Trust inference using implicit influence and projected user network for item recommendation

Abstract

Trust plays a very important role in many existing e-commerce recommendation applications. Social or trust network among users provides additional information along with ratings for improving user reliability on the recommendation. However, in the real world due to the sparse nature of trust data, many algorithms are built for inferring trust. In this work, we propose a new path based trust inference method utilizing the implicit influence information available in the existing trust network. The proposed approach uses the transitivity property of the trust for trust propagation and scale-free complex network property to limit the propagation length in the network. In this regard, we define a new terminology, degree of trustworthiness for a user, which adds the global influence in the inferred trust along a path and considers the maximum trust gaining path between two users. To reduce the sparsity of the network further, we use the projected user network information from user-item feedback history to reconstruct the inferred trust and introduce two methods of reconstruction from the truster and trustee point of view. The proposed reconstruction process can infer the trusted neighbors for a user who has put no trust on others, so far. We have applied the techniques in two real-world datasets and achieved significant performance improvement from the existing trust-based and neighborhood-based methods.

Appendix

Lemma 2

The value of c will always lie within the interval (0, $\frac {1}{d_{max}}$ ] (i.e. $0 < c \leq \frac {1}{d_{max}}$ ).

Proof

From (7), part A is constant for some l, 2 ≤ l ≤ d_max and part B varies according to δ value of the intermediate nodes in the path. Now value of t_i,j will be maximum when ∀u_k ∈{u₁,u₂,…,u_l− 1}, $indeg(u_{k}) = maxindeg(\mathcal {G}_{t})$. Then for 𝜖 = 0, (7) will be

$$\begin{array}{@{}rcl@{}} [t_{i,j}]_{max} &=& \underset{\delta_{k}}{argmax}\left[ \left( \frac{d_{max} - l + 1}{d_{max}}\right) + (l-1)\cdot(\delta_{k})\right]\\ &=& \left( \frac{d_{max} - l + 1}{d_{max}}\right) + (l-1)\cdot \left( c \cdot \frac{indeg(u_{k})}{maxindeg(\mathcal{G}_{t})}\right) \\ &=& 1 - \frac{(l-1)}{d_{max}} + (l-1) \cdot c \\ && \qquad \qquad \hspace{2.7cm}[\because indeg(u_{k}) = maxindeg(\mathcal{G}_{t})] \end{array} $$

Now, as per (2),

$$\begin{array}{@{}rcl@{}} [t_{i,j}]_{max} \leq 1 & \Rightarrow & 1 - \frac{(l-1)}{d_{max}} + (l-1) \cdot c \leq 1 \\ & \Rightarrow & (l-1) \cdot c \leq \frac{(l-1)}{d_{max}} \\ & \Rightarrow & c \leq \frac{1}{d_{max}} \end{array} $$

Again value of t_i,j will be minimum when ∀u_k ∈{u₁,u₂,…,u_l− 1}, indeg(u_k) = 1 [as, u_k lies within the path from u_i to u_j, so u_k must have at least 1 incoming edge]. Then for 𝜖 = 0 minimum vale of t_i,j will be,

$$\begin{array}{@{}rcl@{}} [t_{i,j}]_{min} &=& \underset{\delta_{k}}{argmin}\left[ \left( \frac{d_{max} - l + 1}{d_{max}}\right) + (l-1)\cdot(\delta_{k})\right]\\ & = & \left( \frac{d_{max} - l + 1}{d_{max}}\right) + (l-1) \cdot c \cdot \frac{1}{maxindeg(\mathcal{G}_{t})} \end{array} $$

Now, from the above equation, it is trivial that,

$$\begin{array}{@{}rcl@{}} &&\qquad [t_{i,j}]_{min} > \left( \frac{d_{max} - l + 1}{d_{max}}\right)\\ &&\Rightarrow (l-1) \cdot c \cdot \frac{1}{maxindeg(\mathcal{G}_{t})} > 0 \quad \Rightarrow c > 0 \end{array} $$

Hence, it is proved that for calculating t_i,j by (7) c will always lie within $(0,\frac {1}{d_{max}}]$. □