Abstract
Nowadays, the network data that we need to deal with and make sense of are becoming increasingly large and complex. Small-world networks are a type of complex networks whose underling graphs have small diameter, shorter average path length between nodes, and a high degree of clustering structures and can be found in a wide range of scientific fields, including social networks, sociology, computer science, business intelligence, and biology. However, conventional visualization algorithms for small-work networks lead to a uniform clump of nodes or are restricted to a tree structure, making the network structure difficult to identify and analyze. This work provides a new visual analytical method to improve the situation. Different from previous methods based on spanning trees, this method first generates a weighted planar sub-network based on the measurement of network centrality metrics. A force-directed algorithm based on node-edge repulsion is then applied to visualize this sub-network into a proper layout for better understanding of the data. Finally, the remaining links are placed back to maintain the original network’s integrity. The experimental results show that compared to previous methods, the proposed method can be more effective in differentiating clusters and revealing relationship patterns among individual nodes and clusters in the network. Furthermore, the proposed method is applied to a data of the semiconductor wafer manufacturing industry as a case study. The work shows that this new approach allows users to gain useful insights into the data.
Graphic abstract











Similar content being viewed by others
References
Abbasi A, Hossain L, Leydesdorff L (2012) Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks. J Informetr 6:403–412
Aleardi LC, Devillers O, Fusy É (2013) Canonical ordering for triangulations on the cylinder, with applications to periodic straight-line drawings. In: Proceedings of graph drawing, 2013. Springer, pp 376–387
Archambault D, Munzner T, Auber D (2011) Tugging graphs faster: efficiently modifying path-preserving hierarchies for browsing paths. IEEE Trans Vis Comput Graph 17:276–289
Bassett D, Bullmore E (2017) Small-world brain networks revisited. Neuroscientist 23:499–516
Bender-deMoll S, McFarland DA (2006) The art and science of dynamic network visualization. J Soc Struct 7:1–38
Bertault F (2000) A force-directed algorithm that preserves edge-crossing properties. Inf Process Lett 74:7–13
Bertin J (1983) Semiology of graphics: diagrams, networks, maps. Esri Press, Redlands
Bhandari A, Gupta A, Das D (2017) Betweenness centrality updation and community detection in streaming graphs using incremental algorithm. In: Proceedings of the 6th international conference on software and computer applications, 2017. ACM Press, pp 159–164
Bi C, Fu B, Chen J, Zhao Y, Yang L, Duan Y, Shi Y (2018) Machine learning based fast multi-layer liquefaction disaster assessment. World Wide Web. https://doi.org/10.1007/s11280-018-0632-8
Bi CK, Yang L, Duan YL, Shi Y (2019) A survey on visualization of tensor field. J Vis 22:641–660
Bonchi F, Morales GDF, Riondato M (2016) Centrality measures on big graphs: exact, approximated, and distributed algorithms. In: Proceedings of the 25th international conference companion on world wide web, 2016. ACM Press, pp 1017–1020
Borgatti SP, Mehra A, Brass DJ, Labianca G (2009) Network analysis in the social sciences. Science 323:892–895
Boyer JM, Myrvold WJ (2004) On the cutting edge: simplified O(n) planarity by edge addition. J Graph Algorithms Appl 8:241–273
Brandes U (2001) A faster algorithm for betweenness centrality. J Math Sociol 25:163–177
Brandes U, Indlekofer N, Mader M (2012) Visualization methods for longitudinal social networks and stochastic actor-oriented modeling. Soc Netw 34:291–308
Burger M, Zelazo D, Allgower F (2013) Hierarchical clustering of dynamical networks using a saddle-point analysis. IEEE Trans Autom Control 58:113–124
Correa C, Crnovrsanin T, Ma K-L (2012) Visual reasoning about social networks using centrality sensitivity. IEEE Trans Vis Comput Graph 18:106–120
Crnovrsanin T, Muelder CW, Faris R, Felmlee D, Ma K-L (2014) Visualization techniques for categorical analysis of social networks with multiple edge sets. Soc Netw 37:56–64
Das S, Lee D, Choi W, Doppa JR, Pande PP, Chakrabarty K (2017) VFI-based power management to enhance the lifetime of high-performance 3D NoCs. ACM Trans Des Autom Electron Syst 23:1–26. https://doi.org/10.1145/3092843
Davidson R, Harel D (1996) Drawing graphs nicely using simulated annealing. ACM Trans Graph 15:301–331
Dong J, Horvath S (2007) Understanding network concepts in modules. BMC Syst Biol 1:24
Eades P (1984) A heuristics for graph drawing. Congr Numer 42:146–160
Fößmeier U, Kaufmann M (1996) Drawing high degree graphs with low bend numbers. In: Proceedings of graph drawing, 1996. Springer, pp 254–266
Freeman LC (1979) Centrality in social networks conceptual clarification. Soc Netw 1:215–239
Gibson H, Vickers P (2016) Using adjacency matrices to lay out larger small-world networks. Appl Soft Comput 42:80–92
Gómez D, Figueira JR, Eusébio A (2013) Modeling centrality measures in social network analysis using bi-criteria network flow optimization problems. Eur J Oper Res 226:354–365
Gretarsson B, O’Donovan J, Bostandjiev S, Hall C, Höllerer T (2010) Smallworlds: visualizing social recommendations. Comput Graph Forum 29:833–842
Guan C, Yuen KKF (2013) Towards a hybrid approach of primitive cognitive network process and k-means clustering for social network analysis. In: Proceedings of IEEE international conference on Internet of Things (iThings/CPSCom), 2013. IEEE Press, pp 1267–1271
Gutwenger C, Mutzel P (1997) Grid embedding of biconnected planar graphs. Extended abstract. Max-Planck-Institut f ur Informatik, Saarbrücken
Huang W, Eades P, Hong S-H, Lin C-C (2013) Improving multiple aesthetics produces better graph drawings. J Vis Lang Comput 24:262–272
Huang W, Huang ML, Lin C-C (2016) Evaluating overall quality of graph visualizations based on aesthetics aggregation. Inf Sci 330:444–454
Jia Y, Hoberock J, Garland M, Hart JC (2008) On the visualization of social and other scale-free networks. IEEE Trans Vis Comput Graph 14:1285–1292
Jia Y, Garland M, Hart JC (2011) Social network clustering and visualization using hierarchical edge bundles. Comput Graph Forum 30:2314–2327
Junger M, Leipert S, Mutzel P (1998) A note on computing a maximal planar subgraph using PQ-trees. IEEE Trans Comput Aided Des Integr Circuits Syst 17:609–612
Kant G (1996) Drawing planar graphs using the canonical ordering. Algorithmica 16:4–32
Lee T, Kim C (2014) Statistical comparison of fault detection models for semiconductor manufacturing processes. IEEE Trans Semicond Manuf 28:80–91
Lin C-C, Yen H-C (2012) A new force-directed graph drawing method based on edge–edge repulsion. J Vis Lang Comput 23:29–42
Lin C-C, Huang W, Liu W-Y, Chen W-L (2018) Evaluating aesthetics for user-sketched layouts of symmetric graphs. J Vis Lang Comput 48:123–133
Love M (2007) Genealogy of influence. [Online]. Available: http://mikelove.nfshost.com/genealogy/
Newman ME, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113
Opsahl T, Agneessens F, Skvoretz J (2010) Node centrality in weighted networks: generalizing degree and shortest paths. Soc Netw 32:245–251
Shi L et al. (2009) HiMap: adaptive visualization of large-scale online social networks. In: Proceedings of 2009 IEEE pacific visualization symposium, 2009. IEEE Press, pp 41–48
Shimbel A (1953) Structural parameters of communication networks. Bull Math Biophys 15:501–507
Sohn K, Kim D (2010) Zonal centrality measures and the neighborhood effect. Transp Res Part A: Pol Pract 44:733–743
Valente TW, Foreman RK (1998) Integration and radiality: measuring the extent of an individual’s connectedness and reachability in a network. Soc Netw 20:89–105
van Ham F, Wattenberg M (2008) Centrality based visualization of small world graphs. Comput Graph Forum 27:975–982
Wakita K, Tsurumi T (2007) Finding community structure in mega-scale social networks: [extended abstract]. In: Proceedings of the 16th international conference on world wide web, 2007. ACM Press, pp 1275–1276
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’networks. Nature 393:440–442
Wong PC, Foote H, Mackey P, Chin G, Huang Z, Thomas J (2012) A space-filling visualization technique for multivariate small-world graphs. IEEE Trans Vis Comput Graph 18:797–809
Yang Z, Chen W (2018) A game theoretic model for the formation of navigable small-world networks—the tradeoff between distance and reciprocity. ACM Trans Internet Technol 18:1–38
Zuo X-N, Ehmke R, Mennes M, Imperati D, Castellanos FX, Sporns O, Milham MP (2012) Network centrality in the human functional connectome. Cereb Cortex 22:1862–1875
Acknowledgements
The authors thank the anonymous referees for comments that improved the content as well as the presentation of this paper. This work has been supported in part by Ministry of Science and Technology, Taiwan, under Grants MOST 105-2628-E-009-002-MY3 and MOST 106-2221-E-009-101-MY3, and the Ministry of Education, Taiwan, under the Higher Education Sprout Project.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lin, CC., Huang, W., Liu, WY. et al. A novel centrality-based method for visual analytics of small-world networks. J Vis 22, 973–990 (2019). https://doi.org/10.1007/s12650-019-00582-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12650-019-00582-5