PHYSICAL REVIEW E, VOLUME 64, 047103

Complex structures in generalized small worlds

Marcelo Kuperman1,* and Guillermo Abramson2,†

1Centro Atomico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina2Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Centro Ato´mico Bariloche and Instituto Balseiro,

8400 San Carlos de Bariloche, Argentina~Received 23 February 2001; published 20 September 2001!

We propose a generalization of small world networks, in which the reconnection of links is governed by afunction that depends on the distance between the elements to be linked. An adequate choice of this functionlets us control the clusterization of the system. Control of the clusterization, in turn, allows the generation ofa wide variety of topologies.

DOI: 10.1103/PhysRevE.64.047103 PACS number~s!: 89.75.2k, 87.23.Ge, 05.10.2a

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I. INTRODUCTION

The concept of ‘‘small world’’ was introduced by Milgram @1# in order to describe the topological propertiessocial communities and relationships. Recently, a modethese has been introduced using what was called small w~SW! networks@2#. In the original model of SW networkssingle parameterp, running from 0 to 1, characterizes thdegree of disorder of the network, ranging from a regulattice to a completely random graph@3#. The construction ofthese networks starts from a regular, one-dimensional, podic lattice of N elements and coordination number 2K.Then each of the sites is visited, rewiringK of its links withprobabilityp. Values ofp within the interval@0,1# produce acontinuous spectrum of small world networks. Note thatp isthe fraction of modified regular links. To characterize ttopological properties of the small world networks two manitudes are calculated@2#. The first one,L(p), measures themean distance between any pair of elements in the netwthat is, the shortest path between two vertices, averagedall pairs of vertices. Thus, an ordered lattice hasL(0);N/K, while, for a random network,L(1); ln(N)/ln(K).The second one,C(p), measures the mean clustering ofelement’s neighborhood.C(p) is defined in the followingway: Let us consider the elementi, havingki neighbors con-nected to it. We denote byci(p) the number of neighbors oelementi that are neighbors among themselves, normalito the value that this would have if all of them were conected to one another; namely,ki(ki21)/2. NowC(p) is theaverage, over the system, of the local clusterizationci(p).Ordered lattices are highly clustered, withC(0);3/4, andrandom lattices are characterized byC(1);K/N. Betweenthese extremes small worlds are characterized by a slength between elements, like random networks, and hclusterization, like ordered ones.

Other procedures for developing social networks habeen proposed@4,5#. There, an evolving network is considered. The network grows by the addition of nodes and linEach new link is added according to thesociabilityor popu-larity of the individual it connects to. That is, in the evol

*Electronic address: kuperman@cab.cnea.gov.ar†Electronic address: abramson@cab.cnea.gov.ar

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tionary growth of the network, those nodes with a highnumber of links are granted new links with the highest proability. This process produces topologically different neworks characterized by the site connectivity distribution (d).In @4# three different topologies are shown: scale free nworks whered decays as a power law, broad scale networwith d behaving as a power law with a cutoff, and singscale networks, withd having a fast decaying tail. On thother hand, in@5#, in addition to the scale free networks, thcase withd following an exponential decay is presented.

In the present work, we propose that the reconnectionlink be governed by a distributionQ depending on the distance between nodes instead of their ‘‘popularity.’’ Bchanging the probability distributionQ we can obtain topo-logically different SW networks, displaying features suchfor example, the preservation of high clusterization withcreasing disorder. The clusterization is a statistical or mroscopic magnitude. By observing the microscopic properof the networks a deeper knowledge of what happens isquired.

II. CONSTRUCTION PROCEDURE

As mentioned in the Introduction, we perform a generazation of the original SW network construction@3#. TheSW’s we study are random networks built upon a topologiring with N vertices and coordination number 2K. Each linkconnecting a vertex to a neighbor in the clockwise sensthen rewired at random, with probabilityp, to another vertexof the system, chosen with some criterion. With probabil(12p) the original link is preserved. Self-connections amultiple connections are prohibited. With this procedure,have a regular lattice atp50, and progressively randomgraphs forp.0. The long range links that appear at anyp.0 trigger the small world phenomenon. Atp51 all thelinks have been rewired, and the result is similar to~thoughnot exactly the same as! a completely random network. Thialgorithm should be used with caution, since it can produdisconnected graphs. We have used only connected oneour analysis. Since links are neither destroyed nor creathe resulting network has an average coordination num2K equal to the initial one.

We propose to chose the end point of the reconneclinks according to a probability distributionQ(q) depending

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on the topological distanceq between the two involvednodes in the regular network~before any reconnection takeplace!. The case with uniform probability is thus equivaleto the Strogatz-Watts model. Other distributions may faconnections with nearby or with faraway nodes. The valof q run from 1 to the maximum distance allowed, nameN/2K. When the nodei is to be rewired~according to theprobabilityp) we choose at random one among all the noat a certain distanceq @taken with probabilityQ(q)#. Forexample, as we show below, by favoring reconnection wnearby nodes, we can extend the interval ofp values wherethe small world behavior is found, preserving the regulattice high clusterizationC(p,Q) asL(p,Q) decreases. Butin addition to the macroscopic behavior of SW networksgenerated, it is of interest to explore their intrinsic structuThe results are discussed in the following section.

III. NUMERICAL RESULTS AND DISCUSSION

As an example of a generalization in the spirit of tprevious section, we introduce only one family of SW nworks takingQ}q2m, with 0,m,1. The casem50 givesa uniform Q(q) and is equivalent to already known caseFor m>1 a nonintegrable divergence atq50 limits the pos-sibility of having a normalized probability distribution. Leus start with a study of the average clusterizationC(p,m). InFig. 1, we show two groups of curves.C(p,m) as a functionof the rewiring parameterp for threem values is shown withlines. C(p,m) as a function ofm for three values ofp isshown with lines and symbols. The system under studyN5105 elements andK510. We can see that asm increasesthe region of high clusterization moves toward higher valuof p. This means that the region in the 0<p<1 intervalwhere SW networks can be found is enlarged. The are orelevant aspect derived from this fact that will be analyzlater. This effect is much more evident in the second groof curves,C(p,m) vs m. Moving along one of the curveswhich means fixing a value ofp, for example, p50.5

FIG. 1. Average clusterizationC is displayed as a function othe rewiring probabilityp, and of the exponentm, in the same plot.Data are shown for three values ofp ~lines and symbols!, and forthree values ofm ~lines!, as indicated in the legend.N5105,K510.

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~circles!, we can go from low clusterization atm'0 ~uni-form distributionQ) to high clusterization asm grows. Allthe curves converge to the maximum clusterization whenmapproaches 1.

The other aspect of interest to be discussed is relatethe internal structure of the resulting networks. In recentpers@4,5#, it has been reported that different kinds of behaior can be found in the accumulated connectivity distributiof evolving networks. In the present model, as in Strogand Watts’, the connectivity presents a well defined mevalue, and an exponential decay at high values. This difence is a consequence of the construction method. Atsame time, there is an interesting correlation between loclusterization and connectivity values that is worth analing. In Fig. 2 we show a plot of the connectivity vs thclusterization of each element in the system. It can beserved that for each value of the connectivity there arements with widely different clusterization. This shows thacharacterization of this kind of network in terms of the conectivity alone is incomplete. It is remarkable that, at lovalues ofp, for each value of the connectivity, all~or almostall! the allowed values ofC are present in the system. Icontrast, at high values ofp, an emerging structure can bobserved. The elements fall into classes of clusterizatEach class is characterized by a scale free relationshiptween connectivity and clusterization. This is apparentFig. 2~b!, where each class is represented by the dots falon straight lines in the log-log plot. In both plots, the occpied region is upper bounded by an envelope with powerdecay. That is, highly connected elements are restrictedrange of low clusterizations, while low connected eleme

FIG. 2. Connectivity vs clusterization for each element in tsystem.~a! m50.1, p50.2. ~b! m50.1, p50.9. In both plots,N5105, K510. Except points withc50 ~not shown for reasons oscale!, all points are plotted.

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are responsible for the preservation of the high clusterizaof the network. In qualitative terms we see that the netwis partitioned into small subnetworks composed mainlyelements of low connectivity, while the long distance conection between each subunit is accomplished by highly cnected~popular but otherwise low clustered! elements.

When analyzing the internal structures of the netwoobtained we can see that several kinds of small world stture can arise. An example for small systems (N530, K52) and two kinds of structure is shown in Fig. 3. Bocorrespond to the same value of the rewiring parametep50.5, meaning that approximately one-half of the links habeen reconnected. In Fig. 3~b! we can see an organization othe links in two levels: some of the rewired links haformed local subnetworks of high clusterization, while somothers form connections between these structures. This isto the fact that the distributionQ5q20.9 favors links to nearvertices. In Fig. 3~a!, instead,Q is uniform, there is no pref-erence in the reconnection process, and no local structare formed. Case~b! could, at first sight, be confused with am50 network with a value ofp significantly lower than thecurrent 0.5. However, in that case, many fewer links wohave been rewired. The average clusterization wouldaround 0.5 instead of 0.4. In a system with significangreaterN and K, one would expect a hierarchical organiztion in the connectivity of subnetworks. This organizati

FIG. 3. Two small worlds with 30 elements andK52. Theyhave the same degree of rewiring but differing distributionQ. ~a!p50.5, m50, C50.11. ~b! p50.5, m50.9, C50.41.

FIG. 4. Distribution of the local clusterization. Values ofm andp as indicated in the legends. In all plots,N5105, K510. Exceptpoints with c50 ~not shown for reasons of scale!, all points areplotted.

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can be controlled by the selection of the distributionQ. Ex-amples of each case can be considered. In an evolvingwork with strong attracting or leader nodes, a structure wdevelop showing centralization around these absorbnodes without displaying subnetwork organization. Theternet structure is an example. The existence of providersclients determines nucleation of clients around the providThe same happens with the architecture of tension lines.the contrary, when analyzing the social links of small poplations we find the existence of subnetworks constitutedmembers of the same family. An extreme example is whacalled the connected caveman world, in which people belto almost closed small social environments with only exctional connections among them. Both examples can be meled by takingQ5q2m with m'0 in the first case andm'1 in the second. The central idea is that by choosingappropriateQ it is possible to mimic the particular innestructure of the network of interest.

Let us now turn to the distribution of the clusterizationthe system. In Fig. 4 we show histograms of the values of

FIG. 5. Accumulated probability of finding an element wiclusterizationC or greater. Data are shown form50.1 and threevalues ofp, as indicated in the legend.N5105 andK510.

FIG. 6. Accumulated probability of finding an element wiclusterizationC or greater. Data are shown form50.99 and threevalues ofp, as indicated in the legend.N5105 andK510.

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individual clusterization in the system for four selected vues ofm andp. Each value of the local clusterization presein the system~a discrete magnitude! is represented by a doat a height proportional to its frequency. We can observeeffect of the parameters on the internal organization ofclusterization of the network. The effect of the rewiringp isdifferent at different values ofm. It is interesting to observethat each distribution is composed of a superpositionmany branches. Each one of these comprises a numbelements of close clusterization.

We calculate now the cumulative probability distributioof clusterization Q(C)5(C

1 P(C8), which represents theprobability that an element has clusterizationC or greater. InFigs. 5 and 6 we show the accumulated clusterizationQ(C)for three values of the rewiring parameterp. Figure 5 corre-sponds to a slightly nonuniform distributionQ5q20.1 whileFig. 6 corresponds to a highly nonuniform distributionQ5q20.99. First, observe~in either of the figures! that differentvalues ofp produce different behaviors in the accumulatdistribution. In Fig. 5 we havem50.1. This slight departurefrom uniformity ~Strogatz and Watts’ model! produces a

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slower decay ofQ for growingp, appreciable as a change othe curvature in the logarithmic plots. The fact thatQ5q20.99 generates networks with many highly clustered ements~independently ofp) can be seen in Fig. 6 as a plateathat reaches high values ofC. Even if the three distributionsin Fig. 6 decay in similar ways, they arise from distributioP(C) as different as those shown in Fig. 4~bottom row!.Note also in Figs. 5 and 6 that the effect of aQ that favorshigh clusterization can be seen even at such low levelsdisorder asp50.1. This can be appreciated by comparing tcurves corresponding top50.1 in Figs. 5 and 6~full lines!,where the plateau inQ(C) shows that almost all the valueof clusterization are above 0.4 in the first case and abovein the second.

We have usedQ5q2m as an example of a distributioused to generate the rewiring. The present scheme is raflexible for producing a variety of clusterization distributionand topologies, through the election of a proper distributQ. This fact may be relevant in the modeling of real neworks. At the present time several social phenomena hbeen modeled on the basis of a SW network@6–8#.

rint

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~1998!.@3# D. J. Watts,Small Worlds~Princeton University Press, Princ

eton, NJ, 1999!.@4# L.A.N. Amaral, A. Scala, M. Barthe´lemy, and H.E. Stanley

Proc. Natl. Acad. Sci. U.S.A.97, 11 149~2000!.

@5# R. Albert and A.L. Baraba´si, Phys. Rev. Lett.85, 5234~2000!.@6# G. Abramson and M. Kuperman, Phys. Rev. E63, 030901~R!

~2001!.@7# M. Kuperman and G. Abramson, Phys. Rev. Lett.86, 2909

~2001!.@8# Romualdo Pastor-Satorras and A. Vespignani, e-p

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