Analyzing Time-Decay Effects of Mediating-Objects in Creating Trust-Links

Abstract

We address the problem of modeling trust network evolution through social communications among users in a social media site. In particular, we focus on a social trust-link created between two users having mediating-objects such as mediating-users and mediating-items, and analyze the time-decay effects of mediating-objects on social trust-link creation. To this end, we first introduce the basic TCM model that can be regarded as a conventional link prediction method based on mediating-objects, and propose the TCM model with time-decay by incorporating an appropriate time-decay function into it. We present an efficient learning method of the proposed model, and apply it to an analysis of social trust-link creation for two real item-review sites. We show that the proposed model significantly outperforms the basic TCM model in terms of prediction performance, and clarify several properties of user behavior for social trust-link creation.

Appendix: Learning Algorithm

We consider learning the TCM model with time-decay from the observed data $D_*$. We derive an iterative algorithm for estimating the values of and by maximizing the objective function (see Eq. (5)). Let and be the current estimates of and , respectively. By Jensen’s inequality, we have

$$\begin{aligned}&\log \left( \sum _{k=1}^K e^{\mu _k} \sum _{\alpha \in \mathcal{M}_{k,t} (u,v)} r_{k,t} (\alpha ) f(t - \tau _\alpha (u, v); \lambda _k) \right) \nonumber \\&- \log \left( \sum _{k=1}^K e^{{\bar{\mu }}_k} \sum _{\alpha \in \mathcal{M}_{k,t} (u,v)} r_{k,t} (\alpha ) f(t - \tau _\alpha (u, v); {\bar{\lambda }}_k) \right) \nonumber \\&\ge \ \sum _{k=1}^K \sum _{\alpha \in \mathcal{M}_{k,t} (u,v)} {\bar{q}}_{k, \alpha } (u,v,t) \, \log \left( \frac{e^{\mu _k} \, f(t - \tau _\alpha (u, v); \lambda _k)}{e^{{\bar{\mu }}_k} \, f(t - \tau _\alpha (u, v); {\bar{\lambda }}_k)} \right) , \end{aligned}$$

(7)

for any $(u,v,t) \in D_*$, where

$$\begin{aligned} {\bar{q}}_{k, \alpha } (u,v,t) \ = \ \frac{e^{{\bar{\mu }}_k}_{} \, r_{k,t} (\alpha ) \, f(t - \tau _\alpha (u, v); {\bar{\lambda }}_k)}{\sum _{\ell = 1}^K \sum _{\beta \in \mathcal{M}_{\ell ,t} (u,v)} e^{{\bar{\mu }}_\ell }_{} \, r_{\ell , t} (\beta ) \, f(t - \tau _\beta (u, v); {\bar{\lambda }}_\ell )} \ > \ 0 \end{aligned}$$

(8)

for $k = 1, \dots , K$ and $\alpha \in \mathcal{M}_{k,t} (u,v)$. Note that $\sum _{k=1}^K \sum _{\alpha \in \mathcal{M}^k_{u,v,t}} {\bar{q}}_{k, \alpha } (u,v,t) \ = \ 1.$ Thus, by Eqs. (2), (5) and (7), we have where

(9)

Here, for $(u,v,t) \in D_*$, $w \in V_t(u) \cup \{v \}$, $k = 1, \dots , K$ and $\alpha \in \mathcal{M}_{k,t} (u,w)$, $g_{\alpha } (u, w, t)$ is defined as follows: $g_{\alpha } (u, w, t) = t - \tau _\alpha (u, w)$ if $f(s; \lambda _k) = f_{ex} (s; \lambda _k)$ and $g_{\alpha } (u, w, t) = \log (t - \tau _\alpha (u, w))$ if $f(s; \lambda _k) = f_{pl} (s; \lambda _k)$ (see Eqs. (3) and (4)). Also, const indicates such a constant term that does not depend on and . Note that . Thus, we consider increasing the value of by maximizing . We define by

(10)

for $(u,v,t) \in D$, $k = 1, \dots , K$ and $\alpha \in \mathcal{M}_{k,t} (u,v)$. From Eqs.(9) and (10), we have

(11)

(12)

for $k = 1, \dots , K$. Also, from Eqs.(10), (11) and (12), we have

(13)

(14)

(15)

for $k, \ell = 1, \dots , K$, where $\delta _{k, \ell }$ is the Kronecker delta. We consider a quadratic form

(16)

for , , where $\langle z_{k, \alpha } (w) \rangle$ stands for

for $(u,v,t) \in D_*$, $k = 1, \dots , K$, $w \in V_t (u)$ and $\alpha \in \mathcal{M}_{k,t} (u,w)$. From Eq. (10), note that

Thus, by Eq. (16), we have

for , . This implies that the Hessian matrix of function is negative definite. Hence, we can find the point at which function attains the maximum by solving , for $k = 1, \dots , K$. We employ Newton’s method and obtain an update formula for and (see Eqs. (11), (12), (13), (14) and (15)).