Hierarchical coefficient of a multifractal based network

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Physica A

journal homepage:www.elsevier.com/locate/physa

Hierarchical coefficient of a multifractal based network

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

• There are many classes of networks in the market, each one with their peculiarities.

• We work with the Lucena network—the dual of a multifractal lattice.

• The multifractal is by construction a proper partition of a square according to vertical and horizontal sections.

• The Lucena network shows the hierarchical property, a power-law relation between clustering coefficient and connectivity.

• We work a mathematical demonstration connecting clustering coefficient and connectivity for any scale-free planar network.

a r t i c l e i n f o

Article history:

Received 12 August 2013

Received in revised form 15 October 2013 Available online 26 November 2013

Keywords: Multifractal lattice Hierarchical network Planar graph Apollonius network Space filling network

a b s t r a c t

The hierarchical property for a general class of networks stands for a power-law relation between clustering coefficient, CC and connectivity k: CC∝kβ. This relation is empirically verified in several biologic and social networks, as well as in random and deterministic network models, in special for hierarchical networks. In this work we show that the hierarchical property is also present in a Lucena network. To create a Lucena network we use the dual of a multifractal lattice ML, the vertices are the sites of the ML and links are established between neighbouring lattices, therefore this network is space filling and planar. Besides a Lucena network shows a scale-free distribution of connectivity. We deduce a relation for the maximal local clustering coefficient CCmax

i of a vertex i in a planar graph. This condition expresses that the number of links among neighbour, N△, of a vertex

i is equal to its connectivity ki, that means: N△ = ki. The Lucena network fulfils the condition N△ ≃kiindependent of kiand the anisotropy of ML. In addition, CCmaximplies the thresholdβ =1 for the hierarchical property for any scale-free planar network.

Hierarchical networks are a class of graphs formed by the successive assembling of a same pattern or replication factor [1]. This constructive idea resembles the concept of a fractal but it is not properly the case as pointed in Ref. [2] that has the suggestive title ‘‘Pseudo fractal scale-free web’’. Curiously, the articles that introduced hierarchical networks have not used the hierarchical concept but instead they have explored the deterministic construction process: [2,3]. Indeed, the main appeal of hierarchical networks does not come from their formation algorithm but from a peculiar property that arises from its clustering coefficient distribution.

∗_{Corresponding author. Tel.: +55 84 3215 3422.}

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a

b

c

Fig. 1. The algorithm of Lucena multifractal lattice and network. The thick lines represent the initial square lattice and the vertical and horizontal sections according to the parameterρ =1/2 (the blocks are divided into 3=1+2 segments and the section splits into two parts of length 1 and 2). The thin lines represent the links connecting neighbour sites. Figures (a)–(c) represent steps n=1,2 and 3 respectively. In the passage from n to n+1 each vertex is erased and replaced by four new ones; the number of new connections formed at each step is not trivial. Most junctions among the blocks have a T-like shape (⊤,⊢,⊥or⊣), but some are cross-like (+). The cross-like intersections are related to topological defects.

The small world network as described in Ref. [4] has large average clustering coefficient,

⟨

CC

⟩, and this characteristic

was very important in the beginning of network studies to differentiate it from random networks [5]. However, once we compute the distribution of the cluster coefficient according to connectivity CC

(

k

)

the small world network reveals a plain distribution, that means, the local clustering coefficient of each vertex Vi, CCi does not vary with connectivity. A set of challenging empirical results appears when CC

(

k

)

was computed for cell metabolic networks [1,6], semantic web [7,8] and the World Wide Web [9,7]. All these networks follow the distinctive pattern:

CC

(

k

) ∝

k−β

.

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In addition, the hierarchical network, that is a theoretical model, also presents the same power-law curve, a fact that attracted the attention of the scientific community to this special type of network. Because of Ref. [7] the coefficient

β

was called the hierarchical coefficient, and Eq.(1)the hierarchical property.

On the other hand, hierarchical property(1)does not necessarily imply a hierarchical structure, we cite for instance the scale-free algorithms [10,11] that are not hierarchical. The work [10] proposed a simple and direct model to generate scale-free networks that also shows the clustering property; the model is similar to the usual preferential attachment model, but it introduces an additional step that increases the average clustering coefficient by choosing additional attachment among highly connected vertices.

The paper [12] introduced a partition of the square that forms a multifractal tilling, that means, the area lattice follows a multifractal distribution. A nice review of multifractal lattice properties is [13]. In that paper the multifractal lattice is called a Lucena multifractal in contrast to another bidimensional multifractal object developed in Ref. [14]. In this work we follow the same line and call the Lucena network the graph using the connectivities (neighbourhood) of lattice elements; in this way the Lucena network is the dual of the multifractal lattice. We avoid the term multifractal network because the area of lattice size is multifractal, however the network formed from the lattice topology is not multifractal.

This paper shows that the Lucena network shares the same basic property of hierarchical networks—the power law relation between clustering coefficient and connectivity with

β =

1. In addition, we present evidence that any scale-free planar (non-crossing network) should also obey the hierarchical property. In Section2we review the main properties of a multifractal lattice and explore its basic network characteristics. In Section3we present the curves of the clustering coefficient against connectivity for several values of parameter

ρ

which define the multifractal; in addition a discussion about CCiof bidimensional non-crossing networks is presented. In Section4we conclude the work and discuss similarities between Lucena, Apollonius and hierarchical networks.

2. Construction of the multifractal based network

To show the algorithm of Lucena multifractal we start with a square of size 1 and a given partition parameter 0

< ρ <

1. For convenience we define

ρ =

s

/

r for s and r integers which means that each line segment is divided in r

+

s equal segments

which in the sequence are split in two parts of sizes r and s. The first step, n

=

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

ρ

. In this way the initial square of area one is divided in four blocks of areas:

ρ

2_,

₍

₁

₋

_ρ)

2_{, and two of}

_ρ(

₁

₋

_ρ)

_{. The second step repeats the same procedure inside each of the 4 blocks. Using} this precedence at step n there are 22n_{tiles. A simple picture of this algorithm is shown in}_{Fig. 1}_{for n}

₌

₁

_,

_{2 and 3. A more} involved discussion about the construction of this object is found in Refs. [13,15] the possible rotation of the section cut and, also, lattices with random

ρ

are discussed.

In the Lucena partition of the square the difference between the largest and the smallest lattice areas increases as

ρ →

0, moreover, some lattice elements get more and more stretched in this limit; in this way,

ρ

is a measure of anisotropy. In the opposite limit,

ρ →

1, the partition degenerates into the regular square lattice, a very symmetric object. InFig. 2we show two realisations of Lucena network for

ρ =

1

/

3 (a) and

ρ =

4

/

5 (b). The strong asymmetry of (a), small

ρ

, contrasts with the more balanced lattice area and neighbour connectivity of (b),

ρ

close to one.

The most important property of the Lucena multifractal is that the cutting process segments the initial square into blocks whose area distribution follows a binomial distribution [16]. From this distribution it is possible to calculate the spectrum of

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a

b

Fig. 2. An illustration of Lucena network for two parametersρ. In (a) we show an anisotropic partitionρ =1/3 while in (b) a roughly homogeneous networkρ =4/5; for both situations we use the same network size n=3.

a

b

Fig. 3. Curve of⟨CC(k)⟩versus k for the Lucena network. In (a) we plot the curves for a givenρ =1/12 and many algorithm steps n as indicated in the figure. In (b) we show the graphics of several network parametersρand a fixed size n=6. The property CC(k) ∝k−1_{is illustrated with help of a slope} −1 line.

fractal dimensions for this object: Dk

=

log Cknskrn−k

log(s+r)n/2 where k represents the set of all tiles with same area. The multifractality of the lattice is properly defined in the limit n

→ ∞.

The focus of the present work is the network formed from ML following the rule: the vertices are the lattices and links among vertices are established since lattices touch each other. In this way the Lucena network is formed by vertices that are placed at the centre of the elements of the ML, the links are defined according to the topology: neighbour lattice cells share a connection. For comparison, using this rule the square lattice generates a regular network with all vertices showing four links. The multifractal tilling shows a more intricate topology than regular or Voronoi lattices, in Ref. [17] it is shown that the distribution of connectivity of this object follows a scale free pattern, that means: P

(

k

) ∝

k−γ. Computational experiments show that asymmetric tilling,

ρ →

0, has smaller

γ

, that means, a less sloped distribution that has vertices with large k in the network.

Any tilling of the space that is formed by juxtaposition of blocks without holes between them is a space-filling partition of the space. The definition of a space-filling network is more involved. In the limit n

→ ∞

of multifractal lattice algorithm construction the blocks became infinitesimal and the links among blocks do not only completely cover the space (like a Peano curve), but also they never cross each other. In this sense we call the Lucena network a space-filling and planar graph (non-crossing network).

3. Results: the hierarchical property

In this section we start exploring the clustering coefficient of the Lucena network.Fig. 3depicts the average clustering coefficient for a given connectivity,

⟨

CC

(

k

)⟩

, versus the connectivity itself, k. To perform such statistics we compute for all vertices Viin the lattice two local indices: CCiand ki. After that we average CCiover all vertices with same k to estimate

⟨

CC

(

k

)⟩

. The size effect over the hierarchical property is explored in the next picture.Fig. 3(a) shows

⟨

CC

(

k

)⟩

versus k for

ρ =

1

/

12 and several algorithm steps n. The straight line 2

/

k is depicted in figure indicating an excellent fit. The overall

aspect of this curve points to an absence of size effect in the hierarchical property.

InFig. 3(b) we analyse

⟨

CC

(

k

)⟩

for several values of

ρ

. In this simulation we use a fixed step n

=

6 and display curves of several

ρ

as indicated in the figure legend. The point k

=

2 and

⟨

CC

⟩ =

1 correspond to two vertices in the corner of the

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Fig. 4. Example of a Lucena network, n=4 andρ =2/3, the underlying lattice is now shown. Two vertices and their local communities are highlighted in the figure: one vertex has a maximal number of connections for a planar network N△=k while the other has a topological defectϵ =1.

lattice (upper right and bottom left,Fig. 2). These two vertices have two links that are interconnected, as a consequence, using Eq.(3), the clustering coefficient is maximal. The point k

=

3 corresponds to vertices in the border of the lattice that have three links and, in general, two links between them which produce CCi

=

2

/

3, Eq.(3). The conclusion fromFig. 3 (b) is that hierarchical parameter is not affected by the anisotropic parameter

ρ

. This result is interesting since as

ρ →

0 the network becomes more asymmetric, P

(

k

)

shows a smaller

γ

and as a consequence a larger k is present in the network. In other words, a larger k implies a larger

⟨

CC

(

k

)⟩

disregarding the anisotropy parameter. For both figures we depict an auxiliary straight line with slope

β =

1 to emphasize the curve tendency:

⟨

CC

(

k

)⟩ ∝

k−1

.

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The combined results fromFig. 3(a) and (b) point out that Eq.(2)is a robust property of the Lucena network.

To analyse the hierarchical property in a non-crossing network we analyse a couple of typical vertices and their neighbour communities in a Lucena network. InFig. 4we show a Lucena network with

ρ =

2

/

3 and n

=

4; we select two vertices and their respective closest neighbourhood to illustrate the local clustering coefficient. The definition of local CCi, that means, the clustering coefficient of the vertex Viis done by:

CCi

=

2N△ ki

(

ki

−

1

)

(3) where N△iis the number of connections among the neighbour vertices of Viwhich has a simple topological interpretation:

it is the number of triangles touching vertex V_i. Because the network is non-crossing, a vertex V_iwith connectivity k_iwill have a maximal CCiif N△i

=

ki. In these situations all vertices connected to vertex Viare connected with their neighbours,

additional links among these vertices would produce crossings which are forbidden, a situation illustrated in one of the two highlighted vertices ofFig. 4. In the equation below we show the maximal local clustering coefficient, CCimax, a vertex can have in a planar non-crossing network:

CCimax

=

2N△i ki

(

ki

−

1

)

=

2

(

ki

−

1

)

∝

1 ki

.

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The last proportionality is assumed for large k. We observe that actual vertex Vifollows the condition CCi

<

CCimaxbecause

N△

≤

k. The situation where two vertices connected to Vido not share reciprocal connections, even if they are neighbours, resembles a topological defect. InFig. 4one of the vertices has ki

=

5

=

N△while the other has ki

=

6 but N△

=

5

̸=

6, a

topological defect.

InFig. 4we can check that for most vertices, the number of vertices connected to a given vertex Vi has a number of triangles close to the maximum. Let us assume that the following condition is fulfilled: N△

=

k

−

ϵ

, where

ϵ

is the topological

defect. We interpret

ϵ

as the number of lacking triangles to form the complete sequence of triangles around each vertex. In this case we have

CC

=

2N△ k

(

k

−

1

)

=

2

(

k

−

ϵ)

k

(

k

−

1

)

=

2 k

−

1

−

2

ϵ

k

(

k

−

1

)

(5)

in the limit of k

≫

1 we observe the condition

ϵ ≪

k and as a result we have the equation shown inFig. 3:

CC

=

2

k

.

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The topological defect has a simple geometric interpretation in the underlying ML of Lucena network. In the partition process of the ML most of sites meet at a corner formed by three blocks. However, some blocks meet at a corner formed by four blocks, a pattern that forms a cross in the tilling. It is a standard procedure to look atFig. 1(b) and check for topological defects in the network, we see that there is a four blocks corner behind each topological defect while corners formed by three blocks are not associated with this peculiar asymmetry; indeed, three blocks corners give origin to triangles in the network. In an alternative way: the search for crosses formed at the intersection of blocks is always associated with a topological defect.

4. Final remarks

In this work we show that Lucena network, an object constructed over a multifractal lattice, presents the hierarchical property with

β =

1. The ML is by construction a proper partition of a square according to vertical and horizontal sections. Because links of the Lucena network are defined by lattice neighbourhood they do not cross, in addition, in the limit of infinite partitions the network is space filling. A relation for a maximal local clustering coefficient for vertices in bidimensional non-crossing networks shows the same tendency CCi

∝

k−1. As for each vertex Viwith connectivity kiin a non-crossing network the number of triangles linking Vicannot be larger than kithe consequence of Eq.(4)is that

β

cannot be larger than 1. In this way we conjecture that

β =

1 is the maximal hierarchical parameter for a non-crossing network.

Hierarchical, Apollonius and Lucena networks share in common the hierarchical property with

β =

1 and, also, the fact that all three classes of networks are deterministic by construction. The hierarchical property has a mathematical proof for hierarchical [18] and Apollonius [19] networks while for the Lucena network it lacks a rigorous proof. The rationale behind Eqs.(4)and(5)is, however, a good explanation for hierarchical property in any non-crossing network with scale invariance. In addition, Eq.(5)implies that the network has a constant topological defect disregarding the connectivity. In other words, despite the connectivity of the vertex there is always a fraction lacking triangles that means, in the average, N△

≃

k. In this

context, for the Lucena network, the hierarchical property has a topological interpretation.

The Lucena network shows unique characteristics: it is planar, presents a fat tail and is the dual of a multifractal lattice. Because of these properties it is a promising model to study further network features such as the propagation of gossip [20] and local grow properties [21]. We also intend to analyse the motif distribution in this network that has a large number of triangles and squares [22,23]. A quantitative analysis of this phenomenon is still to be done.

A last note comparing the cited three classes of deterministic networks; the algorithm of the three networks shows an exponential growing number of vertices. The algorithm rule, however is quite diverse in the three situations; hierarchical networks are agglomerative, at each step n

−

1 the same pattern is juxtaposed to the network to form the network at step n. Apollonius and Lucena networks show a divisive algorithm, at step n new links are added inside the network from step n−1. The main difference between these two algorithms is that for the Lucena network, in the passage from n

−

1 to n all vertices are replaced by new ones while the Apollonius algorithm just adds new vertices and links. In the Lucena network algorithm vertices are substituted by new ones, as a consequence the connectivity depends on the neighbourhood in a non-trivial way. For this reason it is more difficult to visualise and describe the Lucena network than their counterparts: the Apollonius and hierarchical networks.

Acknowledgements

Financial support from CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FINEP is acknowl-edged.

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