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Physical realizability of small-world networks. ∗ Thomas Petermann and Paolo De Los Rios Institute of Theoretical Physics, LBS, Ecole Polytechnique Fédérale de Lausanne - EPFL, CH-1015 Lausanne, Switzerland (Dated: February 6, 2008) Supplementingalatticewithlong-rangeconnectionseffectivelymodelssmall-worldnetworkschar- 6 acterized by a high local and global interconnectedness observed in systems ranging from society 0 to the brain. If the links have a wiring cost associated to their length l, the corresponding distri- 0 bution q(l) plays a crucial role. Uniform length distributions have received most attention despite 2 indications that q(l) ∼ l−α exist, e.g. for integrated circuits, the Internet and cortical networks. While length distributions of this typewere previously examined in the context of navigability, we n herediscussforsuchsystemstheemergenceandphysicalrealizability ofsmall-world topology. Our a J simple argument allows to understand under which condition and at what expense a small world results. 9 1 PACSnumbers: 89.75.-k,89.75.Hc ] n The explosion of research activity in the field of com- below,itis the aimofthis paperto investigatethe phys- n - plex networks has led to a novel framework in order to icalrealizabilityofaSWnetwork. Inthe abovemodel,p s describe systems in disciplines ranging from the social allows to interpolate continuously between a fully regu- i d sciences to biology [1]. One feature shared by most real lar (p=0) and an entirely random(p=1) topology, the t. networks is the small-world (SW) property involving a precisenatureofthistransitionbeingdiscussedbelow. If a high degree of interconnectedness both at a local and theshortcutsaremerelyadded(withoutlosinglocalcon- m global level. That is, for every node, most nodes close nections), no significant changes in the emergence of the - to it should also be close to each other and every pair SW topology result. We therefore deal with the model d of nodes is separated, on average, by only a few links where re-wiring is not accompanied by edge removal. n o [2]. More precisely, the latter is usually expressed with In the original formulation of the SW model, which c an at most logarithmic increase of the mean distance as received most of the attention [11], the length distribu- [ a function of the system size. Although the SW phe- tion of the shortcuts is uniform, since a node can choose nomenonhasfirstbeenintroducedinasocialcontext[3], 1 any other node to establish a shortcut, irrespective of v it is also relevant for communication and technological their Euclideandistance. Yet, new interestingproperties 9 systems such as the Internet [4] or electronic circuits [5]. emergeifthisconditionisrelaxed,forexampleifthe dis- 4 Small-worldpropertiesareofgreatrelevanceforcommu- tribution q(l) of connection lengths l decays as a power 4 nication systems: SW networks are particularly efficient law, q(l) ∼ l−α. The navigability in such a small world, 1 for message passing protocols that rely only on the lo- 0 for example, depends on the decay exponent α [12], and cal knowledge of the network available to each node [6]. 6 the nature of random walks and diffusion over the net- It has also been pointed out recently that SW networks 0 work is also affected [13, 14]. It was even conjectured t/ could describe the architecture of neuronal networks: in that the fundamental mechanism behind the SW phe- a vitroneuronalnetworks[7], brainfunctionalnetworks[8] nomenon is neither disorder nor randomness, but rather m as well as the cerebralcortex [9] exhibit SW features. In the presence of multiple length scales [15] in agreement - fact,thetopologyplaysacrucialroleinaneuralnetwork, with q(l)∼ l−α. Here we establish the properties of the d since the high local interconnectedness gives rise to co- wiring mechanism which allows to realize SW networks, n herent oscillations while short global distances ensure a o theimprovednavigabilitybeingaconsequenceoftheSW fast system response [10]. c property. : To model SW networks in Euclidean space, one starts v RealSWnetworksareunlikelytobesuccessfullymod- i with a regular lattice which is highly interconnected lo- eledaccordingtoWattsandStrogatz’recipegivenabove: X callyandthenrewireseverylink(connectingnodesAand if shortcuts have to be physically realized, the cost of a r B) with probability p, that is the edge between the ver- a long-range connection is likely to grow with its length. tices A and B is replaced by a long-range connection (or Since nodes connected by shortcuts can be at any Eu- shortcut) betweenthe nodes A and C,C being chosenat clidean distance from each other, it turns out that the random[2]. Clearly,theshortglobaldistancesaredueto amount of resources that they have to invest in their thepresenceofshortcuts,andasdescribedinmoredetail connections grows linearly with the linear system size, and it is, a priori, unpredictable. This is far from op- timal for systems composed by entities with limited resources (e.g. providers or neurons). Indeed, local (sin- ∗Present address: Unit of Neural Network Physiology, Lab- gle node)and globalwiring cost considerationsare likely oratory of Systems Neuroscience, National Institute of Men- to be key factors in the formation of real SW networks tal Health, Bethesda, Maryland 20892; Electronic address: [email protected] [16, 17, 18, 19, 20, 21]. Regarding connection-lengthdis- 2 theargumentallowingtoderiveα ,letusrecallthatSW c 10-2 α = 2.5 topologyischaracterizedbythefollowingbehaviorofthe α = 1 * mean distance L L*) α = 0 α = 1.5 <d>/ ∗ L hdi=L F , (1) L/ 10-4 α L∗ ( (cid:16) (cid:17) Fα 10-4 10-2 the scaling function obeying [31, 32] = 1 * x if x≪1, L α = 3.5 F (x)∼ (2) >/ α = 2 108 α (ln(x) if x≫1. d L* < >/ <d In other words, SW topology corresponds to a logarith- 106 mic increase of the mean distance with the system size (L≫L∗) whereas in a large world (LW), i.e. if L≪L∗, 1 10 100 106 108 hdi∼L. For α=0, the critical length scale in Eq. (1) is L/L* L/L* givenbyL∗(p)∼p−1/D [32,33]. Ifαispositive,weshall ∗ derive L (as well as α ) through the following indirect c FIG.1: Meandistanceversuslinearsystemsize,bothofthese argument: We look atthe probabilitythat anarbitrarily quantities being rescaled by L∗α(p), for p = 0.001 (◦), p = chosen additional link is a real shortcut, that is, that it 0.002 ((cid:3)), p = 0.004 (△), p = 0.008 (⋄) and p = 0.016 (▽). spans the lattice, The exponent α ranges from 0 to 3.5 as indicated. The data collapsesconfirmEqs.(5)and(1)alsoforα>αc (lowerright L/2 P (L)= q(l)dl, (3) panel). c Z(1−c)L/2 c being small but finite, and require (our ansatz) the tributions q(l) ∼ l−α, such measurements were reported expected number of such connections to be of the order forsystemscreatedthroughself-organization,designand of 1 [31]: evolution,namelyfortheInternet[21],integratedcircuits [22], the humancortex [23] andfor regionsofthe human P (L) p∗(L)LD ≃1. (4) c brain correlated at the functional level [8]. Some modeling effort taking into account the constraint of wiring Here p∗(L)LD is the de(cid:2)sired num(cid:3)ber of additional links, ∗ minimization has been made for systems where the con- implying the emergence of SW topology for p ≫ p (L). nection lengths are [24] or are not distributed according After evaluating the scaling of (3), Eq. (4) reads to a power law [25, 26, 27], and such length distribu- tionsemergequitenaturallywhenwiringcostsalongwith L−D if α<1, shortest paths are minimized [28]. p∗(L)∼ ln(L)/LD if α=1, (5)  In this work we re-analyze the SW phenomenon from Lα−D−1 if α>1. a wiring cost perspective, for networks in D dimensions, built using a power-law decaying distribution of short- Eq.(5)impliesL∗(p)∼p−1/D forα<1,i.e.thebehavior cut lengths. We find, both analytically and numerically, ∗ ofL inthisα-rangeisthesameasthatforα=0. Inthe thatα<D+1is the conditionforthe emergenceofSW case α>1, we have L∗(p)∼p1/(α−D−1), thus becoming behavior. We also found that the local interconnected- infinite at nessincreaseswithαand,givenafixedtotalwiringcost, networks with larger values of α are smaller worlds. α =D+1. (6) c Given a D-dimensional lattice of linear size L, con- sisting of N = LD sites, subject to periodic boundary We therefore have two possible regimes for α<α while c conditions, it shall be supplemented with pN additional LW behavior prevails for α ≥ α . Fig. 1 shows the c connections whose lengths are distributed according to rescaled mean distances as a function of the rescaled q(l)∼l−α as follows: for every link to be added, we first linear system size for different values of α and p = choose its length accordingto the (one-dimensional)dis- 0.001,0.002,...,0.016in eachsetfor the case D =2. The tributionq(l) andthenput it onthe lattice byrandomly observeddatacollapsesforallthechosenvaluesofαcon- choosing one endpoint and the other at the drawn dis- firm Eq. (5) obtained by our simple argument as well as tance l, such that no pair of sites is connected by more Eq.(1). WenumericallyverifiedEq.(2),especiallyinthe than one additional connection. limit L/L∗ ≪1, the logarithmic tail of F further being α Clearly, a certain amount of real shortcuts, i.e. long exhibited best for small α [34]. additional links, is required for SW topology to emerge As outlined above, a SW network is also character- [2, 31]. It can thus be anticipated that the exponent α ized by a high local interconnectedness. This topological hastobe smallerthanacriticalvalueα . Beforewegive property can for example be measured by the clustering c 3 coefficient C which is the probability that two nodes are connected, given that they share a nearest neighbor. In contrast to Watts and Strogatz’ model, our initial lat- tices are characterized by C = 0, but by increasing the exponent of the link-length distribution, the degree of clustering becomes orders of magnitudes larger than for > random networks with the same number of nodes and <d10 links. Let us now examine the wiring costs, which were our (b) prime motivation to look at SW networks with power- lawdecayinglink-lengthdistributions, andanimportant 1 10 100 1000 C /N W ingredient for real SW networks. The moments hli and hl2iplayacrucialroleasfarasthesecostsareconcerned. FIG. 2: (a) Mean distance as a function of the total wiring Indeed, finite hli and hl2i would allow for predictable costs (divided by the number of sites) for 1 dimensional costs for each node, and consequently for a better de- topologies (N = 104). The curves (◦: α = 0, △: α = 1, sign of the network constituents. The total wiring cost ⋄: α = 1.5 and (cid:3): α = 1.75) show that the mean distance C = pLDhli is also an important quantity, its minimi- decreases with α for a fixed value of CW/N. (b) Analogous W sation governing, for example, the evolution of cortical results for D = 2 [N = 500×500 and α = 0 (◦), α = 1 ((cid:3)) and α = 2 (⋄)]. All the points shown here result from networks [17]. We find for the first two moments the averaging over 100 realizations of networks. scaling relations summarised in Tab. I, the expressions for integer α being modified by logarithmic corrections. In 2 dimensions, SW topology can be realized even if fact, it also applies to a version of the SW model where hli = const (that is, for 2 < α < 3 = α ) whereas this c the links are added in a different way: at every site, a is not the case in 1 dimension where hli becomes finite linkisaddedwithprobabilityp-theotherendpointbeing in the L → ∞ limit only above α = 2. Moreover, if c chosenaccordingtothe(D-dimensional)distributionq(l) D =3,itis evenpossibletohavehli=O(1)=hl2iwhile [29]. Thisprocedurediffersfromthepreviousoneinthat still being in the SW regime for 3 < α < 4 = α . An c the site fromwhich the new link willemanate “sees”the appropriatechoiceoftheparametersD andαisthusthe dimensionality of the lattice, giving rise to a different key to modeling networks which are both efficient (SW normalization of q(l) with respect to the version treated topology) and economical (low wiring costs). above. Furthermore, the just described mechanism is It is furthermore interesting to have a closer look at equivalent to adding a link between any pair of sites x the relationship between the wiring costs and the topol- and y with a probability proportional to |x−y|−α [30]. ogy. Asαvaries,onecanaskwhatmeandistanceresults Forthis newconstructionprocedure,the lengthdistri- given a total amount of wiring length for the additional bution reads connections (i.e. the total cost). Fig. 2a reports these dependencies for α = 0,1,1.5 and 1.75 (going from the lD−1−α q(l)= , uppermosttothelowestset)for1dimensionaltopologies L/2˜lD−1−αd˜l of 104 sites. The largestvalue ofhdi (the leftmost circle) 2 corresponds to the length scale L∗ < 103 ≪ 104 = L, where the factor lD−1 Rexplicitly accounts for the nor- thus all the points in the figure represent the system in malization in D-dimensional space. With Eqs. (3) and the SW regime. It can clearly be seen that the mean (4), which do not depend on the details of the “adding” distance decreases with α at fixed wiring costs CW/N, mechanism, we obtain for the critical probability i.e. the larger α the smaller the world. This behavior is qualitatively recoveredwhen expressing Eq. (1) in terms L−D if α<D, of x=CW/N =phli. We made similar observations in 2 p∗(L)∼ ln(L)/LD if α=D,  dimensions (see Fig. 2b). Lα−2D if α>D. Letusnowpointoutthegeneralityofourargumentfor the realizability of SW networks in Euclidean space. In Conversely, this implies L∗ ∼ p1/(α−2D) for α > D, and hence the existence of a SW regime as long as α <2D (7) TABLE I: Behavior of the moments of the shortcut-length c distribution as a function of thelinear system size L(for the in analogy with the previous reasoning. Inequality (7) “adding” procedure1). had already been derived [29, 30], but in a less intuitive 0≤α<1 1<α<2 2<α<3 α>3 framework. hli L L2−α const const In summary, we have givena simple argumentleading hl2i L2 L3−α L3−α const to the precise conditions under which small-worldtopol- ogy emerges, and examined the physical realizability of 4 such networks. Due to the generality of our argument, tegrated circuits, the Internet or the human cortex, we it is also applicable to other small-world models. 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