Analytical estimation of the correlation dimension of integer lattices
AAnalytical estimation of the correlation dimension of integer lattices
Lucas Lacasa ∗ and Jes´us G´omez-Garde˜nes
2, 3, † School of Mathematical Sciences, Queen Mary University of London, Mile End Road, E14NS London, UK Institute for Biocomputation and Physics of Complex System (BIFI) Departamento de Fisica de la Materia Condensada, Universidad de Zaragoza, Spain (Dated: November 21, 2018)Recently [L. Lacasa and J. G´omez-Garde˜nes, Phys. Rev. Lett. , 168703 (2013)], a fractaldimension has been proposed to characterize the geometric structure of networks. This measure isan extension to graphs of the so called correlation dimension , originally proposed by Grassbergerand Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. Thecalculation of the correlation dimension of a graph is based on the local information retrieved froma random walker navigating the network. In this contribution we study such quantity for somelimiting synthetic spatial networks and obtain analytical results on agreement with the previouslyreported numerics. In particular, we show that up to first order the correlation dimension β ofinteger lattices Z d coincides with the Haussdorf dimension of their coarsely-equivalent Euclideanspaces, β = d . PACS numbers:
In this article we address the concept of correla-tion dimension which has been recently extendedto network theory in order to efficiently charac-terize and estimate the dimensionality and ge-ometry of complex networks [1]. This extensionis inspired in the Grassberger-Procaccia method[2–4], originally designed to quantify the frac-tal dimension of strange attractors in dissipativechaotic dynamical systems. When applied to net-works, it proceeds by capturing the trajectory ofa random walker diffusing over a network withwell defined dimensionality. From this trajectory,an estimation of the network correlation dimen-sion is retrieved by looking at the scaling of thewalker’s correlation integral. Here we give an-alytical support to this methodology by obtain-ing the correlation dimension of synthetic net-works representing well-defined limits of real net-works. In particular, we explore fully connectednetworks and integer lattices, these latter beingcoarsely-equivalent [20] to Euclidean spaces. Weshow that their correlation dimension coincideswith the the Haussdorff dimension of the respec-tive coarsely-equivalent Euclidean space.
I. INTRODUCTION
During the last decade the science of networks has shedlight on the importance that the real architecture of theinteractions among the constituents of complex systemshas on the onset of collective behavior [5–7]. In this wayit has contributed to the advance in many branches of ∗ Electronic address: [email protected] † Electronic address: [email protected] science, such as statistical physics and nonlinear dynam-ics, in which the understanding of collective phenomenais fundamental. While the structural aspects of networkshave been largely explored by means of topological mea-sures [8], their geometrical aspects have been ignored,with the remarkable exception of a few attempts to char-acterize the dimensionality of their complex interactionbackbone [9–12]. For instance, the box-counting tech-nique, widely used for estimating the capacity dimen-sion D of an object, was extended in [12–15] as a box-covering algorithm, aimed at characterizing the dimen-sionality of complex networks.Recently [1], we proposed an extension of the conceptof correlation dimension [16] to estimate the dimension-ality of complex networks by using random walkers toexplore the network topology. This extension builds upon the well-known Grassberger-Procaccia method [2–4],originally designed to quantify the fractal dimension ofstrange attractors in dissipative chaotic dynamical sys-tems. This approach relies on embedding a trajectoryof the dynamical system in an m -dimensional space andcalculating a correlation integral over this trajectory.The rationale of the extension of the Grassberger-Procaccia method to the network realm is that the ge-ometrical structure of the network restricts the move-ment of a random walker and, accordingly, a notion ofdimensionality can be extracted through the propertiesof the walker’s trajectory. In particular, if the trajec-tory evolves over some object with well-defined correla-tion dimension, such dimension, β , should be accessibleexperimentally through the scaling of the walker’s cor-relation sum defined in the next section. In addition toits novelty, the use of the Grassberger-Procaccia methodtogether with the machinery of random walks, providesanother nice example of the use of walkers to capture thestructure and organization of a complex network, such asthe centrality of nodes [17], its community structure [18]or the existence of degree correlations [19]. a r X i v : . [ phy s i c s . s o c - ph ] J u l In [1] we showed numerical estimates of the correlationsum for walkers navigating a set of synthetic and real-world networks, finding a range of dimensions 1 < β <
II. CORRELATION DIMENSION FROMRANDOM WALKS IN NETWORKS
Let us start by briefly reviewing the generalizationof the Grassberger-Procaccia method to the computa-tion of the correlation dimension of a complex network.We denote by G a spatially embedded undirected net-work with N nodes and L links, so that each node i of G is labelled by a generic vector v i that uniquely de-termines the location of node i in the underlying space( v ∈ R d , or ∈ Z d when the space is discrete). The net-work topology is given by the so-called N × N adjacencymatrix A , whose elements are defined (for undirected andunweighted graphs) as A ij = A ji = 1 when nodes i and j are connected and A ij = A ji = 0 otherwise.Once the network is defined, we must define the dy-namical evolution of a random walker on network G . Thetime-discrete version of a random walks determines that,at each time step t , the walker at some node i hops to oneof the neighbors j with equal probability. In this way thetransition matrix M of a walker defines the probabilitythat a walker at node i at time t is at a node j at time t + 1 as: M ij = A ij (cid:80) Nl =1 A il = A ij k i , (1)where k i = (cid:80) Nl =1 A il is the degree of node i . Thus, byinitially setting the a walker at some randomly chosennode, one iterates the dynamics prescribed by matrix M and follows the trajectory of the walker (note at thispoint that in practice one does need to store M , as weonly need to have (local) information at each time stepof the neighbors of a given node, rendering this methoduseful for practical situations involving arbitrarily largenetworks, e.g. Internet). Now consider a trajectory of length n generated byan ergodic random walker navigating the network G asdescribed above. The trajectory can be described as thesequence of visited nodes. In the case of spatial networks,the trajectory can be casted in the series { v , v , . . . , v n } ,and embed the series in R m · d (where m is the embeddingdimension) by defining the vector-valued series { V ( t ) } ,where V ( i ) ∈ R m · d is defined as: V ( i ) = [ v i +1 , . . . , v i + m − ] . (2)Finally, the correlation sum function C m ( r ) is de-fined as the fraction of pairs of vectors whose distanceis smaller than some similarity scalar r ∈ R : C m ( r ) = 2 (cid:80) i After introducing the basis for the calculation of thecorrelation dimension of spatial graphs we begin ourstudy with the simple case of a fully connected network.This network, also termed as complete, is a graph G inwhich each node is connected with the rest of the N − A ij = 1 − δ ij (with δ ij = 1 if i (cid:54) = j and δ ii = 0). Note that the fullyconnected network can be understood as the dense-limit( L → N ) of a real network.A fully connected network can be embedded in an Eu-clidean space with diverging dimensionality, where eachnode i is in turn labeled by an infinite dimensional vector: X i = ∞ (cid:88) i =1 α i e i , (5)with α i ∈ N and e i · e j = δ ij . In order to prove that thecorrelation dimension β of such object diverges, we needto find that β m is a monotonically increasing function ofthe embedding dimension m .In what follows we prove the above claim. First, no-tice that the transition matrix M (Eq. (1)) of a randomwalker navigating a fully connected network reads:M ij = 1 − δ ij N − . (6)Showing that the walker can hop between any pair ofnodes i and j with equal probability. This makes the in-finite dimensional labeling above arbitrary for any prac-tical purpose. Thus for convenience and without loss ofgenerality, we label each node by a random number x ex-tracted from a uniform distribution U [0 , n in-dependent and identically distributed random variables, { x (1) , x (2) , x (3) , . . . , x ( n ) } , where each x ( i ) ∈ U [0 , i.e. , (cid:107) V ( i ) − V ( j ) (cid:107) ≡ ξ , (7)extracted from some unknown probability density ξ ∈ ρ ( x ) , x ≥ 0. After dropping irrelevant constants, the cor-relation sum (Eq. 3) reduces to the probability: P ( ξ < r ) = (cid:90) r ρ ( x ) dx . (8)Our program is based on the calculation of ρ ( x ).Let us begin with embedding dimension m = 1. In thiscase V ( i ) = v i = x ( i ) and, according to the L ∞ norm: ξ = | x ( i ) − x ( j ) | , (9)where we recall that x ( i ) and x ( j ) are uniformly dis-tributed random variables. Trivially, ξ is distributedaccording to a triangular distribution f ( x ) = 2(1 − x ).Hence ρ ( x ) = f ( x ) and C ( r ) ∼ P ( ξ < r ) = (cid:90) r (2 − x ) dx = 2 r − r . (10)For small values of r , the scaling is linear, and we obtain:lim r → log[ C m ( r )]log r = 1 + h.o.t. , (11)that is, up to first order we find β = 1. In a second step let us consider the case m ≥ 2, forwhich V ( i ) = ( x ( i ) , x ( i + 1) , . . . , x ( i + m − ξ = max {| x ( i + l ) − x ( i + l + α ) | ; l = 0 , ..., m − } , (12)where each of the random variables of the form | x ( i ) − x ( j ) | is now distributed following a triangular distribu-tion f ( x ). Our problem thus lies in deriving how ξ isdistributed. Note that this problem reduces to an ex-treme value problem, which can be solved using orderstatistics such that: ξ ∼ mf ( x )[ F ( x )] m − ≡ ρ ( x ) , (13)where F ( x ) = (cid:82) x f ( x ) dx is the cumulative distributionfunction of f ( x ). Therefore, in this general case the cor-relation sum yields: C m ( r ) ∼ P ( ξ < r ) = (cid:90) r m (1 − x )(2 x − x ) m − dx ∼ r m + h.o.t. , (14)Thus, we conclude that, up to first order, the correlationsum of a random walker navigating a fully connectednetwork evidences a so called trivial scaling with thesimilarity distance r : the exponent of the scaling β m increases linearly with the embedding dimension withoutsaturation, β m = m . This result is reminiscent of theinfinite dimensional attractor of white noise in theoriginal Grassberger-Procaccia procedure [2–4], and,applied to the network realm, it corresponds to aninfinite correlation dimension. (cid:3) IV. INTEGER LATTICES In what follows we address integer lattices Z d , whichare coarsely-equivalent [20] to Euclidean spaces withHaussdorff dimension d . For d ≤ k i = 2 d ∀ i = 1 , ..., N and are homogeneously locatedin the underlying space, tiling it in a regular way), andfor d ≥ A. 1D Lattice A 1D lattice is simply a chain graph which, intuitively,tends to an object of Haussdorff dimension one as thedistance between nodes shrinks continuously to zero. Inwhat follows we propose two alternative proofs, a ballisticapproximation and a calculation based on the unbiasedmotion of random walkers, both showing that the corre-lation dimension of 1D lattices is β = 1. 1. Ballistic approximation As an approximation (relaxed below), let us first con-sider the case of a ballistic (deterministic) walker in the1D lattice. If this lattice is labeled without loss of gener-ality by integers (where two adjacent nodes are labeledas i and i + 1, and A ij = δ i,i +1 + δ i,i − ), then a typicalwalker produces the string { i, i + 1 , i + 2 , i + 3 , i + 4 , ... } or, by symmetry { i, i − , i − , i − , i − , ... } . Both casesare equivalent and therefore yield equivalent results. Weshall therefore address the former for concreteness.Let us start with embedding dimension m = 1. Then, (cid:107) V ( i ) − V ( j ) (cid:107) = | i − j | (15)is a deterministic variable, and therefore the correlationsum can be explicitly calculated as (cid:88) i 2. Random walker Now we relax the ballistic approximation shown aboveand present address the correlation dimension derivedfrom the motion of a random walker. First, we labelagain without loss of generality the nodes of the 1D lat-tice by consecutive integers, and start by considering anembedding dimension m = 1. In this case the randomwalker performs a simple walk in Z , and (cid:107) V ( i ) − V ( j ) (cid:107) = ξ = | x ( i ) − x ( j ) | . (19)To analyze how ξ is distributed it is easy to notice thatthe distance between x ( i ) − x ( j ) is generated through thesum of j − i random variables, each of which is extractedfrom {− , +1 } , which tends to a normal distribution withzero mean and variance | j − i | by virtue of the central limittheorem. Therefore, ξ is the absolute value of the sumof j − i random variables, whose distribution tends to afolded normal distribution with zero mean and variance | j − i | . Therefore, after dropping irrelevant constants weobtain: ρ ( x ) ∼ e − x / ( j − i ) , (20) so that n (cid:88) j − i =1 Θ( ξ − r ) = n (cid:88) k =1 P ( ξ k < r ) = n (cid:88) k =1 (cid:90) r exp( − x /k ) dx = n (cid:88) k =1 erf (cid:18) r √ k (cid:19) , (21)where erf( x ) is the error function that fulfills:erf( r/ √ n ) ∼ r/ √ n + O ( r ) (22)whose first order is r for r ≤ − n (see the left panel ofFig. 1), and therefore: C ( r ) = n (cid:88) k =1 erf (cid:18) r √ k (cid:19) ≈ n (cid:88) k =1 r √ k ∼ r + h.o.t. , (23) i.e. , up to first order β = 1 for sufficiently large n andsufficiently small r .As a second step, consider an embedding dimension m = 2. In this situation, (cid:107) V ( i ) − V ( j ) (cid:107) = ξ = max {| x ( i ) − x ( j ) | , | x ( i ) − x ( j ) ± |} . Now, the important point is that these three random vari-ables are completely correlated: they are not independentrealizations but, on the contrary, all three depend on asingle realization of the duple { x ( i ) , x ( j ) } . Therefore, wedo not need to apply order statistics in this case: ξ isagain folded-normally distributed. The argument thenproceeds as for m = 1 such that C ( r ) ∼ r + h.o.t..A similar argument holds for a general embeddingdimension, m , and therefore we conclude that for a 1Dlattice, an unbiased random walker generates a correla-tion sum which, in an embedding dimension m reads: C m ( r ) ∼ r + h.o.t. , (24)that is to say, up to first order the predicted correlationdimension of the 1D lattice is again β = 1. (cid:3) B. Lattice D We now consider a random walker in a 2D lattice.This is a regular network where all nodes have degree k i = 4 that tiles Z . In what follows we prove that, upto first order, the correlation dimension of this networkis β = 2.In this case each node of this lattice is labelled by a twodimensional vector ( x, y ), where x, y ∈ Z . Accordingly, arandom walker generates a trajectory of the form (cid:26) (cid:18) x ( i ) y ( i ) (cid:19) , (cid:18) x ( i + 1) y ( i + 1) (cid:19) , (cid:18) x ( i + 2) y ( y + 2) (cid:19) , . . . , (cid:18) x ( n ) y ( n ) (cid:19) (cid:27) − − − − − − (x )/10000 FIG. 1: (Left panel) Error function erf( x/ √ n ) for n = 10 . For small values of x , the function scales linearly with x . (Rightpanel) Log-log plot of erf( x/ √ n ) and x /n for n = 10 . For small values of x , both shapes coincide. where the initial x ( i ) and y ( i ) are uncorrelated randomvariables extracted from a uniform discrete distribution U (1 , n ) and the trajectory is the result of the Markovprocess defined as: x ( i + 1) = (cid:26) x ( i ) + 1 , with probability 1 / x ( i ) − , with probability 1 / y ( i + 1) = (cid:26) y ( i ) + 1 , with probability 1 / y ( i ) − , with probability 1 / m = 1. In this case: (cid:107) V ( i ) − V ( j ) (cid:107) = ξ = max {| x ( i ) − x ( j ) | , | y ( i ) − y ( j ) |} = max { η , η } , (27)where η and η are random variables with a probabil-ity distribution f ( x ) which reduces to the case of a 1Dlattice, i.e. , by dropping irrelevant terms: f ( x ) ∼ exp (cid:18) − x | j − i | (cid:19) , (28) F ( x ) ∼ erf (cid:18) x | j − i | / (cid:19) . (29)Therefore, according to order statistics, we find that: ξ ∼ f ( x ) F ( x ) = exp (cid:18) − x | j − i | (cid:19) erf (cid:18) x | j − i | / (cid:19) , (30)and finally the correlation sum reads: C ( r ) ∼ P ( ξ < r ) = (cid:90) r exp (cid:18) − x | j − i | (cid:19) erf (cid:18) x | j − i | / (cid:19) dx = erf (cid:18) r | j − i | / (cid:19) ∼ r + h.o.t. , (31) i.e. , up to a first order expansion in r , the correlationsum for m = 1, C ( r ), scales quadratically (see the rightpanel of Fig. 1 for a numerical check). Finally, in the general case m > 1, one can triviallyfollow an argument similar to the one used for a randomwalker in a 1D lattice, finding that: C m ( r ) ∼ r + h.o.t. , ∀ m , (32) i.e. , the exponent β m saturates to the correlation dimen-sion β = 2. (cid:3) C. Lattice dD To round off, now we prove that in the general caseof integer lattices (for a general value d ), the correlationdimension of the lattice coincides, up to first order,with the Haussdorff dimension of the coarsely equivalentEuclidean space β = d .First, the trajectory generated by the walker in a d di-mensional lattice, where each node is labelled by a d di-mensional vector ( x , x , . . . , x d ) , x i ∈ Z ∀ i = 1 , , . . . , d ,is: x ( i ) x ( i ) x ( i ) . . .x d ( i ) ; x ( i + 1) x ( i + 1) x ( i + 1) . . .x d ( i + 1) ; . . . ; x ( n ) x ( n ) x ( n ) . . .x d ( n ) (33)and therefore, for a one dimensional embedding ( m = 1)we have (cid:107) V ( i ) − V ( j ) (cid:107) = ξ = max { η , η . . . , η d } , (34)where η l = | x l ( i ) − x l ( j ) | are random variables with aprobability distribution f ( x ). Finding the probabilitydensity of ξ is again an extreme value problem, whereorder statistics predicts: ξ ∼ f ( x ) F ( x ) d − ∼ exp (cid:18) − x | j − i | (cid:19) erf (cid:18) x | j − i | / (cid:19) d − . (35)Therefore, the correlation sum for m = 1 reads: C ( r ) ∼ P ( ξ < r )= (cid:90) r exp (cid:18) − x | j − i | (cid:19) erf (cid:18) x | j − i | / (cid:19) d − dx = erf (cid:18) x | j − i | / (cid:19) d ∼ r d + h.o.t. , (36)up to a first order expansion in r . In the general case m > 1, an argument similar to the one used for a randomwalker in a 1D lattice holds, thus finding that indeed C m ( r ) ∼ r d + h.o.t. , ∀ m , (37) i.e. , the correlation sum scales with D and thus the cor-relation dimension of a dD lattice is β = d . (cid:3) V. CONCLUSION Recently, the notion of fractal dimensionality has beeninvestigated numerically within networks [1, 12, 13, 15].The techniques used have borrowed concepts from mea-sure theory and dynamical systems such as the ca-pacity and correlation dimension respectively. To thisaim the corresponding techniques, such as the classical box-counting algorithm and the Grassberger-Procacciamethod, have been generalized to the network realm.In this manuscript we have focused on the latter ofthese techniques to show that the correlation dimensionof some synthetic networks, as defined in [1] and in equa-tions 3 and 4, coincides with the Haussdorff dimensionof their coarsely equivalent Euclidean spaces [20]. Notethat a network and an Euclidean space are very differentobjects in the small-scale (their topology is entirely dif-ferent) but they resemble each other in the large-scale.Therefore, our results although desired and expected, arenontrivial.In addition, the analytical calculations shown inthis manuscript illustrate the validity of the numericalresults shown in [16] in more sophisticated synthetic andreal-world network. However, finding similar analyticalevidences in the case of empirical networks is quite adifficult task. A slightly easier problem which is leftfor future work is to address the correlation dimensionof spatially embedded complex network ensembles withrobust statistical properties, i.e. , the so-called annealedgraphs [22–25]. Acknowledgments. The authors would like to thankPablo Iglesias for inspiring discussions. J.G.G. is sup-ported by MICINN through the Ramon y Cajal program. [1] L. Lacasa, and J. G´omez-Garde˜nes, Phys. Rev. Lett. ,168703 (2013).[2] P. 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