Importance evaluation method of  complex network nodes based on  information entropy and iteration factor

Wang Ting-Ting; Liang Zong-Wen; Zhang Ruo-Xi

doi:10.7498/aps.72.20221878

Abstract

In the study of complex networks, researchers have long focused on the identification of influencing nodes. Based on topological information, several quantitative methods of determining the importance of nodes are proposed. K-shell is an efficient way to find potentially affected nodes. However, the K-shell overemphasizes the influence of the location of the central nodebut ignores the effect of the force of the nodes located at the periphery of the network. Furthermore, the topology of real networks is complex, which makes the computation of the K-shell problem for large scale-free networks extremely difficult. In order to avoid ignoring the contribution of any node in the network to the propagation, this work proposes an improved method based on the iteration factor and information entropy to estimate the propagation capability of each layer of nodes. This method not only achieves the accuracy of node ordering, but also effectively avoids the phenomenon of rich clubs. To evaluate the performance of this method, the SIR model is used to simulate the propagation efficiency of each node, and the algorithm is compared with other algorithms. Experimental results show that this method has better performance than other methods and is suitable for large-scale networks.

Keywords:

Authors and contacts

School of Computer Science, Southwest Petroleum University, Chengdu 610500, China

Corresponding author: Liang Zong-Wen, zongwen-liang@hotmail.com

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References

[1]	Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104 Google Scholar
[2]	Leskovec J, Adamic L A, Huberman B A 2007 Acm Trans. Web 1 5 Google Scholar
[3]	Freeman L C 1978 Soc. Networks 1 215 Google Scholar
[4]	Freeman L C 1977 Sociometry 40 35 Google Scholar
[5]	Sabidussi G 1966 Psychometrika 31 581 Google Scholar
[6]	Lü L Y, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168 Google Scholar
[7]	Lü L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1 Google Scholar
[8]	Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107 Google Scholar
[9]	Lü L Y, Zhang Y C, Yeung C H, Zhou T 2011 PloS One 6 21202 Google Scholar
[10]	Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888 Google Scholar
[11]	Pei S, Muchnik L, Andrade J S, Zheng Z M, Makse H A 2014 Sci. Rep. 4 5547 Google Scholar
[12]	Montresor A, De Pellegrini F, Miorandi D 2011 Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing San Jose, CA, June 6–8, 2011 p207
[13]	Basaras P, Katsaros D, Tassiulas L 2013 Computer 46 24 Google Scholar
[14]	Wang Z X, Zhao Y, Xi J K, Du C J 2016 Physica A 461 171 Google Scholar
[15]	Zhou S, Mondragon R J 2004 IEEE Commun. Lett. 8 180 Google Scholar
[16]	Wang M, Li W C, Guo Y N, Peng X Y, Li Y X 2020 Physica A 554 124229 Google Scholar
[17]	Zareie A, Sheikhahmadi A, Jalili M, Fasaei M S K 2020 Knowledge-Based Syst. 194 105580 Google Scholar
[18]	Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200 Google Scholar
[19]	Hethcote H W 2000 SIAM Rev. 42 599 Google Scholar
[20]	Ma L I, Ma C, Zhang H F, Wang B H 2016 Physica A 451 205 Google Scholar
[21]	Li Z, Ren T, Ma X Q, Liu S M, Zhang Y X, Zhou T 2019 Sci. Rep. 9 8387 Google Scholar
[22]	Bae J, Kim S 2014 Physica A 395 549 Google Scholar
[23]	Bhat N, Aggarwal N, Kumar S 2020 Procedia Comput Sci. 171 662 Google Scholar
[24]	阮逸润, 老松杨, 汤俊, 白亮 2020 71 176401 Google Scholar Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2020 Acta Phys. Sin. 71 176401 Google Scholar
[25]	Colizza V, Flammini A, Serrano M A, Vespignani A 2006 Nat. Phys. 2 110 Google Scholar
[26]	Rui X B, Meng F R, Wang Z X, Yuan G 2019 Appl. Intell. 49 2684 Google Scholar
[27]	Liu D, Jing Y, Zhao J, Wang W J, Song G J 2017 Sci. Rep. 7 43330 Google Scholar
[28]	Namtirtha A, Dutta A, Dutta B 2018 Physica A 499 310 Google Scholar
[29]	Kim H, Anderson R 2012 Phys. Rev. E 85 026107 Google Scholar
[30]	Takaguchi T, Sato N, Yano K, Masuda N 2012 New J. Phys. 14 093003 Google Scholar
[31]	Qu C Q, Zhan X X, Wang G H, Wu J L, Zhang Z K 2019 Chaos 29 033116 Google Scholar
[32]	胡钢, 许丽鹏, 徐翔 2021 70 108901 Google Scholar Hu G, Xu L P, Xu X 2021 Acta Phys. Sin. 70 108901 Google Scholar
[33]	Newman M E J 2006 Phys. Rev. E 74 036104 Google Scholar
[34]	Yin H, Benson A R, Leskovec J, Gleich D F 2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Halifax, Candana) August 13–17, 2017 p555
[35]	Adamic L A 2005 Glance N Proceedings of the 3rd International Workshop on Link Discovery (New York, USA) 2005 p36
[36]	Mcauley J, Leskovec J 2012 Proceedings of the 25th International Conference on Neural Information Processing Systems (Lake Tahoe, Nevada) 2012 p539
[37]	Leskovec J, Huttenlocher D, Kleinberg J 2010 Proceedings of the 19th International Conference on World Wide Web (New York, USA) 2010 p65
[38]	Rozemberczki B, Davies R, Sarkar R, Sutton C 2019 Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining New York, USA, 2019 p65
[39]	Rocha L, Liljeros F, Holme P 2011 PLoS Comput. Biol. 7 1001109 Google Scholar
[40]	Leskovec J, Kleinberg J, Faloutsos C 2007 ACM Trans. Knowl. Discovery Data 1 2 Google Scholar
[41]	Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521 Google Scholar
[42]	Kenall M G 1938 Biometrika 30 81 Google Scholar
[43]	Zhang J X, Chen D B, Dong Q, Zhao Z D 2016 Sci. Rep. 6 27823 Google Scholar
[44]	Morone F, Makse H 2015 Nature 524 65 Google Scholar
[45]	Goyal A, Lu W, Lakshmanan L 2011 Proceedings of the 20th International Conference on World Wide Web Hyderabad, India, 2011 p47
[46]	Jung K, Heo W, Chen W 2012 IEEE 12th International Conference on Data Mining Brussels, Belgium, December 10–13, 2012 p918

Cited By

图 1 一个简单示例图

Figure 1. A sample graph.

DownLoad: Full-Size Img PowerPoint

图 2 不同方法下不同比例源扩散器的平均最短路径长度L_s　(a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat

Figure 2. Average shortest path length L_s under different proportion of source spreaders by different methods: (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat.

DownLoad: Full-Size Img PowerPoint

图 3 比较在相同时间内感染节点总数的百分比　(a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat

Figure 3. Compare the percentage of the total number of infected nodes over the same time period: (a) NS; (b) EEC; (c) PB; (d) Facebook; (e) WV; (f) Sport; (g) Sex; (h) CondMat.

DownLoad: Full-Size Img PowerPoint

图 4 比较不同传播时间 t 中前 10% 种子节点感染节点的百分比　(a) NS; (b) Facebook; (c) WV; (d) Sex; (f) Sport; (f) CondMat

Figure 4. Compare the percentage of nodes infected by the top 10% of seed nodes in different propagation time t: (a) NS; (b) Facebook; (c) WV; (d) Sex; (f) Sport; (f) CondMat.

DownLoad: Full-Size Img PowerPoint

表 1 节点在每个shell中的信息熵

Table 1. Information entropy of each node.

ks	Node	E
3	1	0.9571
	4	0.8565
	5	0.8099
	2	0.7151
	3	0.6366
2	7/8/9	0.4861
2	6	0.4374
1	21	0.6675
	10	0.4034
	23	0.3720
	20/22/24/25/26	0.2420
	11/12/13/14/16	0.1992
	15	0.1733
	17/18/19	0.1435

DownLoad: CSV

表 2 节点在每个迭代层中的信息熵

Table 2. Information entropy of each node.

Iteration	Node	E₊
7	5	1.3728
6	1	1.4579
	4	1.4378
	2	1.2159
	3	1.0589
5	7/8	0.7839
4	6	0.7189
4	9	0.6430
3	21	0.8084
2	10	0.4818
2	23	0.3720
1	16	0.3400
	15	0.3139
	20/22/24/25/26	0.2420
	17	0.2219
	11/12/13/14	0.1992
	18/19	0.1435

DownLoad: CSV

IE₊算法伪代码
输入: 网络结构G =(V, E)
输出: 网络中节点的排序索引Rank
1: 通过G = (V, E)得出邻接矩阵A
2: 通过K-shell算法得出每个节点的ks值
3: IT ← 1
4: while \|V\| do
5: V_temp←{ }
6: V_i.k ← $\displaystyle\sum\nolimits_{j = 1}^N { {a_{ij} } }$
7: mindegree ← min(V.k)
8: V_temp← find (V. k == mindegree)
9: while V_temp do
10: V_temp. IT←IT
11: V_temp.e₊←$ -{\sum }_{j\in \varGamma \left(i\right)}{I}_{j}\cdot {\rm{ln}}{I}_{j}\cdot {\rm{k}}{{\rm{s}}}_{j} $
12: endwhile
13: delete(V_temp)
14: IT←IT+1
15: V←V-V_temp
16: endwhile
17: ITMax ← IT
18: while length(Rank) < N do
19: for IT←ITMax to 1 do
20: V_temp = find(max(V.IT.e₊))
21: if length (V_temp) > 1
22: 按节点序号从大到小排序
23: end if
24: Rank ← { V_temp, Rank}
25: end for
26: end while
27: return Rank

DownLoad: CSV

表 3 由不同方法得出的排名: DC, CC, ks, C_nc, C_nc+, IKS, IE₊

Table 3. The ranking lists determined by different methonds: DC, CC, ks, C_nc, C_nc+, IKS, IE₊.

Rank	DC	CC	ks	C_nc	C_nc+	IKS	IE₊
1	21	1	1—5	4, 5	1	1	5
2	1, 4, 5, 10	4	6—9	1	4, 5	7	1
3	2, 3	5	10—26	2	2	21	8
4	6—9, 23	21		3	3	4	7
5	11—20	7, 8		21	6—8	9	6
6	22, 24—26	6		6—8	9, 21	10	21
7		9		10	16	5	10
8		23		9	23	8	16
9		16, 20, 22, 24—26		15, 16, 23	15	23	4
10				17	10, 20, 22, 24—26	2	9
11				others		6	23
12						26	15
13						3	2
14						20, 22, 24, 25	20, 22, 24—262
15						11—13, 14, 16	3
16						15	17
17						17—19	11—14
18							18, 19

DownLoad: CSV

表 4 八个常见网络的基本拓扑特征, N和|E|是节点和边的数量, $ \langle d \rangle $和$ \langle k \rangle $是平均距离和平均度, c是聚类系数, β_th和β_c是流行阈值和传播值

Table 4. The basic topological features of the eight real neworks, N and |E|, |E| are the number of nodes and edges, $ \langle d \rangle $ and $ \langle k \rangle $are the average distance and the average degree, c is the clustering coefficient, β_th and β_c are the epidemic threshold and the spread value.

Network	N	\|E\|	$\langle d \rangle$	c	$ \langle k \rangle $	β_th	β_c
NS	379	914	6.0419	0.7981	4.8232	0.1247	0.2494
EEC	986	16064	2.5869	0.4505	32.5842	0.0134	0.0268
PB	1222	16714	2.7375	0.3600	27.3552	0.0123	0.0246
Fecebook	4039	88234	3.6925	0.6170	43.6910	0.0094	0.0188
WV	7066	100736	3.2475	0.2090	28.5129	0.0069	0.0138
Sport	13866	86858	4.2748	0.2761	12.5281	0.0260	0.0520
Sex	15810	38540	7.4630	0	4.8754	0.0365	0.0730
CondMat	23122	93497	5.3523	0.6334	8.0835	0.0450	0.0900

DownLoad: CSV

表 5 SIR模型中节点影响指数R与五个中心性指数之间的Kendall Tau

Table 5. The Kendall Tau between the node influence index R of SIR model and five centrality indices.

Network	DC	CC	C_nc	C_nc+	ks	IKS	IE₊
NS	0.4593	0.3829	0.5604	0.7074	0.4643	0.7301	0.8958
EEC	0.8584	0.8238	0.8999	0.8771	0.8754	0.8963	0.9017
PB	0.8443	0.7956	0.8771	0.8667	0.8653	0.8859	0.9465
Facebook	0.6255	0.4948	0.7416	0.8614	0.6773	0.8926	0.9364
WV	0.8022	0.8583	0.8992	0.8939	0.9171	0.8981	0.9661
Sport	0.6909	0.6891	0.7875	0.8025	0.7437	0.8583	0.9197
Sex	0.4119	0.7329	0.7623	0.8283	0.5151	0.8065	0.8174
CondMat	0.5912	0.7268	0.7303	0.8114	0.6464	0.8565	0.9254

DownLoad: CSV

表 6 不同排序方法的单调性 M

Table 6. The monotonicity M of different ranking methods.

Network	M(DC)	M(CC)	M(C_nc)	M(C_nc+)	M(ks)	M(IKS)	M(IE₊)
NS	0.7642	0.9927	0.9302	0.9593	0.6428	0.8286	0.9221
EEC	0.9571	0.9828	0.9748	0.9998	0.9216	0.9328	0.9881
PB	0.9328	0.9301	0.9433	0.9586	0.9063	0.9266	0.9721
Facebook	0.9398	0.9667	0.9355	0.9646	0.9419	0.9457	0.9898
Sport	0.9032	0.9534	0.9292	0.9377	0.8606	0.9137	0.9818
Sex	0.6001	0.9122	0.9332	0.9581	0.5287	0.9248	0.9989
CondMat	0.8615	0.9544	0.9871	0.9864	0.8032	0.9069	0.9996

DownLoad: CSV

map

[1]	Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104 Google Scholar
[2]	Leskovec J, Adamic L A, Huberman B A 2007 Acm Trans. Web 1 5 Google Scholar
[3]	Freeman L C 1978 Soc. Networks 1 215 Google Scholar
[4]	Freeman L C 1977 Sociometry 40 35 Google Scholar
[5]	Sabidussi G 1966 Psychometrika 31 581 Google Scholar
[6]	Lü L Y, Zhou T, Zhang Q M, Stanley H E 2016 Nat. Commun. 7 10168 Google Scholar
[7]	Lü L Y, Chen D B, Ren X L, Zhang Q M, Zhang Y C, Zhou T 2016 Phys. Rep. 650 1 Google Scholar
[8]	Brin S, Page L 1998 Comput. Netw. ISDN Syst. 30 107 Google Scholar
[9]	Lü L Y, Zhang Y C, Yeung C H, Zhou T 2011 PloS One 6 21202 Google Scholar
[10]	Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888 Google Scholar
[11]	Pei S, Muchnik L, Andrade J S, Zheng Z M, Makse H A 2014 Sci. Rep. 4 5547 Google Scholar
[12]	Montresor A, De Pellegrini F, Miorandi D 2011 Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing San Jose, CA, June 6–8, 2011 p207
[13]	Basaras P, Katsaros D, Tassiulas L 2013 Computer 46 24 Google Scholar
[14]	Wang Z X, Zhao Y, Xi J K, Du C J 2016 Physica A 461 171 Google Scholar
[15]	Zhou S, Mondragon R J 2004 IEEE Commun. Lett. 8 180 Google Scholar
[16]	Wang M, Li W C, Guo Y N, Peng X Y, Li Y X 2020 Physica A 554 124229 Google Scholar
[17]	Zareie A, Sheikhahmadi A, Jalili M, Fasaei M S K 2020 Knowledge-Based Syst. 194 105580 Google Scholar
[18]	Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200 Google Scholar
[19]	Hethcote H W 2000 SIAM Rev. 42 599 Google Scholar
[20]	Ma L I, Ma C, Zhang H F, Wang B H 2016 Physica A 451 205 Google Scholar
[21]	Li Z, Ren T, Ma X Q, Liu S M, Zhang Y X, Zhou T 2019 Sci. Rep. 9 8387 Google Scholar
[22]	Bae J, Kim S 2014 Physica A 395 549 Google Scholar
[23]	Bhat N, Aggarwal N, Kumar S 2020 Procedia Comput Sci. 171 662 Google Scholar
[24]	阮逸润, 老松杨, 汤俊, 白亮 2020 71 176401 Google Scholar Ruan Y R, Lao S Y, Tang J, Bai L, Guo Y M 2020 Acta Phys. Sin. 71 176401 Google Scholar
[25]	Colizza V, Flammini A, Serrano M A, Vespignani A 2006 Nat. Phys. 2 110 Google Scholar
[26]	Rui X B, Meng F R, Wang Z X, Yuan G 2019 Appl. Intell. 49 2684 Google Scholar
[27]	Liu D, Jing Y, Zhao J, Wang W J, Song G J 2017 Sci. Rep. 7 43330 Google Scholar
[28]	Namtirtha A, Dutta A, Dutta B 2018 Physica A 499 310 Google Scholar
[29]	Kim H, Anderson R 2012 Phys. Rev. E 85 026107 Google Scholar
[30]	Takaguchi T, Sato N, Yano K, Masuda N 2012 New J. Phys. 14 093003 Google Scholar
[31]	Qu C Q, Zhan X X, Wang G H, Wu J L, Zhang Z K 2019 Chaos 29 033116 Google Scholar
[32]	胡钢, 许丽鹏, 徐翔 2021 70 108901 Google Scholar Hu G, Xu L P, Xu X 2021 Acta Phys. Sin. 70 108901 Google Scholar
[33]	Newman M E J 2006 Phys. Rev. E 74 036104 Google Scholar
[34]	Yin H, Benson A R, Leskovec J, Gleich D F 2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (Halifax, Candana) August 13–17, 2017 p555
[35]	Adamic L A 2005 Glance N Proceedings of the 3rd International Workshop on Link Discovery (New York, USA) 2005 p36
[36]	Mcauley J, Leskovec J 2012 Proceedings of the 25th International Conference on Neural Information Processing Systems (Lake Tahoe, Nevada) 2012 p539
[37]	Leskovec J, Huttenlocher D, Kleinberg J 2010 Proceedings of the 19th International Conference on World Wide Web (New York, USA) 2010 p65
[38]	Rozemberczki B, Davies R, Sarkar R, Sutton C 2019 Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining New York, USA, 2019 p65
[39]	Rocha L, Liljeros F, Holme P 2011 PLoS Comput. Biol. 7 1001109 Google Scholar
[40]	Leskovec J, Kleinberg J, Faloutsos C 2007 ACM Trans. Knowl. Discovery Data 1 2 Google Scholar
[41]	Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521 Google Scholar
[42]	Kenall M G 1938 Biometrika 30 81 Google Scholar
[43]	Zhang J X, Chen D B, Dong Q, Zhao Z D 2016 Sci. Rep. 6 27823 Google Scholar
[44]	Morone F, Makse H 2015 Nature 524 65 Google Scholar
[45]	Goyal A, Lu W, Lakshmanan L 2011 Proceedings of the 20th International Conference on World Wide Web Hyderabad, India, 2011 p47
[46]	Jung K, Heo W, Chen W 2012 IEEE 12th International Conference on Data Mining Brussels, Belgium, December 10–13, 2012 p918

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