 Research
 Open Access
Partial differential equation modeling with Dirichlet boundary conditions on social networks
 Bo Du^{1}Email author,
 Xiuguo Lian^{1} and
 Xiwang Cheng^{1}
 Received: 6 January 2018
 Accepted: 20 March 2018
 Published: 4 April 2018
Abstract
The being a wide range of applications of the Internet, social networks have become an effective and convenient platform for information communication, propagation and diffusion. Most of information exchange and spreading exist in social networks. The issue of information diffusion in social networks is getting more and more attention by government and individuals. The researchers investigated either empirical studies or focused on ordinary differential equation (ODE) models with only consideration of temporal dimension in most prior work. As is well known, partial differential equations (PDEs) can describe temporal and spatial patterns of information diffusion over online social networks; however, until now, results for understanding information propagation of social networks over both temporal and spatial dimensions are few. This paper is devoted to investigating a nonautonomous diffusive logistic model with Dirichlet boundary conditions to describe the process of information propagation in social networks. By constructing upper and lower solutions we obtain the dynamic behavior of the solution to the nonautonomous diffusive logistic model. Our results show that information diffusion is greatly affected by the diffusion coefficient \(d(t)\) and the intrinsic growth rate \(r(t)\).
Keywords
 Social networks
 Spatiotemporal patterns
 Logistic model
 Stability
1 Introduction
With the rapid development of Internet technology, a new platform for information communication and diffusion has been constructed by online social networking which has established a wider range of social relations [1–5]. Online social networks have better features, including the speed and range of information transmission, rather than the traditional ways of information exchange. Moreover, social networks also play a significant role in business negotiation and information sharing. A lot of people used some new social media sites such as Twitter, Facebook and Wechat as they first appeared. We believe that these new social media sites will have indepth development and play a key role in contemporary society. Facing large amounts of data, the laws of communication for big data on online social networks should be studied by the researchers. For reducing unwanted information over social media, studying information diffusion process is necessary. However, due to the complexity of the network structure and the rapid change of social network platforms, for studying information spreading on online social networks there exist many difficulties.
The mechanism of information diffusion on online social networks including characterizing user behavior, characterizing social cascades in Flickr, network level footprints of Facebook, applications etc. [6–15] has been studied by many authors. Recently, some results have been obtained by using mathematical models to predict information diffusion over a time period in online social networks; see e.g. [16–20]. Inspired by empirical studies of networked systems, Newman [21] studied the structure and function of complex networks. Banavar, Maritan and Rinaldo [22] derived a general relationship between size and flow rates in general networks with local connectivity which can predict scaling relations applicable to all efficient transportation networks.
A number of concerns in modeling information diffusion in online social networks have been put forward on the basis of PDEbased models. Specially, PDEs can establish many complex models. However, the results are uncommonly poor for the diffusion models by PDEs in online social networks. Recently, both temporal and spatial patterns of information diffusion process on social networks were studied by Wang et al. [23–25] through constructing an intuitive cyberdistance among online users. Using real data coming from Digg (an online social network), Wang verified the reliability of the PDE model. Zhu, Zhao and Wang [26–28] studied several reaction–diffusion malware propagation models and obtained some results for stability and bifurcation of positive equilibrium points. Dai et al. [29] studied a partial differential equation with a Robin boundary condition in online social networks and discussed temporal and spatial properties of social networks. Since PDEs have complex natures and no standard characteristic equations, it is difficult to study PDE models by using matrix theory. Hence, the authors only have begun to study propagation models with PDEs in online social networks, and there are still many problems to be solved.
As is well known, studying information diffusion in online social networks by PDEbased models is very difficult, and this presents a new opportunity and challenge for mathematicians. In the ecological and physics models, the researchers focused on different boundary conditions, eigenvalue analysis and dynamic properties of diffusion. Cantrell and Cosner [30] studied a diffusive logistic equation with spatially varying growth rate. The authors obtained the theorem for the principal eigenvalue of the corresponding linearized equation, which is significant for the research of the dynamics of a population inhabiting a heterogeneous environment. Afrozi and Brown [31] discussed the existence of principal eigenvalues for Robin boundary conditions with an indefinite weight function.
 (1)
We develop a nonautonomous model with Dirichlet boundary conditions by using PDEs on social networks. To the best of our knowledge, few results have been obtained for our model. Our new model is more accurate as regards the actual situation; hence, it will be more important for the applications.
 (2)
We study the sensitivity of some parameters in the present model, guaranteeing the controllability of the model. Thus, we can control the stability interval by sensitive parameters which have important implications for practical applications.
 (3)
It makes a lot of sense to establish a unified framework to handle the reaction–diffusion terms and the influence of the variable coefficients. We develop some mathematical techniques (including upper and lower solutions method, comparison principle, and the like) for overcoming these difficulties.
 (4)
Our model is based on nonautonomous partial differential equations which are proposed to characterize temporal and spatial patterns of information diffusion over online social networks.
The remaining structure of this paper is arranged as follows. In Sect. 2, a nonautonomous PDE model with Dirichlet boundary conditions is developed in ndimensional space \(\mathbb{R}^{n}\). In Sect. 3, a number of dynamic properties for the present model are presented. Finally, conclusions are drawn in Sect. 4.
2 Nonautonomous PDE model in social networks
In view of the PDE model, by constructing an intuitive cyberdistance among online users, Wang et al. [23] studied both temporal and spatial patterns of information diffusion process in social media. In generally, we divide the information diffusion into two sections: contentbased and structurebased processes in an online social network. Myers, Zhu and Leskovec [20] pointed out that the contentbased and structurebased processes are analogous to the external and internal influences, respectively. It is challenging to reflect the above two process of information propagation in a mathematical model. PDEs can combine with time scale and space scale effectively and characterize the spacetime developing properties of the complex models.
Symbols and their meanings of system (2.4)
Parameters  Meaning 

d(t)  The popularity of information which promotes the spread of the information 
r(t)  The intrinsic growth rate of influenced users with the same distance 
b  The carrying capacity which is the maximum possible density of users 
a  The intraspecific competition rate with influenced users at a given distance 
Remark 2.1
3 Dynamic analysis for a diffusion logistic model
 (H_{1}):

\(d(t)\) is continuous differentiable decreasing positive function on \([0,+\infty )\) and satisfies$$ \lim_{t\rightarrow +\infty }d(t)=d_{\infty }. $$
 (H_{2}):

\(r(t)\) is continuous differentiable function on \([0,+\infty)\) and satisfieswhere \(r_{1}\), \(r_{2}\) are positive constants.$$ r_{1}\leq \lim_{t\rightarrow +\infty }r(t)\leq r_{2}, $$
Definition 3.1
Lemma 3.1
(Comparison principle)
Let \(v(t,x)\) be a solution of (2.4), \(\hat{v}(t,x)\) and \(\check{v}(t,x)\) are upper and lower solutions of (2.4) respectively, then \(\check{v}(t,x)\leq v(t,x)\leq \hat{v}(t,x)\) in \(\bar{\Omega }\times [0,+\infty )\).
Lemma 3.2
Proof
Theorem 3.1
Proof
Theorem 3.2
Proof
Remark 3.1
If \(r(t)=r\) in (2.4) is a positive constant, then \(\check{u}^{*}(x)= \bar{u}^{*}(x):=u^{*}(x)\). Hence, for (2.4) there exists a unique positive steady state solution \(u^{*}(x)\). It is obvious that the results of [33] are special results of the present paper in the case of \(r(t)=r\) in (2.4) being a positive constant. On the other hand, by upper and lower solutions, we cannot find that for (2.4) there exists a unique positive steady state solution \(u^{*}(x)\) and we only obtain the scope of the solution to (2.4). We hope that some new methods can be developed by the researchers for obtaining an unique positive solution of (2.4).
4 Conclusions
In this article, we study a nonautonomous reaction–diffusion system on social networks. It is noted that the diffusion rate d and intrinsic growth rate r are not constants, which is different from the past results [25, 26, 33, 34].
How the nonautonomous case affects information propagation and the dynamic properties of solutions over social networks is obtained by upper and lower solutions methods and comparison principle. More importantly, we investigate the effects of the variable diffusion rate and the intrinsic growth rate on the scale of information spreading in networks. Our results show that a nonautonomous reaction–diffusion system has more complex asymptotic stability than an autonomous reaction–diffusion system.
Declarations
Acknowledgements
The authors would like to thanks the editor and the referees for their valuable comments and suggestions, which improved the quality of our paper.
Funding
The work is supported by Natural Science Foundation of Jiangsu High Education Institutions of China (Grant No. 17KJB110001).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Xia, L., Jiang, G., Song, B., Song, Y.: Rumor spreading model considering hesitating mechanism in complex social networks. Physica A 437, 295–303 (2015) MathSciNetView ArticleGoogle Scholar
 Zhao, J., Wu, J., Xu, K.: Weak ties: subtle role of information diffusion in online social networks. Phys. Rev. E 82, 016105 (2010) View ArticleGoogle Scholar
 Zheng, M., Lu, L., Zhao, M.: Spreading in online social networks: the role of social reinforcement. Phys. Rev. E 88, 012818 (2013) View ArticleGoogle Scholar
 Huberman, B.A.: The Laws of the Web. MIT Press, Cambridge (2001) Google Scholar
 Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabási, A.L.: The largescale organization of metabolic networks. Nature (London) 411, 651–654 (2000). https://doi.org/10.1038/35036627 Google Scholar
 Benevenuto, F., Rodrigues, T., Cha, Y., Almeida, V.: Characterizing user behavior in online social networks. In: Proceedings of the 9th ACM SIGCOMM Conference on Internet Measurement (IMC’09), pp. 49–62. ACM, New York (2009). https://doi.org/10.1145/1644893.1644900 View ArticleGoogle Scholar
 Cha, Y., Mislove, A., Adams, B., Gummadi, K.: Characterizing social cascades in Flickr. In: Proceedings of the First Workshop on Online Social Networks (WOSN’08), pp. 13–18 (2008) View ArticleGoogle Scholar
 Lerman, K., Ghosh, R.: 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 (2010) Google Scholar
 Nazir, A., Raza, S., Gupta, D., Chuah, C., Krishnamurthy, B.: Network level footprints of Facebook applications. In: Proceedings of the 9th ACM SIGCOMM Conference on Internet Measurement (IMC’09), pp. 63–75. ACM, New York (2009). https://doi.org/10.1145/1644893.1644901 View ArticleGoogle Scholar
 Du, B., Hu, M., Lian, X.: Dynamical behavior for a stochastic predatorprey model with HV type functional response. Bull. Malays. Math. Soc. 40, 487–503 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Tang, S., Blenn, N., Doerr, C., Van Mieghem, P.: Digging in the digg social news website. IEEE Trans. Multimed. 13(5), 1163–1175 (2011). https://doi.org/10.1109/TMM.2011.2159706 View ArticleGoogle Scholar
 Du, B.: Stability analysis of periodic solution for a complexvalued neural networks with bounded and unbounded delays. Asian J. Control 20, 1–12 (2018) View ArticleGoogle Scholar
 Yu, B., Fei, H.: Modeling social cascade in the Flickr social network. In: Sixth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD’09). IEEE, New York (2009). https://doi.org/10.1109/FSKD.2009.719 Google Scholar
 Schneider, F., Feldmann, A., Krishnamurthy, B., Willinger, W.: Understanding online social network usage from a network perspective. In: Proceedings of the 9th ACM SIGCOMM Conference on Internet Measurement (IMC’09), pp. 35–48. ACM, New York (2009). https://doi.org/10.1145/1644893.1644899 View ArticleGoogle Scholar
 Du, Y., Lin, Z.: Spreadingvanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Barrat, A., Barthelemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008) View ArticleMATHGoogle Scholar
 Ghosh, R., Lerman, K.: A framework for quantitative analysis of cascades on network. In: Proceedings of the Fourth ACM International Conference on Web Search and Data Mining, pp. 665–674. ACM, New York (2011). https://doi.org/10.1145/1935826.1935917 View ArticleGoogle Scholar
 Jin, F., Dougherty, E., Saraf, P., Cao, Y., Ramakrishnan, N.: Epidemiological modeling of news and rumors on Twitter. In: Proceedings of the 7th Workshop on Social Network Mining and Analysis (SNAKDD’13), Article No. 8. ACM, New York (2013). https://doi.org/10.1145/2501025.2501027 Google Scholar
 Kumar, R., Novak, J., Tomkins, A.: Structure and evolution of online social networks. In: Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 611–617 (2006) View ArticleGoogle Scholar
 Myers, S., Zhu, C., Leskovec, J.: Information diffusion and external influence in networks. In: Proceedings of the 18th ACM, SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD’12), pp. 33–41 (2012) View ArticleGoogle Scholar
 Newman, M.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003) MathSciNetView ArticleMATHGoogle Scholar
 Banavar, J., Maritan, A., Rinaldo, A.: Size and form in efficient transportation networks. Nature 399, 130–132 (1999) View ArticleGoogle Scholar
 Wang, F., Wang, H., Xu, K.: Diffusive logistic model towards predicting information diffusion in online social networks. In: Distributed Computing Systems Workshops (ICDCSW), 2012 32nd International Conference, pp. 133–139 (2012) View ArticleGoogle Scholar
 Wang, F., Wang, H., Xu, K.: Characterizing information diffusion in online social networks with linear diffusive model. In: 2013 IEEE 33rd International Conference on Distributed Computing Systems (ICDCS), pp. 307–316 (2013) View ArticleGoogle Scholar
 Lei, C., Lin, Z., Wang, H.: The free boundary problem describing information diffusion in online social networks. J. Differ. Equ. 254, 1326–1341 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Zhu, L., Zhao, H., Wang, X.: Stability and bifurcation analysis in a delayed reactiondiffusion malware propagation model. Comput. Math. Appl. 69, 852–887 (2015) MathSciNetView ArticleGoogle Scholar
 Zhu, L., Zhao, H., Wang, X.: Bifurcation analysis of a delay reaction–diffusion malware propagation model with feedback control. Commun. Nonlinear Sci. Numer. Simul. 22, 747–768 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Zhu, L., Zhao, H., Wang, X.: Complex dynamic behavior of a rumor propagation model with spatialtemporal diffusion terms. Inf. Sci. 349–350, 119–136 (2016) View ArticleGoogle Scholar
 Dai, G., Ma, R., Wang, H., Wang, F., Xu, K.: Partial differential equations with Robin boundary condition in online social networks. Discrete Contin. Dyn. Syst. 20, 1609–1624 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Cantrell, R., Cosner, C.: The effects of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315–338 (1991) MathSciNetView ArticleMATHGoogle Scholar
 Afrozi, G., Brown, K.: On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions. Proc. Am. Math. Soc. 127, 125–130 (1999) MathSciNetView ArticleGoogle Scholar
 http://en.wikipedia.org/wiki/Sixdegreesofseparation
 Tang, Q., Lin, Z.: The asymptotic analysis of an insect dispersal model on a growing domain. J. Math. Anal. Appl. 378, 649–656 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Crampin, E.J., Gaffney, E.A., Maini, P.K.: Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120 (1999) View ArticleMATHGoogle Scholar
 Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33–76 (1978) MathSciNetView ArticleMATHGoogle Scholar
 Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992) MATHGoogle Scholar
 Ludwig, D., Aronson, D.G., Weinberger, H.F.: Spatial patterning of the spruce budworm. J. Math. Biol. 8, 217–258 (1979) MathSciNetView ArticleMATHGoogle Scholar
 RodriguezBernal, A., VidalLopez, A.: Existence, uniqueness and attractivity properties of positive complete trajectories for nonautonomous reaction–diffusion problem. Discrete Contin. Dyn. Syst. 18, 537–567 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Murray, J.D.: Mathematical Biology. Springer, Berlin (1993) View ArticleMATHGoogle Scholar