Abstract
Kleinberg’s small world model [20] simulates social networks with both strong and weak ties. In his original paper, Kleinberg showed how the distribution of weak-ties, parameterized by \(\gamma \), influences the efficacy of myopic routing on the network. Recent work on social influence by k-complex contagion models discovered that the distribution of weak-ties also impacts the spreading rate in a crucial manner on Kleinberg’s small world model [15]. In both cases the parameter of \(\gamma = 2\) proves special: when \(\gamma \) is anything but 2 the properties no longer hold.
In this work, we propose a natural generalization of Kleinberg’s small world model to allow node heterogeneity: instead of a single global parameter \(\gamma \), each node has a personalized parameter \(\gamma \) chosen independently from a distribution \(\mathcal {D}\). In contrast to the original model, we show that this model enables myopic routing and k-complex contagions on a large range of the parameter space, improving the robustness of the model. Moreover, we show that our generalization is supported by real-world data. Analysis of four different social networks shows that the nodes do not show homogeneity in terms of the variance of the lengths of edges incident to the same node.
J. Gao would like to acknowledge support through NSF DMS-1418255, CCF-1535900, CNS-1618391, DMS-1737812 and AFOSR FA9550-14-1-0193. G. Schoenebeck and F. Yu gratefully acknowledge the support of the National Science Foundation under Career Award 1452915 and AitF Award 1535912.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In order to eliminate the boundary effect, we wrap up the grid into a torus – i.e., the top boundary is identified with the bottom boundary and the left boundary is identified with the right boundary.
- 2.
For discrete distribution, the probability density function exists if we allow using Dirac delta function.
- 3.
The scalar depends on the constants \(k, \eta , \alpha , K\).
References
Adler, J.: Bootstrap percolation. Phys. A: Stat. Theor. Phys. 171(3), 453–470 (1991)
Amini, H.: Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electr. J. Comb. 17(1), R25 (2010)
Amini, H., Fountoulakis, N.: What i tell you three times is true: bootstrap percolation in small worlds. In: Goldberg, P.W. (ed.) WINE 2012. LNCS, vol. 7695, pp. 462–474. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35311-6_34
Balogh, J., Pittel, B.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30, 257–286 (2007)
Boguna, M., Krioukov, D., Claffy, K.C.: Navigability of complex networks. Nat. Phys. 5, 74–80 (2009)
Bollobás, B., Chung, F.R.K.: The diameter of a cycle plus a random matching. SIAM J. Discret. Math. 1(3), 328–333 (1988)
Burt, R.S.: Structural Holes: The Social Structure of Competition. Cambridge University Press, Cambridge (1992)
Burt, R.S.: Structural Holes: The social structure of competition. Harvard University Press, Cambridge (1995)
Chalupa, J., Leath, P.L., Reich, G.R.: Bootstrap percolation on a Bethe lattice. J. Phys. C: Solid State Phys. 12(1), L31 (1979)
Dodds, P.S., Muhamad, R., Watts, D.J.: An experimental study of search in global social networks. Science 301, 827 (2003)
Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 409–410 (1998)
Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: How complex contagions in preferential attachment models and other time-evolving networks. IEEE Trans. Netw. Sci. Eng. PP(99), 1 (2017). https://doi.org/10.1109/TNSE.2017.2718024. ISSN 2327–4697
Ebrahimi, R., Gao, J., Ghasemiesfeh, G., Schoenebeck, G.: Complex contagions in Kleinberg’s small world model. In: Proceedings of the 6th Innovations in Theoretical Computer Science (ITCS 2015), pp. 63–72. January 2015
Gao, J., Ghasemiesfeh, G., Schoenebeck, G., Yu, F.-Y.: General threshold model for social cascades: analysis and simulations. In: Proceedings of the 2016 ACM Conference on Economics and Computation, pp. 617–634. ACM (2016)
Ghasemiesfeh, G., Ebrahimi, R., Gao, J.: Complex contagion and the weakness of long ties in social networks: revisited. In: Proceedings of the fourteenth ACM conference on Electronic Commerce, pp. 507–524. ACM (2013)
Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol. 83(6), 1420–1443 (1978)
Jackson, M.O.: Social and Economic Networks. Princeton University Press, Princeton (2008). ISBN 0691134405, 9780691134406
Janson, S., Luczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph \({G}_{n, p}\). Ann. Appl. Probab. 22(5), 1989–2047 (2012)
Jeong, H., Mason, S.P., Barabasi, A.-L., Oltvai, Z.N.: Lethality and centrality in protein networks. Nature 411, 41–42 (2001)
Kleinberg, J., The small-world phenomenon: an algorithm perspective. In: Proceedings of the 32-nd Annual ACM Symposium on Theory of Computing, pp. 163–170 (2000)
Krioukov, D., Papadopoulos, F., Boguna, M., Vahdat, A.: Greedy forwarding in scale-free networks embedded in hyperbolic metric spaces. In: ACM SIGMETRICS Workshop on Mathematical Performance Modeling and Analysis (MAMA) June 2009
Kumar, R., Liben-Nowell, D., Tomkins, A.: Navigating low-dimensional and hierarchical population networks. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 480–491. Springer, Heidelberg (2006). https://doi.org/10.1007/11841036_44. ISBN 3-540-38875-3
Milgram, S.: The small world problem. Phychol. Today 1, 61–67 (1967)
Newman, M.E.J., Moore, C., Watts, D.J.: Mean-field solution of the small-world network model. Phys. Rev. Lett. 84, 3201–3204 (2000)
Schoenebeck, G., Yu, F.-Y.: Complex contagions on configuration model graphs with a power-law degree distribution. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 459–472. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-54110-4_32
Travers, J., Milgram, S.: An experimental study of the small world problem. Sociometry 32, 425 (1969)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)
Williams, R.J., Berlow, E.L., Dunne, J.A., Barabasi, A.L., Martinez, N.D.: Two degrees of separation in complex food webs. Proc. Nat. Acad. Sci. 99(20), 12913–12916 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Gao, J., Schoenebeck, G., Yu, FY. (2017). Cascades and Myopic Routing in Nonhomogeneous Kleinberg’s Small World Model. In: R. Devanur, N., Lu, P. (eds) Web and Internet Economics. WINE 2017. Lecture Notes in Computer Science(), vol 10660. Springer, Cham. https://doi.org/10.1007/978-3-319-71924-5_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-71924-5_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-71923-8
Online ISBN: 978-3-319-71924-5
eBook Packages: Computer ScienceComputer Science (R0)