1. Measuring Partial Balance in Signed Networks
Samin Aref and Mark C. Wilson
Department of Computer Science
The University of Auckland
CMSS Summer Workshop, Auckland, 20 February 2016
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2. Measuring partial balance in signed networks
Theory of Structural Balance (Heider 1944) (Cartwright and Harary 1956) Enemy of a friend: enemy Enemy of an enemy: friend Friend of a friend: friend Friend of an enemy: enemy Signed graphs exhibiting such relations are “balanced”.
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Measuring partial balance in signed networks

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4. Measuring partial balance in signed networks
Balanced cycles: even number of negative edges
Unbalanced cycles: odd number of negative edges
Sufficient condition
Networks containing no cycles with an odd number of negative edges are balanced.
[2]
Figures: D. Easley and J. Kleinberg, Networks, crowds, and markets: Reasoning about a highly connected world. pp 122 Cambridge University Press, 2010
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Measuring partial balance in signed networks

5. Measuring partial balance in signed networks
Networks containing no cycles with an odd number of negative edges are balanced.
[2]
Figures: D. Easley and J. Kleinberg, Networks, crowds, and markets: Reasoning about a highly connected world. pp 122 Cambridge University Press, 2010
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Measuring partial balance in signed networks

6. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

7. Measures of Partial Balance
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Measuring partial balance in signed networks

8. Measuring partial balance in signed networks
Four main approaches
Cycles
D(G)
C(G)
Closed walks
W(G)
Triads
T(G)
S(G)
Edge removal
L(G)
F(G)
Degree of balance (Cartwright and Harary 1956)
Weighted degree of balance (Norman and Roberts 1972)
Walk-based measure of balance (Estrada and Benzi 2014)
Triangle index (Cartwright and Harary 1956) (Terzi and Winkler 2011)
Relative signed clustering coefficient (Kunegis 2014)
Frustration index (Harary 1959)
Normalised frustration index
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Measuring partial balance in signed networks

9. Undirected signed graph
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Measuring partial balance in signed networks

10. Measures based on cycles
Balanced cycles of length k
Unbalanced cycles of length k
Degree of balance (Cartwright and Harary 1956)
Weighted degree of balance (Norman and Roberts 1972)
decreasing functions of length
e.g.
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Measuring partial balance in signed networks

11. Measures based on closed-walks
Triangle index
(Cartwright and Harary 1956)
(Terzi and Winkler 2011)
Balanced closed-walks of length k
Unbalanced closed-walks of length k
Walk-based measure of balance
(Estrada and Benzi 2014)
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Measuring partial balance in signed networks

12. Measures based on frustration index
Line index of balance (Harary 1959)
Number of edges to negate and make the network balanced
Upper bound
Normalised frustration index
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Measuring partial balance in signed networks

13. Measuring partial balance in signed networks
Axioms
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Measuring partial balance in signed networks

14. Measuring partial balance in signed networks
Axiomatic properties
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Measuring partial balance in signed networks

15. What Do We Expect on Synthetic Data?
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16. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

17. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

18. Results on Real Networks
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Measuring partial balance in signed networks

19. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

20. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

21. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

22. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

23. Measuring partial balance in signed networks
Senate bill co-sponsorship network [14]
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Measuring partial balance in signed networks

24. Partially Balanced Networks
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Measuring partial balance in signed networks

25. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

26. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

27. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

28. Measuring partial balance in signed networks
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Measuring partial balance in signed networks

29. Measuring partial balance in signed networks
Recommendations
Weighted cycles
Sampling triads
Closed walks
Edge removal
Cycles
Triads
Axiomatic properties
Exactness
Detecting closeness to total balance
Differentiating partial balance
Large networks applicability
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Measuring partial balance in signed networks

30. References:
F. Heider, “Social perception and phenomenal causality.” Psychological review, vol. 51, no. 6, p. 358, 1944.
D. Easley and J. Kleinberg, Networks, crowds, and markets: Reasoning about a highly connected world. Cambridge University Press, 2010.
D. Cartwright and F. Harary, “Structural balance: a generalization of Heider’s theory.” Psychological review, vol. 63, no. 5, p. 277, 1956.
F. Harary, “On the measurement of structural balance,” Behavioral Science, vol. 4, no. 4, pp. 316–323, Oct. 1959
R. Z. Norman and F. S. Roberts, “A derivation of a measure of relative balance for social structures and a characterization of extensive ratio systems,” Journal of mathematical psychology, vol. 9, no. 1, pp. 66–91, 1972.
E. Terzi and M. Winkler, “A spectral algorithm for computing social balance,” in Algorithms and models for the web graph. Springer, 2011, pp. 1–13.
E. Estrada and M. Benzi, “Walk-based measure of balance in signed networks: Detecting lack of balance in social networks,” Physical Review E, vol. 90, no. 4, p , 2014.
J. Kunegis, “Applications of Structural Balance in Signed Social Networks,” arXiv: [physics], Feb. 2014, arXiv:
D. McCandless, Information is Beautiful by Univers Labs, source: multiple news reports
K. E. Read, “Cultures of the central highlands, New Guinea,” Southwestern Journal of Anthropology, pp. 1–43, 1954.
S. F. Sampson, “A Novitiate in a Period of Change,” An Experimental and Case Study of Social Relationships (PhD thesis). Cornell University, Ithaca, 1968.
T. Newcomb, The Acquaintance Process. New York: Holt, Rinehart and Winston.(1966)." The General Nature of Peer Group Influence", pps in College Peer Groups, edited by TM Newcomb and EK Wilson. Chicago: Aldine Publishing Co, 1961.
T. B. Lemann and R. L. Solomon, “Group characteristics as revealed in sociometric patterns and personality ratings,” Sociometry, pp. 7–90, 1952.
Z. Neal, “The backbone of bipartite projections: Inferring relationships from co-authorship, co-sponsorship, co-attendance and other co-behaviors,” Social Networks, vol. 39, pp. 84–97, 2014.
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31. Measuring partial balance in signed networks
Questions Thank You
Aref, S., Wilson, M.C.: Measuring partial balance in signed networks. arXiv preprint arXiv: (2015)
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Measuring partial balance in signed networks

32. Measuring partial balance in signed networks
Appendix
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Measuring partial balance in signed networks

33. Application in International Relations
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Measuring partial balance in signed networks

34. Alliance network (giant component=25 n=39)
[9]
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Figure: D. McCandless, Information is Beautiful by Univers Labs, source: multiple news reports

35. Enmity network (almost connected)
[9]
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Figure: D. McCandless, Information is Beautiful by Univers Labs, source: multiple news reports

36. Middle East signed network
[9]
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Figure: D. McCandless, Information is Beautiful by Univers Labs, source: multiple news reports