On Stochastic Confidence of Information Spread in Opportunistic Networks

Predicting spreading patterns of information or virus has been a popular research topic for which various mathematical tools have been developed. These tools have mainly focused on estimating the average time of spread to a fraction (e.g., <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq1-2431711.gif"/></alternatives></inline-formula>) of the agents, i.e., so-called average <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq2-2431711.gif"/> </alternatives></inline-formula>-completion time <inline-formula><tex-math notation="LaTeX">$E(T_{\alpha})$</tex-math> <alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq3-2431711.gif"/></alternatives></inline-formula>. We claim that understanding stochastic confidence on the time <inline-formula><tex-math notation="LaTeX">$T_{\alpha}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq4-2431711.gif"/></alternatives></inline-formula> rather than only its average gives more comprehensive knowledge on the spread behavior and wider engineering choices. Obviously, the knowledge also enables us to effectively accelerate or decelerate a spread. To demonstrate the benefits of understanding the distribution of spread time, we introduce a new metric <inline-formula><tex-math notation="LaTeX">$G_{\alpha, \beta}$ </tex-math> <alternatives><inline-graphic xlink:type="simple" xlink:href="lee-ieq5-2431711.gif"/></alternatives></inline-formula> that denotes the time required to guarantee <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq6-2431711.gif"/></alternatives></inline-formula> completion (i.e., penetration) with probability <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq7-2431711.gif"/></alternatives></inline-formula>. Also, we develop a new framework characterizing <inline-formula><tex-math notation="LaTeX">$G_{\alpha, \beta}$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="lee-ieq8-2431711.gif"/></alternatives></inline-formula> for various spread parameters such as number of seeders, contact rates between agents, and heterogeneity in contact rates. We apply our technique to a large-scale experimental vehicular trace and show that it is possible to allocate resources for acceleration of spread in a far more elaborated way compared to conventional average-based mathematical tools.

[1]  H. Andersson,et al.  Stochastic Epidemic Models and Their Statistical Analysis , 2000 .

[2]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[3]  Timur Friedman,et al.  Characterizing pairwise inter-contact patterns in delay tolerant networks , 2007, Autonomics.

[4]  W. Stewart,et al.  A numerical study of large sparse matrix exponentials arising in Markov chains 1 1 This work has ben , 1999 .

[5]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[6]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[7]  Yousef Saad,et al.  Efficient Solution of Parabolic Equations by Krylov Approximation Methods , 1992, SIAM J. Sci. Comput..

[8]  Jari Saramäki,et al.  Small But Slow World: How Network Topology and Burstiness Slow Down Spreading , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  S. Cornell,et al.  Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape , 2001, Science.

[10]  Qinghua Li,et al.  Multicasting in delay tolerant networks: a social network perspective , 2009, MobiHoc '09.

[11]  Donald F. Towsley,et al.  Performance modeling of epidemic routing , 2006, Comput. Networks.

[12]  Injong Rhee,et al.  Max-Contribution: On Optimal Resource Allocation in Delay Tolerant Networks , 2010, 2010 Proceedings IEEE INFOCOM.

[13]  Injong Rhee,et al.  Providing probabilistic guarantees on the time of information spread in opportunistic networks , 2013, 2013 Proceedings IEEE INFOCOM.

[14]  Brian Gallagher,et al.  MaxProp: Routing for Vehicle-Based Disruption-Tolerant Networks , 2006, Proceedings IEEE INFOCOM 2006. 25TH IEEE International Conference on Computer Communications.

[15]  Tao Zhou,et al.  Impact of Heterogeneous Human Activities on Epidemic Spreading , 2011, ArXiv.

[16]  P. Van Mieghem,et al.  Virus Spread in Networks , 2009, IEEE/ACM Transactions on Networking.

[17]  Saurabh Bagchi,et al.  Modeling and automated containment of worms , 2005, 2005 International Conference on Dependable Systems and Networks (DSN'05).

[18]  Donald F. Towsley,et al.  Code red worm propagation modeling and analysis , 2002, CCS '02.

[19]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[20]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[21]  Do Young Eun,et al.  Characterizing link connectivity for opportunistic mobile networking: Does mobility suffice? , 2013, 2013 Proceedings IEEE INFOCOM.

[22]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[23]  Stratis Ioannidis,et al.  Optimal and scalable distribution of content updates over a mobile social network , 2009, IEEE INFOCOM 2009.

[24]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[25]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .