Fractality of multiple colored substitution networks

Accepted Manuscript Fractality of multiple colored substitution networks Ziyu Li, Jialing Yao, Qin Wang

PII: DOI: Reference:

S0378-4371(19)30314-0 https://doi.org/10.1016/j.physa.2019.03.079 PHYSA 20714

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Physica A

Received date : 25 January 2019 Please cite this article as: Z. Li, J. Yao and Q. Wang, Fractality of multiple colored substitution networks, Physica A (2019), https://doi.org/10.1016/j.physa.2019.03.079 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights (for review)

1. We construct substitution networks of multiple colors. 2. By ergodic theory a constant exists as the growing speed of the networks. 3. The fractality of multiple colored substitution networks is obtained.

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FRACTALITY OF MULTIPLE COLORED SUBSTITUTION NETWORKS ZIYU LI, JIALING YAO, AND QIN WANG Abstract. For multiple colored initial graphs, we can construct evolving selfsimilar networks by substitution rules. In this paper, we obtain the fractality of our substitution networks through the estimates on the cardinality of the minimal covering of boxes.

1. Introduction The research of complex networks derives from Watts and Strogatz [1] and Barab´ asi and Albert [2]-[4], who presented dynamics of small-world networks and scale-free effect of random networks respectively. Afterwards Newman researched the structure of complex networks [5]-[9]. As one of the most influential founders, Mandelbrot initiated the study of fractals [10]. Song et al. brought notion of fractals into the study of complex networks [11]-[14]. Rozenfeld and Havlin et al. [15] explored the self-similarity and fractal dimension of fractal networks. A great number of applications of substitution [16] have been proven to benefit several subjects such as chemistry [17], physics [18], bioinformatics [19], information science [20] and so on. In particular, the self-similar substitution has became a powerful tool to study complex networks [21]-[23]. It is of great interest to study the self-similar and fractal properties of complex networks generated by substitution rule. Xi et al. [24] started the research of scalefree and fractality for substitution networks based on one-colored initial graph. Moreover Li and Xi [25] obtained the scale-free effect by setting up a new system of multiple colored substitution. We begin with the notation of multiple colored substitution networks in [25]. (1) Suppose there are initial colored graphs {Gi = (Vi , Ei )}m i=1 and fix {Ai , Bi } ⊂ Vi (1) for every i. Let di denote the geodesic distance between Ai and Bi in Gi . The (t) substitution network {Gi }m i=1 are constructed inductively by replacing every i(1) (t−1) m }j=1 with Gi , where nodes of this edge are replaced by Ai colored edge in {Gj and Bi . To investigate the fractality, we do not need the directed graphs as in [24] and [25]. Sun et al. [26] researched the fractality of evolving substitution networks, in the case that d1 = d2 = · · · = dm .

(1.1)

2000 Mathematics Subject Classification. Primary 28A80. Key words and phrases. fractal networks, fractality, substitution networks. Qin Wang is the corresponding author. The work is supported by National Natural Science Foundation of China (Nos. 11831007, 11771226, 11371329, 11471124). 1

2

ZIYU LI, JIALING YAO, AND QIN WANG

G 1(1)

color 1

A1

B1

A1

G1

A2

B2

A2

B1

A1

B1

G 2(2)

G 2(1)

color 2

(2)

A2

B2

B2

Figure 1. An example of substitution networks In this paper, we will deal with the fractality in the general case without (1.1). (t) Suppose {Gi }m i=1 are substitution networks of m colors, where the geodesic dis(t) tance between Ai and Bi is di . Moreover, we let di be the geodesic distance (t) between Ai and Bi in Gi for each i. For calculating the cardinality of edges in (t) Gi , we introduce the color matrix Mm×m = (aij )1≤i,j≤m where aij represents (1) the cardinality of j-colored edges in Gi . Let γ be the spectral radius or PerronFrobenius eigenvalue of M . To use box counting method, we recall that the l-box B(l) is a node subset such that the geodesic distance between any two nodes in (t) the subset is strictly less than l, and at the same time we let Ni (l) denote the (t) (t) (t) minimum cardinality of l-boxes needed to cover Gi = (Vi , Ei ). In fact we have the following results. Proposition 1. Given any general colored graphs, there exists a constant α such that for any ε, (t) αt ≤ max di ≤ (α + ε)t i

for all t ≥ tε where tε depends on ε.

Theorem 1. If Mm×m is irreducible with spectral radius γ > 1 and α ≥ 2, then (t) substitution networks {Gi }t have the fractality, that is (t)

♯Vi

(t) Ni (l)

log γ

∼ l− log α .

Remark 1. When di ≥ 2, then α ≥ 2 (see Lemma 4). Example 1. In Fig. 1, we demonstrate substitution rules of two colors and rela(t) tionship between the number of boxes Ni (l) and size l of boxes by volume-greedy  5 4 ball-covering algorithm [28]. With the definitions we have M = and as 2 1 shown in Fig. 2, we obtain the fractality.

FRACTALITY OF MULTIPLE COLORED SUBSTITUTION NETWORKS t=4

slope= - 1.3349

slope= - 1.3383

minimum number of boxes needed to cover

minimum number of boxes needed to cover

t=3

3

size of box

size of box

Figure 2. Fractal dimension by VGBC algorithm for t = 3, 4 2. Preliminaries 2.1. Coding of edges and nodes. (t) (t) (t) Suppose Gi = (Ei , Vi ). Let c be the color mapping such that c(e) ∈ (t) {1, · · · , m} is the color of edge e. As in [25], for any edge in Ei , we have a unique admissible coding e = ei1 ei2 · · · eit . (1)

In the admissible coding, we note that eij belongs to the graph Gc(ei ) , which j−1 implies that the j-th edge belongs to the the initial graph determined by the color of the (j − 1)-th edge. (t) For any node in Gi , we also have a unique admissible coding v = ei1 ei2 · · · eit−1 v with v ∈ Vu(1) \{Au , Bu } for u = c(eit−1 ). 2.2. Cardinality of edges and nodes. Recall the color matrix M and its spectral radius γ. Owing to matrix multiplica(t) (t) tion we acquire that the element aij in the matrix M t = (aij )1≤i,j≤m represents (t)

(t)

(t)

the cardinality of j-colored edges in Gi . For simplicity let ♯Ei and ♯Vi be the (t) number of edges and nodes in Gi . We let εi denote the vector with a 1 in the i-th coordinate and 0’s elsewhere. We obtain that

(t) (2.1) ♯Ei = εi M t 1 , (1)

where k·k1 is the 1-norm. Set V = (♯V1 (t)

(t−1)

♯{Vi /Vi

(1)

− 2, ♯V2

(1)

− 2, · · · , ♯Vm − 2), we have

} = εi M t−1 VT .

In terms of above results, we have Xt Xt (j−1) (j) (t) )+2=2+ ♯(Vi /Vi ♯Vi =

j=1

j=1

εi M j−1 VT .

(2.2)

(t)

Therefore a further lemma is needed to estimate the ♯Vi .

Lemma 1. Suppose M is irreducible with spectral radius γ and for any positive vector w > 0, we have

t t

c−1 ≤ cw γ t for all t, w γ ≤ wM 1

where cw is a constant dependent on w.

4

ZIYU LI, JIALING YAO, AND QIN WANG

Proof. Let u > 0 with kuk1 = 1 be the left Perron-Frobenius eigenvector of M , that is, uM = γu. Then there is a constant cw such that c−1 w u ≤ w ≤ cw u and

t

uM = γ t , 1

which implies





−1 t t t t t

c−1 w γ = cw (uM ) 1 ≤ wM 1 ≤ cw (uM ) 1 = cw γ for all t.

The lemma follows.



In conclusion, we have Proposition 2. For all t, we have (t)

(t)

c−1 γ t ≤ ♯Ei , ♯Vi

≤ cγ t for all t and i,

where c > 0 is a constant. P Proof. Let w = i εi . It follows from (2.1) and (2.2) that (t)

≤ ≤ cw γ t ,

♯Ei

(t)

≤ 2 + ||VT ||∞

♯Vi for some constant

c′w .

Xt

j=1



wM t ≤ c′ γ t , w 1

On the other hand, we only need to show

εi M t ≥ c−1 γ t for all t and i. 1

Now, since M is irreducible, there is an positive integer bi,j such that εi M bi,j ≥ εj .

Hence

||εj M t ||1 ≤ ||(εi M bi,j )M t ||1 = ||(εi M t )M bi,j ||1 ≤ ||M bi,j || · ||εi M t ||1 P where the matrix norm ||(cij )|| = i,j |cij |. That means X  X

t t

c−1 ||εj M t ||1 = ||M bi,j || · ||εi M t ||1 . w γ ≤ wM 1 = j

j

The proposition follows.



2.3. Estimate of geodesic distance. (t) (t) Now, we shall estimate the geodesic distance di between Ai and Bi in Gi . Let (t)

D(t) = maxi di . We have Lemma 2. D(k1 +k2 ) ≤ D(k1 ) · D(k2 ) for all k1 and k2 . (k )

Proof. Suppose in the graph Gi 1 , there is a path η connecting Ai and Bi such that the length of η is equal or less than D(k1 ) . After k2 times of substitution (or (k ) iteration), every edge e in η with color j is turned into a copy of Gj 2 whose two (k2 ) entrances has geodesic distance equal or less than D . Link these two entrances with a geodesic path for every edge e, we obtain a path connecting Ai and Bi , and its length is equal or less than D(k1 ) · D(k2 ) . 

FRACTALITY OF MULTIPLE COLORED SUBSTITUTION NETWORKS

5

The following lemma comes from the standard result in ergodic theory, e.g. see [29]. Lemma 3. If ak1 +k2 ≤ ak1 + ak2 , then there exists a constant β such that ak ak = inf = β. lim k k k→∞ k Now we return to Proof of Proposition 1. Let ak = log D(k) , then by Lemmas 2-3, we have log D(k) log D(k) = inf =α k→∞ k k k for some α. In other words, we obtain that lim

(t)

αt ≤ max di ≤ (α + ε)t i



for all t ≥ tε where tε depends on ε. 3. Proof of Theorem 1 (t)

In this section, according to the estimate on ♯Vi (t) we only need to estimate Ni (l).

in Proposition 2, to estimate

♯Vi,t (t) Ni (l)

At first, using the structure of graph, we have

Lemma 4. α ≥ mini di .

(t)

We also get the estimate of the diameter of Gi . Lemma 5. Fix ε > 0. If α ≥ 2, then (t)

αt ≤ diamGi ≤ cε (α + ε)t for all i and all t ≥ tε ,

where cε > 0 is a constant dependent on ε. Proof. In fact, we obtain that (t)

(t)

diamGi ≥ di ≥ αt .

On the other hand, using the similarity we obtain that        (t−1) (t) (t) (t) . + 2 max diamGi max di ≤ max diamG1 max diamGi i i i i   (t) Now let ct = maxi diamGi . Suppose D(k) ≤ (α + ε)k for any k ≥ nε , we have ct

≤

≤ ≤ ≤ ≤

≤

c1 D(t) + 2ct−1

c1 D(t) + 2c1 D(t−1) + 22 ct−1  Xt  X 2t−k D(k) + 2t−k D(k) + 2t c1 c1 k=nε k
cε (α + ε)t

6

ZIYU LI, JIALING YAO, AND QIN WANG



when t ≥ tε is large enough since α + ε > 2.

Proof of Theorem 1. (t) (1) We first give the upper bound of Ni (l). For two admissible edge sequence e and f , we say e ≺ f if e is the prefix of f if e = ei1 ei2 · · · eik and f = ei1 ei2 · · · eik · · · eik′ with k < k ′ . Given 1 ≤ i ≤ m, we let Ai,k = {e : e is an admissible edge sequence of length k}

and the box (1)

Bi,e = {f v: e ≺ f =ei1 ei2 · · · eit−1 v and ei1 ∈ Gi }, (t)

then {Be }e∈Ai,k is a box covering of Gi . Suppose e =ei1 ei2 · · · eik and the color (t−k) ) satisfying of eik is j, then the diameter of Be is exactly diam(Gj (t−k)

αt−k ≤ diam(Gj

) ≤ cε (α + ε)t−k .

Now for l = 2cε (α + ε)t−k , we have (k)

diam(Be ) < l and ♯Ai,k = ♯Ei

≤ cγ k

by Proposition 2. Hence (t)

Ni0 (l) ≤ ♯Ai,k ≤ cγ k for l = (2cε )(α + ε)t−k .

(3.1)

(t)

(2) We will obtain a lower bound of Ni (l). (k) (t−k) (t−k) = di0 . Consider the graph Gi0 and iterate (t − k) times Suppose maxi di (t)

(k)

(k)

for every edge of Gi0 , then we obtain Gi0 composed of ♯Ei0 small worlds such (t−k)

that every small world has two entrances with geodesic distance di0 (t−k)

with

≥ αt−k .

di0

(k)

Given a small world We with e ∈ Vi0 , consider its two entrances, we can take a geodesic path in the small world with vertices sequence a0 a1 · · · aue with ue = (t−k) di0 , select an approximately middle point  aue /2 if ue is odd, (e) a = a(ue +1)/2 if ue is even. (t−k)

Since the geodesic distance between a(e) and any entrance is larger than di0 t−k

α

/3, then {a

(e)

}e∈V (k) is subset of i0

(t) Vi0

such that

/3 ≥

′

dt (a(e) , a(e ) ) ≥ αt−k /3 (t)

where dt is geodesic distance on Gi0 . Hence (t)

(k)

Ni0 (l) ≥ ♯Vi0

≥ c−1 γ k for l = αt−k /3.

Since the color matrix M is irreducible, given i ∈ {1, · · · , m}, there is a positive (bi,i ) integer bi,i0 such that Gi 0 contains at least one edge of color i0 , hence we obtain that (k−bi,i0 ) (t) (3.2) ≥ (cγ bi,i0 )−1 γ k for l = αt−k /(3αbi,i0 ). Ni (l) ≥ ♯Vi0

FRACTALITY OF MULTIPLE COLORED SUBSTITUTION NETWORKS (t)

(3) According to the estimate on ♯Vi

7

in Proposition 2, we have (t)

c−1 γ k ≤ ♯Vi

≤ cγ k .

(3.3)

Using (3.1)-(3.3) and letting ε → 0, we get the fractality of the multiple colored substitution networks without the assumption (1.1). 

4. Conclusion In multiple colored substitution networks, we obtain the self-similarity and fractality. The core of fractality is to estimate the growing speed of diameters of networks. Using ergodic theory we claim that there exists a constant α satisfying log D(t) → α (as t → ∞) and consequently obtain Theorem 1. As a conclusion we t extend substitution networks to multiple colored substitution networks, which can simulate more general phenomena in bioinformatics, computer science and physics in that there are usually more than one laws occurring during physical or chemical process. With proper models, multiple colored substitution networks can be applied widely. References [1] D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442. [2] A. L. Barab´ asi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509–512. [3] A. L. Barab´ asi, R. Albert, H. Jeong, Mean-field theory for scale-free random networks, Phys. A 272(1-2) (1999) 173-187. [4] A. L. Barab´ asi, R. Albert, and H. Jeong, Scale-free characteristics of random networks: The topology of the World Wide Web, Phys. A 281 (2000) 69–77. [5] M. E. J. Newman, The structure and function of complex networks, SIAM review 45 (2003) 167–256. [6] M. E. J. Newman, Analysis of weighted networks, Physical review E, 70 (2004) 056131. [7] M. E. J. Newman, M. Girvan, Finding and evaluating community structure in networks, Physical. Review. E 69 (2) (2004) 026113. [8] M. E. J. Newman, Modularity and community structure in networks, Proceedings of the national academy of sciences 103.23 (2006) 8577–8582. [9] M. E. J. Newman, Networks: An Introduction. Oxford: Oxford University Press 2010. [10] B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 156 (1967) 636–638. [11] C. Song, S. Havlin, H. A. Makse, Complex networks are self-similar, Nature 433 (2004) . [12] C. Song, S. Havlin, H. A. Makse, Self-similarity of complex networks, Nature 433 (2005) 392–395. [13] C. Song, H. A. Makse, Origins of fractality in the growth of complex networks, Nature Physics 2 (2006) 275–281. [14] C. Song, L. K. Gallos, S. Havlin, A. M. Hern´ an, How to calculate the fractal dimension of a complex network: the box covering algorithm, Journal of Statistical Mechanics 3 (2007) P03006. [15] H. D. Rozenfeld, S. Havlin, D. Ben-Avraham, Fractal and transfractal recursive scale-free nets, New J. Phys. 9(6) (2007) 175–175. [16] V. Berth, M. Rigo (Eds.), Combinatorics Automata and Number Theory. Cambridge: Cambridge University Press 2010. [17] F. Leroux, A. Mar A, D. Guyomard, Y. Piffard, Cation Substitution in the Alluaudite Structure Type: Synthesis and Structure of AgMn3 (PO4 )(HPO4 )2 , Journal of Solid State Chemistry, 117(1) (1995) 206–212.

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