Skip to main content
SpringerLink
Log in

Structural comparisons of networks and model-based detection of small-worldness

  • Regular Article
  • Published:
Journal of Economic Interaction and Coordination Aims and scope Submit manuscript

Abstract

In this paper, we consider the problem of assessing the “level of small-worldness” of a graph and of detecting small-worldness features in real networks. After discussing the limitations of classical approaches, based on the computation of network indicators, we propose a new procedure, which involves the comparison of network structures at different “observation scales”. This allows small-world features to be caught, even if “hidden” deeply into the network structure. Applications of the procedure to both simulated and real data show the effectiveness of the proposal, also in distinguishing between different small-world models and in detecting emerging small-worldness in dynamical networks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. By “possibly strategic” we refer to changes that may impact on the functioning of the system represented by the network; typically, this is connected to the way information flow across small-world networks (Braha 2004; Costa and Barros 2006), affecting agents’ behavior and potentially leading to systemic modifications and macroscopic effects (e.g. see Uzzi et al. 2007, p. 81, on firm performance in the area of patenting rates).

  2. This inconsistency is less evident in Walsh (1999), as the author associates the small-worldness with \(\mu \) much larger than 1.

  3. We select this interval of probability values, since in this case the networks usually display small-world characteristics.

  4. \(\bar{C}\) assumes values between 0.36 and 0.29 over the time period.

  5. Similar values are obtained by computing the proximity ratio \(\mu \) proposed in Walsh (1999).

  6. In other words, we assure that the number of rewired edges in the WS model is equal to the number of links added in the NW model.

References

  • Adamic LA (1999) The small world web. In: Abiteboul S, Vercoustre AM (eds) Research and advanced technology for digital libraries. ECDL 1999. Lecture Notes in Computer Science, vol 1696. Springer, Berlin, Heidelberg

  • Andrade RFS, Miranda JGV, Pinho STR, Petit Lobão T (2008) Measuring distances between complex networks. Phys Lett A 372(32):5265–5269

    Article  Google Scholar 

  • Axenovich M, Kézdy A, Martin R (2008) On the editing distance of graphs. J Graph Theory 58(2):123–138

    Article  Google Scholar 

  • Barmpoutis D, Murray RM (2010) Networks with the smallest average distance and the largest average clustering. arXiv:1007.4031

  • Barrat A, Weigt M (2000) On the properties of small-world network models. Eur Phys J B 13:547–560

    Article  Google Scholar 

  • Barthélemy M, Amaral LAN (1999) Small-World networks: evidence for a crossover picture. Phys Rev Lett 82(15):3180–3183

    Article  Google Scholar 

  • Battiston S, Catanzaro M (2004) Statistical properties of board and director networks. Eur Phys J B 38:345–352

    Article  Google Scholar 

  • Bellenzier L, Grassi R (2014) Interlocking directorates in Italy: persistent links in network dynamics. J Econ Interact Coord 9(2):183–202

    Article  Google Scholar 

  • Bertoni F, Randone P (2006) The small-world of Italian finance: ownership interconnections and board interlocks amongst Italian listed companies. SSRN Electron J. doi:10.2139/ssrn.917587

  • Braha D (2004) Information flow structure in large-scale product development organizational networks. J Inf Technol 19(4):244–253

    Article  Google Scholar 

  • Braha D, Bar-Yam Y (2006) From centrality to temporary fame:dynamic centrality in complex networks. Complexity 12(2):59–63

    Article  Google Scholar 

  • Caldarelli G, Catanzaro M (2004) The corporate boards networks. Phys A 338:98–106

    Article  Google Scholar 

  • Carley KM (2003) Dynamic network analysis. In: Breiger R, Carley K, Pattison P (eds) Dynamic social network modeling and analysis: workshop summary and papers, committee on human factors, National Research Council. National Research Council, Washington, pp 133–0145

  • Černý V (1985) Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J Optim Theory App 45:41–51

    Article  Google Scholar 

  • Costa RA, Barros J (2006) Network information flow in small-world networks. Available on www.arXiv.org

  • Erdős P, Rényi A (1959) On random graphs I. Publ Math (Debrecen) 6:290–297

    Google Scholar 

  • Erdős P, Rényi A (1960) On the evolution of random graphs. Publ Math Inst Hung Acad Sci 5:17–61

    Google Scholar 

  • Fattore M, Grassi R (2014) Measuring dynamics and structural change of time-dependent socio-economic networks. Qual Quant 48(4):1821–1834

    Article  Google Scholar 

  • Gao X, Xiao B, Tao D, Li X (2010) A survey of graph edit distance. Pattern Anal Appl 13(1):113–129

    Article  Google Scholar 

  • Granville V, Krivanek M, Rasson JP (1994) Simulated annealing: a proof of convergence. IEEE T Pattern Anal 16(6):652–656

    Article  Google Scholar 

  • Harary F (1969) Graph theory. Perseus Books, Cambridge

    Book  Google Scholar 

  • Holme P, Saramaki J (2012) Temporal networks. Phys Rep 519(3):97–125

    Article  Google Scholar 

  • Humphries MD, Gurney K (2008) Network “small-world-ness”: a quantitative method for determining canonical network equivalence. PlosOne 3:4

    Article  Google Scholar 

  • Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680

    Article  Google Scholar 

  • Kleinberg J (2000) The small-world phenomenon: an algorithmic perspective. In: Proceedings of 32nd ACM symposium on theory of computing, pp 163–170

  • König MD, Tessone CJ (2011) Network evolution based on centrality. Phys Rev E 84(5):11

    Article  Google Scholar 

  • Lindner I, Strulik H (2014) From tradition to modernity: economic growth in a small world. J Dev Econ 109:17–29

    Article  Google Scholar 

  • Newman MEJ (2000) Models of the small world: a review. J Stat Phys 101(3–4):819–841

    Article  Google Scholar 

  • Newman MEJ (2001) The structure of scientific collaboration networks. Proc Natl Acad Sci USA 98(2):404–409

    Article  Google Scholar 

  • Newman MEJ (2010) Networks: an introduction. Oxford University Press, Oxford

    Book  Google Scholar 

  • Newman MEJ, Watts DJ (1999a) Renormalization group analysis of the small-world network model. Phys Lett A 263(4–6):341–346

    Article  Google Scholar 

  • Newman MEJ, Watts DJ (1999b) Scaling and percolation in the small-world network model. Phys Rev E 60(6):7332–7342

    Article  Google Scholar 

  • R Development Core Team (2006) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna

    Google Scholar 

  • Raddant M, Milakovic M, Birg L (2015) Persistence in corporate networks. J Econ Interact Coord. doi:10.1007/s11403-015-0165-5

    Google Scholar 

  • Santella P, Drago C, Polo A, Gagliardi E (2009) A comparison among the director networks in the main listed companies in France, Germany, Italy and the United Kingdom, MPRA Paper 16397. University Library of Munich, Germany

  • Telesford QK, Joyce KE, Hayasaka S, Burdette JH, Laurienti PJ (2011) The ubiquity of small-world networks. Brain Connect 1(5):367–75

    Article  Google Scholar 

  • Uzzi B, Amaral LA, Reed-Tsochas F (2007) Small-world networks and management science research: a review. Eur Manag Rev 4:77–91

    Article  Google Scholar 

  • Wagner A, Fell DA (2001) The small world inside large metabolic networks. Proc Biol Sci 268:1803–1810

    Article  Google Scholar 

  • Walsh T (1999) Search in a small world. In: Proceedings of the 16th international joint conference on artificial intelligence, Morgan Kaufmann Publishers Inc., San Francisco, pp 1172–1177

  • Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–444

    Article  Google Scholar 

  • Wilhite A (2001) Bilateral trade and “small-world networks”. Comput Econ 18(1):49–64

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the editor and the anonymous referees for their careful reviews on an earlier version of this paper, and all the attendants to the Workshop on the Economic Science with Heterogeneous Interacting Agents 2015 for their very constructive comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Finance and Econometrics, Catholic University of Milan, Milan, Italy

    Gian Paolo Clemente

  2. Department of Statistics and Quantitative Methods, University of Milano-Bicocca, Via Bicocca degli Arcimboldi, 8, 20126, Milan, Italy

    Marco Fattore & Rosanna Grassi

Authors
  1. Gian Paolo Clemente
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Fattore
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rosanna Grassi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rosanna Grassi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clemente, G.P., Fattore, M. & Grassi, R. Structural comparisons of networks and model-based detection of small-worldness. J Econ Interact Coord 13, 117–141 (2018). https://doi.org/10.1007/s11403-017-0202-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11403-017-0202-7

Keywords

JEL Classification

Search

Navigation