- Email: [email protected]
Correct degree distribution of apollonian networks
Available online at www.sciencedirect.com
Physics Procedia 00 (2009) 1–3 Physics Procedia 3 (2010) 1791–1793 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Correct Degree Distribution of Apollonian Networks Jin-Li Guo* Wang Li-Na Business School, University of Shanghai for Science and Technology, Shanghai 200093, China Elsevier use only: Received date here; revised date here; accepted date here
Abstract In this paper, we point out that there is a shortcoming of the degree distribution and the analyzing approach of the Apollonian network in [Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 94, 018702 (2005).]. Because the Apollonian network is a deterministic network, its degree distribution can be directly calculated. We correct the degree distribution of the Apollonian network. We also give a numerical simulation of network evolution. The analytical result agrees with the simulation well. The results show that there is the shortcoming of the results of Herrmann et al.
c 2009
2010 Published Publishedby byElsevier ElsevierB.V. Ltd © Key words: scale-free; apollonian networks; power-law distribution
In 2005, Andrade and Herrmann et al. presented a perfect network model: Apollonian network [1], which is a scale-free and displays small-world effect. Unfortunately, the degree distribution in Ref.[1] is faulty. A shortcoming in Ref.[1] is that the analyzing approach is inappropriate . Our analysis is as follows. The structure of an Apollonian network has been given in Ref.[1]. The network grows in a geometric series with t 1 a common ratio 3. At time step t , there are 3 new nodes created. For convenience, nodes of the i th batch are i 1 assigned to numbers. k 0 j t ( j 1,2,3) and k ij t ( j 1,2, ,3 ; i 1,2, , t ) denote the degree of the j th node of the 0th batch and the i th batch respectively. For sufficiently large t , the initial three nodes are neglected. Then one gets k ij t 3 u 2 t i ( j 1,2, ,3i 1 ; i 1,2,, t ) . (1)
For an arbitrary 0 i h , from Eq. (1), one can obtain
k ij (t )
3 u 2 t i ! 3 u 2 t h
k hj (t ) , (2) which means that the degree of the i th batch is larger than the degree of the h th batch. For sufficiently large t , the i 1 possible value of degree k is kij (t ) , j 1,2, ,3 , i 1,2, , t , the number of nodes with degree k k ij (t ) * Project supported by the National Natural Science Foundation of China (Grant No 70871082) and the Shanghai Leading Academic Discipline Project (Grant No S30504). .Corresponding author. E-mail address: [email protected].
c 2010 Published by Elsevier Ltd 1875-3892 doi:10.1016/j.phpro.2010.07.020
1792
J.-L. Guo, L.-N. Wang / Physics Procedia 3 (2010) 1791–1793 J. L, Guo/ Physics Procedia 00 (2010) 1–3
t
¦
i 1
l 1
3 . We define p(k ) to be the is equal to 3 , and the total number of nodes in the network is equal to l 1 i 1 fraction of nodes in the network that have degree k . Therefore, p ( k k ( t )) is the ratio between 3 and ij t l 1 3 . We obtain the degree distribution of the network as follows l 1
¦
p(k )
1
3i 1 t
¦
l
l 1
3 1
ln 3
3
2 ln 2 ln kij | u3 3
2 § 3 · ln 2 ¨ ¸ , k 3©k¹
3,3 u 2,3 u 2 2 , ,3 u 2t 1 .
(3)
Although the degree spectrum of the network is discrete, the above equation indicates that it can be regarded as a “scale-free” network with degree exponent J ln 3 ln 2 . The analytical result agrees with the numerical simulation well (see Fig.1). The authors of in Ref.[1] acknowledge their mistake and the degree exponent is modified by J ln 3 ln 2 (see Ref.[3]). However, their mistake is not an erratum, but it is not appropriate that they used the approach for analyzing the Apollonian network. i 1 For sufficiently large t , the value of degree k is possible kij (t ) ( j 1,2, ,3 , i 1,2, , t ), from (2), we know that the cumulative degree distribution can be obtained as follows: ln 3
pcum (k )
¦ p(k
2 §¨ 3 ¦ ¨ l 1 3 © k lj i
i
)
lj
l 1
· ln 2 ¸ | 3i t ¸ ¹
ln 3
§ 3 · ln 2 (4) ¨ ¸ . ©k¹ ln 3 ln 2 , that is, for the Apollonian
That shows the exponent of the cumulative degree distribution is also J network, the exponent of the degree distribution is equal to the exponent of the cumulative degree distribution. Therefore, in Ref. [1] there is the shortcoming of the analyzing approach and the degree distribution of the Apollonian network, resulting in the similar error in [2,4].
0
10
-2
10
p k
slope = - ln3/ln2 -4
10
-6
10
-8
10
0
10
2
10
4
k
10
6
10
Fig.1. Numerical simulations of network evolution: Degree distribution of the Apollonian network, with t=19 and 193710247 nodes. The analytical result agrees with the simulation well.
Acknowledgments. This paper is firstly submitted as a comment on Ref. [1] before the Publishing Note [3]. This work is supported by the National Natural Science Foundation of China under Grant No. 70871082 and the Foundation of Shanghai Leading Academic Discipline Project (Grant No. S30504).
J.-L. Guo, L.-N. Wang / Physics Procedia 3 (2010) 1791–1793 J. L. Guo/ Physics Procedia 00 (2010) 1–3
References [1] Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 94, 018702 (2005). [2] Soares D J B, Andrade J S, Herrmann H J, et al. Int. J. Mod. Phys. C 17, 1219 (2006) [3] Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 102, 079901 (2009). [4] Doye J P K and Massen C P. Phy. Rev. E 71 016128(2005).
1793
Physics Procedia 00 (2009) 1–3 Physics Procedia 3 (2010) 1791–1793 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Correct Degree Distribution of Apollonian Networks Jin-Li Guo* Wang Li-Na Business School, University of Shanghai for Science and Technology, Shanghai 200093, China Elsevier use only: Received date here; revised date here; accepted date here
Abstract In this paper, we point out that there is a shortcoming of the degree distribution and the analyzing approach of the Apollonian network in [Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 94, 018702 (2005).]. Because the Apollonian network is a deterministic network, its degree distribution can be directly calculated. We correct the degree distribution of the Apollonian network. We also give a numerical simulation of network evolution. The analytical result agrees with the simulation well. The results show that there is the shortcoming of the results of Herrmann et al.
c 2009
2010 Published Publishedby byElsevier ElsevierB.V. Ltd © Key words: scale-free; apollonian networks; power-law distribution
In 2005, Andrade and Herrmann et al. presented a perfect network model: Apollonian network [1], which is a scale-free and displays small-world effect. Unfortunately, the degree distribution in Ref.[1] is faulty. A shortcoming in Ref.[1] is that the analyzing approach is inappropriate . Our analysis is as follows. The structure of an Apollonian network has been given in Ref.[1]. The network grows in a geometric series with t 1 a common ratio 3. At time step t , there are 3 new nodes created. For convenience, nodes of the i th batch are i 1 assigned to numbers. k 0 j t ( j 1,2,3) and k ij t ( j 1,2, ,3 ; i 1,2, , t ) denote the degree of the j th node of the 0th batch and the i th batch respectively. For sufficiently large t , the initial three nodes are neglected. Then one gets k ij t 3 u 2 t i ( j 1,2, ,3i 1 ; i 1,2,, t ) . (1)
For an arbitrary 0 i h , from Eq. (1), one can obtain
k ij (t )
3 u 2 t i ! 3 u 2 t h
k hj (t ) , (2) which means that the degree of the i th batch is larger than the degree of the h th batch. For sufficiently large t , the i 1 possible value of degree k is kij (t ) , j 1,2, ,3 , i 1,2, , t , the number of nodes with degree k k ij (t ) * Project supported by the National Natural Science Foundation of China (Grant No 70871082) and the Shanghai Leading Academic Discipline Project (Grant No S30504). .Corresponding author. E-mail address: [email protected].
c 2010 Published by Elsevier Ltd 1875-3892 doi:10.1016/j.phpro.2010.07.020
1792
J.-L. Guo, L.-N. Wang / Physics Procedia 3 (2010) 1791–1793 J. L, Guo/ Physics Procedia 00 (2010) 1–3
t
¦
i 1
l 1
3 . We define p(k ) to be the is equal to 3 , and the total number of nodes in the network is equal to l 1 i 1 fraction of nodes in the network that have degree k . Therefore, p ( k k ( t )) is the ratio between 3 and ij t l 1 3 . We obtain the degree distribution of the network as follows l 1
¦
p(k )
1
3i 1 t
¦
l
l 1
3 1
ln 3
3
2 ln 2 ln kij | u3 3
2 § 3 · ln 2 ¨ ¸ , k 3©k¹
3,3 u 2,3 u 2 2 , ,3 u 2t 1 .
(3)
Although the degree spectrum of the network is discrete, the above equation indicates that it can be regarded as a “scale-free” network with degree exponent J ln 3 ln 2 . The analytical result agrees with the numerical simulation well (see Fig.1). The authors of in Ref.[1] acknowledge their mistake and the degree exponent is modified by J ln 3 ln 2 (see Ref.[3]). However, their mistake is not an erratum, but it is not appropriate that they used the approach for analyzing the Apollonian network. i 1 For sufficiently large t , the value of degree k is possible kij (t ) ( j 1,2, ,3 , i 1,2, , t ), from (2), we know that the cumulative degree distribution can be obtained as follows: ln 3
pcum (k )
¦ p(k
2 §¨ 3 ¦ ¨ l 1 3 © k lj i
i
)
lj
l 1
· ln 2 ¸ | 3i t ¸ ¹
ln 3
§ 3 · ln 2 (4) ¨ ¸ . ©k¹ ln 3 ln 2 , that is, for the Apollonian
That shows the exponent of the cumulative degree distribution is also J network, the exponent of the degree distribution is equal to the exponent of the cumulative degree distribution. Therefore, in Ref. [1] there is the shortcoming of the analyzing approach and the degree distribution of the Apollonian network, resulting in the similar error in [2,4].
0
10
-2
10
p k
slope = - ln3/ln2 -4
10
-6
10
-8
10
0
10
2
10
4
k
10
6
10
Fig.1. Numerical simulations of network evolution: Degree distribution of the Apollonian network, with t=19 and 193710247 nodes. The analytical result agrees with the simulation well.
Acknowledgments. This paper is firstly submitted as a comment on Ref. [1] before the Publishing Note [3]. This work is supported by the National Natural Science Foundation of China under Grant No. 70871082 and the Foundation of Shanghai Leading Academic Discipline Project (Grant No. S30504).
J.-L. Guo, L.-N. Wang / Physics Procedia 3 (2010) 1791–1793 J. L. Guo/ Physics Procedia 00 (2010) 1–3
References [1] Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 94, 018702 (2005). [2] Soares D J B, Andrade J S, Herrmann H J, et al. Int. J. Mod. Phys. C 17, 1219 (2006) [3] Andrade J S, Herrmann H J, Andrade R F S, et al. Phys. Rev. Lett. 102, 079901 (2009). [4] Doye J P K and Massen C P. Phy. Rev. E 71 016128(2005).
1793