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Partial differential equations with Robin boundary condition in online social networks
1. | Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China, China |
2. | School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069, United States, United States |
References:
[1] |
G. A. Afrouzi and K. J. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions,, Proc. Amer. Math. Soc., 127 (1999), 125.
doi: 10.1090/S0002-9939-99-04561-X. |
[2] |
W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, Nonlinear Anal., 32 (1998), 819.
doi: 10.1016/S0362-546X(97)00530-0. |
[3] |
A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks,, Cambridge University Press, (2008).
doi: 10.1017/CBO9780511791383. |
[4] |
F. Benevenuto, T. Rodrigues, M. Cha and V. Almeida, Characterizing user behavior in online social networks,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 49.
doi: 10.1145/1644893.1644900. |
[5] |
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments,, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293.
doi: 10.1017/S030821050001876X. |
[6] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons Ltd, (2003).
doi: 10.1002/0470871296. |
[7] |
M. Cha, A. Mislove, B. Adams and K. Gummadi, Characterizing social cascades in Flickr,, in Proceeding WOSN '08 Proceedings of the First Workshop on Online Social Networks, (2008), 13.
doi: 10.1145/1397735.1397739. |
[8] |
L. C. Evans, Partial Differential Equations,, AMS, (1998).
|
[9] |
P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991).
|
[10] |
R. Ghosh and K. Lerman, A framework for quantitative analysis of cascades on networks,, in ACM International Conference on Web Search and Data Mining, (2011), 665.
doi: 10.1145/1935826.1935917. |
[11] |
A. Guille, H. Hacid, C. Favre and D. Zighed, Information diffusion in online social networks: A survey,, SIGMOD Record, 42 (2013), 17.
doi: 10.1145/2503792.2503797. |
[12] |
E. L. Ince, Ordinary Differential Equation,, Dover, (1944).
|
[13] |
F. Jin, E. Dougherty, P. Saraf, Y. Cao and N. Ramakrishnan, Epidemiological modeling of news and rumors on Twitter,, in Proceedings of the 7th Workshop on Social Network Mining and Analysis, (2013).
doi: 10.1145/2501025.2501027. |
[14] |
K. Kreith, Picone's identity and generalizations,, Rend. Mat., 8 (1975), 251.
|
[15] |
R. Kumar, J. Novak and A. Tomkins, Structure and evolution of online social networks,, in Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2006), 611.
doi: 10.1145/1150402.1150476. |
[16] |
J. Langa, J. Robinson, A. Rodriguez-Bernal and A. Suarez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179.
doi: 10.1137/080721790. |
[17] |
J. A. Langa, A. R. Bernal and A. Suárez, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414.
doi: 10.1016/j.jde.2010.04.001. |
[18] |
C. Lei, Z. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks,, J. Differential Equations, 254 (2013), 1326.
doi: 10.1016/j.jde.2012.10.021. |
[19] |
K. Lerman and R. Ghosh, Information contagion: An empirical study of spread of news on Digg and Twitter social networks,, in Proceedings of 4th International Conference on Weblogs and Social Media (ICWSM), (2010). Google Scholar |
[20] |
J. D. Logan, Applied Partial Differential Equations,, Springer (2015)., (2015).
|
[21] |
Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics,, in Tutorials in Mathematical Biosciences. IV, (1922), 171.
doi: 10.1007/978-3-540-74331-6_5. |
[22] |
A. Madzvamuse, E. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion,, J. Math. Biol., 61 (2010), 133.
doi: 10.1007/s00285-009-0293-4. |
[23] |
J. Mierczyn'ski, The principal spectrum for linear nonautonomous parabolic PDEs of second order: Basic properties,, J. Differential Equations, 168 (2000), 453.
doi: 10.1006/jdeq.2000.3893. |
[24] |
S. Myers, C. Zhu and J. Leskovec, Information diffusion and external influence in networks,, KDD '12 Proceedings of the 18th ACM, (2012), 33.
doi: 10.1145/2339530.2339540. |
[25] |
J. D. Murray, Mathematical Biology I. An Introduction,, Springer-Verlag, (2002).
|
[26] |
A. Nazir, S. Raza, D. Gupta, C.-N. Chuah and B. Krishnamurthy, Network level footprints of facebook applications,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 63.
doi: 10.1145/1644893.1644901. |
[27] |
M. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.
doi: 10.1137/S003614450342480. |
[28] |
M. E. J. Newman, Networks: An Introdution,, Oxford University Press, (2010).
doi: 10.1093/acprof:oso/9780199206650.001.0001. |
[29] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992).
|
[30] |
M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine,, Ann. Scuola Norm. Sup. Pisa, 11 (1910). Google Scholar |
[31] |
A. Rodriguez-Bernal and A. Vidal-López, Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problem,, Discrete Contin. Dyn. Syst., 18 (2007), 537.
doi: 10.3934/dcds.2007.18.537. |
[32] |
A. Rodriguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, J. Differential Equations, 244 (2008), 2983.
doi: 10.1016/j.jde.2008.02.046. |
[33] |
D. Romero, C. Tan and J. Ugander, On the Interplay between Social and Topical Structure,, Proc. 7th International AAAI Conference on Weblogs and Social Media (ICWSM), (2013). Google Scholar |
[34] |
F. Schneider, A. Feldmann, B. Krishnamurthy and W. Willinger, Understanding online social network usage from a network perspective,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 35.
doi: 10.1145/1644893.1644899. |
[35] |
H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Amer. Math. Soc., (1995).
|
[36] |
C. A. Swanson, Picone's identity,, Rend. Mat., 8 (1975), 373.
|
[37] |
S. Tang and N. Blenn, Christian Doerr and Piet Van Mieghem,, Digging in the Digg Social News Website, (2011). Google Scholar |
[38] |
Q. Tang and Z. Lin, The asymptotic analysis of an insect dispersal model on a growing domain,, J. Math. Anal. Appl., 378 (2011), 649.
doi: 10.1016/j.jmaa.2011.01.057. |
[39] |
Z. Tufekci, Big Data: Pitfalls, Methods and Concepts for an Emergent Field (March 7, 2013)., Available at SSRN: , ().
doi: 10.2139/ssrn.2229952. |
[40] |
F. Wang, H. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks,, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), (2012), 133.
doi: 10.1109/ICDCSW.2012.16. |
[41] |
H. Wang, F. Wang and K. Xu, Modeling information diffusion in online social networks with partial differential equations,, , (). Google Scholar |
[42] |
F. Wang, K. Xu and H. Wang, Discovering shared interests,, in 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), (2012), 163. Google Scholar |
[43] |
F. Wang, H. Wang, K. Xu, J. Wu and J. Xia, Characterizing information diffusion in online social networks with linear diffusive model,, in 33nd International Conference on Distributed Computing Systems (ICDCS), (2013), 307.
doi: 10.1109/ICDCS.2013.14. |
[44] |
J. Yang and S. Counts, Comparing Information Diffusion Structure in Weblogs and Microblogs,, 4th Int'l AAAI Conference on Weblogs and Social Media, (2010). Google Scholar |
[45] |
J. Yang and J. Leskovec, Modeling information diffusion in implicit networks,, in 2010 IEEE 10th International Conference on Data Mining (ICDM), (2010), 599.
doi: 10.1109/ICDM.2010.22. |
[46] |
B. Yu and H. Fei, Modeling Social Cascade in the Flickr Social Network,, Fuzzy Systems and Knowledge Discovery, (2009).
doi: 10.1109/FSKD.2009.719. |
[47] | |
[48] |
show all references
References:
[1] |
G. A. Afrouzi and K. J. Brown, On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions,, Proc. Amer. Math. Soc., 127 (1999), 125.
doi: 10.1090/S0002-9939-99-04561-X. |
[2] |
W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, Nonlinear Anal., 32 (1998), 819.
doi: 10.1016/S0362-546X(97)00530-0. |
[3] |
A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks,, Cambridge University Press, (2008).
doi: 10.1017/CBO9780511791383. |
[4] |
F. Benevenuto, T. Rodrigues, M. Cha and V. Almeida, Characterizing user behavior in online social networks,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 49.
doi: 10.1145/1644893.1644900. |
[5] |
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments,, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293.
doi: 10.1017/S030821050001876X. |
[6] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons Ltd, (2003).
doi: 10.1002/0470871296. |
[7] |
M. Cha, A. Mislove, B. Adams and K. Gummadi, Characterizing social cascades in Flickr,, in Proceeding WOSN '08 Proceedings of the First Workshop on Online Social Networks, (2008), 13.
doi: 10.1145/1397735.1397739. |
[8] |
L. C. Evans, Partial Differential Equations,, AMS, (1998).
|
[9] |
P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991).
|
[10] |
R. Ghosh and K. Lerman, A framework for quantitative analysis of cascades on networks,, in ACM International Conference on Web Search and Data Mining, (2011), 665.
doi: 10.1145/1935826.1935917. |
[11] |
A. Guille, H. Hacid, C. Favre and D. Zighed, Information diffusion in online social networks: A survey,, SIGMOD Record, 42 (2013), 17.
doi: 10.1145/2503792.2503797. |
[12] |
E. L. Ince, Ordinary Differential Equation,, Dover, (1944).
|
[13] |
F. Jin, E. Dougherty, P. Saraf, Y. Cao and N. Ramakrishnan, Epidemiological modeling of news and rumors on Twitter,, in Proceedings of the 7th Workshop on Social Network Mining and Analysis, (2013).
doi: 10.1145/2501025.2501027. |
[14] |
K. Kreith, Picone's identity and generalizations,, Rend. Mat., 8 (1975), 251.
|
[15] |
R. Kumar, J. Novak and A. Tomkins, Structure and evolution of online social networks,, in Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2006), 611.
doi: 10.1145/1150402.1150476. |
[16] |
J. Langa, J. Robinson, A. Rodriguez-Bernal and A. Suarez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179.
doi: 10.1137/080721790. |
[17] |
J. A. Langa, A. R. Bernal and A. Suárez, On the long time behavior of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414.
doi: 10.1016/j.jde.2010.04.001. |
[18] |
C. Lei, Z. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks,, J. Differential Equations, 254 (2013), 1326.
doi: 10.1016/j.jde.2012.10.021. |
[19] |
K. Lerman and R. Ghosh, Information contagion: An empirical study of spread of news on Digg and Twitter social networks,, in Proceedings of 4th International Conference on Weblogs and Social Media (ICWSM), (2010). Google Scholar |
[20] |
J. D. Logan, Applied Partial Differential Equations,, Springer (2015)., (2015).
|
[21] |
Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics,, in Tutorials in Mathematical Biosciences. IV, (1922), 171.
doi: 10.1007/978-3-540-74331-6_5. |
[22] |
A. Madzvamuse, E. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion,, J. Math. Biol., 61 (2010), 133.
doi: 10.1007/s00285-009-0293-4. |
[23] |
J. Mierczyn'ski, The principal spectrum for linear nonautonomous parabolic PDEs of second order: Basic properties,, J. Differential Equations, 168 (2000), 453.
doi: 10.1006/jdeq.2000.3893. |
[24] |
S. Myers, C. Zhu and J. Leskovec, Information diffusion and external influence in networks,, KDD '12 Proceedings of the 18th ACM, (2012), 33.
doi: 10.1145/2339530.2339540. |
[25] |
J. D. Murray, Mathematical Biology I. An Introduction,, Springer-Verlag, (2002).
|
[26] |
A. Nazir, S. Raza, D. Gupta, C.-N. Chuah and B. Krishnamurthy, Network level footprints of facebook applications,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 63.
doi: 10.1145/1644893.1644901. |
[27] |
M. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.
doi: 10.1137/S003614450342480. |
[28] |
M. E. J. Newman, Networks: An Introdution,, Oxford University Press, (2010).
doi: 10.1093/acprof:oso/9780199206650.001.0001. |
[29] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992).
|
[30] |
M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare ordinaria del second'ordine,, Ann. Scuola Norm. Sup. Pisa, 11 (1910). Google Scholar |
[31] |
A. Rodriguez-Bernal and A. Vidal-López, Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problem,, Discrete Contin. Dyn. Syst., 18 (2007), 537.
doi: 10.3934/dcds.2007.18.537. |
[32] |
A. Rodriguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, J. Differential Equations, 244 (2008), 2983.
doi: 10.1016/j.jde.2008.02.046. |
[33] |
D. Romero, C. Tan and J. Ugander, On the Interplay between Social and Topical Structure,, Proc. 7th International AAAI Conference on Weblogs and Social Media (ICWSM), (2013). Google Scholar |
[34] |
F. Schneider, A. Feldmann, B. Krishnamurthy and W. Willinger, Understanding online social network usage from a network perspective,, in Proceedings of ACM SIGCOMM International Measurement Conference, (2009), 35.
doi: 10.1145/1644893.1644899. |
[35] |
H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Amer. Math. Soc., (1995).
|
[36] |
C. A. Swanson, Picone's identity,, Rend. Mat., 8 (1975), 373.
|
[37] |
S. Tang and N. Blenn, Christian Doerr and Piet Van Mieghem,, Digging in the Digg Social News Website, (2011). Google Scholar |
[38] |
Q. Tang and Z. Lin, The asymptotic analysis of an insect dispersal model on a growing domain,, J. Math. Anal. Appl., 378 (2011), 649.
doi: 10.1016/j.jmaa.2011.01.057. |
[39] |
Z. Tufekci, Big Data: Pitfalls, Methods and Concepts for an Emergent Field (March 7, 2013)., Available at SSRN: , ().
doi: 10.2139/ssrn.2229952. |
[40] |
F. Wang, H. Wang and K. Xu, Diffusive logistic model towards predicting information diffusion in online social networks,, in 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), (2012), 133.
doi: 10.1109/ICDCSW.2012.16. |
[41] |
H. Wang, F. Wang and K. Xu, Modeling information diffusion in online social networks with partial differential equations,, , (). Google Scholar |
[42] |
F. Wang, K. Xu and H. Wang, Discovering shared interests,, in 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), (2012), 163. Google Scholar |
[43] |
F. Wang, H. Wang, K. Xu, J. Wu and J. Xia, Characterizing information diffusion in online social networks with linear diffusive model,, in 33nd International Conference on Distributed Computing Systems (ICDCS), (2013), 307.
doi: 10.1109/ICDCS.2013.14. |
[44] |
J. Yang and S. Counts, Comparing Information Diffusion Structure in Weblogs and Microblogs,, 4th Int'l AAAI Conference on Weblogs and Social Media, (2010). Google Scholar |
[45] |
J. Yang and J. Leskovec, Modeling information diffusion in implicit networks,, in 2010 IEEE 10th International Conference on Data Mining (ICDM), (2010), 599.
doi: 10.1109/ICDM.2010.22. |
[46] |
B. Yu and H. Fei, Modeling Social Cascade in the Flickr Social Network,, Fuzzy Systems and Knowledge Discovery, (2009).
doi: 10.1109/FSKD.2009.719. |
[47] | |
[48] |
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