Skip to main content
SpringerLink
Log in

Threshold conditions for arbitrary cascade models on arbitrary networks

  • Regular Paper
  • Published:
Knowledge and Information Systems Aims and scope Submit manuscript

Abstract

Given a network of who-contacts-whom or who-links-to-whom, will a contagious virus/product/meme spread and ‘take over’ (cause an epidemic) or die out quickly? What will change if nodes have partial, temporary or permanent immunity? The epidemic threshold is the minimum level of virulence to prevent a viral contagion from dying out quickly and determining it is a fundamental question in epidemiology and related areas. Most earlier work focuses either on special types of graphs or on specific epidemiological/cascade models. We are the first to show the G2-threshold (twice generalized) theorem, which nicely de-couples the effect of the topology and the virus model. Our result unifies and includes as special case older results and shows that the threshold depends on the first eigenvalue of the connectivity matrix, (a) for any graph and (b) for all propagation models in standard literature (more than 25, including H.I.V.). Our discovery has broad implications for the vulnerability of real, complex networks and numerous applications, including viral marketing, blog dynamics, influence propagation, easy answers to ‘what-if’ questions, and simplified design and evaluation of immunization policies. We also demonstrate our result using extensive simulations on real networks, including on one of the biggest available social-contact graphs containing more than 31 million interactions among more than 1 million people representing the city of Portland, Oregon, USA.

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

Similar content being viewed by others

References

  1. Albert R, Jeong H, Barabási AL (2000) Error and attack tolerance of complex networks. Nature 407(6794): 378–482

    Article  Google Scholar 

  2. Anderson RM, May RM (1991) Infectious diseases of humans. Oxford University Press, Oxford

    Google Scholar 

  3. Barrat A, Barthélemy M, Vespignani A (2010) Dynamical processes on complex networks. Cambridge University Press, Cambridge

    Google Scholar 

  4. Barrett CL, Bisset KR, Eubank SG, Feng X, Marathe MV (2008) Episimdemics: an efficient algorithm for simulating the spread of infectious disease over large realistic social networks. In: ACM/IEEE conference on supercomputing

  5. Bass FM (1969) A new product growth for model consumer durables. Manag Sci 15(5): 215–227

    Article  MATH  Google Scholar 

  6. Bikhchandani S, Hirshleifer D, Welch I (1992) A theory of fads, fashion, custom, and cultural change in informational cascades. J Political Econ 100(5): 992–1026

    Article  Google Scholar 

  7. Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (2010) Catastrophic cascade of failures in interdependent networks. Nature 464(7291): 1025–1028

    Article  Google Scholar 

  8. Castellano C, Pastor-Satorras R (2010) Thresholds for epidemic spreading in networks. Phys Rev Lett 105

  9. Chakrabarti D, Leskovec J, Faloutsos C, Madden S, Guestrin C, Faloutsos M (2007) Information survival threshold in sensor and P2P networks. In: IEEE INFOCOM

  10. Chakrabarti D, Wang Y, Wang C, Leskovec J, Faloutsos C (2008) Epidemic thresholds in real networks. ACM TISSEC 10(4)

  11. Chen Q, Chang H, Govindan R, Jamin S, Shenker SJ, Willinger W (2002) The origin of power laws in internet topologies revisited. In: IEEE INFOCOMM. http://topology.eecs.umich.edu/data.html

  12. Chen W, Wang C, Wang Y (2010) Scalable influence maximization for prevalent viral marketing in large-scale social networks. In: KDD

  13. Chung F, Lu L, Vu V (2003) Eigenvalues of random power law graphs. Ann Comb 7(1)

  14. Cohen R, Havlin S, ben Avraham D (2003) Efficient immunization strategies for computer networks and populations. Phys Rev Lett 91(24): 247901

    Article  Google Scholar 

  15. Dodds PS, Watts DJ (2004) A generalized model of social and biological contagion. J Theor Biol 232: 587–604

    Article  MathSciNet  Google Scholar 

  16. Easley D, Kleinberg J (2010) Networks, crowds, and markets: reasoning about a highly connected world. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  17. Eubank S, Guclu H, Anil Kumar VS, Marathe MV, Srinivasan A, Toroczkai Z, Wang N (2004) Modelling disease outbreaks in realistic urban social networks. Nature 429(6988): 180–184

    Article  Google Scholar 

  18. Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. In: SIGCOMM, pp 251–262

  19. Ganesh A, Massoulié L, Towsley D (2005) The effect of network topology on the spread of epidemics. In: INFOCOM

  20. Goldenberg J, Libai B, Muller E (2001) Talk of the network: a complex systems look at the underlying process of word-of-mouth. Mark Lett 3(12): 211–223

    Article  Google Scholar 

  21. Gomez Rodriguez M, Leskovec J, Krause A (2010) Inferring networks of diffusion and influence. In: Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, New York, NY, USA (KDD ’10), pp 1019–1028

  22. Granovetter M (1978) Threshold models of collective behavior. Am J Sociol 83(6): 1420–1443

    Article  Google Scholar 

  23. Gruhl D, Guha R, Liben-Nowell D, Tomkins A (2004) Information diffusion through blogspace. In: WWW ’04. http://www2004.org/proceedings/docs/1p491.pdf

  24. Hayashi Y, Minoura M, Matsukubo J (2003) Recoverable prevalence in growing scale-free networks and the effective immunization. arXiv:cond-mat/0305549 v2

  25. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42

  26. Hethcote HW, Yorke JA (1984) Gonorrhea transmission dynamics and control. Springer Lecture notes in biomathematics 46

  27. Hirsch MW, Smale S (1974) Differential equations, dynamical systems and linear algebra. Academic Press, New York

    MATH  Google Scholar 

  28. Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  29. Kempe D, Kleinberg J, Tardos E (2003) Maximizing the spread of influence through a social network. In: KDD

  30. Kephart JO, White SR (1993) Measuring and modeling computer virus prevalence. In: IEEE computer society symposium on research in security and privacy

  31. Kleinberg J (2007) The wireless epidemic. Nature 449

  32. Kumar R, Novak J, Raghavan P, Tomkins A (2003) On the bursty evolution of blogspace. In: WWW

  33. Lad M, Zhao X, Zhang B, Massey D, Zhang L (2003) Analysis of BGP update surge during slammer worm attack. In: 5th International workshop on distributed computing (IWDC). http://citeseer.ist.psu.edu/lad03analysis.html

  34. Lappas T, Terzi E, Gunopoulos D Mannila H (2010) Finding effectors in social networks. In: SIGKDD

  35. Leskovec J, Adamic LA, Huberman BA (2006) The dynamics of viral marketing. In: EC. ACM Press, New York, NY, USA, pp 228–237

  36. Leskovec J, Backstrom L, Kleinberg J (2009) Meme-tracking and the dynamics of the news cycle. In: ACM SIGKDD

  37. Maymounkov P, Mazières D (2002) Kademlia: a peer-to-peer information system based on the XOR metric. In: Revised papers from the first international workshop on peer-to-peer systems, Springer, London, UK (IPTPS ’01), pp 53–65. http://dl.acm.org/citation.cfm?id=646334.687801

  38. McCuler CR (2000) The many proofs and applications of perron’s theorem. SIAM Rev 42

  39. McKendrick AG (1926) Applications of mathematics to medical problems. In: Proceedings of Edin. Math. Society, vol 14, pp 98–130

  40. Milnes HW (Aug–Sept 1963) Conditions that the zeros of a polynomial lie in the interval [−1, 1] when all zeros are real. Am Math Mon 70(7)

  41. NDSSL (2007) Synthetic data products for societal infrastructures and protopopulations: data set 2.0. NDSSL-TR-07-003. http://ndssl.vbi.vt.edu/Publications/ndssl-tr-07-003.pdf

  42. Newman MEJ (2002) Spread of epidemic disease on networks. Phys Rev E 66(1): 016128. doi:10.1103/PhysRevE.66.016128

    Article  MathSciNet  Google Scholar 

  43. Newman MEJ (2005) Threshold effects for two pathogens spreading on a network. Phys Rev Lett 95(10): 108701

    Article  Google Scholar 

  44. Pastor-Santorras R, Vespignani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett 86: 14

    Article  Google Scholar 

  45. Pathak N, Banerjee A, Srivastava J (2010) A generalized linear threshold model for multiple cascades. In: ICDM

  46. Prakash BA, Tong H, Valler N, Faloutsos M, Faloutsos C (2010) Virus propagation on time-varying networks: theory and immunization algorithms. In: ECML-PKDD

  47. Prakash BA, Beutel A, Rosenfeld R, Faloutsos C (2012) Winner takes all: competiting viruses or ideas on fair-play networks. In: WWW

  48. Richardson M, Domingos P (2002) Mining knowledge-sharing sites for viral marketing. In: SIGKDD

  49. Rogers EM (2003) Diffusion of innovations, 5th ed. Free Press. http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&path=ASIN/0743222091

  50. Saito K, Kimura M, Ohara K, Motoda H (2012) Efficient discovery of influential nodes for sis models in social networks. Knowl Inf Syst (KAIS) 30(3): 613–635

    Article  Google Scholar 

  51. Vojnovic M, Gupta V, Karagiannis T, Gkantsidis C (2008) Sampling strategies for epidemic-style information dissemination. In: IEEE INFOCOM

  52. Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393: 440–442

    Article  Google Scholar 

  53. Website (November 15, 2011) Mainline bittorrent website. http://www.bittorrent.com/

  54. Zhao J, Wu J, Feng X, Xiong H, Xu K (2011) Information propagation in online social networks: a tie-strength perspective. Knowl Inf Syst 1–20. doi:10.1007/s10115-011-0445-x

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    B. Aditya Prakash

  2. Yahoo! Research, Santa Clara, CA, USA

    Deepayan Chakrabarti

  3. University of California, Riverside, CA, USA

    Nicholas C. Valler & Michalis Faloutsos

  4. Carnegie Mellon University, Pittsburgh, PA, USA

    Christos Faloutsos

Authors
  1. B. Aditya Prakash
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deepayan Chakrabarti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nicholas C. Valler
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michalis Faloutsos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Christos Faloutsos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Aditya Prakash.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prakash, B.A., Chakrabarti, D., Valler, N.C. et al. Threshold conditions for arbitrary cascade models on arbitrary networks. Knowl Inf Syst 33, 549–575 (2012). https://doi.org/10.1007/s10115-012-0520-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10115-012-0520-y

Keywords

Search

Navigation