A planar network between short and long range behaviour

Darlan A. Moreira

a

, Liacir dos Santos Lucena

b

, Gilberto Corso

c,∗

a_{Escola de Ciências e Tecnologia - Campus Central, Universidade Federal do Rio Grande do Norte, 59078-970 Natal-RN, Brazil} b_{Departamento de Física Teorica e Experimental, International Center for Complex Systems, Universidade Federal do Rio Grande do} Norte, 59078-970 Natal-RN, Brazil

c_{Departamento de Biofísica e Farmacologia, Centro de Biociências, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN,} Brazil

h i g h l i g h t s

• The Lucena Network is the dual of a multifractal partition of the square. The Projected Lucena Network PLN is the Lucena Network projected into the square lattice.

• By construction the PLN is a planar network that is embedded in the plane.

• The PLN follows the relation P(r) ∝r−δ_{for P(r)the probability of a node be connected to another node at distance r.}

• The fractal dimension, df, of the PLN depends on its internal asymmetry, we identify two regimes: short range,δ ≃3.9, and long range,

δ ≃3.2.

• For the short range limit df ≃2, the dimension of the embedding. For long range df ≃2.5.

a r t i c l e i n f o

Article history:

Received 19 December 2014

Received in revised form 27 February 2015 Available online 25 April 2015

Keywords:

Multifractal lattice

Spatially embedded networks Planar graph

Fractal dimension

a b s t r a c t

We study the Projection of Lucena Network PLN: a planar network whose nodes coincide with the sites of the square lattice. The PLN has one internal parameterρ, forρ →1 the network resembles a symmetric regular lattice with almost none highly connected node whileρ →0 is more asymmetric and there are several highly connected nodes. We esti-mate P(r), the probability of a node be connected to another node at distance r. The PLN follows the fractal scaling P(r) ∝r−δwith 3< δ <4 according toρ. Forρ →1 we haveδ close to 4 which is a signature of short range interactions, whileρ →0 showsδ →3 and has long range interactions. In addition, the fractal dimension, df, of the PLN behaves as:

forρ →1, df =2, the dimension of the embedding space while forρ →0, df increases

and the PLN shares some similarities with a small world graph.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

In the last decade the study of networks has been a fruitful research avenue [1,2], scientists realized that networks are useful for modelling a large class of physical, biological and social systems [3–5]. Despite the fact that network theory was originally formulated in a geometric free framework, many real networks are spatially embedded, for instance, airline networks [6], social networks [7] or human travel networks [8]. In this way several attempts to model spatially constrained networks have been proposed [9–11]. In this work instead of suggesting a new spatial embedded model we explore fractal characteristics of the planar Lucena network which is naturally a bidimensional structure.

∗_{Corresponding author.}

E-mail address:[email protected](G. Corso).

http://dx.doi.org/10.1016/j.physa.2015.03.088

sections following the parameterρ. In figures (a) and (b) we show both the multifractal and the Lucena networks for n=1 and n=2 respectively. At each step the pattern shown in (a) is repeated inside all new rectangles. We use thick lines in (b) to highlight the original pattern. In the transition from n

to n+1 each vertex is erased and replaced by four new ones; the number and the topology of new connections at each step depend onρ.

Before the analysis of Lucena network we discuss the Kosmidis, Havlin, Bunde model KHB [12] that has inspired much of our work, Ref. [13] also employed a similar model. The KHB model is an algorithm developed to study spatially embedded networks for one and two dimensions [14–16]. In the KHB model links are created randomly over nodes of a square lattice and the probability of linking nodes decays as a power-law with their distance. In a mathematical formulation we have:

P

(

r

) ∝

r−δ (1)

for P

(

r

)

the probability to have a connection between two nodes situated at distance r. The regime of small

δ

characterizes a network with connections extended to long distances while large

δ

defines the short range regime. The standard example of long range interactions is the Erdös–Rényi model while the short range is typically of a regular lattice with only local connections. We work in this paper with a special subset of the embedded networks that fulfil condition(1)—the Projected Lucena Network PLN. This network, besides following(1)is non crossing, or planar, but before we examine its properties we will present some of the story of the Lucena network.

The Lucena network starts in the paper [17] that introduces a peculiar multifractal partition of the square, that means, the area lattice distribution of the blocks of the partition follows a multifractal distribution. A review of the properties of multifractal lattice is found in Ref. [18]. The Lucena network is by construction a graph that uses the connectivities (neigh-bourhood) of the multifractal lattice elements; in this sense Lucena network is the dual of a multifractal lattice. In contrast to the KHB network the PLN is planar. Indeed, the links in the KHB network can cross each other while by construction PLN links never cross. In addition, the PLN follows a deterministic algorithm while the KHB network is essentially aleatory. The geometric characteristics of the PLN make this object an interesting candidate to integrated circuits design [19].

In this paper we study some fractal properties of the PLN model. Our intention is to study short and long range behaviour in the PLN model and its implications with fractal properties of the network. As we shall see the PLN is a transition model between these two limits. In Section2we present in some detail the multifractal lattice, the Lucena network and its projection into the square lattice. In Section3we show the main results of our manuscript: we compute

δ

of relation

P

(

r

) ∝

r−δ_{and estimate the fractal dimension d}

ffor several

ρ

, the internal parameter of the PLN model. Finally in Section4

we discuss our results in the light of other spatial constraint networks of the literature.

2. The construction of the Projected Lucena Network

Initially we present the algorithm of construction of the Lucena multifractal. In the sequence we show the Lucena network which is the dual of the multifractal lattice and its projection into the square lattice. The algorithm of the multifractal lattice starts with a square and a section ratio 0

< ρ <

1, a free parameter of the partition. The first step of the algorithm, n

=

1, consists of two sections of the square: a vertical and a horizontal following the same ratio

ρ

. At the second step the same operation is repeated inside each one of the four blocks. In this way, at step n there are 22ntiles. A schematic picture of the algorithm is illustrated inFig. 1for n

=

1 and 2. Refs. [20,18] show a detailed discussion about internal symmetries, algebraic properties and topology of the multifractal lattice. InFig. 1(b) we notice that the patterns ofFig. 1(a) are repeated inside each one of the four blocks, however a rotational freedom in this operation is possible. The multifractal lattice can be deterministic and follow one specific rotation pattern or assume a random rotation in the interactive process of the algorithm. In this work we assume, otherwise stated, a constant rotational pattern in the algorithm. In contrast, Refs. [21,22] explore a full random version of this algorithm for an aleatory parameter

ρ

.

c

_d

Fig. 2. A picture of the Lucena network (a) and (c) together with its projection into the square lattice (b) and (d). We illustrate the spatial constrained

networks for two parametersρto highlight differences between symmetric and asymmetric networks. In (a) and (b) we show a more homogeneous

networkρ =5/6 (a) and (b) while in (c) and (d) it is shown a marked anisotropic partitionρ =1/4; for both cases it is used n=4. Despite the Lucena

network and the PLN have the same topology we explore the fractal properties of the PLN that are dependent on the embedding space.

The section ratio

ρ

regulate the asymmetry of the multifractal. The disparity between the largest and the smallest lattice areas increases as

ρ →

0 where lattice elements get increasingly stretched. The other limit,

ρ →

1, corresponds to a more symmetric object, indeed, in this limit the partition degenerates into the regular square lattice.Fig. 2illustrates two realizations of Lucena network for

ρ =

5

/

6 (a) and

ρ =

1

/

4 (c). The homogeneous connectivity of (a),

ρ →

1, contrasts with the asymmetry shown in (c),

ρ →

0.

The asymmetry of the multifractal is also reflected in the distribution of connectivity of Lucena network. Moreover, Ref. [18] brings the distribution of connectivity of the multifractal lattice that has the same connectivity pattern of the Lucena network. The analysis in Ref. [18] includes several lattice sizes and ratio parameters

ρ

. For any

ρ

most of lattice blocks have four or five connections while a minority of nodes follows a fat-tail distribution of connectivity. The case

ρ →

1 shows a cut off in the fat tail in opposition to

ρ →

0 for which the fat tail is extended over several orders of magnitude. To illustrate the density of highly connected nodes we compute the number of nodes with connectivity above 6 and above 12. The index

fk>k⋆is defined as the fraction, f , between the number of highly connected nodes (k⋆

>

6 and 12 in our simulations) by the

total number of nodes of the network;Table 1shows this quantity.

In the algorithm of construction of Lucena network, the number of nodes evolves with step n as 4

,

16

, . . . ,

22n, seeFig. 1. This underlying geometry suggests a straightforward projection of Lucena network into a regular square lattice with size

L

=

2n

₋

_{1. We call Projected Lucena Network PLN the geometric transformation of the Lucena network into the square}

lattice. In this way, the PLN is an embedded network whose vertices coincide with the vertices of a square lattice, the links, however, follow a singular distribution. Therefore, the dimension of the embedding space of the PLN is always 2. In the subsequent section we show that the PLN follows surprising dimensional properties. For sake of clarityFig. 2presents a couple of Lucena networks together with their projections into the square lattice. It is worth of note that the PLN in the limit of

ρ →

1 resembles a regular pattern while in the limit

ρ →

0 it shows long range interactions along preferential symmetric lines of columns with high density of connections.

Table 1

The asymmetry parameterρ, the fractal coefficient of the

distribution of distance parameterδ, the fractal distance df,

and the fraction of highly connected nodes fk>6and fk>12.

ρ δ df fk>6 fk>12 1/24 3.38±0.016 2.47±0.02 0.103 0.01 1/12 3.27±0.016 2.43±0.02 0.103 0.01 1/4 3.30±0.03 2.09±0.01 0.105 0.01 1/2 3.37±0.05 2.06±0.01 0.124 0.002 3/4 3.60±0.08 2.04±0.01 0.073 0.00 8/10 3.87±0.10 2.03±0.01 0.080 0.00 9/10 3.91±0.11 2.0±0.09 0.063 0.00

3. Fractal properties of the Projected Lucena Network

In this section we explore the fractal properties of PLN, we perform an analysis of the distribution of Euclidean distances of the nodes and in the sequence we study its fractal dimension. We start analysing the behaviour of the distribution of distances P

(

r

)

.Fig. 3(a) shows, for

ρ =

9

/

10 and 1

/

4, the graphics of P

(

r

)

versus distance r. The exponents

δ

of the power-law depends on the asymmetry of the network and the consequent node connections distribution. InFig. 3(d) we explore the size scale effects on the computation of

δ

. As we can see in this figure, increasing lattice size extends the range of r in the analysis and as a consequence the estimation of

δ

becomes more robust. InTable 1we display a couple of

δ

values for several

ρ

. In this analysis, to compute the error and to improve the statistics we use a random internal rotation of the pattern, that means, in the algorithm of the multifractal lattice we keep constant the section

ρ

, but randomly rotate the section pattern inside the blocks.

To obtain the fractal dimension of the PLN we use a rather general relation. We compute the mass M, in our case the number of nodes, within a hemisphere of radius r. A fractal object, by definition, follows the scale:

M

∝

rdf ₍₂₎

where the exponent d_f represents the dimension of the network. It is important to note that in the computation of M we take into account only the connected nodes inside a radius r, if we count any node df would be trivially the dimension of

the embedding space. Ref. [15] shows a detailed presentation of this algorithm, including an explanatory figure.

Fig. 4illustrates the computation of df. InFig. 4(a), (b) and (c) we show the graphic of M against r for three distinct

ρ

as

illustrated in the legend. The right panel,Fig. 4(d), (e) and (f) shows the finite size effect of this computation. In the limit of

ρ →

1, the PLN assumes a regular lattice pattern and df goes to 2, seeFig. 4(a). In addition the power law range in

this regime extends over the full distance r. The opposite limit,

ρ →

0, the range for estimation of the fractal dimension decreases in r, as illustrated inFig. 4(c).

InTable 1we show all studied values of

ρ

together with the estimated

δ

and df, for all simulations we use n

=

11.

In addition, we depicted the fraction of nodes with high connectivity or fk>6and fk>12to illustrate the presence of highly

connected nodes for diverse

ρ

. All these parameters:

δ

, dfand fk>k⋆are correlated to the asymmetry parameter

ρ

. 4. Final remarks

The PLN has an intrinsic parameter

ρ

that controls its asymmetry. In the

ρ →

1 limit the PLN resembles a regular lattice with df

=

2 and

δ =

4, the topological dimension of the embedding space and a typical short range interaction. On the

Fig. 4. Curves for the estimation of the fractal dimension df of the PLN. The coefficient df is the slope of M(r), the quantity of connected sites inside a

radius of size r. In (a), (b) and (c) we plot M(r)versus r, in a log–log scale, forρ = 9 10,ρ =

1 2andρ =

1

6, respectively. For this particular picture we use

n=11. In (d), (e) and (f) we analyse finite size effect for n=9,10 and n=11, respectively; in this set of figures we useρ = 1₃.

opposite limit,

ρ →

0, the asymmetry increases and the PLN becomes a long range like network with

δ

assuming values close to 3 and the topologic dimension rising up to 2.5. This behaviour is associated with the presence of hubs that goes to zero in the limit of

ρ →

1 limit and increases dramatically for

ρ →

0. In this way, by fitting

ρ

, the PLN discloses a rich behaviour ranging from an almost regular lattice to a complex network that follows the small world principle: a rich class of short-range connections with a few long range interactions [23].

The original algorithm of the KHB model that created long and short range connection by tuning

δ

in relation(1)uses an embedded square lattice and keeps the connectivity as almost constant. This strict condition over the connectivity is not necessary, in this way we can figure out that the PLN is a singular class of graphs among the embedded networks that follow relation(1). Besides the PLN be originated from the dual of a multifractal tilling, what makes PLN special is that it is a planar realization of Eq.(1). If another class of planar networks fulfil Eq.(1)remains an open question. Moreover the PLN is a rich structure that opens many questions in planar networks, we intend to explore in a future work the optimal

ρ

that maximize transport properties [24–27] in the PLN.

The relation(1)was introduced to create models that express a power law distribution for the probability of finding con-nected node according to the distance. Indeed, Eq.(1)is verified for some real embedded networks, for instance, in global airline network the probability that two airports have a direct link at distance r is

δ =

3, [6], and in the mobile phone con-nection network, the probability P

(

r

)

to have a friend decays with

δ =

2, [28]. In the KHB model, the relation P

(

r

) ∝

r−δ_{is a}

key ingredient to tune long and short range interaction in embedded networks. In this work we verified that fractal relation

(1)naturally appears in a network that follows a construction algorithm originated from a lattice partition. For the best of our knowledge this is the first time this fractal propriety is not trivially verified in a theoretical network.

Acknowledgements

Financial support from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FINEP is acknowl-edged. Research carried out with the aid of the Computer System of High Performance of the International Institute of Physics UFRN, Natal, Brazil.

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