# American Institute of Mathematical Sciences

August  2015, 20(6): 1609-1624. doi: 10.3934/dcdsb.2015.20.1609

## 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

Received  February 2014 Revised  January 2015 Published  June 2015

In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
Citation: Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609
##### 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), 125-130. 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), 819-830. 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, 49-62. 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), 293-318. 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, 13-18. doi: 10.1145/1397735.1397739. [8] L. C. Evans, Partial Differential Equations, AMS, Rhode Island, 1998. [9] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 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, 665-674. 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), 17-28. doi: 10.1145/2503792.2503797. [12] E. L. Ince, Ordinary Differential Equation, Dover, New York, 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, Article No. 8. doi: 10.1145/2501025.2501027. [14] K. Kreith, Picone's identity and generalizations, Rend. Mat., 8 (1975), 251-262. [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, 611-617. 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), 2179-2216. 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), 414-445. 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), 1326-1341. 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. [20] J. D. Logan, Applied Partial Differential Equations, Springer (2015). [21] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008, 171-205. 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), 133-164. 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), 453-476. 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, SIGKDD International Conference on Knowledge Discovery and Data Mining, 2012, 33-41. doi: 10.1145/2339530.2339540. [25] J. D. Murray, Mathematical Biology I. An Introduction, Springer-Verlag, New York, 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, 63-75. doi: 10.1145/1644893.1644901. [27] M. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. 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, New York and London, 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), p144. [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), 537-567. 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), 2983-3030. 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. [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, 35-48. doi: 10.1145/1644893.1644899. [35] H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995. [36] C. A. Swanson, Picone's identity, Rend. Mat., 8 (1975), 373-397. [37] S. Tang and N. Blenn, Christian Doerr and Piet Van Mieghem, Digging in the Digg Social News Website, IEEE Transaction on Multimediam, 2011. [38] Q. Tang and Z. Lin, The asymptotic analysis of an insect dispersal model on a growing domain, J. Math. Anal. Appl., 378 (2011), 649-656. 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: http://ssrn.com/abstract=2229952. 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, 133-139. 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, arXiv:1310.0505. [42] F. Wang, K. Xu and H. Wang, Discovering shared interests, in 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), 2012, 163-168. [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, 307-316. 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. [45] J. Yang and J. Leskovec, Modeling information diffusion in implicit networks, in 2010 IEEE 10th International Conference on Data Mining (ICDM), 2010, 599-608. 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] , http://en.wikipedia.org/wiki/Six_degrees_of_separation. [48]

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##### 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), 125-130. 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), 819-830. 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, 49-62. 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), 293-318. 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, 13-18. doi: 10.1145/1397735.1397739. [8] L. C. Evans, Partial Differential Equations, AMS, Rhode Island, 1998. [9] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 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, 665-674. 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), 17-28. doi: 10.1145/2503792.2503797. [12] E. L. Ince, Ordinary Differential Equation, Dover, New York, 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, Article No. 8. doi: 10.1145/2501025.2501027. [14] K. Kreith, Picone's identity and generalizations, Rend. Mat., 8 (1975), 251-262. [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, 611-617. 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), 2179-2216. 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), 414-445. 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), 1326-1341. 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. [20] J. D. Logan, Applied Partial Differential Equations, Springer (2015). [21] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences. IV, Lecture Notes in Math., 1922, Springer, Berlin, 2008, 171-205. 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), 133-164. 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), 453-476. 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, SIGKDD International Conference on Knowledge Discovery and Data Mining, 2012, 33-41. doi: 10.1145/2339530.2339540. [25] J. D. Murray, Mathematical Biology I. An Introduction, Springer-Verlag, New York, 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, 63-75. doi: 10.1145/1644893.1644901. [27] M. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. 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, New York and London, 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), p144. [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), 537-567. 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), 2983-3030. 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. [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, 35-48. doi: 10.1145/1644893.1644899. [35] H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995. [36] C. A. Swanson, Picone's identity, Rend. Mat., 8 (1975), 373-397. [37] S. Tang and N. Blenn, Christian Doerr and Piet Van Mieghem, Digging in the Digg Social News Website, IEEE Transaction on Multimediam, 2011. [38] Q. Tang and Z. Lin, The asymptotic analysis of an insect dispersal model on a growing domain, J. Math. Anal. Appl., 378 (2011), 649-656. 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: http://ssrn.com/abstract=2229952. 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, 133-139. 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, arXiv:1310.0505. [42] F. Wang, K. Xu and H. Wang, Discovering shared interests, in 2012 32nd International Conference on Distributed Computing Systems Workshops (ICDCSW), 2012, 163-168. [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, 307-316. 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. [45] J. Yang and J. Leskovec, Modeling information diffusion in implicit networks, in 2010 IEEE 10th International Conference on Data Mining (ICDM), 2010, 599-608. 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] , http://en.wikipedia.org/wiki/Six_degrees_of_separation. [48]
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