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How small is it? Comparing indices of small worldliness

  • ZACHARY P. NEAL (a1)


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.



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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), 147165.
Bagler, G. (2008). Analysis of the airport network of India as a complex weighted network. Physica A, 387 (12), 29722980.
Bassett, D. S., et al. (2008). Hierarchical organization of human cortical networks in health and schizophrenia. Journal of Neuroscience, 28 (37), 92399248. doi: 10.1523/jneurosci.1929–08.2008.
Bullmore, E., & Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10 (3), 186198. doi: 10.1038/nrn2575.
Cardillo, A., Scellato, S., Latora, V., & Porta, S. (2006). Structural properties of planar graphs of urban street patterns. Physica A, 73, 066107.
Davis, G. F., Yoo, M., & Baker, W. E. (2003). The small world of the American corporate elite, 1982–2001. Strategic Organization, 1 (3), 301326.
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), 28252830.
Grannis, R. (2010). Six degrees of “who cares?”. American Journal of Sociology, 115 (4), 9911017.
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.
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), 77947799.
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), 409421. doi: 10.1007/s10334-010-0205-z.
Han, D. D., Qian, J. H., and Liu, J. G. (2009). Network topology and correlation features affiliated with European airline companies. Physica A, 388, 7181.
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.
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), 503511. doi:10.1098/rspb.2005.3354.
Kogut, B., & Walker, G. (2001). The small world of Germany and the durability of national networks. American Sociological Review, 66 (3), 317335.
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), 426436.
Liu, Y., et al. (2008). Disrupted small-world networks in schizophrenia. Brain, 131 (4), 945961. doi: 10.1093/brain/awn018.
Lusseau, D. (2003). The emergent properties of a dolphin social network. Proceedings of the Royal Society of London B (Suppl.), 270, S186S188.
Masucci, A. P., Smith, D., Crooks, A., & Batty, M. (2009). Random planar graphs and the London street network. European Physics Journal B, 71 (2), 259271.
Milgram, S. (1967). The small world problem. Psychology Today, 1 (1), 6167.
Neal, Z. P. (2014). The devil is in the details: Differences in air traffic networks by scale, species, and season. Social Networks, 38, 6373. doi: 10.1016/j.socnet.2014.03.003.
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), 369380. doi: 10.1007/s10464-015-9720-4.
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), 177184.
Newman, M. E. J. (2001). The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 98 (2), 404409.
Newman, M. E. J. (2010). Networks: An Introduction. New York: Oxford University Press.
Schilling, M. A., & Phelps, C. C. (2007). Interfirm collaboration networks: The impact of large-scale network structure on firm innovation. Management Science, 53 (7), 11131126.
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.
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.
Sporns, O., & Zwi, J. D. (2004). The small world of the cerebral cortex. Neuroinformatics, 2 (2), 145162.
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), 367375. doi: 10.1089/brain.2011.0038.
Uzzi, B., & Spiro, J. (2005). Collaboration and creativity: The small world problem. American Journal of Sociology, 111 (2), 447504. doi: 10.1086/432782.
Wagner, A., & Fell, D. A. (2001). The small world inside large metabolic networks. Proceedings of the Royal Society B, 268 (1478), 18031810.
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440442. doi:10.1038/30918.



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