Expected Nodes : a quality function for the detection of link - PowerPoint PPT Presentation

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Expected Nodes : a quality function for the detection of link communities e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and No´ Matthieu Latapy LIP6 - CNRS & UPMC , Universit´ e Pierre et Marie Curie – Sorbonne Universit´ es, Paris, France. CompleNet 2015 25 March 2015

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Summary 1 Link community 2 Expected Nodes : a new quality function 3 Tests with LF benchmark 4 Conclusion and perspectives No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 2/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Community Detection Node community Link community No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 3/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Node community Input : A graph, G = ( V , E ). Output : A partition P of V . S. Fortunato. Community detection in graphs. Example in a email dataset. Communities : groups of people. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 4/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Link community Input : A graph, G = ( V , E ). Output : A partition P of E . T.S. Evans et R. Lambiotte. Line graphs, link partitions, and overlapping communities. Y.-Y. Ahn, J. P. Bagrow, et S. Lehmann. Link communities reveal multiscale complexity in networks. Example in a email dataset. Communities : threads. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 5/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Expected Nodes : a new quality function No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 6/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Outline of the quality function Why is the group of blue links relevant ? No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 7/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Outline of the quality function Why is the group of blue links relevant ? Dense blue links and sparse pink links compare to what could be expected in the configuration model . No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 7/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 The idea behind Expected Nodes Compare observed nodes to expected one : No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 8/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 The idea behind Expected Nodes Compare observed nodes to expected one : No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 8/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 The idea behind Expected Nodes Compare observed nodes to expected one : No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 8/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Internal quality function L : set of links, V ( L ) : internal nodes of L . The internal quality of group L is : Q in ( L ) = E [ V ( L )] − | V ( L ) | E [ V ( L )] E [ V ( L )] : sum of random variable with hypergeometric distribution. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 9/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 External quality function Compare adjacent nodes to expected ones : No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 10/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 External quality function Compare adjacent nodes to expected ones : No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 10/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Combining both quality functions | L out | : set of adjacent links to L. Q ext ( L out ) computed in a similar way as Q in ( L ). Good internal quality Bad interal quality Bad external quality Good external quality Q ∗ ( L ) = 2 | L | Q in ( L ) + | L out | Q ext ( L out ) | L | + | L out | No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 11/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Tests with LF benchmark No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 12/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Test method No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 13/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Ground truth generation LF generation No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 14/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Results for Evans et al. 0 . 8 0 0 . 7 5 Evans function 0 . 7 0 Highlight : 0 . 6 5 Q ( TA ) < Q ( E 2) 0 . 6 0 Q ( TA ) = Q ( TB ) 0 . 5 5 0 . 5 0 Ep TA TB LC T A T B E 2 L C Figure – Evaluation of Ef from Evans et al. on several partitions. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 15/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Results for the Partition density 0 . 6 Partition density 0 . 5 0 . 4 Highlight : 0 . 3 • Q ( TA ) ≤ Q ( LC ) 0 . 2 • Q ( TA ) = Q ( TB ) 0 . 1 0 . 0 Ep TA TB LC T A T B E 2 L C Figure – Evaluation of the partition density from Ahn et al. on several partitions. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 16/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Results for Expected Nodes 1 . 4 1 . 2 Expected Nodes 1 . 0 0 . 8 Highlight : 0 . 6 • Q ( TA ) > Q ( X ) 0 . 4 • Q ( TA ) � = Q ( TB ) 0 . 2 0 . 0 Ep TA TB LC T A T B E 2 L C Figure – Evaluation of Expected Nodes on several partitions. No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 17/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Conclusion and perspectives To sum up : • Consider community of links instead of nodes. • Definition of Expected Nodes to evaluate link partitions. • On the tests, the ground truth is the best choice only for Expected Nodes . Perspectives • Design an algorithm for maximizing Expected Nodes . • More detailed comparisons between quality functions No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 18/19

Questions ?

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 LF generation example Green group : a community in the ground truth No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 20/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Partition Ep No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 21/19

u n i v e r s i t e p i e r r e e t m a r i e c u r i e - l i p 6 Partition LC No´ e Gaumont , Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 22/19