Abstract
As a mathematical scaffold for network science, graph theory abstracts complex systems into complex networks. However, graphs ignore the multiplicity of combinatorial relationships in network systems, leading to limitations in graph-based metrics reflecting the importance of nodes. To address the shortcomings of graphs in describing network complexity, this study proposes the use of co-occurrence pattern truth tables to represent the combinations of multiple nodes in a network. Based on this, the concept of positive sensitivity is proposed to measure one aspect of the importance of nodes in a network. In addition, network sensitivity is proposed to depict the robustness of the network. The proposed approach is verified to be workable with Monte Carlo simulations and a real network exemplified by the high-speed rail network, constructed with provincial capitals of China as nodes. The results in comparison with traditional graph theory-based indices show that both the nodes and the network are assessed with reasonable results different from those of the graph-derived metrics. This study focuses on the combinatorial relationships of nodes in networks, providing a new perspective for the analysis of complex networks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The source codes and data are available for downloading at the link: https://doi.org/10.6084/m9.figshare.19255103.
References
Agryzkov T et al (2014) A new betweenness centrality measure based on an algorithm for ranking the nodes of a network. Appl Math Comput 244:467–478. https://doi.org/10.1016/j.amc.2014.07.026
Aksoy SG et al (2020) Hypernetwork science via high-order hypergraph walks. EPJ Data Sci 9(1):16. https://doi.org/10.1140/epjds/s13688-020-00231-0
Albert R et al (2000) Error and attack tolerance of complex networks. Nature 406(6794):378–382
Alon N, Boppana RB (1987) The monotone circuit complexity of boolean functions. Combinatorica 7(1):1–22. https://doi.org/10.1007/BF02579196
Amaral LAN, Ottino JM (2004) Complex networks. Eur Phys J Condens Matter 38(2):147–162. https://doi.org/10.1140/epjb/e2004-00110-5
Ambainis A et al. (2016) Sensitivity versus certificate complexity of boolean functions. In: Computer science – theory and applications. edited. Springer International Publishing, Cham. pp 16–28 https://doi.org/10.1007/978-3-319-34171-2_2
Babai L et al (1990) Lower bounds to the complexity of symmetric boolean functions. Theoret Comput Sci 74(3):313–323. https://doi.org/10.1016/0304-3975(90)90080-2
Barabási A, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512. https://doi.org/10.1126/science.286.5439.509
Barceló JM et al (2004) Study of internet autonomous system interconnectivity from bgp routing tables. Comput Netw 45(3):333–344. https://doi.org/10.1016/j.comnet.2004.03.011
Ben-Or M, Linial N (1985) Collective coin flipping, robust voting schemes and minima of Banzhaf values. Paper presented at 26th annual symposium on foundations of computer science (sfcs 1985), IEEE
Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw ISDN Syst 30(1–7):107–117
Calderoni F et al (2017) Communities in criminal networks: a case study. Soc Netw 48:116–125. https://doi.org/10.1016/j.socnet.2016.08.003
Cavallaro L et al (2020) Disrupting resilient criminal networks through data analysis: the case of Sicilian Mafia. PLoS ONE 15(8):e236476
Chakraborty S (2005) Sensitivity, block sensitivity and certificate complexity of Boolean functions, Citeseer
Chen D et al (2012) Identifying influential nodes in complex networks. Physica A 391(4):1777–1787. https://doi.org/10.1016/j.physa.2011.09.017
Crucitti P et al (2004a) Error and attack tolerance of complex networks. Physica A 340(1–3):388–394. https://doi.org/10.1016/j.physa.2004.04.031
Crucitti P et al (2004b) Model for cascading failures in complex networks. Phys Rev E 69(4 Pt 2):45104. https://doi.org/10.1103/PhysRevE.69.045104
De Montis A et al (2011) Time evolution of complex networks: commuting systems in insular Italy. J Geogr Syst 13(1):49–65. https://doi.org/10.1007/s10109-010-0130-8
Demšar U et al (2008) Identifying critical locations in a spatial network with graph theory. Trans GIS 12(1):61–82. https://doi.org/10.1111/j.1467-9671.2008.01086.x
Estrada E (2010) Randic index, irregularity and complex biomolecular networks. Acta Chim Slov 57:597–603
Estrada E, Rodríguez-Velázquez JA (2006) Subgraph centrality and clustering in complex hyper-networks. Physica A 364:581–594. https://doi.org/10.1016/j.physa.2005.12.002
Euler L (1741) Solutio problematis ad geometriam situs pertinentis. Commentarii Acad Scientiarum Petropolitanae 128–140
Feng S et al (2021) Hypergraph models of biological networks to identify genes critical to pathogenic viral response. BMC Bioinform 22(1):287. https://doi.org/10.1186/s12859-021-04197-2
Freeman LC et al (1991) Centrality in valued graphs: A measure of betweenness based on network flow. Soc Netw 13(2):141–154. https://doi.org/10.1016/0378-8733(91)90017-N
Golomb S (1959) On the classification of boolean functions. IRE Trans Circuit Theory 6(5):176–186. https://doi.org/10.1109/TCT.1959.1086595
Guillaume J et al (2005) Comparison of failures and attacks on random and scale-free networks. edited. Springer Berlin Heidelberg, Berlin, Heidelberg. p 186–196. https://doi.org/10.1007/11516798_14
Ha D et al (2011) Hyper networks. Imperial College Press, London
Hâncean M et al (2020) The impact of human mobility networks on the global spread of COVID-19. J Complex Netw. https://doi.org/10.1093/comnet/cnaa041
Hernando A et al (2010) Unravelling the size distribution of social groups with information theory in complex networks. Eur Phys J B 76(1):87–97. https://doi.org/10.1140/epjb/e2010-00216-1
Iyer SAKT (2013) Attack robustness and centrality of complex networks. PLoS ONE 8(4):1–17. https://doi.org/10.1371/journal.pone.0059613
Jia T et al (2021) Identification and analysis of urban influential regions using spatial interaction networks. Trans GIS 25(6):2821–2839. https://doi.org/10.1111/tgis.12806
Kaluza P et al (2010) The complex network of global cargo ship movements. J R Soc Interface 7(48):1093–1103
Klamt S et al (2009) Hypergraphs and cellular networks. PLoS Comput Biol 5(5):e1000385. https://doi.org/10.1371/journal.pcbi.1000385
Lacasa L et al (2008) From time series to complex networks: the visibility graph. Proc Natl Academ Sci PNAS 105(13):4972–4975. https://doi.org/10.1073/pnas.0709247105
Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87(19):198701. https://doi.org/10.1103/PhysRevLett.87.198701
Liu F et al (2020) GMM: a generalized mechanics model for identifying the importance of nodes in complex networks. Knowl Based Syst 193:105464. https://doi.org/10.1016/j.knosys.2019.105464
Lloyd EK (1980) Graph theory—an introductory course. Math Gaz 64(429):217
Madar G et al (2020) Examining the robustness of the Ontario truck road network. J Geogr Syst 22(3):309–333. https://doi.org/10.1007/s10109-020-00320-8
Moreno Y et al (2004) Dynamics of rumor spreading in complex networks. Phys Rev E 69(6 Pt 2):66130. https://doi.org/10.1103/PhysRevE.69.066130
Murray AT et al (2007) Critical network infrastructure analysis: interdiction and system flow. J Geogr Syst 9(2):103–117. https://doi.org/10.1007/s10109-006-0039-4
Newman ME (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256
Newman ME (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74(3 Pt 2):36104. https://doi.org/10.1103/PhysRevE.74.036104
Papo D et al (2014) Complex network theory and the brain. Philosophical Trans R Soc Lond Ser b Biol Sci. https://doi.org/10.1098/rstb.2013.0520
Ramadan E et al (2004) A hypergraph model for the yeast protein complex network, IEEE. https://doi.org/10.1109/IPDPS.2004.1303205
Sabidussi G (1966) The centrality index of a graph. Psychometrika 31(4):581–603
Schneider CM et al (2011) Mitigation of malicious attacks on networks. Proc Natl Acad Sci 108(10):3838–3841. https://doi.org/10.1073/pnas.1009440108
Shaw ME (1954) Some effects of unequal distribution of information upon group performance in various communication nets. J Abnormal Soc Psychol 49((4,pt.1)):547–553. https://doi.org/10.1037/h0053638
Stephenson K, Zelen M (1989) Rethinking centrality: methods and examples. Soc Netw 11(1):1–37. https://doi.org/10.1016/0378-8733(89)90016-6
Sun G, Bin S (2017) Router-level internet topology evolution model based on multi-subnet composited complex network model. J Internet Technol 18(6):1275–1283
Wang J et al (2008) Attack vulnerability of scale-free networks due to cascading failures. Physica A 387(26):6671–6678. https://doi.org/10.1016/j.physa.2008.08.037
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–442. https://doi.org/10.1038/30918
Xu Z, Sui DZ (2007) Small-world characteristics on transportation networks: a perspective from network autocorrelation. J Geogr Syst 9(2):189–205. https://doi.org/10.1007/s10109-007-0045-1
Yang R et al (2007) Epidemic spreading on heterogeneous networks with identical infectivity. Phys Lett A 364(3–4):189–193. https://doi.org/10.1016/j.physleta.2006.12.021
Yang L et al (2020) On multiplexity-aware influence spread in social networks. IEEE Access 8:106705–106713. https://doi.org/10.1109/ACCESS.2020.2999312
Zhang Z, Liu C (2010) A hypergraph model of social tagging networks. J Stat Mech Theory Exp 2010(10):P10005. https://doi.org/10.1088/1742-5468/2010/10/P10005
Zhang H, Zhong S, Deng Y, Cheong KH (2021) LFIC: identifying influential nodes in complex networks by local fuzzy information centrality. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2021.3112226
Zhou T et al (2006) Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Phys Rev E 74(5 Pt 2):56109. https://doi.org/10.1103/PhysRevE.74.056109
Zou X et al (2020) An efficient method of advertising on online social networks. Springer, Cham
Acknowledgements
Supported by the National Natural Science Foundation of China, (42271476) & Wuhan University 351 Talents Program for Young Scholars, (2020007).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Luo, J., Fei, T., Tian, M. et al. Sensitivity metrics of complex network based on co-occurrence truth table: exemplified by a high-speed rail network. J Geogr Syst 25, 519–538 (2023). https://doi.org/10.1007/s10109-023-00419-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10109-023-00419-8