Abstract
Optimal design of hydrometric networks has been a long-standing problem in hydrology. Evaluation of the importance (or influence) of the individual monitoring stations is key to achieve an optimal design of a hydrometric network. The present study employs the concepts of complex networks towards assessing the importance of individual stations in a hydrometric network. For implementation, a streamflow network of 218 stations in Australia is studied, and monthly streamflow data of 26 years (1981–2006) are analyzed. Each station is considered as a node in the network and the connections between any pair of nodes are identified based on mutual information in the streamflow values. Six different node ranking measures are used to examine the importance of nodes in the network: degree centrality, betweenness centrality, closeness centrality, degree and influence of line, weighted degree betweenness, and clustering coefficient. Different threshold values of mutual information are also considered to examine the influence of threshold on the best node ranking measure. The six node ranking measures are evaluated using the decline rate of network efficiency. The results indicate that different node ranking measures identify different stations as the most important and least important in the network. Betweenness centrality and weighted degree betweenness generally perform the best in identifying the most important stations across the thresholds. The weighted degree betweenness measure outperforms the others in the identification of the least important stations, especially at higher thresholds. The clustering coefficient performs the worst in identifying the importance of stations in the streamflow monitoring network.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig9_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00477-022-02340-w/MediaObjects/477_2022_2340_Fig10_HTML.png)
Similar content being viewed by others
Data Availability and Code
The streamflow data used in this study were obtained from the Australian Bureau of Meteorology. The code may be obtained from the authors upon request.
References
Adhikary SK, Yilmaz AG, Muttil N (2015) Optimal design of rain gauge network in the Middle Yarra River catchment, Australia. Hydrol Process 29:2582–2599. https://doi.org/10.1002/hyp.10389
Agarwal A, Maheswaran R, Marwan N, Caesar L, Kurths J (2018a) Wavelet-based multiscale similarity measure for complex networks. Eur Phys J B 91:296. https://doi.org/10.1140/epjb/e2018-90460-6
Agarwal A, Marwan N, Maheswaran R, Merz B, Kurths J (2018b) Quantifying the roles of single stations within homogeneous regions using complex network analysis. J Hydrol 563:802–810. https://doi.org/10.1016/j.jhydrol.2018.06.050
Agarwal A, Caesar L, Marwan N, Maheswaran R, Merz B, Kurths J (2019) Network-based identification and characterization of teleconnections on different scales. Sci Rep 9:1–12. https://doi.org/10.1038/s41598-019-45423-5
Agarwal A, Marwan N, Maheswaran R, Ozturk U, Kurths J, Merz B (2020) Optimal design of hydrometric station networks based on complex network analysis. Hydrol Earth Syst Sci 24:2235–2251. https://doi.org/10.5194/hess-24-2235-2020
Agarwal A, Guntu RK, Banerjee A, Gadhawe MA, Marwan N (2022) A complex network approach to study the extreme precipitation patterns in a river basin. Chaos Interdiscip J Nonlinear Sci 32:013113. https://doi.org/10.1063/5.0072520
Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74(1):47–97. https://doi.org/10.1103/RevModPhys.74.47
Bae J, Kim S (2014) Identifying and ranking influential spreaders in complex networks by neighborhood coreness. Phys A 395:549–559. https://doi.org/10.1016/j.physa.2013.10.047
Bao Z-K, Ma C, **ang B-B, Zhang H-F (2017) Identification of influential nodes in complex networks: method from spreading probability viewpoint. Phys A 468:391–397. https://doi.org/10.1016/j.physa.2016.10.086
Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512. https://doi.org/10.1126/science.286.5439.509
Boers N, Bookhagen B, Marwan N, Kurths J, Marengo J (2013) Complex networks identify spatial patterns of extreme rainfall events of the South American Monsoon System. Geophys Res Lett. https://doi.org/10.1002/grl.50681
Braga AC, Alves LGA, Costa LS, Ribeiro AA, De Jesus MMM, Tateishi AA, Ribeiroc HV (2016) Characterization of river flow fluctuations via horizontal visibility graphs. Phys A 444:1003–1011. https://doi.org/10.1016/j.physa.2015.10.102
Brown SC, Versace VL, Lester RE, Walter MT (2015) Assessing the impact of drought and forestry on streamflows in south-eastern Australia using a physically based hydrological model. Environ Earth Sci 74(7):6047–6063. https://doi.org/10.1007/s12665-015-4628-8
CSIRO and Bureau of Meteorology (2007) Climate Change in Australia. Technical Report. www.climatechangeinaustralia.gov.au
Cetinkaya CP, Harmancioglu NB (2014) Reduction of streamflow monitoring networks by a reference point approach. J Hydrol 512:263–273. https://doi.org/10.1016/j.jhydrol.2014.03.006
Chebbi A, Bargaoui ZK, Abid N, Cunha MC (2017) Optimization of a hydrometric network extension using specific flow, kriging and simulated annealing. J Hydrol 555:971–982. https://doi.org/10.1016/j.jhydrol.2017.10.076
Chen D, Lü L, Shang M-S, Zhang Y-C, Zhou T (2012) Identifying important nodes in complex networks. Phys Stat Mech Appl 391:1777–1787. https://doi.org/10.1016/j.physa.2011.09.017
Cleugh H, Smith MS, Battaglia M, Graham P (2011) Climate change: science and solutions for Australia. CSIRO Publishing, Victoria
Cover TM, Thomas JA (2006) Elements of information theory. Wiley-Interscience, New York
Curadoa M, Rodriguez R, Tortosac L, Vicent JF (2022) A new centrality measure in dense networks based on two-way random walk betweenness. Appl Math Comput 412:126560. https://doi.org/10.1016/j.amc.2021.126560
Davar Z, Brimley W (1990) Hydrometric network evaluation: audit approach. J Water Resour Plann Manage 116(1):134–146. https://doi.org/10.1061/(ASCE)0733-9496(1990)116:1(134)
Deepthi B, Sivakumar B (2022) General circulation models for rainfall simulations: performance assessment using complex networks. Atmos Res 278:106333. https://doi.org/10.1016/j.atmosres.2022.106333
Eagleson PS (1967) Optimum density of rainfall networks. Water Resour Res 3(4):1021–1033. https://doi.org/10.1029/WR003i004p01021
Elmezain M, Othman EA, Ibrahim HM (2021) Temporal degree–degree and closeness–closeness: a new centrality metrics for social network analysis. Mathematics 9:2850. https://doi.org/10.3390/math9222850
Estrada E (2012) The structure of complex networks: theory and applications. Oxford University Press, Oxford
Fang K, Sivakumar B, Woldemeskel FM (2017) Complex networks, community structure, and catchment classification in a large-scale river basin. J Hydrol 545:478–493. https://doi.org/10.1016/j.jhydrol.2016.11.056
Freeman L (1977) A set of measures of centrality based on betweenness. Sociometry 40(1):35–41. https://doi.org/10.2307/3033543
Freeman LC (1979) Centrality in social networks conceptual clarification. Soc Netw 1:215–239. https://doi.org/10.1016/0378-8733(78)90021-7
Gadhawe MA, Guntu RK, Agarwal A (2021) Network-based exploration of basin precipitation based on satellite and observed data. Eur Phys J Spec Top 230:3343–3357. https://doi.org/10.1140/epjs/s11734-021-00017-z
Ganapathy A, Agarwal A (2022) Customized sea-surface temperature indicators linking to streamflow at different timescales. Int J Climatol. https://doi.org/10.1002/joc.7853
Gao S, Ma J, Chen Z, Wang G, **ng C (2014) Ranking the spreading ability of nodes in complex networks based on local structure. Phys A 403:130–147. https://doi.org/10.1016/j.physa.2014.02.032
Gao C, Wei DJ, Hu Y, Mahadevan S, Deng Y (2013) A modified evidential methodology of identifying important nodes in weighted networks. Phys A 392:5490–5500
Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99:7821–7826. https://doi.org/10.1073/pnas.122653799
Halverson MJ, Fleming SW (2015) Complex network theory, streamflow, and hydrometric monitoring system design. Hydrol Earth Syst Sci 19:3301–3318. https://doi.org/10.5194/hess-19-3301-2015
Han X, Sivakumar B, Woldemeskel FM, Guerra de Aguilar M (2018) Temporal dynamics of streamflow: application of complex networks. Geosci Lett 5:10. https://doi.org/10.1186/s40562-018-0109-8
Hou B, Yao Y, Liao D (2012) Identifying all-around nodes for spreading dynamics in complex networks. Phys Stat Mech Its Appl 391:4012–4017. https://doi.org/10.1016/j.physa.2012.02.033
Istalkar P, Unnithan SLK, Biswal B, Sivakumar B (2021) A Canberra distance-based complex network classification framework using lumped catchment characteristics. Stoch Environ Res Risk Assess 35:1293–1300. https://doi.org/10.1007/s00477-020-01952-4
Jha SK, Sivakumar B (2017) Complex networks for rainfall modeling: spatial connections, temporal scale, and network size. J Hydrol 554:482–489. https://doi.org/10.1016/j.jhydrol.2017.09.030
Joo H, Kim HS, Kim S, Sivakumar B (2021) Complex networks and integrated centrality measure to assess the importance of streamflow stations in a river basin. J Hydrol 598:126280. https://doi.org/10.1016/j.jhydrol.2021.126280
Kitsak M, Gallos LK, Havlin S, Liljeros F, Muchnik L, Stanley HE, Makse HA (2010) Identification of influential spreaders in complex networks. Nat Phys 6:888–893. https://doi.org/10.1038/nphys1746
Kotikot SM, Omitaomu OA (2021) Spatial–temporal patterns of historical, near-term, and projected drought in the conterminous United States. Hydrology 8:136. https://doi.org/10.3390/hydrology8030136
Krstanovic PF, Singh VP (1992) Evaluation of rainfall networks using entropy: 1. Theoretical development. Water Resour Manag 6:279–293. https://doi.org/10.1007/BF00872281
Latora V, Nicosia V, Russo G (2017) Complex networks: principles, methods and applications. Cambridge University Press, Cambridge
Leach JM, Kornelsen KC, Samuel J, Coulibaly P (2015) Hydrometric network design using streamflow signatures and indicators of hydrologic alteration. J Hydrol 529(3):1350–1359. https://doi.org/10.1016/j.jhydrol.2015.08.048
Li J, Bárdossy A, Guenni L, Liu M (2011) A copula-based observation network design approach. Environ Model Softw 26:1349–1357. https://doi.org/10.1016/j.envsoft.2011.05.001
Li C, Singh VP, Mishra AK (2012) Entropy theory-based criterion for hydrometric network evaluation and design: maximum information minimum redundancy. Water Resour Res 48:W05521. https://doi.org/10.1029/2011WR011251
Liu J, **ong Q, Shi W, Shi X, Wang K (2016) Evaluating the importance of nodes in complex networks. Phys Stat Mech Appl 452:209–219. https://doi.org/10.1016/j.physa.2016.02.049
Malik N, Bookhagen B, Marwan N, Kurths J (2012) Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks. Clim Dyn 39:971–987. https://doi.org/10.1007/s00382-011-1156-4
Mishra AK, Coulibaly P (2010) Hydrometric network evaluation for Canadian watersheds. J Hydrol 380:420–437. https://doi.org/10.1016/J.JHYDROL.2009.11.015
Namtirtha A, Dutta B, Dutta A (2022) Semi-global triangular centrality measure for identifying the influential spreaders from undirected complex networks. Expert Syst Appl 206:117791. https://doi.org/10.1016/j.eswa.2022.117791
Naufan I, Sivakumar B, Woldemeskel FM, Raghavan SV, Vu MT, Liong SY (2018) Spatial connections in regional climate model rainfall outputs at different temporal scales: application of network theory. J Hydrol 556:1232–1243. https://doi.org/10.1016/j.jhydrol.2017.05.029
Rodriguez-Iturbe I, Mejia JM (1974) The design of rainfall networks in time and space. Water Resour Res 10(4):713–728. https://doi.org/10.1029/WR010i004p00713
Sabidussi G (1966) The centrality index of a graph. Psychometrika 31:581–603. https://doi.org/10.1007/BF02289527
Scarsoglio S, Laio F, Ridolfi L (2013) Climate dynamics: a network-based approach for the analysis of global precipitation. PLoS ONE 8:e71129. https://doi.org/10.1371/journal.pone.0071129
Serinaldi F, Kilsby CG (2016) Irreversibility and complex network behavior of stream flow fluctuations. Phys A 450:585–600. https://doi.org/10.1016/j.physa.2016.01.043
Sivakumar B, Woldemeskel FM (2014) Complex networks for streamflow dynamics. Hydrol Earth Syst Sci 18:4565–4578. https://doi.org/10.5194/hess-18-4565-2014
Sivakumar B, Woldemeskel FM (2015) A network-based analysis of spatial rainfall connections. Environ Model Softw 69:55–62. https://doi.org/10.1016/j.envsoft.2015.02.020
Sreeparvathy V, Srinivas VV (2020) A fuzzy entropy approach for design of hydrometric monitoring networks. J Hydrol 586:124797. https://doi.org/10.1016/j.jhydrol.2020.124797
Stedinger JR, Tasker GD (1985) Regional hydrologic analysis: 1. Ordinary, weighted, and generalized least squares compared. Water Resour Res 21(9):1421–1432. https://doi.org/10.1029/WR021i009p01421
Steuer R, Kurths J, Daub C, Weise J, Selbig J (2002) The mutual information: detecting and evaluating dependencies between variables. Bioinformatics 18(2):S231–S240
Stolbova V, Martin P, Bookhagen B, Marwan N, Kurths J (2014) Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka. Nonlin Processes Geophys 21:901–917. https://doi.org/10.5194/npg-21-901-2014
Stosic T, Stosic B, Singh VP (2017) Optimizing streamflow monitoring networks using joint permutation entropy. J Hydrol 552:306–312. https://doi.org/10.1016/j.jhydrol.2017.07.003
Tiwari S, Jha SK, Singh A (2020) Quantification of node importance in rain gauge network: influence of temporal resolution and rain gauge density. Sci Rep 10:9761. https://doi.org/10.1038/s41598-020-66363-5
Tiwari S, Jha S, Sivakumar B (2019) Reconstruction of daily rainfall data using the concepts of complex networks: accounting for spatial connections in neighborhood selection. J Hydrol 579:124185
Tongal H, Sivakumar B (2017) Cross-entropy clustering framework for catchment classification. J Hydrol 552:433–446. https://doi.org/10.1016/j.jhydrol.2017.07.005
Tumiran SA, Sivakumar B (2021) Catchment classification using community structure concept: application to two large regions. Stoch Environ Res Risk Assess 35:561–578. https://doi.org/10.1007/s00477-020-01936-4
Tumiran SA, Sivakumar B (2021) Community structure concept for catchment classification: a modularity density-based edge betweenness (MDEB) method. Ecol Ind 124:107346. https://doi.org/10.1016/j.ecolind.2021.107346
Wang S, Du Y, Deng Y (2017) A new measure of identifying important nodes: efficiency centrality. Commun Nonlinear Sci Numer Simulat 47:151–163. https://doi.org/10.1016/j.cnsns.2016.11.008
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393(6684):440–444
Xu P, Wang D, Singh VP, Wang Y, Wu J, Wang L, Zou X, Liu J, Zou Y, He R (2018) A kriging and entropy-based approach to raingauge network design. Environ Res 161:61–75. https://doi.org/10.1016/j.envres.2017.10.038
Yasmin N, Sivakumar B (2018) Temporal streamflow analysis: coupling nonlinear dynamics with complex networks. J Hydrol 564:59–67. https://doi.org/10.1016/j.jhydrol.2018.06.072
Yasmin N, Sivakumar B (2021a) Study of temporal streamflow dynamics with complex networks: network construction and clustering. Stoch Environ Res Risk Assess 35:579–595. https://doi.org/10.1007/s00477-020-01931-9
Yasmin N, Sivakumar B (2021b) Spatio-temporal connections in streamflow: a complex networks-based approach. Stoch Environ Res Risk Assess 35:2375–2390. https://doi.org/10.1007/s00477-021-02022-z
Zhang XS, Amirthanathan GE, Bari MA, Laugesen RM, Shin D, Kent DM, MacDonald AM, Turner ME, Tuteja NK (2016) How streamflow has changed across Australia since the 1950s: evidence from the network of hydrologic reference stations. Hydrol Earth Syst Sci 20:3947–3965. https://doi.org/10.5194/hess-20-3947-2016
Zhang X, Zhu J, Wang Q, Zhao H (2013) Identifying influential nodes in complex networks with community structure. Knowl Based Syst 42:74–84. https://doi.org/10.1016/j.knosys.2013.01.017
Acknowledgements
The work is partially supported by the IIT Bombay seed Grant (RD/0519-IRCCSH0-027) provided to Bellie Sivakumar. The authors would like to thank the two reviewers for their constructive comments and useful suggestions on an earlier version of this manuscript.
Funding
The work is partially supported by the IIT Bombay seed grant (RD/0519-IRCCSH0-027) provided to Bellie Sivakumar.
Author information
Authors and Affiliations
Contributions
Conceptualization: B. Deepthi, Bellie Sivakumar; Methodology: B. Deepthi, Bellie Sivakumar; Formal analysis and investigation: B. Deepthi, Bellie Sivakumar; Writing - original draft preparation: B. Deepthi; Writing - review and editing: Bellie Sivakumar; Funding acquisition: Bellie Sivakumar; Supervision: Bellie Sivakumar.
Corresponding author
Ethics declarations
Competing Interests
The authors have no relevant financial or non-financial interests to disclose.
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
Deepthi, B., Sivakumar, B. Towards assessing the importance of individual stations in hydrometric networks: application of complex networks. Stoch Environ Res Risk Assess 37, 1333–1352 (2023). https://doi.org/10.1007/s00477-022-02340-w
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-022-02340-w