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: Thomas.Petermann@alumni.ethz.ch [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 / ∗ 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* < >/ 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 thatα1, 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 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. We believe this work to have intriguing implications in their further showed that small-world networks can be con- modeling. structed in a very economical way if the parameters D We thank Marc Barthélemy and Francesco Piazza for and α are chosen appropriately (although of course in their valuable comments, as well as to the EC-Fet Open realsystemsDisseldomatunableparameter). 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Physical realizability of small-world networks

Similar Physical realizability of small-world networks