Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-27T11:25:32.312Z Has data issue: false hasContentIssue false
  • English
  • Français

How small is it? Comparing indices of small worldliness

Published online by Cambridge University Press:  03 April 2017

ZACHARY P. NEAL
ZACHARY P. NEAL*
Affiliation:
Psychology Department, Michigan State University, 316 Physics Room 262, East Lansing, MI 4882, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Many studies have attempted to determine whether an observed network exhibits a so-called “small-world structure.” Such determinations have often relied on a conceptual definition of small worldliness proposed by Watts and Strogatz in their seminal 1998 paper, but recently several quantitative indices of network small worldliness have emerged. This paper reviews and compares three such indices—the small-world quotient (Q), a small-world metric (ω), and the small-world index(SWI)—in the canonical Watts–Strogatz re-wiring model and in four real-world networks. These analyses suggest that researchers should avoid Q, and identify considerations that should guide the choice between ω and SWI.

Keywords

small worldclusteringpath length
Type
Research Article
Information
Network Science , Volume 5 , Issue 1 , March 2017 , pp. 30 - 44
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldrich, H. E., & Kim, P. H. (2007). Small worlds, infinite possibilities? How social networks affect entrepreneurial team formation and search. Strategic Entrepreneurship Journal, 1 (1–2), 147–165.Google Scholar
Bagler, G. (2008). Analysis of the airport network of India as a complex weighted network. Physica A, 387 (12), 2972–2980.CrossRefGoogle Scholar
Bassett, D. S., et al. (2008). Hierarchical organization of human cortical networks in health and schizophrenia. Journal of Neuroscience, 28 (37), 9239–9248. doi: 10.1523/jneurosci.1929–08.2008.Google Scholar
Bullmore, E., & Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10 (3), 186–198. doi: 10.1038/nrn2575.CrossRefGoogle ScholarPubMed
Cardillo, A., Scellato, S., Latora, V., & Porta, S. (2006). Structural properties of planar graphs of urban street patterns. Physica A, 73, 066107.Google ScholarPubMed
Davis, G. F., Yoo, M., & Baker, W. E. (2003). The small world of the American corporate elite, 1982–2001. Strategic Organization, 1 (3), 301–326.Google Scholar
Gallos, L. K., Makse, H. A., & Sigman, M. (2012). A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proceedings of the National Academy of Sciences, 109 (8), 2825–2830.Google Scholar
Grannis, R. (2010). Six degrees of “who cares?”. American Journal of Sociology, 115 (4), 991–1017.Google Scholar
Guimera, R., Danon, L., Diaz-Guilera, A., Giralt, F., & Arenas, A. (2003). Self-similar community structure in a network of human interactions. Physical Review E, 68, 065103.Google Scholar
Guimera, R., Mossa, S., Turtschi, A., & Amaral, L. A. N. (2005). The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Procedings of the National Academy of Sciences, 102 (22), 7794–7799.Google Scholar
Guye, M., Bettus, G., Bartolomei, F., & Cozzone, P. (2010). Graph theoretical analysis of structural and functional connectivity MRI in normal and pathological brain networks. MAGMA, 23 (5–6), 409–421. doi: 10.1007/s10334-010-0205-z.Google Scholar
Han, D. D., Qian, J. H., and Liu, J. G. (2009). Network topology and correlation features affiliated with European airline companies. Physica A, 388, 71–81.Google Scholar
Humphries, M. D., & Gurney, K. (2008). Network ‘small-world-ness’: A quantitative method for determining canonical network equivalence. PLoS ONE, 3, e0002051. doi:10.1371/journal.pone.0002051.Google Scholar
Humphries, M. D., Gurney, K., & Prescott, T. J. (2006). The brainstem reticular formation is a small-world, not scale-free, network. Proceedings of the Royal Society B, 273 (1585), 503–511. doi:10.1098/rspb.2005.3354.Google Scholar
Kogut, B., & Walker, G. (2001). The small world of Germany and the durability of national networks. American Sociological Review, 66 (3), 317–335.CrossRefGoogle Scholar
Lawlor, J. A., & Neal, Z. P. (2016). Networked community change: Understanding community systems change through the lens of social network analysis. American Journal of Community Psychology, 57 (3–4), 426–436.CrossRefGoogle ScholarPubMed
Liu, Y., et al. (2008). Disrupted small-world networks in schizophrenia. Brain, 131 (4), 945–961. doi: 10.1093/brain/awn018.Google Scholar
Lusseau, D. (2003). The emergent properties of a dolphin social network. Proceedings of the Royal Society of London B (Suppl.), 270, S186–S188.Google Scholar
Masucci, A. P., Smith, D., Crooks, A., & Batty, M. (2009). Random planar graphs and the London street network. European Physics Journal B, 71 (2), 259–271.Google Scholar
Milgram, S. (1967). The small world problem. Psychology Today, 1 (1), 61–67.Google Scholar
Neal, Z. P. (2014). The devil is in the details: Differences in air traffic networks by scale, species, and season. Social Networks, 38, 63–73. doi: 10.1016/j.socnet.2014.03.003.CrossRefGoogle Scholar
Neal, Z. P. (2015). Making big communities small: Using network science to understand the ecological and behavioral requirements for community social capital. American Journal of Community Psychology, 55 (3–4), 369–380. doi: 10.1007/s10464-015-9720-4.Google Scholar
Neal, Z. P., Neal, J. W., Lawlor, J. A., & Mills, K. J. (2015). Small worlds or worlds apart? Using network theory to understand the research-practice gap. Psychosocial Interventions, 24 (3), 177–184.Google Scholar
Newman, M. E. J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 98 (2), 404–409.Google Scholar
Newman, M. E. J. (2010). Networks: An Introduction. New York: Oxford University Press.Google Scholar
Schilling, M. A., & Phelps, C. C. (2007). Interfirm collaboration networks: The impact of large-scale network structure on firm innovation. Management Science, 53 (7), 1113–1126.Google Scholar
Sen, P., Dasgupa, S., Chatterjee, A., Sreeram, P. A., Mukherjee, G., & Manna, S. S. (2003). Small-world properties of the Indian rail network. Physica A, 67, 036106.Google Scholar
Singh, P. V. (2010). The small-world effect: The influence of macro-level properties of developer collaboration networks on open-source project success. ACM Transactions on Software Engineering and Methodology, 20 (2), 6.Google Scholar
Sporns, O., & Zwi, J. D. (2004). The small world of the cerebral cortex. Neuroinformatics, 2 (2), 145–162.Google Scholar
Telesford, Q. K., Joyce, K. E., Hayasaka, S., Burdette, J. H., & Laurenti, P. J. (2011). The ubiquity of small-world networks. Brain Connectivity, 1 (5), 367–375. doi: 10.1089/brain.2011.0038.Google Scholar
Uzzi, B., & Spiro, J. (2005). Collaboration and creativity: The small world problem. American Journal of Sociology, 111 (2), 447–504. doi: 10.1086/432782.Google Scholar
Wagner, A., & Fell, D. A. (2001). The small world inside large metabolic networks. Proceedings of the Royal Society B, 268 (1478), 1803–1810.Google Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442. doi:10.1038/30918.Google Scholar