Skip to main content

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

  • Conference paper
  • First Online:
New Frontiers in Mining Complex Patterns (NFMCP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10312))

Included in the following conference series:

  • 547 Accesses


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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

  2. 2.

  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.


  1. Barabási, A.L.: The origin of bursts and heavy tails in human dynamics. Nature 435, 207–211 (2005)

    Article  Google Scholar 

  2. Barbieri, N., Bonchi, F., Manco, G.: Who to follow and why: link prediction with explanations. In: Proceedings of KDD 2014, pp. 1266–1275 (2014)

    Google Scholar 

  3. Chen, W., Lakshmanan, L., Castillo, C.: Information and influence propagation in social networks. Synth. Lect. Data Manag. 5, 1–177 (2013)

    Article  Google Scholar 

  4. Crandall, D., Cosley, D., Huttenlocher, D., Kleinberg, J., Suri, S.: Feedback effects between similarity and social influence in online communities. In: Proceedings of KDD 2008, pp. 160–168 (2008)

    Google Scholar 

  5. Gomez-Rodriguez, M., Leskovec, J., Krause, A.: Inferring networks of diffusion and influence. In: Proceedings of KDD 2010, pp. 1019–1028 (2010)

    Google Scholar 

  6. Guha, R., Kumar, R., Raghavan, P., Tomkins, A.: Propagation of trust and distrust. In: Proceedings of WWW 2004, pp. 403–412 (2004)

    Google Scholar 

  7. Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of KDD 2003, pp. 137–146 (2003)

    Google Scholar 

  8. Kimura, M., Saito, K., Nakano, R., Motoda, H.: Extracting influential nodes on a social network for information diffusion. Data Min. Knowl. Disc. 20, 70–97 (2010)

    Article  MathSciNet  Google Scholar 

  9. Koren, Y.: Collaborative filtering with temporal dynamics. In: Proceedings of KDD 2009, pp. 447–456 (2009)

    Google Scholar 

  10. Leskovec, J., Backstrom, I., Kumar, K., Tomkins, A.: Microscopic evolution of social networks. In: Proceedings of KDD 2008, pp. 462–470 (2008)

    Google Scholar 

  11. Leskovec, J., Huttenlocher, D., Kleinberg, J.: Predicting positive and negative links in online social networks. In: Proceedings of WWW 2010, pp. 641–650 (2010)

    Google Scholar 

  12. Liben-Nowell, D., Kleinberg, J.: The link-prediction problem for social networks. J. Am. Soc. Inf. Sci. Technol. 58, 1019–1031 (2007)

    Article  Google Scholar 

  13. Liu, H., Lim, E., Lauw, H., Le, M., Sun, A., Srivastava, J., Kim, Y.: Predicting trusts among users of online communities: an epinion case study. In: Proceedings of EC 2008, pp. 310–319 (2008)

    Google Scholar 

  14. Nguyen, V., Lim, E., Jiang, J., Sun, A.: To trust or not to trust? Predicting online trusts using trust antecedent framework. In: Proceedings of ICDM 2009, pp. 896–901 (2009)

    Google Scholar 

  15. Oliveira, J.G., Barabási, A.L.: Dawin and Einstein correspondence patterns. Nature 437, 1251 (2005)

    Article  Google Scholar 

  16. Tang, J., Chang, S., Aggarwal, C., Liu, F.: Negative link prediction in social media. In: Proceedings of WSDM 2015, pp. 87–96 (2015)

    Google Scholar 

  17. Tang, J., Gao, H., Hu, X., Liu, H.: Exploiting homophily effect for trust prediction. In: Proceedings of WSDM 2013, pp. 53–62 (2013)

    Google Scholar 

  18. Tang, J., Gao, H., Liu, H., Sarma, A.D.: eTrust: understanding trust evolution in an online world. In: Proceedings of KDD 2012, pp. 253–261 (2012)

    Google Scholar 

  19. Weng, L., Ratkiewicz, J., Perra, N., Goncalves, B., Castillo, C., Bonchi, F., Schifanella, R., Menczer, F., Flammini, A.: The role of information diffusion in the evolution of social networks. In: Proceedings of KDD 2013, pp. 356–364 (2013)

    Google Scholar 

  20. Zhang, J., Yu, P.S., Zhou, Z.: Meta-path based multi-network collective link prediction. In: Proceedings of KDD 2014, pp. 1286–1295 (2014)

    Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to Masahiro Kimura .

Editor information

Editors and Affiliations

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

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

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

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

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


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


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


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


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


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)).

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

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.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-61460-1

  • Online ISBN: 978-3-319-61461-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics