A topic-biased user reputation model in rating systems

Abstract

In rating systems like Epinions and Amazon’s product review systems, users rate items on different topics to yield item scores. Traditionally, item scores are estimated by averaging all the ratings with equal weights. To improve the accuracy of estimated item scores, user reputation [a.k.a., user reputation (UR)] is incorporated. The existing algorithms on UR, however, have underplayed the role of topics in rating systems. In this paper, we first reveal that UR is topic-biased from our empirical investigation. However, existing algorithms cannot capture this characteristic in rating systems. To address this issue, we propose a topic-biased model (TBM) to estimate UR in terms of different topics as well as item scores. With TBM, we develop six topic-biased algorithms, which are subsequently evaluated with experiments using both real-world and synthetic data sets. Results of the experiments demonstrate that the topic-biased algorithms effectively estimate UR across different topics and produce more robust item scores than previous reputation-based algorithms, leading to potentially more robust rating systems.

Appendix

In this appendix, we give the proof that Lemma 1 holds for TB-$L_1$-MAX and TB-$L_1$-MIN. Theorem 1 and 2 for them and other topic-biased algorithms are similar to the proof of TB-$L_1$-AVG.

1.1 TB-$L_1$-MAX

Proof

If $s=1$, let

$$\begin{aligned} o_h=\left\{ \begin{array}{ll} \displaystyle {\arg \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^1|}, &{}\quad if \,\, c_{ik}^1 > c_{ik}^0\\ \displaystyle {\arg \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^0|}, &{}\quad otherwise,\\ \end{array} \right. \end{aligned}$$

then

$$\begin{aligned} |r_j^2-r_j^1|&= |\frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}c_{ik}^1b_{jk}}-\frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}c_{ik}^0b_{jk}}| \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|c_{ik}^1-c_{ik}^0|} \\&= \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|\lambda \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^1|-\lambda \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^0||} \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|\lambda b_{hk}|r_h^1-r_h^0||}\\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}|r_\alpha ^1-r_\alpha ^0|}\\&\le \lambda \cdot |r_\alpha ^1-r_\alpha ^0|. \end{aligned}$$

Assume when $s=t$ the lemma holds, then we show $s=t+1$ the lemma still holds. Let

$$\begin{aligned} o_x=\left\{ \begin{array}{ll} \displaystyle {\arg \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^t|}, &{}\quad if\,\, c_{ik}^t > c_{ik}^{t-1}\\ \displaystyle {\arg \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^{t-1}|}, &{}\quad otherwise,\\ \end{array} \right. \end{aligned}$$

then

$$\begin{aligned} |r_j^{t+1}-r_j^t|&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|c_{ik}^t-c_{ik}^{t-1}|} \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}\cdot |\lambda \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^t|-\lambda \max _{o_j\in N_i}b_{jk}|R_{ij}-r_j^{t-1}||} \\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}\cdot b_{xk}|r_x^t-r_x^{t-1}|} \\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}\cdot \lambda ^t|r_\alpha ^1-r_\alpha ^0|} \\&\le \lambda ^t|r_\alpha ^1-r_\alpha ^0|. \end{aligned}$$

This completes the proof. $\square$

1.2 TB-$L_1$-MIN

Proof

If $s=1$, let

$$\begin{aligned} o_h=\left\{ \begin{array}{ll} \displaystyle {\arg \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^0|}, &{}\quad if\,\, c_{ik}^1 > c_{ik}^0\\ \displaystyle {\arg \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^1|}, &{}\quad otherwise,\\ \end{array} \right. \end{aligned}$$

then

$$\begin{aligned} |r_j^2-r_j^1|&= |\frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}c_{ik}^1b_{jk}}-\frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}c_{ik}^0b_{jk}}| \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|c_{ik}^1-c_{ik}^0|} \\&= \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|\lambda \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^1|-\lambda \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^0||} \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|\lambda b_{hk}|r_h^1-r_h^0|}\\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|r_\alpha ^1-r_\alpha ^0|}\\&\le \lambda \cdot |r_\alpha ^1-r_\alpha ^0|. \end{aligned}$$

Assume when $s=t$ the lemma holds, then we show $s=t+1$ the lemma still holds. Let

$$\begin{aligned} o_x=\left\{ \begin{array}{ll} \displaystyle {\arg \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^{t-1}|}, &{}\quad if\,\, c_{ik}^t > c_{ik}^{t-1}\\ \displaystyle {\arg \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^t|}, &{}\quad otherwise,\\ \end{array} \right. \end{aligned}$$

then

$$\begin{aligned} |r_j^{t+1}-r_j^t|&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}|c_{ik}^t-c_{ik}^{t-1}|} \\&\le \frac{1}{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}\cdot |\lambda \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^t|-\lambda \min _{o_j\in N_i}b_{jk}|R_{ij}-r_j^{t-1}||} \\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\sum _{t_k\in T}b_{jk}\cdot b_{xk}|r_x^t-r_x^{t-1}|} \\&\le \frac{\lambda }{|M_j|}\displaystyle {\sum _{u_i\in M_j}R_{ij}\lambda ^{t-1}|r_\alpha ^1-r_\alpha ^0|} \\&\le \lambda ^t|r_\alpha ^1-r_\alpha ^0|. \end{aligned}$$

This completes the proof. $\square$