Social-Affiliation Networks: Patterns and the <Emphasis Type="SmallCaps">SOAR</Emphasis> Model

Dhivya Eswaran; Reihaneh Rabbany; Artur W. Dubrawski; Christos Faloutsos

doi:10.1007/978-3-030-10928-8_7

Joint European Conference on Machine Learning and Knowledge Discovery in Databases

ECML PKDD 2018: Machine Learning and Knowledge Discovery in Databases pp 105-121 | Cite as

Social-Affiliation Networks: Patterns and the SOAR Model

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Authors and affiliations

Dhivya Eswaran
Reihaneh Rabbany
Artur W. Dubrawski
Christos Faloutsos

Conference paper

First Online: 23 January 2019

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Part of the Lecture Notes in Computer Science book series (LNCS, volume 11052)

Abstract

Given a social-affiliation network – a friendship graph where users have many, binary attributes e.g., check-ins, page likes or group memberships – what rules do its structural properties such as edge or triangle counts follow, in relation to its attributes? More challengingly, how can we synthetically generate networks which provably satisfy those rules or patterns? Our work attempts to answer these closely-related questions in the context of the increasingly prevalent social-affiliation graphs. Our contributions are two-fold: (a) Patterns: we discover three new rules (power laws) in the properties of attribute-induced subgraphs, substructures which connect the friendship structure to affiliations; (b) Model: we propose SOAR– short for SOcial-Affiliation graphs via Recursion– a stochastic model based on recursion and self-similarity, to provably generate graphs obeying the observed patterns. Experiments show that: (i) the discovered rules are useful in detecting deviations as anomalies and (ii) SOAR is fast and scales linearly with network size, producing graphs with millions of edges and attributes in only a few seconds. Code related to this paper is available at: www.github.com/dhivyaeswaran/soar.

Keywords

Graph mining Attributes Patterns Anomalies Generator

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Notes

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grants No. CNS-1314632, IIS-1408924 and by DARPA under award FA8750-17-2-0130. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, or other funding parties. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.

Appendix (Proofs from Sect. 5)

First, recall the following properties of the Kronecker product [26] for any four suitably sized matrices A, B, C and D: \((A\otimes B)^T = A^T \otimes B^T\); \((A\otimes B)(C\otimes D) = AB \otimes CD\); \(\text {Tr}[A\otimes B] = \text {Tr}[A] \text {Tr}[B]\); \(\sigma (A\otimes B) = \sigma (A) \sigma (B)\) where \(\sigma (\cdot )\) is the spectral radius.

Next, observe that edge count, node count, triangle count and spectral radius of AIS for an attribute \(a\) can be explicitly expressed as a function of adjacency matrix \(\mathbf {A}\) and the \(a^{th}\) column in \(\mathbf {F}\) (call it \(f_a\)) as follows: (i) Node count of AIS, \(n_{a}= f_a^Tf_a\); (ii) Edge count of AIS, \(m_{a}= \frac{1}{2}f_a^T\mathbf {A}f_a\); (iii) Triangle count of AIS, \(\varDelta _{a}= \frac{1}{6}\text {Tr}[\left( \mathcal {D}(f_a)\mathbf {A}\mathcal {D}(f_a)\right) ^3]\), assuming no self loops – here, \(\mathcal {D}(f_a)\) denotes the diagonalization of vector \(f_a\); (iv) Spectral radius of AIS, \(\sigma _{a}= \sigma (\mathcal {D}(f_a)\mathbf {A}\mathcal {D}(f_a))\).

Let the compact notation, \(\bigotimes _{j=1}^{n}A_j\) denote \(A_1\otimes A_2 \ldots \otimes A_n\). Accordingly, every column of \(\bigotimes _{j=1}^{n}A_j\) can be expressed as the Kronecker product of a column from each \(A_j\), \(j \in \{1,\ldots ,n\}\). We are now ready to state our proofs.

Proof

(Lemma 3). Any column \(f_a\) in \(\mathbf {F}_1\otimes \mathbf {F}_2\) is a Kronecker product of columns \(f_{i,1}\) in \(\mathbf {F}_1\) and \(f_{j,2}\) in \(\mathbf {F}_2\) for some i, j. The node count of AIS of a is \(f_a^Tf_a = (f_{i,1} \otimes f_{j,2})^T(f_{i,1}\otimes f_{j,2})\) which simplifies to \((f_{i,1}^Tf_{i,1})(f_{j,2}^Tf_{j,2})\) i.e., \(n_{a}= n_{i,1}n_{j,2}\). Similarly, the edge count of AIS of a is \(m_{a}= \frac{1}{2}(f_{i,1}\otimes f_{j,2})^T(\mathbf {A}_1\otimes \mathbf {A}_2)(f_{i,1}\otimes f_{j,2})\) which can be written as \( 2(\frac{1}{2}f_{i,1}^T\mathbf {A}_1f_{i,1})(\frac{1}{2}f_{j,2}^T\mathbf {A}_2f_{j,2}) \propto m_{i,1}m_{j,2}\). Now, as \(\mathcal {G}_1,\mathcal {G}_2\) follow [P1] with exponent \(\alpha \) (given), \(m_{i,1} \propto n_{i,1}^\alpha \text { and }m_{j,2} \propto n_{j,2}^\alpha \). Hence, \(m_{a}\propto n_{a}^\alpha \).

Proof

(Lemma 4). Again, let \(f_a = f_{i,1}\otimes f_{j,2}\) for attributes i, j, a in \(\mathcal {G}_1,\mathcal {G}_2\) and \(\mathcal {G}_1\otimes \mathcal {G}_2\) respectively. The node count of AIS of a, again, is \(n_{a}\propto n_{i,1}n_{j,2}\). The triangle count of AIS of a is \(\varDelta _{a}= \frac{1}{6}\text {Tr}[(\mathcal {D}(f_a)(\mathbf {A}_1\otimes \mathbf {A}_2)\mathcal {D}(f_a))^3]\nonumber \) which can be simplified as \(\frac{1}{6}\left( \text {Tr}[(\mathcal {D}(f_{i,1})\mathbf {A}_1\mathcal {D}(f_{i,1}))^3]\right) \left( \text {Tr}[(\mathcal {D}(f_{j,2})\mathbf {A}_2\mathcal {D}(f_{j,2}))^3]\right) \propto \varDelta _{i,1}\varDelta _{j,2}\) using the first, second and third Kronecker properties stated above. Now, as i and j follow [P2] with exponent \(\beta \) (given), \(\varDelta _{i,1}\propto n_{i,1}^\beta \), and \(\varDelta _{j,2}\propto n_{j,2}^\beta \). This results in \(\varDelta _{a}\propto n_{a}^\beta \).

Proof

(Lemma 5). Once again, let \(f_a = f_{i,1}\otimes f_{j,2}\) for attributes i, j, a in \(\mathcal {G}_1,\mathcal {G}_2\) and \(\mathcal {G}_1\otimes \mathcal {G}_2\) respectively. We know from the previous proof that triangle count of AIS of a follows \(\varDelta _{a}\propto \varDelta _{i,1}\varDelta _{j,2}\). Now, spectral radius of AIS of a is \(\sigma _{a}= \sigma (\mathcal {D}(f_a)(\mathbf {A}_1\otimes \mathbf {A}_2)\mathcal {D}(f_a))\) which is \(\sigma (\mathcal {D}(f_{i,1})\mathbf {A}_1\mathcal {D}(f_{i,1}))\sigma (\mathcal {D}(f_{j,2})\mathbf {A}_2\mathcal {D}(f_{j,2})) = \sigma _{i,1}\sigma _{j,2}\) due to the second and fourth Kronecker properties stated above. As graphs \(\mathcal {G}_1,\mathcal {G}_2\) follow [P3] with exponent \(\gamma \) (given), i.e., \(\sigma _{i,1}\propto \varDelta _{i,1}^\gamma \) and \(\sigma _{j,2}\propto \varDelta _{j,2}^\gamma \). Therefore, \(\sigma _{a}\propto \varDelta _{a}^\gamma \).

References

1.
Power law. https://en.wikipedia.org/wiki/Power_law
2.
Akoglu, L., Faloutsos, C.: RTG: a recursive realistic graph generator using random typing. Data Min. Knowl. Discov. 19(2), 194–209 (2009)MathSciNetCrossRefGoogle Scholar
3.
Akoglu, L., Tong, H., Koutra, D.: Graph based anomaly detection and description: a survey. Data Min. Knowl. Discov. 29(3), 626–688 (2015)MathSciNetCrossRefGoogle Scholar
4.
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar
5.
Bastian, M., Heymann, S., Jacomy, M.: Gephi: an open source software for exploring and manipulating networks. In: ICWSM. The AAAI Press (2009)Google Scholar
6.
Chakrabarti, D., Faloutsos, C.: Graph mining: laws, generators, and algorithms. ACM Comput. Surv. 38(1), 2 (2006)CrossRefGoogle Scholar
7.
Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-MAT: a recursive model for graph mining. In: SDM, pp. 442–446. SIAM (2004)Google Scholar
8.
Cho, E., Myers, S.A., Leskovec, J.: Friendship and mobility: user movement in location-based social networks. In: KDD, pp. 1082–1090. ACM (2011)Google Scholar
9.
Crandall, D.J., Cosley, D., Huttenlocher, D.P., Kleinberg, J.M., Suri, S.: Feedback effects between similarity and social influence in online communities. In: KDD, pp. 160–168. ACM (2008)Google Scholar
10.
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: SIGCOMM, pp. 251–262 (1999)Google Scholar
11.
Fond, T.L., Neville, J.: Randomization tests for distinguishing social influence and homophily effects. In: WWW, pp. 601–610. ACM (2010)Google Scholar
12.
Goh, K.I., Kahng, B., Kim, D.: Spectra and eigenvectors of scale-free networks. Phys. Rev. E 64(5), 051903 (2001)CrossRefGoogle Scholar
13.
Goldenberg, A., Zheng, A.X., Fienberg, S.E., Airoldi, E.M.: A survey of statistical network models. Found. Trends Mach. Learn. 2(2), 129–233 (2009)CrossRefGoogle Scholar
14.
Gong, N.Z., Xu, W., Huang, L., Mittal, P., Stefanov, E., Sekar, V., Song, D.: Evolution of social-attribute networks: measurements, modeling, and implications using Google+. In: IMC, pp. 131–144. ACM (2012)Google Scholar
15.
Kim, M., Leskovec, J.: Multiplicative attribute graph model of real-world networks. Internet Math. 8(1–2), 113–160 (2012)MathSciNetCrossRefGoogle Scholar
16.
Koutra, D., Koutras, V., Prakash, B.A., Faloutsos, C.: Patterns amongst competing task frequencies: super-linearities, and the Almond-DG model. In: Pei, J., Tseng, V.S., Cao, L., Motoda, H., Xu, G. (eds.) PAKDD 2013. LNCS (LNAI), vol. 7818, pp. 201–212. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37453-1_17CrossRefGoogle Scholar
17.
Leskovec, J., Backstrom, L., Kumar, R., Tomkins, A.: Microscopic evolution of social networks. In: KDD, pp. 462–470. ACM (2008)Google Scholar
18.
Leskovec, J., Chakrabarti, D., Kleinberg, J.M., Faloutsos, C., Ghahramani, Z.: Kronecker graphs: an approach to modeling networks. JMLR 11, 985–1042 (2010)MathSciNetzbMATHGoogle Scholar
19.
Leskovec, J., Kleinberg, J.M., Faloutsos, C.: Graph evolution: densification and shrinking diameters. TKDD 1(1), 2 (2007)CrossRefGoogle Scholar
20.
McGlohon, M., Akoglu, L., Faloutsos, C.: Weighted graphs and disconnected components: patterns and a generator. In: KDD, pp. 524–532. ACM (2008)Google Scholar
21.
Mislove, A., Marcon, M., Gummadi, P.K., Druschel, P., Bhattacharjee, B.: Measurement and analysis of online social networks. In: IMC, pp. 29–42. ACM (2007)Google Scholar
22.
Newman, M.E.J., Watts, D.J., Strogatz, S.H.: Random graph models of social networks. Proc. Natl. Acad. Sci. 99(Suppl. 1), 2566–2572 (2002)CrossRefGoogle Scholar
23.
Perozzi, B., Akoglu, L., Sánchez, P.I., Müller, E.: Focused clustering and outlier detection in large attributed graphs. In: KDD, pp. 1346–1355. ACM (2014)Google Scholar
24.
Pfeiffer III, J.J., Moreno, S., La Fond, T., Neville, J., Gallagher, B.: Attributed graph models: modeling network structure with correlated attributes. In: WWW, pp. 831–842. ACM (2014)Google Scholar
25.
Rabbany, R., Eswaran, D., Dubrawski, A.W., Faloutsos, C.: Beyond assortativity: proclivity index for attributed networks (ProNe). In: Kim, J., Shim, K., Cao, L., Lee, J.-G., Lin, X., Moon, Y.-S. (eds.) PAKDD 2017. LNCS (LNAI), vol. 10234, pp. 225–237. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57454-7_18CrossRefGoogle Scholar
26.
Schacke, K.: On the Kronecker product. Master’s thesis, University of Waterloo (2004)Google Scholar
27.
Titchmarsh, E.C., Heath-Brown, D.R.: The Theory of the Riemann Zeta-Function. Oxford University Press, Oxford (1986)Google Scholar
28.
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)CrossRefGoogle Scholar
29.
Zheleva, E., Sharara, H., Getoor, L.: Co-evolution of social and affiliation networks. In: KDD, pp. 1007–1016. ACM (2009)Google Scholar

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Authors and Affiliations

Dhivya Eswaran
- 1
Email author
Reihaneh Rabbany
- 2
Artur W. Dubrawski
- 1
Christos Faloutsos
- 1

1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA
2.School of Computer ScienceMcGill UniversityMontrealCanada

Social-Affiliation Networks: Patterns and the SOAR Model

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