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

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New Frontiers in Mining Complex Patterns (NFMCP 2016)

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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.

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Notes

  1. 1.

    http://www.epinions.com/.

  2. 2.

    http://www.cosme.net/.

  3. 3.

    We also evaluated its prediction capability in terms of the area under the ROC curve (AUC) for trust-link prediction, and confirmed that the results for AUC were similar to those for PLR.

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Acknowledgements

This work was partly supported by JSPS Grant-in-Aid for Scientific Research (C) (No. 26330352), Japan.

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Correspondence to Masahiro Kimura .

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Appendix: Learning Algorithm

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)).

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Takahashi, H., Kimura, M. (2017). Analyzing Time-Decay Effects of Mediating-Objects in Creating Trust-Links. In: Appice, A., Ceci, M., Loglisci, C., Masciari, E., Raś, Z. (eds) New Frontiers in Mining Complex Patterns. NFMCP 2016. Lecture Notes in Computer Science(), vol 10312. Springer, Cham. https://doi.org/10.1007/978-3-319-61461-8_7

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