 49.1 13 744 0.52 0.29 0.00700.0025 149.2 99 436 0.63 0.43 0.021

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assigned links. Thus, C0 provides an upper bound on theaverage clustering which one would expect solely from thedegree distribution while neglecting transitive linking ofthe model. Using the generating function approach forgraphs with arbitrary degree distributions [21] and assum-ing that fluctuations of the average degree of the neighbor-hood of a node can be neglected, an upper bound C0 canbe derived in terms of the first two moments:

C0 �1

�k�N

µ�k2��k�

2 1

∂2

. (3)

This result holds exactly in the case of the Poissonian de-gree distribution of a random network �Crand � C0�. AsTable I shows, the network generated by the model ex-hibits an even higher average clustering coefficient thana network with links distributed randomly according tothe same degree distribution. In particular, the network ismuch more clustered than a random network with a Poisso-nian distribution as required for the small-world property.Furthermore, the clustering is not as strongly dependenton the mean degree �k� as C0 of a network with the samedegree distribution but no transitive linking. This observa-tion is directly explained by Eq. (2).

The scaling of the average path length with system sizeis shown in Fig. 2. The data are consistent with a loga-rithmic behavior and, thus, our model meets the secondrequirement for a small-world behavior as well. Also, thisis what one expects in the framework of a random graphwith arbitrary degree distribution [21]. Moreover, we cancompare the average path length of our model � with thepath length �0 of a network with the same degree distribu-tion and randomly assigned links. Applying the generatingfunction approach, we obtain

n2 � �k2� 2 �k� , (4)

FIG. 2. The average path length ��N� as a function of systemsize N on a semilogarithmic scale �p � 0.04�. The data are ingood agreement with a logarithmic fit (straight line).

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i.e., the mean number of second neighbors n2 of a nodeis the difference between the second and the first momentof the degree distribution. Equation (4), in combinationwith the results in [21], leads to the following estimate ofthe average shortest path length:

�0 �log� N

�k� �

log� �k2�2�k��k� �

1 1 . (5)

For the Poissonian distribution of a random network, oneobtains

�rand �logN

log�k�. (6)

Note that only N , �k�, and �k2� are used to estimate � asalso used in the derivation of (3). With the help of Table I,one finds that �0 � 1.59 and �rand � 1.77 for p � 0.0025.Numerical simulations of our model similarly yield a veryshort path length of � � 2.38 which is further evidencethat the simple linking rule of our model leads to small-world behavior. Intuitively, �0 , �rand follows from thehighly connected “hubs” present in the scale-free networksfor p � 0.0025. The fact that � is slightly larger than �rand

is due to the fact that, in step (i) of the model, many linksare used to build highly clustered neighborhoods. Thisprice to pay for clustering, however, only slightly affectsthe small overall mean path length.

One example for an observed small-world effect is thenetwork of coauthorships between physicists in high en-ergy physics [17]. Nodes are researchers who are con-nected if they have coauthored a paper. In a recent studyof the publications in the SPIRES database over the fiveyear period 1995–1999, a graph was reconstructed fromthe data and analyzed [17,22]. The resulting network con-sists of 55 627 nodes with a mean degree �kS� � 173, amean shortest path length �S � 4.0, and a very high clus-tering coefficient CS � 0.726. The degree distribution isconsistent with a power law of exponent 21.2 [17]. Fromthese data, one can derive C0 and �0 for a network with thesame degree distribution but random links to C0 � 0.19and �0 � 1.81. These numbers show that the real networkexhibits clustering which is very much higher than wouldbe expected from the degree distribution alone. Also, thepath length is short but still larger than for a randomlylinked network of the same degree distribution. Using thelogarithmic scaling for �, the data of the example agreewith the values of the above model. The basic assumptionsmade in the model are met by the data set, as the numberof researchers in the sample is, to a good approximation,stationary, and as the small rate of researchers entering orleaving the system during the time frame of the samplejustifies the regime of small p ø 1.

The previous example demonstrates how our model canbe applied to a social network in the dynamically station-ary state. Moreover, the model studied here is also able to

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accommodate other small-world scenarios, e.g., without ascale-free degree distribution [12] as the particular shapeof the distribution varies, depending on the turnover ratep. The question of the origin of small-world behavior insocial systems has led to other approaches as well. In aninteresting model Mathias and Gopal [23] showed that asmall-world topology can arise from the combined opti-mization of network distance and physical distance. Ap-plications of this principle, however, are more likely to befound in the field of transportation networks rather thanacquaintance networks considered here. A more similarapproach to the study presented here has been taken by Jinet al. [24], who study a model which shares a mechanismsimilar to transitive linking as defined here. Otherwise,however, it is more complicated than we feel it needs tobe, at least for some classes of social networks. Also itsupper limit on the degree of a node, motivated by one spe-cific trait of some acquaintance networks, makes the modelless suitable to meet experimental data of social networkswhich exhibit broad degree distributions.

Even though our work has focused on social networks,possible applications also include biological systems. Forexample, the neural network of the fresh-water polyp hy-dra self-organizes by the continuous addition and removalof neurons maintaining a stationary size [25,26]. It isconceivable that some form of transitive linking may oc-cur when new neurons form connections into an exist-ing dynamical neural network, e.g., when linking betweentwo neurons is guided by the mutual correlation of theiractivities.

In conclusion, a simple dynamical model for the emer-gence of small-world network structures has been stud-ied. It is based on a local linking rule which connectsnodes who share a common neighbor, as well as on a slowturnover of nodes and links in the system. The networkapproaches a dynamically stationary state with high clus-tering and small average path lengths which scale loga-rithmically with system size. Depending on a single freeparameter, the turnover rate of nodes, this model inter-polates between networks with a scale-free and an expo-nential degree distribution, both of which are observed inexperimental data of social networks.

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*Email address: bornholdt@theo-physik.uni-kiel.de[1] S. H. Strogatz, Nature (London) 410, 268 (2001), and ref-

erences therein.[2] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 47, 74

(2002).[3] S. N. Dorogovtsev and J. F. F. Mendes, cond-mat/0106144.[4] D. J. Watts and S. H. Strogatz, Nature (London) 393, 440

(1998).[5] D. J. Watts, Small worlds: The Dynamics of Networks be-

tween Order and Randomness (Princeton University Press,Princeton, NJ, 1998).

[6] M. E. J. Newman, J. Stat. Phys. 101, 819 (2000).[7] M. Barthélémy and L. A. N. Amaral, Phys. Rev. Lett. 82,

3180 (1999).[8] M. E. J. Newman and D. J. Watts, Phys. Rev. E 60, 7332

(1999).[9] R. Monasson, Eur. Phys. J. B 12, 555 (1999).

[10] A. Barrat and M. Weigt, Eur. Phys. J. B 13, 547 (2000).[11] M. E. J. Newman, C. Moore, and D. J. Watts, Phys. Rev.

Lett. 84, 3201 (2000).[12] L. A. N. Amaral, A. Scala, M. Barthélémy, and H. E. Stan-

ley, Proc. Natl. Acad. Sci. U.S.A. 97, 11 149 (2000), andreferences therein.

[13] S. Milgram, Psych. Today 2, 60 (1967).[14] The Small World, edited by M. Kochen (Ablex, Norwood,

1989).[15] R. Albert, H. Jeong, and L.-A. Barabási, Nature (London)

406, 378 (2000).[16] R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86,

3200 (2001).[17] M. E. J. Newman, Phys. Rev. E 64, 016131 (2001).[18] A.-L. Barabási, H. Jeong, Z. Néda, E. Ravasz, A. Schubert,

and T. Vicsek, cond-mat/0104162.[19] S. Bornholdt and H. Ebel, Phys. Rev. E 64, 035104(R)

(2001).[20] K. Klemm and V. Eguíluz, cond-mat/0107607.[21] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys.

Rev. E 64, 026118 (2001).[22] M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).[23] N. Mathias and V. Gopal, Phys. Rev. E 63, 021117 (2001).[24] E. M. Jin, M. Girvan, and M. E. J. Newman, Phys. Rev. E

64, 046132 (2001).[25] U. Technau et al., Proc. Natl. Acad. Sci. U.S.A. 97, 12 127

(2000).[26] G. Hager and C. N. David, Development 124, 569 (1997).

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