Temporal Topic-Based Multi-Dimensional Social Influence Evaluation in Online Social Networks

Abstract

Analytical applications in online social networks can be generalized as the influence evaluation problem, which targets at finding most influential users. Nowadays social influence evaluation is still an open and challenging issue. Most influence evaluation models focus on the single dimensional evaluation factor but fail to research on the multi-dimensional factors. In this paper, we propose a novel influence evaluation model: the temporal topic influence (TTI) evaluation model, which is a time-aware, content-aware and structure-aware evaluation model. For the aim of multi-dimensional evaluation, we incorporate multi-dimensional measure factors into our model, including the time factor, the topological information and the topic distribution information, etc. We propose a novel concept of user gravitational ability which is inspired by Newton’s law of universal gravitation. It can integrate multi-dimensional factors in an appropriate way. Our experiments are conducted on the Sina Weibo data set. Through the experimental analysis, we prove TTI model can calculate users’ influences effectively and efficiently. The TTI model can distinguish the value of users’ influences. And the TTI model identifies the top-k influential users with higher quality. We also validate the effect of time and topic measure factors in the influence evaluation process.

Appendix

In this part, we show the explanation of $G^s$ at first, and then we analyze and prove the time complexity of our algorithms.

1.1 Explanation of $G^s$

We conduct the experiments on our data sets and consult the Stanley Milgram’s experiments which called six degrees of separation [35]. The experiments both show the fact that two arbitrary users in the social network could become connected through six other users at most. We can get the further prove from previous research works. In model TTI, $G^s$ denotes the user gravitation ability constant. The user gravitation ability always exists between two users, user u can attract user v under $G^s$ in online social networks. In other words, user u can make an attraction to user v in any condition. Therefore, through the theory of six degrees of separation, we set the user gravitation ability constant 6.0 in this paper.

1.2 Proof of Theorem 1

Proof

Suppose n denotes the total number of users, m denotes the number of topics, e denotes the number of linked edges, $|N_u|$ denotes the number of friends of user u, $|N_v|$ denotes the number of friends of user v, and $|C_{uv}|$ denotes the number of common friends between user u and v, at last a denotes the average degree of all users.

It is shown in Algorithm 1 at line 4, all topics should be considered for calculating the dynamic topic distribution P. It takes the complexity of O(m). At line 5, the shortest path between user u and v should be found. It will be time-consuming if we search p(u, v) from the the whole edge list P(u, v). We use the Bellman–Ford algorithm to calculate the shortest path and it takes the complexity of O(ne). For the connection distance at line 6, the time complexity is $O(|C_{uv}|)\cdot (O(|N_u|)+O(|N_v|))$. The last three steps from line 7 to line 9 take constant complexity. In general, the complexity of lines 3–9 is $(O(m)+O(ne)+O(|C_{uv}|)\cdot (O(|N_u|)+O(|N_v|)))$. It is much larger than the real complexity, because users at the end of the calculation section have already finished some calculation parts such as, the topic distribution similarity, the shortest distance and the connection distance. In this case, we just take a time cost of $O(|N_u|)$ for searching those values.

It can be easily understood that the average degree of all users a is much bigger than the number of average common friends. The size of the network is very big compared to the number of topic m. Thus, we can use the average degree a to replace $O(|N_u|)$ and $O(|N_v|)$, and O(m) and $O(|C_{uv}|)$ can be neglected without loss of generality. As a result, the time complexity of Algorithm 1 is $O(ne+a)$ per iteration.

Figure 5 shows that the distribution of user’s degree fits with the power-law distribution. So we can take the average degree a as a constant. We can deem the total time complexity of Algorithm 1 as O(ne) per iteration. $\square$

1.3 Proof of Theorem 2

Proof

In Algorithm 2 at line 3–6, we consider the adjustment process of TTI model. At line 5, we need to calculate the proportion $p_{vu}$ that from user v to user u. But we have already get the results from Algorithm 1, it only need to search the results in the neighbor set of user v. Therefore, the time complexity is $O(|N_v|)$ at line 5. Line 3 and line 6 both take constant time complexity of O(1). It takes a total of time complexity $O(|N_u|\cdot |N_v|)$ for line 3–6.

The time complexity of each adjustment iteration (line 2–8) is $O(n\cdot |N_u|\cdot |N_v|)$. As the reason shows in the proof of theorem 1, we use the average degree a to replace $O(|N_u|)$ and $O|N_v|)$. The time complexity of the adjustment calculation process is $O(a^2 n)$ per iteration.

Since the same reason in the proof of theorem 1, we can take the average degree a as a constant. So the time complexity of Algorithm 2 is O(n) per iteration. $\square$