Multi-relational reinforcement for computing credibility of nodes

Abstract

Nowadays, growing number of social networks are available on the internet, with which users can conveniently make friends, share information, and exchange ideas with each other. As the result, large amount of data are generated from activities of those users. Such data are regarded as valuable resources to support different mining tasks, such as predicting friends for a user, ranking users in terms of their influence on the social network, or identifying communities with common interests. Traditional algorithms for those tasks are often designed under the assumption that a user selects another user as his friend based on their common interests. As a matter of fact, users on a social network may not always develop their friends with common interest. For example, a user may randomly select other users as his friends just in order to attract more links reversely from them. Therefore, such links may not indicate his influence. In this paper, we study the user rank problem in terms of their ‘real’ influences. For this sake, common interest relationships among users are established besides their friend relationships. Then, the credible trust link from one node to another is on account of their similarities, which means the more similar the two users, the more credible their trust relation. So the credibility of a node is high if its trust inlinks are credible enough. In this work, we propose a framework that computes the credibility of nodes on a multi-relational network using reinforcement techniques. To the best of our knowledge, this is the first work to assess credibility exploited knowledge on multi-relational social networks. The experimental results on real data sets show that our framework is effective.

Appendix

1.1 Proof of convergence

In this Appendix, we present the technical details of the proof of general case of a ≠ b, N(a) ≠ ∅ and N(b) ≠ ∅, 0 ≤ s(a, b) − S i m ^(ϕ)(a, b) ≤ D ^(ϕ+1).

Induction Basis Let us prove that (10) holds for ϕ = 0, i.e. that for every two nodes a, b:

$$ s(a,b) - Sim^{(0)}(a,b) \leq D $$

(13)

Since a ≠ b, N(a) ≠ ∅ and N(b) ≠ ∅, then S i m ⁽⁰⁾(a, b) = 0 by definition, s(a, b) is defined by the general recursive (1) and (8), and consequently

$$ \begin{array}{llll} &Sim^{((\phi+1))}(u_{i}, u_{j}) = \alpha Sim_{T}^{(\phi+1)}(u_{i}, u_{j}) + (1-\alpha)Sim_{S}^{(\phi+1)}(u_{i}, u_{j})\\ &= \alpha \left( \frac{D\cdot\beta}{|d^{i}(u_{i})||d^{i}(u_{j})|}\underset{u_{p} \in d^{i}(u_{i})}{\sum}\underset{u_{q} \in d^{i}(u_{j})}{\sum}{Sim^{(\phi)}(u_{p}, u_{q})}\right. \\ &~~~~~~~~~\left.+\frac{D\cdot(1-\beta)}{|d^{o}(u_{i})||d^{o}(u_{j})|}\underset{u_{p} \in d^{o}(u_{i})}{\sum}\underset{u_{q} \in d^{o}(u_{j})}{\sum}{Sim^{(\phi)}(u_{p}, u_{q})}\right) \\ &+ (1 - \alpha) \cdot (D \cdot evidence(u_{i}, u_{j}) \cdot \underset{(u_{i},o_{i})\in E_{b}}{\sum}\underset{(u_{j},o_{j})\in E_{b}}{\sum}{(\textbf{W}(u_{i}, o_{i})\textbf{W}(u_{j}, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}){Sim^{(\phi)}(u_{i}, u_{j})})}) \\ &= \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(u_{i})||d^{i}(u_{j})|}\underset{u_{p} \in d^{i}(u_{i})}{\sum}\underset{u_{q} \in d^{i}(u_{j})}{\sum} {Sim^{(\phi)}(u_{p}, u_{q})} \right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(u_{i})||d^{o}(u_{j})|}\underset{u_{p} \in d^{o}(u_{i})}{\sum}\underset{u_{q} \in d^{o}(u_{j})}{\sum} {Sim^{(\phi)}(u_{p}, u_{q})}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(u_{i}, u_{j}) \cdot \underset{(u_{i},o_{i})\in E_{b}}{\sum}\underset{(u_{j},o_{j})\in E_{b}}{\sum}{(\textbf{W}(u_{i}, o_{i})\textbf{W}(u_{j}, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}) \cdot {Sim^{(\phi)}(u_{i}, u_{j})})}) \\ \end{array} $$

$$ \begin{array}{llllll} &s(a,b) - Sim^{(0)}(a,b) = s(a,b) \\ &= \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum}\underbrace{s(u_{p},u_{q})}_{\leq 1} \right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum}\underbrace{s(u_{p},u_{q})}_{\leq 1}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j})\underbrace{s(u_{i},u_{j})}_{\leq 1})}) \\ \end{array} $$

$$ \begin{array}{llllll} &\leq \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum} 1 \right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum} 1\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}) \cdot 1)}) \\ &\leq \alpha \cdot D (\beta + (1 - \beta)) + (1 - \alpha) \cdot D = D \\ \end{array} $$

Which proves (13). Other iterative algorithms such as [17, 24, 29] also exhibit similar convergence property.

Inductive step Provided that (10) holds for a given ϕ for all node pairs, let us prove that (10) holds for ϕ + 1 as well:

$$ \begin{array}{lllllll} &s(a,b) - Sim^{(\phi + 1)}(a,b) \\ &= \left( \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum}{s(u_{p},u_{q})} \right.\right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum}{s(u_{p},u_{q})}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}){s(u_{i},u_{j})})})) \\ &- \left( \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum}{Sim^{(\phi)}{(u_{p},u_{q})}} \right.\right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum}{Sim^{(\phi)}{(u_{p},u_{q})}}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}){Sim^{(\phi)}{(u_{i},u_{j})}})})) \\ &= \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum}{\underbrace{\left( s(u_{p},u_{q}) - Sim^{(\phi)}{(u_{p},u_{q})}\right)}_{\leq D^{\phi+1} by~inductive~hypothesis}} \right.\\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum}{\underbrace{\left( s(u_{p},u_{q}) - Sim^{(\phi)}{(u_{p},u_{q})}\right)}_{\leq D^{\phi+1} by~inductive~hypothesis}}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot}\\ &~~~~~~~~~\!\!\!\!evidence(o_{i}, o_{j}) \cdot\!\!\!\! \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}\!\!\!\!{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}){\underbrace{(s(u_{i},u_{j}) - Sim^{(\phi)}{(u_{i},u_{j})})}_{\leq D^{\phi+1} by~inductive~hypothesis}}})) \end{array} $$

$$ \begin{array}{lllllll} &\leq \alpha \cdot D \cdot \left( \frac{\beta}{|d^{i}(a)||d^{i}(b)|}\underset{u_{p} \in d^{i}(a)}{\sum}\underset{u_{q} \in d^{i}(b)}{\sum} D^{\phi+1}\right. \\ &~~~~~~~~~\left.+\frac{1-\beta}{|d^{o}(a)||d^{o}(b)|}\underset{u_{p} \in d^{o}(a)}{\sum}\underset{u_{q} \in d^{o}(b)}{\sum} D^{\phi+1}\right) \\ &+ (1 - \alpha) \cdot D \cdot (evidence(a, b) \cdot \underset{(a,o_{i})\in E_{b}}{\sum}\underset{(b,o_{j})\in E_{b}}{\sum}{(\textbf{W}(a, o_{i})\textbf{W}(b, o_{j}) \cdot} \\ &~~~~~~~~~evidence(o_{i}, o_{j}) \cdot \underset{(o_{i},u_{i})\in E_{b}}{\sum}\underset{(o_{j},u_{j})\in E_{b}}{\sum}{\textbf{W}(o_{i}, u_{i})\textbf{W}(o_{j}, u_{j}) \cdot D^{\phi+1})}) \\ &\leq D \cdot (\alpha \cdot (\beta \cdot D^{\phi+1} + (1 - \beta) \cdot D^{\phi+1}) + (1 - \alpha)\cdot D^{\phi+1}) = D \cdot D^{\phi+1} = D^{(\phi+1)+1} \\ \end{array} $$

The latter finally proves (10).