An improvement method for degree and its extending centralities in directed networks

Abstract Vital nodes identification in social networks keeps attracting lots of attention in recent years because of its great theoretical and practical significance for many applications. Degree centrality is one of the most efficient neighborhood-based metrics and has been widely used because of its simplicity and low computation complexity. Many other centralities derived from degree have been proposed to try to further improve the accuracy on ranking performance by extending the coverage area of neighbors. But a common problem of degree and its extensional centralities is that the directivity of edges between nodes is not considered. In directed network, nodes’ importance are not only related with their neighbor numbers but also related with the relationships between the nodes and their neighbors. In this paper, we propose a modification method with an adjustable parameter α on degree to improve its ranking accuracy in directed networks. This method redefines the degree of a node by separately taking into account its out-degree and in-degree and uses parameter α to flexibly set the relative weights between out-degree and in-degree when evaluating nodes’ importance in different scenes, which can effectively avoid the situation that too many neighbors from one direction could lead to extreme high value of centrality. We also apply this method on another two centralities: semi-local and k-hop, and for the two centralities that considering multi-step neighbors, the corresponding improvement centralities could eliminate the irrelevant multi-step neighbors when evaluating nodes’ importance. Experimental results in 3 real social networks and 1 artificial network indicate that, setting α to be appropriate value the improvement centralities can better estimate nodes’ connection abilities, propagation abilities and immunization abilities while keeping the same compute complexity with the corresponding original centralities, especially in the network with low percentage of bidirectional edges. Moreover, the improvement method proposed in this paper also can be applied to other centralities, such as eigenvector centrality, neighborhood centrality, Exdegree centrality and percolation based centrality, to improve their performances in directed networks.

[1]  Fang-Xiang Wu,et al.  United Complex Centrality for Identification of Essential Proteins from PPI Networks , 2017, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[2]  Ping Hu,et al.  Individual-based optimal weight adaptation for heterogeneous epidemic spreading networks , 2018, Commun. Nonlinear Sci. Numer. Simul..

[3]  Lucas C Parra,et al.  Finding influential nodes for integration in brain networks using optimal percolation theory , 2018, Nature Communications.

[4]  Tao Li,et al.  The analysis of an SEIR rumor propagation model on heterogeneous network , 2017 .

[5]  M. Prokopenko,et al.  Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks , 2013, PloS one.

[6]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.

[7]  S. Borgatti,et al.  Betweenness centrality measures for directed graphs , 1994 .

[8]  Jianwei Niu,et al.  K-hop centrality metric for identifying influential spreaders in dynamic large-scale social networks , 2014, 2014 IEEE Global Communications Conference.

[9]  B. Kahng,et al.  Percolation transitions in scale-free networks under the Achlioptas process. , 2009, Physical review letters.

[10]  Yi Pan,et al.  Identification of Essential Proteins Based on Edge Clustering Coefficient , 2012, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[11]  Jianfeng Ma,et al.  Identifying critical nodes’ group in complex networks , 2019, Physica A: Statistical Mechanics and its Applications.

[12]  Peng Zhang,et al.  DPRank centrality: Finding important vertices based on random walks with a new defined transition matrix , 2017, Future Gener. Comput. Syst..

[13]  Yong Deng,et al.  Identifying influential spreaders by weight degree centrality in complex networks , 2016 .

[14]  Tore Opsahl,et al.  Clustering in weighted networks , 2009, Soc. Networks.

[15]  M. Zelen,et al.  Rethinking centrality: Methods and examples☆ , 1989 .

[16]  Esmaeil Bagheri,et al.  FSIM: A Fast and Scalable Influence Maximization Algorithm Based on Community Detection , 2018, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[17]  Jon Kleinberg,et al.  Authoritative sources in a hyperlinked environment , 1999, SODA '98.

[18]  Yi-Cheng Zhang,et al.  Leaders in Social Networks, the Delicious Case , 2011, PloS one.

[19]  Chao Liu,et al.  Modeling the spread of malware with the influence of heterogeneous immunization , 2016 .

[20]  Xinchu Fu,et al.  Immunization and epidemic threshold of an SIS model in complex networks , 2016 .

[21]  James Moody,et al.  Peer influence groups: identifying dense clusters in large networks , 2001, Soc. Networks.

[22]  Wenbin Li,et al.  Local Search in Weighted and Directed Social Networks: The Case of Enron Email Networks , 2009, 2009 IEEE/WIC/ACM International Joint Conference on Web Intelligence and Intelligent Agent Technology.

[23]  Steffen Dereich,et al.  Random networks with sublinear preferential attachment: The giant component , 2010, 1007.0899.

[24]  Shouming Zhong,et al.  Web malware spread modelling and optimal control strategies , 2017, Scientific Reports.

[25]  Frank Bentley,et al.  The Composition and Use of Modern Mobile Phonebooks , 2015, CHI.

[26]  Liang Liu,et al.  Modeling and analyzing malware propagation in social networks with heterogeneous infection rates , 2018, Physica A: Statistical Mechanics and its Applications.

[27]  Lev Muchnik,et al.  Identifying influential spreaders in complex networks , 2010, 1001.5285.

[28]  Hui Gao,et al.  Identifying Influential Nodes in Large-Scale Directed Networks: The Role of Clustering , 2013, PloS one.

[29]  John Skvoretz,et al.  Node centrality in weighted networks: Generalizing degree and shortest paths , 2010, Soc. Networks.

[30]  Ming Tang,et al.  Identify influential spreaders in complex networks, the role of neighborhood , 2015, ArXiv.

[31]  Hongyong Zhao,et al.  Dynamical behaviours and control measures of rumour-spreading model with consideration of network topology , 2017, Int. J. Syst. Sci..

[32]  Liu Zhi,et al.  Evaluating influential spreaders in complex networks by extension of degree , 2015 .

[33]  Gert Sabidussi,et al.  The centrality index of a graph , 1966 .

[34]  Ulrik Brandes,et al.  Maintaining the duality of closeness and betweenness centrality , 2016, Soc. Networks.

[35]  Yaming Zhang,et al.  Rumor and authoritative information propagation model considering super spreading in complex social networks , 2018, Physica A: Statistical Mechanics and its Applications.

[36]  Yuanyuan Zhang,et al.  High-Order Degree and Combined Degree in Complex Networks , 2018, Mathematical Problems in Engineering.

[37]  Duanbing Chen,et al.  Vital nodes identification in complex networks , 2016, ArXiv.

[38]  M. Kendall.  A NEW MEASURE OF RANK CORRELATION , 1938 .

[39]  M. N. Doja,et al.  Group based Centrality for Immunization of Complex Networks , 2018, Physica A: Statistical Mechanics and its Applications.

[40]  W. O. Kermack,et al.  Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity , 1932 .

[41]  Jérôme Kunegis,et al.  KONECT: the Koblenz network collection , 2013, WWW.

[42]  Tianqing Zhu,et al.  Evolutionary virus immune strategy for temporal networks based on community vitality , 2017, Future Gener. Comput. Syst..

[43]  Gang Xie,et al.  Efficient identification of node importance in social networks , 2016, Inf. Process. Manag..

[44]  Fuzhong Nian,et al.  The epidemic spreading on the multi-relationships network , 2018, Appl. Math. Comput..

[45]  Stephen P. Borgatti,et al.  Centrality and network flow , 2005, Soc. Networks.

[46]  Lei Gao,et al.  Effective information spreading based on local information in correlated networks , 2016, Scientific Reports.

[47]  Linyuan Lu,et al.  Effective spreading from multiple leaders identified by percolation in social networks , 2015, ArXiv.

[48]  Han-Xin Yang,et al.  Suppressing traffic-driven epidemic spreading by use of the efficient routing protocol , 2015, ArXiv.

[49]  Guojing Zhang,et al.  Bi-directional h-index: A new measure of node centrality in weighted and directed networks , 2018, J. Informetrics.

[50]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[51]  Qiang Liu,et al.  Leveraging local h-index to identify and rank influential spreaders in networks , 2017, Physica A: Statistical Mechanics and its Applications.

[52]  Tianqing Zhu,et al.  Wormhole: The Hidden Virus Propagation Power of the Search Engine in Social Networks , 2019, IEEE Transactions on Dependable and Secure Computing.

[53]  W. Fan,et al.  Online social networks—Paradise of computer viruses☆ , 2011 .

[54]  Ping Hu,et al.  Epidemic spreading with awareness diffusion on activity-driven networks , 2018, Physical Review. E.

[55]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[56]  Yicheng Zhang,et al.  Identifying influential nodes in complex networks , 2012 .

[57]  Lada A. Adamic,et al.  The political blogosphere and the 2004 U.S. election: divided they blog , 2005, LinkKDD '05.

[58]  Guanhua Yan,et al.  Malware propagation in online social networks: nature, dynamics, and defense implications , 2011, ASIACCS '11.