1. Multidimensional Network Analysis Foundations of multidimensional Network Analysis, Berlingerio, Coscia, Giannotti, Monreale, Pedreschi. WWW Journal 2012

2. Classical Network Representation Only one kind of relation Different connections indistinguishable

3. Multidimensionality as different types (or different values)

4. Real world examples

5. Possible Models Tensors For each dimension we define a new matrix Complex in space and time, first framework are being studied Cfr. T. G. Kolda and B. W. Bader. Tensor Decompositions and Applications. SIAM Review 51(3):455-500, September 2009. Multigraphs Multiple different relations among entities Misses some analytical powers of tensor decomposition Extension of some graph mining algorithms Cfr. Foundations of multidimensional Network Analysis, Berlingerio, Coscia, Giannotti, Monreale, Pedreschi. ASONAM 2011.

6. Extending the monodimensional settings Is it possible to generalize known measures (i.e. degree?) Are there novel measures meaningful only in the multidimensional setting?

7. Relevance of a dimension for a node Intuition: Twitter dimension for User5 should be low, Flickr dimension for User4 should be high

8. Generalizing the Degree Degree = 8 Neighbors = 4

9. Neighbor XOR XOR(4, Flickr) = 2 XOR(4, Faceb) = 0 XOR(4, Linkedin) = 0 XOR(7, Flickr) = 1 XOR(7, Faceb) = 0 XOR(7, Linkedin) = 0

10. Dimension Relevance DR(4, Flickr) = 1 DR(4, Faceb) = 0 DR(4, Linkedin) =.25 DR(7, Flickr) = 1 DR(7, Faceb) = 0 DR(7, Linkedin) =.5 DR(2,Flickr) =.5 DR(2,Faceb) = 1 DR(2, Linkedin) =.5

11. Weighted Dimension Relevance W(4, Flickr) =.6875 W(4, Faceb) = 0 W(4, Linkedin) =.0625 Captures number of neighbors reachable through D, weighted by the number of alternative connections.

12. Conclusions The extension of basic statistics in multidimensional network may lead to a novel analytical power We started to unveil this analytical power We show possible applications in more general research settings Other work on this topic: Link prediction Strength of the tie