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Theoretical Information Reuse and Integration pp 173–196Cite as

Concurrent Alignment of Multiple Anonymized Social Networks with Generic Stable Matching

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 446))

Abstract

Users nowadays are normally involved in multiple (usually more than two) online social networks simultaneously to enjoy more social network services. Some of the networks that users are involved in can share common structures either due to the analogous network construction purposes or because of the similar social network characteristics. However, the social network datasets available in research are usually pre-anonymized and accounts of the shared users in different networks are mostly isolated without any known connections. In this paper, we want to identify such connections between the shared users’ accounts in multiple social networks (which are called the anchor links), and the problem is formally defined as the M-NASA (Multiple Anonymized Social Networks Alignment) problem. M-NASA is very challenging to address due to (1) the lack of known anchor links to build models, (2) the studied networks are anonymized, where no users’ personal profile or attribute information is available, and (3) the “transitivity law” and the “one-to-one property” based constraints on anchor links. To resolve these challenges, a novel two-phase network alignment framework UMA (Unsupervised Multi-network Alignment) is proposed in this paper. Extensive experiments conducted on multiple real-world partially aligned social networks demonstrate that UMA can perform very well in solving the M-NASA problem.

This paper is an extended version of PNA: Partial Network Alignment with Generic Stable Matching accepted by IEEE IRI 2015 [32].

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Notes

  1. 1.

    http://stackoverflow.com.

  2. 2.

    http://superuser.com.

  3. 3.

    http://programmers.stackexchange.com.

  4. 4.

    http://www.quora.com.

  5. 5.

    http://stackexchange.com/users/11683/jon-skeet?tab=accounts.

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Acknowledgments

This work is supported in part by NSF through grants III-1526499, CNS-1115234, and OISE-1129076, Google Research Award, and the Pinnacle Lab at Singapore Management University.

Author information

Authors and Affiliations

  1. University of Illinois at Chicago, Chicago, IL, USA

    Jiawei Zhang & Philip S. Yu

  2. National Laboratory for Novel Software Technology, Nanjing University, Nanjing, China

    Qianyi Zhan

  3. Institute for Data Science, Tsinghua University, Beijing, China

    Philip S. Yu

Authors
  1. Jiawei Zhang
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  2. Qianyi Zhan
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  3. Philip S. Yu
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Corresponding author

Correspondence to Jiawei Zhang .

Editor information

Editors and Affiliations

  1. d'Informatique, Ecole Nationale Supérieure, Alger, Algeria

    Thouraya Bouabana-Tebibel

  2. Code 71730, BS, SPAWAR Systems Center Pacific, San Diego, California, USA

    Stuart H. Rubin

Appendix: New Objective Function

Appendix: New Objective Function

Based on the above relaxations used in Sect. 3.3, the new objective function can be represented as

$$\begin{aligned}&\bar{\mathbf {T}}^{(i,j)}, \bar{\mathbf {T}}^{(j,k)}, \bar{\mathbf {T}}^{(k,i)} \\&={\arg \min }_{\mathbf {T}^{(i,j)}, \mathbf {T}^{(j,k)}, \mathbf {T}^{(k,i)}} \left\| (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)} \mathbf {T}^{(i,j)} - \mathbf {S}^{(j)} \right\| ^2_F\\&+ \left\| (\mathbf {T}^{(j,k)})^\top \mathbf {S}^{(j)} \mathbf {T}^{(j,k)} - \mathbf {S}^{(k)} \right\| ^2_F + \left\| (\mathbf {T}^{(k,i)})^\top \mathbf {S}^{(k)} \mathbf {T}^{(k,i)} - \mathbf {S}^{(i)} \right\| ^2_F\\&+ \alpha \cdot \left\| (\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)} \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)} - \mathbf {T}^{(k,i)} \mathbf {S}^{(i)} (\mathbf {T}^{(k,i)})^\top \right\| ^2_F\\&+ \beta \cdot \left\| \mathbf {T}^{(i,j)} \right\| _0 + \gamma \cdot \left\| \mathbf {T}^{(j,k)} \right\| _0 + \theta \cdot \left\| \mathbf {T}^{(k,i)} \right\| _0\\&s.t.\ \mathbf {0}^{|\mathcal {U}^{(i)}| \times |\mathcal {U}^{(j)}|} \preccurlyeq \mathbf {T}^{(i,j)} \preccurlyeq \mathbf {1}^{|\mathcal {U}^{(i)}| \times |\mathcal {U}^{(j)}|},\\&\ \ \ \ \ \mathbf {0}^{|\mathcal {U}^{(j)}| \times |\mathcal {U}^{(k)}|} \preccurlyeq \mathbf {T}^{(j,k)} \preccurlyeq \mathbf {1}^{|\mathcal {U}^{(j)}| \times |\mathcal {U}^{(k)}|},\\&\ \ \ \ \ \mathbf {0}^{|\mathcal {U}^{(k)}| \times |\mathcal {U}^{(i)}|} \preccurlyeq \mathbf {T}^{(k,i)} \preccurlyeq \mathbf {1}^{|\mathcal {U}^{(k)}| \times |\mathcal {U}^{(i)}|}. \end{aligned}$$

The Lagrangian function of the objective function can be represented as

$$\begin{aligned}&\mathcal {L}(\mathbf {T}^{(i,j)}, \mathbf {T}^{(j,k)}, \mathbf {T}^{(k,i)}, \beta , \gamma , \theta ) = \left\| (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)} \mathbf {T}^{(i,j)} - \mathbf {S}^{(j)} \right\| ^2_F\\&+\left\| (\mathbf {T}^{(j,k)})^\top \mathbf {S}^{(j)} \mathbf {T}^{(j,k)} - \mathbf {S}^{(k)} \right\| ^2_F + \left\| (\mathbf {T}^{(k,i)})^\top \mathbf {S}^{(k)} \mathbf {T}^{(k,i)} - \mathbf {S}^{(i)} \right\| ^2_F\\&+\alpha \cdot \left\| (\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)} \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)} - \mathbf {T}^{(k,i)} \mathbf {S}^{(i)} (\mathbf {T}^{(k,i)})^\top \right\| ^2_F\\&+ \beta \cdot \left\| \mathbf {T}^{(i,j)} \right\| _0 + \gamma \cdot \left\| \mathbf {T}^{(j,k)} \right\| _0 + \theta \cdot \left\| \mathbf {T}^{(k,i)} \right\| _0. \end{aligned}$$

The partial derivatives of function \(\mathcal {L}\) with regard to \(\mathbf {T}^{(i,j)}\), \(\mathbf {T}^{(j,k)}\), and \(\mathbf {T}^{(k,i)}\) will be:

$$\begin{aligned} (1)&\frac{\partial \mathcal {L}\left( \mathbf {T}^{(i,j)}, \mathbf {T}^{(j,k)}, \mathbf {T}^{(k,i)}, \beta , \gamma , \theta \right) }{\partial \mathbf {T}^{(i,j)}}\\&= 2 \cdot \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}(\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\\&+ 2 \cdot (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}(\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)} \\&+ 2 \alpha \cdot \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top \\&+ 2 \alpha \cdot (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top \\&- 2 \cdot \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}(\mathbf {S}^{(j)})^\top - 2 \cdot (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {S}^{(j)}\\&- 2 \alpha \cdot (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}\mathbf {S}^{(i)}(\mathbf {T}^{(k,i)})^\top (\mathbf {T}^{(j,k)})^\top \\&-2 \alpha \cdot \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top (\mathbf {T}^{(k,i)})^\top (\mathbf {T}^{(j,k)})^\top - \beta \cdot \mathbf {1}\mathbf {1}^\top .\\ (2)&\frac{\partial \mathcal {L}\left( \mathbf {T}^{(i,j)}, \mathbf {T}^{(j,k)}, \mathbf {T}^{(k,i)}, \beta , \gamma , \theta \right) }{\partial \mathbf {T}^{(j,k)}}\\&= 2 \cdot \mathbf {S}^{(j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top (\mathbf {S}^{(j)})^\top \mathbf {T}^{(j,k)}\\&+ 2 \cdot (\mathbf {S}^{(j)})^\top \mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top \mathbf {S}^{(j)}\mathbf {T}^{(j,k)} \\&+ 2 \alpha \cdot (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\\&+ 2 \alpha \cdot (\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}(\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\\&- 2 \cdot \mathbf {S}^{(j)}\mathbf {T}^{(j,k)}(\mathbf {S}^{(k)})^\top - 2 \cdot (\mathbf {S}^{(j)})^\top \mathbf {T}^{(j,k)}\mathbf {S}^{(k)}\\&- 2 \alpha \cdot (\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}\mathbf {S}^{(i)}(\mathbf {T}^{(k,i)})^\top \\&-2 \alpha \cdot (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top (\mathbf {T}^{(k,i)})^\top - \gamma \cdot \mathbf {1}\mathbf {1}^\top .\\ (3)&\frac{\partial \mathcal {L}\left( \mathbf {T}^{(i,j)}, \mathbf {T}^{(j,k)}, \mathbf {T}^{(k,i)}, \beta , \gamma , \theta \right) }{\partial \mathbf {T}^{(k,i)}}\\&= 2 \cdot \mathbf {S}^{(k)}\mathbf {T}^{(k,i)}(\mathbf {T}^{(k,i)})^\top (\mathbf {S}^{(k)})^\top \mathbf {T}^{(k,i)} \\&+ 2 \cdot (\mathbf {S}^{(k)})^\top \mathbf {T}^{(k,i)}(\mathbf {T}^{(k,i)})^\top \mathbf {S}^{(k)}\mathbf {T}^{(k,i)} \\&+ 2 \alpha \mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top (\mathbf {T}^{(k,i)})^\top \mathbf {T}^{(k,i)}\mathbf {S}^{(i)} \\&+ 2 \alpha \mathbf {T}^{(k,i)}\mathbf {S}^{(i)}(\mathbf {T}^{(k,i)})^\top \mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top \\&- 2 \cdot \mathbf {S}^{(k)}\mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top - 2 \cdot (\mathbf {S}^{(k)})^\top \mathbf {T}^{(k,i)}\mathbf {S}^{(i)}\\&- 2 \alpha \cdot (\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top (\mathbf {S}^{(i)})^\top \mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}\mathbf {S}^{(i)}\\&- 2 \alpha \cdot (\mathbf {T}^{(j,k)})^\top (\mathbf {T}^{(i,j)})^\top \mathbf {S}^{(i)}\mathbf {T}^{(i,j)}\mathbf {T}^{(j,k)}\mathbf {T}^{(k,i)}(\mathbf {S}^{(i)})^\top - \theta \cdot \mathbf {1}\mathbf {1}^\top . \end{aligned}$$

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Zhang, J., Zhan, Q., Yu, P.S. (2016). Concurrent Alignment of Multiple Anonymized Social Networks with Generic Stable Matching. In: Bouabana-Tebibel, T., Rubin, S. (eds) Theoretical Information Reuse and Integration. Advances in Intelligent Systems and Computing, vol 446. Springer, Cham. https://doi.org/10.1007/978-3-319-31311-5_8

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  • DOI: https://doi.org/10.1007/978-3-319-31311-5_8

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