Can Everybody Sit Closer to Their Friends Than Their Enemies?

Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected through a positive edge than its neighbors connected through a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given l- dimensional Euclidean space. We focus on the 1-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and provides it when applicable.

[1]  F. Harary.  On the notion of balance of a signed graph. , 1953 .

[2]  Avrim Blum,et al.  Correlation Clustering , 2004, Machine Learning.

[3]  S Redner,et al.  Dynamics of social balance on networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Jure Leskovec,et al.  Predicting positive and negative links in online social networks , 2010, WWW '10.

[5]  Sahin Albayrak,et al.  Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization , 2010, SDM.

[6]  Frank Harary,et al.  Counting balanced signed graphs using marked graphs , 1981 .

[7]  Jure Leskovec,et al.  Governance in Social Media: A Case Study of the Wikipedia Promotion Process , 2010, ICWSM.

[8]  J. Davis.  Clustering and Structural Balance in Graphs , 1967 .

[9]  Michael Szell,et al.  Multirelational organization of large-scale social networks in an online world , 2010, Proceedings of the National Academy of Sciences.

[10]  Ulrik Brandes,et al.  Summarizing Dynamic Bipolar Conflict Structures , 2006, IEEE Transactions on Visualization and Computer Graphics.

[11]  Debra Lauterbach,et al.  Surfing a Web of Trust: Reputation and Reciprocity on CouchSurfing.com , 2009, 2009 International Conference on Computational Science and Engineering.

[12]  Jure Leskovec,et al.  Signed networks in social media , 2010, CHI.

[13]  Laurent Viennot,et al.  Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing , 2000, Theor. Comput. Sci..

[14]  F Harary,et al.  On the number of balanced signed graphs. , 1967, The Bulletin of mathematical biophysics.

[15]  F. Harary,et al.  STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY1 , 1977 .

[16]  Frank Harary,et al.  A simple algorithm to detect balance in signed graphs , 1980, Math. Soc. Sci..