Correlation between the electrical transport performance and the communicability sequence entropy in complex networks

Chen Dan; Shi Dan-Dan; Pan Gui-Jun

doi:10.7498/aps.68.20190230

Abstract

Optimization of the network’s electrical transport properties not only conduces to understanding the relationship between structure and network function, but also can improve the electrical engineering technology. The effective way to solve this problem is to treat the network from the information viewpoint and seek the information structure measure which affects crucially the network electrical transport performance. Recent studies have shown that the communicability sequence entropy of complex networks can effectively quantify the global structural information of networks. Based on this measure, the difference between networks can be quantified effectively, and the connotation of communicability sequence entropy is explained. In this paper, we predict that the electrical transport performance of complex networks has a strong correlation with the communicability sequence entropy. For this reason, we mainly study the correlation characteristics of the electrical transport performance and communicability sequence entropy of small-world networks, scale-free networks, degree-correlated scale-free networks, community networks, and IEEE57 and other electrical node networks. The results show that the electrical transport performances of these networks are all a monotonically increasing function of communicability sequence entropy, namely, the communicability sequence entropy, and the electrical transport properties have a positive correlation. Specifically, in the process evolving from a regular network to a small-world network, the communicability sequence entropy and electrical transport performance of the network increase gradually. For scale-free networks, in the process of increasing degree distribution exponent, communicability sequence entropy and electrical transport performance of the network increase gradually. For degree-correlated scale-free networks, during the evolution from assortative to disassortative topology, communicability sequence entropy and electrical transport performance both decrease gradually. For networks with community structure, the communicability sequence entropy and electrical transport performance decrease with the increase of the number of communities. Finally, the correlation between communicability sequence entropy and electrical transport performance of two classical node power supply networks and corresponding randomization network models are also studied. The results show that as the order of d increases, both communicability sequence entropy and electrical transport performance decrease. And both are getting closer to the original network's communicability sequence entropy and electrical transport performance. The rule is beneficial to providing an effective strategy for designing a high transmission efficiency of the power network, that is, we can optimize the electrical transport performance by improving the network communicability sequence entropy.

Keywords:

Authors and contacts

Faculty of Physics and Electronic Science, Hubei University, Wuhan 430062, China

Corresponding author: Pan Gui-Jun, pangj8866@hubu.edu.cn

Funds: Project supported by the Education Foundation of Hubei Province, China (Grant No. D20120104).

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References

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[2]	汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用(北京: 清华大学出版社) 第1页 Wang X F, Li X, Chen G R 2006 Complex Network Theory and Application (Beijing: Tsinghua University Press) p1 (in Chinese)
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[26]	黄丽亚, 霍宥良, 王青, 成谢锋 2019 物理学报 68 018901 Google Scholar Huang L Y, Huo Y L, Wang Q, Cheng X F 2019 Acta Phys. Sin. 68 018901 Google Scholar
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[37]	Wu J J, Gao Z Y, Sun H J 2006 Phys. Rev. E 74 066111 Google Scholar
[38]	Gao J, Barzel B, Barabási A L 2016 Nature 530 307 Google Scholar
[39]	Newman M E J 2002 Phys. Rev. Lett. 89 208701 Google Scholar
[40]	Johnson S, Torres J, Marro J, Munñoz M A 2010 Phys. Rev. Lett. 104 108702 Google Scholar
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Cited By

图 1 完全网络任意一对节点之间等效电导的计算过程

Figure 1. The calculation of equivalent conductance between any pair of nodes in a complete network.

DownLoad: Full-Size Img PowerPoint

图 2 WS小世界网络的(a)通信序列熵$S_N$和(b) 平均全局电导$\left\langle G \right\rangle$与重连概率$P_{\rm rew}$的依赖关系; (c), (d)相应的$S_N$与$\left\langle G \right\rangle$之间的关系

Figure 2. Dependence of (a) communicability sequence entropy $S_N$ and (b) mean global conductance $\left\langle G \right\rangle$ on rewiring probability of WS small-world network; (c) , (d) the relation between $S_N$ and $\left\langle G \right\rangle$ of WS small-world network.

DownLoad: Full-Size Img PowerPoint

图 3 无标度网络的(a)通信序列熵$S_N$和(b)平均全局电导$\left\langle G \right\rangle$与度分布指数$\gamma$的依赖关系; (c), (d) 将$\left\langle G \right\rangle$与$S_N$映射在同一坐标系下的结果

Figure 3. The dependence of (a) communicability sequence entropy $S_N$ and (b) mean global conductance $\left\langle G \right\rangle$ on degree distribution exponent $\gamma$ of scale-free network; (c), (d) the results of mapping $\left\langle G \right\rangle$ with $S_N$ in the same coordinate system.

DownLoad: Full-Size Img PowerPoint

图 4 度分布指数$\gamma$分别等于(a) 2.5和(d) 3.0时的关联无标度网络的通信序列熵$S_N$与度-度关联系数$r$的依赖关系; 相应的平均全局电导$\left\langle G \right\rangle$与关联系数$r$的依赖关系, $\gamma$分别等于(b) 2.5和(e) 3.0; (c), (f) $\left\langle G \right\rangle$与$S_N$的关系曲线

Figure 4. Dependence of communicability sequence entropy $S_N$ of the scale-free network on the degree-degree correlation coefficient $r$, here, the degree distribution exponent $\gamma$ is equal to (a) 2.5 and (d) 3.0, respectively; (b), (e) the dependence of the mean global conductance $\left\langle G \right\rangle$ on the correlation coefficient $r$; (c), (f) the $\left\langle G \right\rangle$ and $S_N$ relation curve.

DownLoad: Full-Size Img PowerPoint

图 5 以BA网络为例构建的社团网络可视化图, 社团个数为1—6, 分别表示为C1—C6. 图中每一个网络中社团网络的生成方式都是按照文献[7]的方法生成. 网络规模$N = 900$, 网络边数$E \approx 2700$. 具体算法如下: 1)首先生成一个含有900个节点, 2700条边的BA网络(图C1); 2)生成两个含有450个节点、1350条边的BA网络, 然后在每个社团中随机断开少量的边, 并将这些断开的边连接到其他社团中断开的边的端点上, 形成一个含有两个社团的网络C2; 3)以此类推, 便可生成含有3, 4, 5, 6个社团的BA网络C3, C4, C5, C6

Figure 5. A visualization of the community network based on the BA network. The number of communities is 1 to 6, which are denoted as C1 to C6. The generation method of the community network in each network in the figure is generated according to the method of Ref. [7]. The network size is $N = 900$, the number of network edges is $E \approx 2700$. The specific algorithm is as follows: 1) First, a BA network with 900 nodes and 2700 edges is generated, as shown in figure C1; 2) two BA networks containing 450 nodes and 1350 edges are generated, and then a small number of edges are randomly disconnected in each community, and these disconnected edges are connected to the endpoints of the interrupted edges of other communities to form a network C2 containing two communities; 3) in this way, BA network C3, C4, C5 and C6 containing 3, 4, 5 and 6 communities can be generated.ntaining 3, 4, 5 and 6 communities can be generated.

DownLoad: Full-Size Img PowerPoint

表 1 含有社团结构的网络[BA (左), ER (右)]通信序列熵$S_N$、平均全局电导$\left\langle G \right\rangle$与社团个数的关系

Table 1. Relationship between communicability sequence entropy $S_N$, mean global conductance $\left\langle G \right\rangle$ and number of communities in networks [BA (left), ER (right)] containing communities.

BA	$S_N$	$\left\langle G \right\rangle$	ER	$S_N$	$\left\langle G \right\rangle$
1(C1)	0.9363	1.9380	1	0.9704	2.2106
2 (C2)	0.8925	1.6168	2	0.9256	1.8006
3 (C3)	0.8703	1.5026	3	0.9010	1.6625
4 (C4)	0.8512	1.4401	4	0.8832	1.5728
5 (C5)	0.8440	1.3929	5	0.8704	1.5203
6 (C6)	0.8409	1.3653	6	0.8642	1.4813

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表 2 电力供需网络以及对应的随机化参考模型的$S_N$和平均电导$\left\langle G \right\rangle$, IEEE57 (左), IEEE118 (右)

Table 2. Power supply network and corresponding randomized reference model $S_N$ and mean global conductance $\left\langle G \right\rangle$, IEEE57 (left), IEEE118 (right).

IEEE57	$S_N$	$\left\langle G \right\rangle$	IEEE118	$S_N$	$\left\langle G \right\rangle$
0阶	0.8744	0.6404	0阶	0.8884	0.7418
1阶	0.8502	0.6302	1阶	0.8692	0.7388
2阶	0.8391	0.6225	2阶	0.8689	0.7086
原始	0.8116	0.5616	原始	0.8029	0.4981

DownLoad: CSV

[1]	Dorogovtsev S N, Mendes J F F 2003 Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford: Oxford University Press) p2
[2]	汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用(北京: 清华大学出版社) 第1页 Wang X F, Li X, Chen G R 2006 Complex Network Theory and Application (Beijing: Tsinghua University Press) p1 (in Chinese)
[3]	Barabási A L 2016 Network Science (Cambridge: Cambridge University Press) p5
[4]	Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47 Google Scholar
[5]	Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175 Google Scholar
[6]	Watts D J, Strogatz S H 1998 Nature 393 440 Google Scholar
[7]	Barabási A L, Albert R 1999 Science 286 509 Google Scholar
[8]	Cohen R, Havlin S 2010 Complex Networks: Structure, Robustness and Function (Cambridge: Cambridge University Press) p26, p181
[9]	Newman M E J 2010 Networks: An Introduction (Oxford: Oxford University Press) p23
[10]	Kleinberg J M 2000 Nature 406 845 Google Scholar
[11]	Roberson M R, Ben-Avraham D 2006 Phys. Rev. E 74 017101 Google Scholar
[12]	Li G, Reis S D S, Moreira A A, Havlin S, Stanley H E, Andrade J S 2010 Phys. Rev. Lett. 104 018701 Google Scholar
[13]	Li G, Reis S D S, Moreira A A, Havlin S, Stanley H E, Andrade J S 2013 Phys. Rev. E 87 042810 Google Scholar
[14]	Pan G J, Niu R W 2016 Physica A 463 509 Google Scholar
[15]	Niu R W, Pan G J 2016 Physica A 461 9 Google Scholar
[16]	Lopez E, Buldyrev S V, Havlin S, Stanley H E 2005 Phys. Rev. Lett. 94 248701 Google Scholar
[17]	Lopez E, Carmib S, Havlin S, Buldyrev S V, Stanley H E 2006 Physica D 224 69 Google Scholar
[18]	Carmi1 S, Wu Z, Lopez E, Havlin S, Stanley H E 2007 Eur. Phys. J. B 57 165 Google Scholar
[19]	Baba A O, Bamaarouf O, Rachadi A, Ez-Zahraouy H 2017 Int. J. Mod. Phys. C 28 1750064 Google Scholar
[20]	Xue Y H, Wang J, Li L, He D, Hu B 2010 Phys. Rev. E 81 037101 Google Scholar
[21]	Asztalos A, Sreenivasan S, Szymanski B K, Korniss G 2012 Eur. Phys. J. B 85 288 Google Scholar
[22]	Oliveira C L N, Morais P A, Moreira A A, Andrade J S 2014 Phys. Rev. Lett. 112 148701 Google Scholar
[23]	蔡萌, 杜海峰, 任义科, 费尔德曼 M 2011 物理学报 60 110513 Google Scholar Cai M, Du H F, Ren Y K, Feldman M 2011 Acta Phys. Sin. 60 110513 Google Scholar
[24]	Hu Y, Wang Y, Li D, Havlin S, Di Z 2011 Phys. Rev. Lett. 106 108701 Google Scholar
[25]	李勇军, 尹超, 于会, 刘尊 2016 物理学报 65 020501 Google Scholar Li Y J, Yin C, Yu H, Liu Z 2016 Acta Phys. Sin. 65 020501 Google Scholar
[26]	黄丽亚, 霍宥良, 王青, 成谢锋 2019 物理学报 68 018901 Google Scholar Huang L Y, Huo Y L, Wang Q, Cheng X F 2019 Acta Phys. Sin. 68 018901 Google Scholar
[27]	Wang B, Tang H W, Guo C H, Xiu Z L 2006 Physica A 363 591 Google Scholar
[28]	吴俊, 谭跃进, 郑宏钟, 朱大智 2007 系统工程理论与实践 27 101 Google Scholar Wu J, Tan Y J, Zheng H Z, Zhu D Z 2007 System Eng. Theor. Prac. 27 101 Google Scholar
[29]	蔡萌, 杜海峰, 费尔德曼 M 2014 物理学报 63 060504 Google Scholar Cai M, Du H F, Feldman M W 2014 Acta Phys. Sin. 63 060504 Google Scholar
[30]	Cai M, Cui Y, Stanley H E 2017 Sci. Rep. 7 9340 Google Scholar
[31]	Braunstein S L, Ghosh S, Severini S 2006 Ann. Comb. 10 291 Google Scholar
[32]	de Domenico M, Biamonte J 2016 Phys. Rev. X 6 041062
[33]	Chen D, Shi D D, Qin M, Xu S M, Pan G J 2018 Phys. Rev. E 98 012319
[34]	Estrada E, Hatano N 2008 Phys. Rev. E 77 036111 Google Scholar
[35]	Estrada E, Hatano N, Benzi M 2012 Phys. Rep. 514 89 Google Scholar
[36]	Estrada E 2012 Linear Algebra Appl. 436 4317 Google Scholar
[37]	Wu J J, Gao Z Y, Sun H J 2006 Phys. Rev. E 74 066111 Google Scholar
[38]	Gao J, Barzel B, Barabási A L 2016 Nature 530 307 Google Scholar
[39]	Newman M E J 2002 Phys. Rev. Lett. 89 208701 Google Scholar
[40]	Johnson S, Torres J, Marro J, Munñoz M A 2010 Phys. Rev. Lett. 104 108702 Google Scholar
[41]	Erdös P, Rényi A 1959 Publ. Math. Debrecen 6 290
[42]	Orsini C, Dankulov M M, Colomer-de-Simón P, Jamakovic A, Mahadevan P, Vahdat A, Bassler K E, Toroczkai Z, Boguñá M, Caldarelli G, Fortunato S, Krioukov D 2015 Nat. Commun. 6 8627 Google Scholar

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