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

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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 ,

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Expected Nodes: a quality function for the detection of link communities

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy

LIP6 - CNRS & UPMC , Universit´ e Pierre et Marie Curie – Sorbonne Universit´ es, Paris, France.

CompleNet 2015

25 March 2015

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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

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Community Detection

Node community Link community

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 3/19

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Node community

Example in a email dataset. Communities : groups of people. Input : A graph, G = (V , E). Output : A partition P of V .

S. Fortunato.

Community detection in graphs.

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 4/19

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Link community

Example in a email dataset. Communities : threads. 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.

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 5/19

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Expected Nodes : a new quality function

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 6/19

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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

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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

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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

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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

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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

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Internal quality function

L : set of links, V (L) : internal nodes of L. The internal quality of group L is : Qin(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

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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

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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

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Combining both quality functions

|Lout| : set of adjacent links to L. Qext(Lout) computed in a similar way as Qin(L).

Good internal quality Bad external quality Good external quality Bad interal quality

Q∗(L) = 2|L|Qin(L) + |Lout|Qext(Lout) |L| + |Lout|

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 11/19

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Tests with LF benchmark

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 12/19

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Test method

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 13/19

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Ground truth generation

LF generation

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 14/19

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Results for Evans et al.

T A T B E 2 L C . 5 . 5 5 . 6 . 6 5 . 7 . 7 5 . 8

Ep TA TB LC Evans function

Figure – Evaluation of Ef from Evans et al. on several partitions.

Highlight : Q(TA) < Q(E2) Q(TA) = Q(TB)

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 15/19

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Results for the Partition density

T A T B E 2 L C . 0 . 1 . 2 . 3 . 4 . 5 . 6

TA TB Ep LC Partition density

Figure – Evaluation of the partition density from Ahn et al. on several partitions.

Highlight :

Q(TA) ≤ Q(LC)
Q(TA) = Q(TB)

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 16/19

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Results for Expected Nodes

T A T B E 2 L C . 0 . 2 . 4 . 6 . 8 1 . 0 1 . 2 1 . 4

TA TB Ep LC Expected Nodes

Figure – Evaluation of Expected Nodes on several partitions.

Highlight :

Q(TA) > Q(X)
Q(TA) = Q(TB)

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 17/19

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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

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Questions ?

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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

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Partition Ep

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 21/19

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Partition LC

No´ e Gaumont, Fran¸ cois Queyroi, Cl´ emence Magnien and Matthieu Latapy — 25 March 2015 22/19