# Partial differential equation modeling with Dirichlet boundary conditions on social networks

## 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 non-autonomous 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 non-autonomous 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 Spatio-temporal 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, 2, 3, 4, 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 in-depth 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, 7, 8, 9, 10, 11, 12, 13, 14, 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, 17, 18, 19, 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 PDE-based 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, 24, 25] through constructing an intuitive cyber-distance 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, 27, 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 PDE-based 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 non-autonomous 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 non-autonomous 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 non-autonomous PDE model with Dirichlet boundary conditions is developed in *n*-dimensional 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 Non-autonomous PDE model in social networks

In view of the PDE model, by constructing an intuitive cyber-distance 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: content-based and structure-based processes in an online social network. Myers, Zhu and Leskovec [20] pointed out that the content-based and structure-based 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 space-time developing properties of the complex models.

*x*-axis and the density at the location

*x*is defined by \(U_{x}\). Let \(I(t,x)\) be the density of influenced users at distance

*x*during time

*t*. The flux of the influenced users at distance

*x*during time

*t*can be denoted by \(J=-c\frac{\partial I}{\partial x}\), where

*c*represents the popularity of information. In Twitter, marking keywords or topics can be described by a hashtag. Since online users in social media show heterogeneity,

*c*can be controlled by the distance

*x*from the source in the content-based process. The distance metric can characterize the shortest friendship hops, and the average distance on Twitter is 4.67 [32]. Wang et al. [23] studied the following diffusion model:

*d*,

*a*and

*b*are positive constants, Ω is an open subset of \(\mathbb{R}^{n}\) with smooth boundary

*∂*Ω. Then they changed (2.1) into a non-autonomous reaction–diffusion problem:

*q*is positive constant, \(u(t,x)\) represents the density of influenced users with a distance of

*x*at time

*t*. For the other parameters’ meanings, see Table 1.

Symbols and their meanings of system (2.4)

Parameters | Meaning |
---|---|

| The popularity of information which promotes the spread of the information |

| The intrinsic growth rate of influenced users with the same distance |

| The carrying capacity which is the maximum possible density of users |

| The intra-specific 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

*ǔ*is called a lower solution of (2.4) if it satisfies all the reversed inequalities in (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

*Let*\(u(t,x)\)

*be a nonnegative nontrivial solution of the following problem*:

*If*\(u_{0}(x)\in C^{2}(\bar{\Omega }(0))\), \(u_{0}(x)=\Delta u(0,x)=0\)

*for*\(x\in \partial \Omega (0)\)

*and*\(\Delta u(0,x)\leq 0\)

*for*\(x\in \bar{ \Omega }(0)\),

*then*\(u(t,x)\in C^{2,1}(\Omega (0)\times (0,\infty ))\)

*and*

### Proof

### Theorem 3.1

*If the assumptions*(H

_{1})

*and*(H

_{2})

*hold and*\(a\leq \frac{d _{\infty }\lambda_{1}}{r_{2}}\),

*then the solution of problem*(2.4)

*satisfies*

### Proof

*M*is a positive constant. We choose

*M*so large that \(M\phi (x) \geq u_{0}(x)\), For any solution \(\hat{u}(t,x)\) of (3.3), we can see that \(\hat{u}(t,x)\) is an upper solution of (2.4). Let

*u*be any solution of (2.4). It follows from Lemma 3.1 that

_{1}), \(d(t)\) tends decreasingly to \(d_{\infty }\) as \(t\rightarrow \infty \), then \(d(t)\geq d_{\infty }\) for \(t\geq 0\). By assumption (H

_{2}), \(r_{1}< r(t)\leq r_{2}\) for \(t\geq 0\). Thus, \(\hat{u}(t,x)\) satisfies

### Theorem 3.2

*If the assumptions*(H

_{1})

*and*(H

_{2})

*hold and*\(a> \frac{d_{ \infty }\lambda_{1}}{r_{1}}\),

*then the solution of problem*(3.14)

*satisfies*

*where*\(u^{*}(x)\)

*is a unique positive solution of*(3.8), \(\check{u} ^{*}(x)\)

*is a unique positive solution of*(3.15).

### Proof

_{1}), we have \(\lim_{t\rightarrow \infty } {d}(t)=d_{\infty }\) and \(d(t)\) is decreasing for \(t>0\), there exists a \(T_{1}>0\) such that \(d_{\infty }\leq d(t)\leq d_{\infty }+\varepsilon \) for \(t>T_{1}\). Let \(\hat{u}(t,x)\) be the unique solution of the problem:

*M*is a positive constant. We choose

*M*so large that \(M\phi (x) \geq u_{0}(x)\), For any solution \(\hat{u}(t,x)\) of (3.5), we can see that \(\hat{u}(t,x)\) is an upper solution of (2.4). Let

*u*be any solution of (2.4). It follows from Lemma 3.1 that

*δ*is a sufficiently small constant such that \(\delta \phi (x)\leq u_{0}\). It is easy to see that \(\check{u}(t,x)\) is a lower solution of (2.4) in \([T_{1},\infty )\times \bar{\Omega }(0)\). From \(\Delta \check{u}(T_{1},x)=-\delta \lambda_{1}\phi (x)\leq 0\) and Lemma 3.1, we have \(\Delta \check{u}(t,x)\leq 0\) in \([T_{1},\infty )\times \Omega (0)\). Thus,

*ε*, obviously,

### 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 non-autonomous 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 non-autonomous 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 non-autonomous reaction–diffusion system has more complex asymptotic stability than an autonomous reaction–diffusion system.

## Notes

### 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.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Funding

The work is supported by Natural Science Foundation of Jiangsu High Education Institutions of China (Grant No. 17KJB110001).

### Competing interests

The authors declare that they have no competing interests.

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