Topologic distance in the Lucena network

DOI:10.1140/epjb/e2016-60669-6

Regular Article

PHYSICAL

JOURNAL

B

Topologic distance in the Lucena network

Darlan A. Moreira1 and Gilberto Corso2,a

1 _{Escola de Ciˆencias e Tecnologia, Campus Central Universidade Federal do Rio Grande do Norte, 59078-970 Natal-RN, Brazil} 2 _{Departamento de Biof´ısica e Farmacologia, Centro de Biociˆencias, Universidade Federal do Rio Grande do Norte, 59072-970,}

Natal-RN, Brazil

Received 12 August 2015 / Received in final form 20 November 2015

Published online 12 May 2016 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2016

Abstract. The Lucena network (LN) is the dual of a multifractal partition of the square. We analyze the

relation between the typical topologic distancel and the number of vertices N of the LN. The multifractal partition has one parameterρ which controls the geometrical asymmetry of the multifractal. In the limit ofρ → 1 the blocks of the partition are squared, the connections amont the blocks are short range, the LN is more regular and the relation l ∝ √N is observed. For the limit ρ → 0 the blocks are strongly asymmetric, long range connections appear, and the topologic distance followsl ∝ (log N)α, a weak small world phenomenon. For any network size we calculate analytically the size of the minimum distancelmin

(ρ → 0) and the maximal distance lmax(ρ → 1). The distance in the weak small world regime is calculated using the number of vertices inside a radius of length l and taking into account the network average connectivity and the exponentα.

1 Introduction

The topological distance has been a hot issue in the XXI century network theory [1,2]. The dependence of the typ-ical distance l with the number of vertices N is impor-tant to classify networks [2,3]. Much of the appeal of net-work science came from the small world phenomenon and the six degrees of separation experiment [4,5]. Indeed, the small world relation in networks arises already in random networks. Moreover, the study of topologic distance is cru-cial to understand the velocity of spreading of a signal in the network, for instance the propagation of a epi-demic [6,7] in a social network or transport features in networks [8,9].

Regular networks [1] have a constant connectivity for all vertices and have large l; the square lattice, for in-stance, showsl ∝√N. On the other side, the vertices of a Random Networks (RN) follow a Poisson like distribu-tion of connectivity. RN display the reladistribu-tionl ∝ log N [2]. The increase ofl with the logarithm of N characterize the small world phenomenon [2,3]. Spatially embedded net-works [10,11] may disclose a relationl ∝ (log N)α, with

α > 1, which shows a grow of l with N situated in

tween regular and small world networks, we call this be-havior as weak small world phenomenon. The three cases above presented are also used to investigate the number of verticesN(l) inside a radius of topological distance l:

N(l) ∝

⎧ ⎨ ⎩

lp _{regular graph (1)}

exp(l) small world (2) exp(l1α) weak small world. (3)

a _{e-mail:[email protected]}

In general, a regular lattice follows a algebraic relation with coefficient p. In this manuscript we work with net-works on a plane. A regular lattice on a plane can be represented over a metric space with their vertices equally spaced. In this case if we take a circle of distance l the number of vertices inside the circle followsN(l) ∝ l2, the exponent two is related to the dimensionality of the space. If we consider a regular lattice defined over a line, like the Strogatz model [3], we haveN(l) ∝ l. In both cases the dis-tances inside the network grow in an algebraic way which characterize a large world network model.

The standard case of small world network is the RN that shows an exponential relation between N(l) and l, equation (2). For a RN each vertex has an average connec-tivityk, in this case we can figure out that any vertex is connected to k others. For the case l = 2, for instance, each vertex is connected to another k vertices, in this way, the number of vertices connected to the original ver-tex at distancel = 2 is k2. Following the same reasoning, for a distancel, the number of connected vertices is kl which expresses an exponential behavior. Following this line, equation (2) is written as:

N(l)  kl_{. (4)}

Furthermore, the third case, the stretched exponential, equation (3), represents an intermediate situation in which the grow in the number of connections is not so fast as the exponential case, neither as slow as for regular networks. This relation was explored in reference [12] to understand the relation (3) observed in random networks embedded in a plane and following metric constraints [10,13]. In this

(a)

(b) (c)

Fig. 1. The three first steps of the construction of the

mul-tifractal lattice (a)n = 1, (b) n = 2 and (c) n = 3. In these figures we use alwaysρ = 2/3.

manuscript we explore the Lucena network, a planar ob-ject that share, depending onρ, either relations (1) or (3), depending on the value of its single paramater− the par-tition parameter, ρ ≡ r/s, where r and s are related to multifractal block size; in Figure1,r = 2 and s = 3.

The Lucena network is developed from a multifrac-tal lattice. It is worthy of note that the multifracmultifrac-tal lat-tice was studied for many years before the introduction of Lucena network. The first work about this subject de-scribes a partition of the square used to model oil reser-voirs [14], afterwards the partition was explored in the context of percolation theory [15]. This partition of the square, is simple in the sense it has just one parameterρ, nonetheless it is extraordinary because it gives origin to a multifractal lattice [15,16]. Recently, the dual of the multi-fractal lattice, called the Lucena network LN, was studied in the context of network theory [16,17]. For ρ → 0 the LN is strongly asymmetric with long range connections while forρ → 1 LN becomes regular-like with short range connections among the vertices [18].

In this work we explore the relation between the topo-logic distancel and network size N of the LN. As we shall see, the LN has two limits according to ρ: at one side LN follows the relation l ∝ √N, while in the opposite limitl ∝ (log N)α. In addition, we present an analytical estimation for the maximal and minimal topologic dis-tances for limit cases ofρ and any network size. The rest of the paper is outlined as follows, in Section 2 we review the main properties of multifractal lattice and its dual− the Lucena network. In Section 3 we explore specifically the scaling of the topological distance with network size and perform analytical estimations. In Section 4 we con-clude the work and discuss similarities between Lucena and other complex networks.

2 Construction of the multifractal lattice

and the Lucena network

To present the algorithm of the multifractal lattice we start with a unit square and a given partition 0< ρ < 1. In the first step a vertical segment divide the square in two pieces according to a ratio ρ. In the sequence two new horizontal segments divide each one of the rectangular blocks into new blocks using the same ratioρ. Following

this procedure, at stepn = 1, four blocks are formed from the square, this schema is exemplified in Figure1a.

In the second step of the algorithm,n = 2, each block is divided into new four blocks producing 42 rectangular objects. In the stepn the number of blocks is N = 4n. Fig-ure 1b illustrates the step n = 2 and Figure1c the step

n = 3 of the algorithm. In the sequence of Figures1a−1c

we useρ = 2/3 and the with the same vertical and hor-izontal section symmetry [15,16]. In this work, to as well as in previous papers [14–18], we explore the pattern of lattice partition presented in Figure1.

It was previously explored that this partition produces a multifractal object [15]. The partition of the square above defined generates an area distribution that follows a binomial distribution. Besides, the spectrum of fractal dimensions for this object is given by:

Dk= log C

n kskrn−k

log(s + r)n/2 ,

wherek represents the set of all tiles with same area and

ρ = r/s [15]. The multifractality is in the limit n → ∞;

in this limit the square (Euclidean dimension 2) is di-vided in an infinite k-sets each one with a given fractal dimensionD_k.

The dual of the multifractal lattice is the LN. By def-inition, in the LN the blocks of the multifractal are the vertices and connection among vertices are established us-ing the neighborhood condition. In this way the LN has 4n vertices at stepn, however the number of connections of each vertex depends onρ and the choice of horizontal and vertical sections. The dependence of the distribution of connections, and an eventual, fat tail in the distribution of connectivity is explored in reference [16]. We note that, by definition, the LN is planar, that means, their links do not cross. In addition, in the limit of n → ∞ the LN is space filling [17].

The value of ρ is crucial for characterizing the LN, the limits ρ → 0 and ρ → 1 display opposite differences.

For ρ → 1, the blocks of the partition resemble squares,

the multifractal lattice becomes almost regular. In opposi-tion, forρ → 0 the rectangles are strongly asymmetric and the LN presents long-range connections and strong hetero-geneity [18]. In Figure2 we illustrate these two cases (a)

ρ = 4/5 is regular-like while (b) ρ = 1/5 shows strong

asymmetry. For both cases we evolve the algorithm until

n = 3. In these two figures we show the multifractal

lat-tice with thin lines, we add points in the middle of the rectangles to better visualize the vertices of the LN, the connection among vertices are depicted by dotted lines.

To explore in more detail the role ofρ in the properties of Lucena network we show in Figure3the clustering coef-ficientC. In a previous paper [17] we have analyzed the hierarchical properties of the Lucena network, that means,

Ci∝ ki−1forCi andk_i the local clustering coefficient and connectivity. However, the global behavior ofC was not introduced in that paper. In fact,C shows two limit cases

forρ → 0 and ρ → 1. We see in Figure3 that forρ  0.6

(a)

(b)

Fig. 2. An illustration of both the multifractal lattice and

Lucena network for two distinct values of ρ. The thin solid lines indicate the blocks of the partition while the dotted lines the connections among vertices of the LN. In (a) we show a roughly homogeneous networkρ = 4/5 with short connections and in while in (b) an asymmetric partitionρ = 1/5 with some long-range interactions; for both cases we evolve the algorithm untiln = 3.

The topological distance, or chemical distance, be-tween two vertices i and j is the length of the minimal pathl_i,j between these vertices [19]. In this work we use the algorithm of Dijkstra [20] to estimatel_i,j. The average topological distance of the networkl is the arithmetic av-erage among all pairs ofl_i,j. The diameter of the network is the maximuml_i,j. The analysis of typical distances in networks uses either the diameter, orl, to characterize the typical distance of a network. Indeed, the diameter is pro-portional to l and these two measures are equivalent; in this manuscript we usel.

3 Results

In this section we investigate the relation between the topological distance versus network size according to ρ. In Figure4we plotl versus ρ for several network sizes N. The network size depends on the stepn of the algorithm

asN = 4n, for simplicity we employn in the legend of

Fig-ure4. This picture points out that in the limit ofρ → 1 the typical distancel is much larger than in the opposite limitρ → 0. In addition, in the limit LN is regular like l is larger, a quite intuitive result as we compare distances in regular and random (or small-world) networks. In this figure we employ a normalized version of the topologic

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.355 0.360 0.365 0.370 0.375 0.380 ρ <C> ● n=7 n=9 n=11

Fig. 3. The clustering coefficient C versus ρ. The curve

presents two limit cases for ρ → 0 and ρ → 1. In the leg-end is indicated three lattice sizes corresponding to n = 7, 9 and 11. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 l/l_max ρ n=7 n=8 n=9 n=10 n=11

Fig. 4. Curve of normalized topological distance l/lmaxversus

ρ for several network sizes indicated in the figure. The two

extreme ρ → 0 and ρ → 1 show minimal lmin and maximal

lmax topologic distances. The difference betweenlminandlmax

increases with network size.

distance: l/lmax. The maximum l corresponds to the

al-most regular network limitρ → 1; as a result in this limit

l/lmax = 1 for any network size. This notation draw

at-tention to the relation between maximum and minimall for any network size.

In a regular lattice the typical distance between any two points follows an algebraic relation, a standard exam-ple is the square lattice with size L and N = L2 vertices. In this case the typical distance scales as L ∝ √N. A similar scaling is observed in the Lucena network for the regular limit ρ → 1, see Figures 5a (ρ = 9/10) and 5b

(ρ = 1/2). We use here the more general scaling L ∝ Nβ.

In the set of Figure5 we plotl versus N for 2 ≤ n ≤ 11.

As ρ approaches to zero the exponent β → 0.5, we

100 101 102 103 101 102 103 104 105 106 107 l N ρ=9/10 β=0.493 100 101 102 103 101 102 103 104 105 106 107 l N ρ=1/2 β=0.442 100 101 102 103 101 102 103 104 105 106 107 l N ρ=1/12 α=2.399 100 101 102 103 101 102 103 104 105 106 107 l N ρ=1/24 α=2.389

Fig. 5. Fitting of topological distance l versus network size N

for severalρ. For ρ = 9/10 (a) and ρ = 1/2 (b) the fitting was performed withl ∝ Nβ while for ρ = 1/12 (c) and ρ = 1/24 (d) the adequate fitting isl ∝ (log N)α.

respectivelyβ = 0.445, 0.478, 0.488; for all cases the error calculated for the curve fitting is 0.001.

The opposite limit of the LN,ρ → 0, presents a differ-ent scaling, Figures 5c and 5d show the computational data and the fitting l ∝ (log N)α. We indicate in the figure the exponent α obtained by standard minimum square fitting method:α = 2.399 ± 0.037 for ρ = ₁₂1 and

α = 2.389 ± 0.034 for 1

24. Forρ > 121 the l ∝ (log N)α

fitting shows systematic deviations from the simulated data and it is not acceptable. The intermediary range

1

12 < ρ < 12 does not properly fit neither a

√

N equation

nor (logN)α.

In Figure 4 we distinguish for each curve two typi-cal distances: the maximum topologitypi-cal distancelmax, for

ρ → 1, and the minimum topological distance lmin, for

ρ → 0. The difference between lmax andlmin depends on

network size, see Figure4. We perform an analytical esti-mation ofκ = lmin

lmax using the diameter of the network, that

means, we calculatel corresponding the the total number of vertices N in the web. We approximate lmax by the

size of the square:lmax=√N =√4n = 2n. The analysis

oflmin is done combining equations (3) and (4):

N(l) = klα1

min. ₍₅₎

Further we usek and α from independent simulations. The inversion of equation (5) produces

lmin=  log N logk α =  n log 4 logk α . (6)

In this way we produce the following estimation of the ratio: κ = _llmin max = 1 2n  n log 4 logk α . (7)

The comparison between the observed results from simula-tionκ_simuland theoreticalκ_theovalues for several network

sizen is presented in Table1that summarizes the

quanti-tative results. The values ofκ_simul are directly estimated from Figure4that plotsl/lmax, in this caseκ = lmin/lmax

Table 1. Comparison between theoretical results κtheo

ob-tained from equation (7) and simulational resultsκ_simul. The error between these two results is not above 10%.

n 7 9 11

κtheo 0.50 0.20 0.092 κsimul 0.46 0.21 0.089

corresponds to the value ofl/lmax in the limit of smallρ that corresponds to the smallest topologic distance. On the other side, κ_theor is estimated from equation (7). To apply equation (7) we use the best fitting ofα from Fig-ure 5. In addition, the value of k = 5.5 is estimated empirically. In agreement with [17,18] k is independent

of ρ and decreases weakly with network size. We notice

that, despite the presence of long range connections and an eventual fat tail, the quantityk is invariant over ρ. We check in Table1that the difference between simulation and theory is bellow 10% and decreases for large lattice size.

4 Final remarks

In this work we explore the topologic distance l in the LN, that means, we analyze the relation between l and the number of nodesN. The LN has a free parameter ρ with two distinct limits, forρ → 0, the LN has a marked heterogeneity and shows the scalingl ∝ (log N)αwhile in the opposite limit ρ → 1 the LN is almost regular and

l ∝ √N, a typical regular network scaling. For a given

network size we estimate analytically the maximum topo-logical distancelmax(ρ → 1) and the minimum topological

distance lmin (ρ → 0). In addition, we compute the ratio between these two lengthsκ = lmin

lmax and successfully

com-pare with simulations. We find thatκ grows as κ ∝ _n2n_α, or using network size N, κ ∝

√ N

log N. To summarize,

inde-pendent of network size the behavior of l displays three distinct regimes: (I) 0< ρ ≤ ₁₂1, (II) ₁₂1 < ρ < 1₂ and (III)

1

2 ≤ ρ < 1. Regime (I) follows a regular trend l ∝

√ N

while regime (III) shows a weak small world behavior. A comparison of our result with the model presented in reference [11] is enlightening. In that paper the authors generate a random network model embedded in a square lattice that obeys the following metric constraint:P (r) ∝

r−δ _{for P (r) the probability of a node be connected to}

another node at distancer. In that model, for large δ the nodes have short range connections while for small δ the long range connections dominate and the network becomes random like. For an intermediary δ their model follows equation (3) − the weak small world phenomenon. The LN shows similarities and differences with the cited model, we find in a previous paper [18] that the relationP (r) ∝

r−δ _{naturally arises in the LN. Otherwise, despite the LN}

is also defined on a plane it is a non crossing network. For comparison, in the cited model α > 4 a value larger than presented in the LN. To conclude, in both case the networks have metric constraints and display regular and weak small world phenomenon according to a parameter that control the range of connectivity.

To finalize, suppose we have a plane with a metric mea-sure and a set of randomly spaced points. These points can be a set of vertices of a non crossing network on the plane. If distances among points follow a Poisson like dis-tribution with a average distancer the number of points inside an imaginary circle of radiusR should increases as

N(R) ∝ R2_{, this relation is a natural consequence of being}

immersed in a metric bidimensional plane. Otherwise, for a Random Network with average connectivityk the

rela-tionN(l) ∝ kl is a topologic propriety, a metric free

as-sociation among elements with similar connectivity. In this context the relationN(l) ∝ kl1/α can be interpreted as a topologic relation with a metric constraint. We are aware that the mathematical treatment presented in our work is incomplete. Indeed, a theoretical model that encompasses regular and weak small world into a single framework is necessary to fill the gap between pure topology and metric spaces.

Financial support from CNPq (Conselho Nacional de

Desen-volvimento Cient´ıfico e Tecnol´ogico) is acknowledged.

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