Modeling and Analysis of Dynamic Social Ties in D2D Collaborative Video Transmission

Social networks are one of the main carriers of information diffusion. Changes in social ties will affect the quality of Device-toDevice (D2D) communications especially the video transmission. For further improving the communication utility of users, it is of great significance to effectively integrate D2D communications and social networks. To this end, this paper utilizes a stochastic approach to modeling and analysis of dynamic social ties in D2D collaborative video transmission. Specifically, a stochastic mathematical model is established and analyzed, in which the combined effect of many factors such as interest, geographical position, career, social class, value system, and interaction is considered. Based on the Brownian motion theory, the strength of social ties among social individuals with time is studied. Next, the reliability function and adaptive parameter estimation are performed. Finally, some examples are conducted to illustrate the main results of this paper, from which one can see that the proposed model has a good predictive ability of the changing trend of social ties.


Introduction
With the development of mobile Internet and wireless communication technology, Device-to-Device (D2D) communications have become one of the key technologies of the future wireless communication and social networks have become one of the main carriers of information diffusion [1]. rough D2D communications and social networks, people can not only communicate with friends but also share pictures and videos faster and more conveniently. Since the communication equipment are usually carried by people, the dynamic social environment and a wide range of social applications require D2D communications, especially the D2D collaborative video transmission, to be more selfadapting and to meet more general communication needs. When users share videos locally through the wireless shortdistance D2D communications, if users successfully establish a D2D link, the video streams will spread rapidly over social networks. However, there are users' mobility and the occurrence of random events in social networks, and changes in users behavior will lead to dynamic interactions between users, which will affect the success rate of users to establish D2D links, thereby affecting the information diffusion [2]. erefore, it is essential to study the D2D collaborative video transmission in conjunction with social networks.
In [3], a D2D communication-assisted caching framework for video multicast was proposed, which considers the social trust and social reciprocity to encourage effective collaboration between users. Wang et al. [4] studied an Expected Available Duration (EAD) indicator to measure the chance of D2D users' downloading video packets from neighbors. Zhang et al. [5] considered the D2D pairing for cooperative video transmission. e social tie in previous work was usually calculated according to the personal information of users and the interaction information among different users. However, the social communication is affected by many factors, including social topology, user behavior, and inherent content characteristics, and the social ties are a dynamic process that changes with time.
e study of social ties is conducive to understanding human behavior, which can be applied in viral spread [6], information recommendation [7], traffic planning, and complex socioeconomic phenomena [8]. Granovetter [9] first defined the social ties that are a combination of time, emotional intensity, intimacy, and reciprocity, which provides a theoretical basis for many researchers to analyze the linear combination of the four elements of social ties.
With the development of social network, the research of modeling the structure of social ties has been paid attention increasingly. In [10], the authors pointed out that strangers established social ties based on shared interests, career, or activities. Xiang et al. [11] proposed an unsupervised model, in which the social ties between users were regarded as a latent variable that caused interactive behaviors. Based on this idea, Zhao et al. [12] presented a probabilistic generative model, which considers life activities and moving patterns. However, Xiang et al. and Zhao et al. [11,12] neglect the assignment of activity topics. Meanwhile, Xiong et al. [13] proposed a general framework to measure social ties by similarities, interaction activities, and the co-occurrence of users' names. In [14], the authors calculated social ties based on users' profile information and interaction activities in different activity fields. In [15], a language model based on sentiment classification, similarity, and interactivity was applied to compute social ties before adopting K-means clustering method to cluster the users. It is noteworthy that Zhao et al. and Ju and Tao [14,15] both assume that social ties are a static constant, resulting that the dynamic behavior of social network cannot be analyzed.
In addition to the above theoretical studies on social ties, there are also some application scenarios that describe social ties [16][17][18][19]. In traditional social networking websites, modeling the social ties has a wide spectrum of applications [16,17]. In [18], a graphical probabilistic model and Topical Affinity Propagation (TAP) approach were applied to study social ties and social influence, respectively. In [19], the spatiotemporal patterns of social ties were represented by the factors of the tensors. Yi et al. [20] proposed a dynamic model with social ties and self-confirmation mechanism, and verified the key role of social connections in information diffusion.
Inspired by the abovementioned work and based on the fact that the real world is a dynamic environment and it is difficult to gather accurate real-time date to track the changes of social ties, this paper attempts to utilize stochastic processes to study social ties. A stochastic mathematical model, which incorporates the combined effect of many factors such as interest, geographical position, career, social class, value system, and interaction, is proposed and analyzed. Specifically, the reliability function and adaptive parameter estimation are conducted. To illustrate the main results, some numerical examples are given at the end of this paper.
e subsequent materials of this paper are organized as follows. Section 2 formulates the stochastic model. Section 3 makes a mathematical analysis of this model. Some numerical examples are given in Section 4. Finally, Section 5 outlines this work.

Scenario Description.
Social tie is a key factor for the successful spread of video streaming among users in D2D communications. Since there are many influencing factors in social networks that affect the social tie between users, the social tie between users is dynamic, which will affect the success rate of D2D link establishment between users, thereby affecting communication performance. Based on the above description, the purpose of this paper is to build a dynamic model of social ties among social individuals for better understanding the D2D collaborative video transmission. As shown in Figure 1, considering a single-cell cellular network, . . , C M in the coverage area of the base station. At time t � t k , users D 1 and D 2 establish a D2D communication link and reuse the spectrum resources of cellular user C 2 . Between time t � t k and t � t k+1 , if D 1 and D 2 have a negative interaction, resulting that the strength of the social ties decreases and is below the trust threshold of user D 1 . en, at time t � t k+1 , users D 1 and D 2 cannot successfully communicate. Similarly, at time t � t k , user D 4 needs to obtain the required video resources from D 3 , and user D 3 is willing to share the data packet with D 4 if the physical conditions are met, then a trust relationship is established between D 4 and D 3 at time t � t k+1 . Based on this background, starting from the social level of users, a model of dynamic social ties between users is established. e strength of the social ties between any two users i and j can be regarded as a set of random variables S ij (t), t ≥ 0 that related to the time t.

Model Assumption.
At any time t, S ij (t) can be defined as a random representation of the strength of social ties between individuals, then S ij (t), t > 0 can be understood as a random process of social ties strength between individuals. To modeling and analysis of dynamic social ties S ij (t), t ≥ 0 between users i and j at any given time t, the following assumptions are imposed: (A1) As the social ties between users i and j can be regarded as a random variable S ij (t), then for τ > 0,  Proof. Assume that S ij (t k ) � S t k , for any time t � t k , k ∈ N + . en, (1) According to the above formula, one can get that, given the present state of the process, the future state is independent of the past. From the definition of Markov process in [21], the random variable S ij (t) satisfies the Markov property, and S ij (t), t > 0 is called the Markov process.
us, the proof is complete. □ Lemma 2. e random variable ΔS ij (t) obeys the normal distribution.
Collecting the foregoing assumptions and Lemmas 1 and 2, at time t � t k , k ∈ N + , the social ties S ij (t) can be modeled by the Wiener process with an adaptive drift, which can be expressed by the following stochastic system [23]: where is the standard Brownian motion) and W ∼ N(0, Q), Q is a constant, u(t k ) is the drift parameter at t k , and η and α are diffusion coefficients of the adaptive drift and social ties, respectively.

Reliability Function.
is section considers a liner model of system (2) based on a Wiener process as follows: where S t 0 is the observed social tie at t � t 0 , u is the drift parameter and u t k ∼ N(u k , σ k ) at t � t k , k ∈ N + , and B t is the standard Brownian motion. During the interaction, any two users i and j trust each other whether their relationship reaches to a given threshold S T ij or not, T � t | S ij (t) ≥ S T ij is the time that users trust each other for the first time. It is well known that the Probability Density Function (PDF) of T follows inverse Gaussian distribution [24]. en, it can be expressed as follows [25]: According to statistic characteristics and conditional distribution of the Wiener process, the cumulative distribution function (CDF) can be expressed as the following reliability function [24,25]: where Φ(·) expresses the CDF of the standard normal random variable. Combined the foregoing analysis, one of main results of this paper can be obtained as follows.

Theorem 1.
For the social tie process S ij (t), t > 0 given by system (3), the PDF and CDF can be expressed as equations (4) and (5), respectively.

Adaptive Parameter Estimation.
In this section, the Kalman filtering [26] is applied to estimate the mean and variance of drift parameter u. On this basis, parameters α and Q are estimated by the expectation-maximization (EM) algorithm [27].
For a given observation sample S � S t 1 , S t 2 , . . . , S t k , the observation equation of drift parameter u forms a Kalman filtering framework. Let u t k and σ t k stand for the updated drift parameter and variance, respectively. Let u t k |t k−1 and σ t k |t k−1 represent the estimated mean and variance based on the previous moment, respectively. erefore, the mean and variance of drift parameter u can be derived from the following Kalman filtering equations for k ∈ N + : Figure 1: D2D collaborative video transmission scenario.

Discrete Dynamics in Nature and Society
With initial conditions u t 0 � u 0 and σ t 0 � σ 0 , K k is the Kalman gain. e EM algorithm includes E-step and M-step, and it is necessary to take the expectation of the complete log-likelihood function in the first step, and then maximize the expectation to obtain estimated parameters until a convergence is achieved.
e backward iteration rules are as follows: where u t k|n , σ t k|n , and D t k−1 are the RTS smoothing state estimation, variance, and gain function on the basis of the current estimated parameters, respectively. en, Let E u L(Θ, S; u) { } denote the conditional expectation of the complete log-likelihood function given u t k , C t k t k−1 |n � E(u t k u t k−1 | S, Θ), and C t k |n � E(u 2 t k | S, Θ). en, Finally, the estimation of parameters can be obtained by maximizing the complete log-likelihood function in (12).

Numerical Examples
Some numerical examples are given to illustrate the main results of this paper in this section. Example 1. Consider system (2) with initial conditions u 0 � 0.1, σ 0 � 0.1, n � 50, and S T ij � 0.7. e social tie sample S is randomly produced by obeying normal distribution. Figure 2 shows the estimated mean and variance of the drift parameter u and the square of diffusion coefficients α 2 and Q.
Example 2. Consider system (2) with initial conditions u 0 � 0.1, σ 0 � 0.1, n � 3, and S T ij � 0.7. e social tie sample S is randomly produced by obeying normal distribution. Figure 3 displays the probability density function of achieving social trust for the first time.
Example 3. Consider system (2) with initial conditions u 0 � 0.1, σ 0 � 0.1, n � 3, and S T ij � 0.7. e social tie sample S is randomly produced by obeying normal distribution. Figure 4 shows the reliability of social tie over time. As time increases, the reliability between users gradually increases with time. In addition, at the same time point, the greater the drift coefficient, the greater the reliability.
Example 4. Consider system (2) with initial conditions u 0 � 0.1, σ 0 � 0.1, and n � 50. e social tie sample S is randomly produced by obeying normal distribution. Figure 5 reveals the variation curve of the estimated social tie strength under adaptive and fixed drift coefficient, respectively. e changes of social tie over time are not monotonous, and the interaction process is positive or negative.
e estimated values obtained from the two methods have the same trend as the observed values, and the model has a good predictive ability. In addition, the predicted value 4 Discrete Dynamics in Nature and Society    obtained by updating the drift parameter is closer to the real value than that of the fixed drift parameter.

Conclusions and Future Work
In this paper, a stochastic mathematical model describing the social ties, which incorporates the combined effect of many factors, such as interest, geographical position, career, social class, value system, and interaction, has been proposed and analyzed. e reliability function and adaptive parameter estimation have both been determined. To illustrate the main results, some numerical examples have been given at the end of this paper. is work contributes to the understanding of social phenomena.
As research based on the background of big data and artificial intelligence gradually enters people's vision [30,31], our proposed model can also open up new research directions under this background. For example, using artificial intelligence methods to abstract and analyze multidimensional features such as network behavior, content attributes, positional relationships, structural features, and privacy protection policies. In addition, the data in the social network is dynamic and transmitted in the form of data streams. e linking and generation of relationships are constantly changing, and the popularity of the video will also change over time. For large-scale dynamic networks, it is of great significance to further study efficient dynamic models and algorithms for video transmission.

Data Availability
Data sharing is not applicable to this article as no datasets were generated.