Temporal Distance Metrics for Social Networks Analysis John Tang 1 , Mirco Musolesi 1 , Cecilia Mascolo 1 and Vito Latora 2 Computer Laboratory, University of Cambridge 2 INFN/Dept of Physics, University of Catania
Credit: Mark Newman mirco.musolesi@cl.cam.ac.uk 2
hhhCredit: kc claffy, CAIDA hhhHyperbolic view of BGP tables mirco.musolesi@cl.cam.ac.uk 3
Reality Mining Dataset mirco.musolesi@cl.cam.ac.uk 4
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Credit: Flutracker.com mirco.musolesi@cl.cam.ac.uk 6
Problem : existing metrics do not capture the inherent dynamism of networks over time. We need new temporal metrics defined over temporal graphs for studying dynamic processes over these networks. mirco.musolesi@cl.cam.ac.uk 7
An Example of Temporal Graph A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 8
…and the Corresponding Static Graph A B C D E F mirco.musolesi@cl.cam.ac.uk 9
INFOCOM (2nd day) mirco.musolesi@cl.cam.ac.uk 10
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Calculating the Temporal Distance A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 12
Calculating the Temporal Distance (t = 1) D at distance 1 A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 13
Calculating the Temporal Distance (t = 2) A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 14
Calculating the Temporal Distance (t = 3) B and C at distance 3 A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 15
Calculating the Temporal Distance (t = 4) F at distance 4 A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 16
What about the Static Distance? A B A B A B A B A B C D C D C D C D C D E F E F E F E F E F E is statically reachable but in reality it is not dynamically reachable! A‐> F requires 2 transmissions (hops), but in reality it requires 3 No information about the duration of the process mirco.musolesi@cl.cam.ac.uk 17
What about the Symmetric Distance (F to A)? A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 18
Calculating the Inverse Temporal Distance (t = 1) A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 19
Calculating the Inverse Temporal Distance (t = 2) A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 20
Calculating the Inverse Temporal Distance (t = 3) A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 21
Calculating the Inverse Temporal Distance (t = 4) A is not reachable [infinite distance] A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 22
Let’s Get a Bit More Formal… • Characteristic temporal path length: • Defined considering the horizon of the infection • Possible problem due to the potential divergence due to pairs of nodes that are not temporally connected mirco.musolesi@cl.cam.ac.uk 23
Let’s Get a Bit More Formal… • Characteristic temporal path length: • Defined considering the horizon of the infection • Possible problem due to the potential divergence due to pairs of nodes that are not temporally connected mirco.musolesi@cl.cam.ac.uk 24
Impact of the Horizon Parameter (F ‐> A, h = 1) A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 25
Impact of the Horizon Parameter (F ‐> A, h = 2) A was not reachable at all with h = 1 (in 4 time windows), but with h = 2 it is a distance 1! A B A B A B A B C D C D C D C D E F E F E F E F t = 2 t = 1 t = 3 t = 4 mirco.musolesi@cl.cam.ac.uk 26
Let’s Get a Bit More Formal… • Characteristic temporal path length: • Defined considering the horizon of the infection • Possible problem related to the potential divergence due to pairs of nodes that are not temporally connected mirco.musolesi@cl.cam.ac.uk 27
Let’s Get a Bit More Formal… • Solution: definition of temporal efficiency: • High value of E (low value of L) means that the nodes of the graphs can communicate efficiently mirco.musolesi@cl.cam.ac.uk 28
Local Temporal Efficiency Node i A B D C E F mirco.musolesi@cl.cam.ac.uk 29
Temporal Clustering Coefficient Node i A B D C E F mirco.musolesi@cl.cam.ac.uk 30
INFOCOM Dataset: Static vs Temporal Metrics H=wfwf h min = 12am, h max = 12pm, w = 5 min Static metrics underestimate L mirco.musolesi@cl.cam.ac.uk 31
Reality Mining Dataset h = 1, tmin =12 am, tmax = 12pm, w = 5 min h = 1, t min =12 am, t max = 12pm, w = 5 min mirco.musolesi@cl.cam.ac.uk 32
Variation of the Clustering Coefficient over Time Reality Mining Dataset (t min = 12 am, t max = 12 pm, w = 5 min) mirco.musolesi@cl.cam.ac.uk 33
Temporal Efficiency vs Static Efficiency MIT Dataset mirco.musolesi@cl.cam.ac.uk 34
Current Research Agenda • Centrality measures • Study of dynamic processes over the networks: – Message dissemination – Epidemics – Information propagation • Analysis of new and larger datasets • Small‐world behavior in temporal networks mirco.musolesi@cl.cam.ac.uk 35
Summary of the Talk • New temporal metrics for studying dynamic processes over dynamic networks – Temporal distance – Temporal efficiency – Temporal clustering • Analysis using real datasets mirco.musolesi@cl.cam.ac.uk
Questions? Mirco Musolesi mirco.musolesi@cl.cam.ac.uk [soon to be: mirco@cs.st‐andrews.ac.uk] http://www.cl.cam.ac.uk/~mm753 mirco.musolesi@cl.cam.ac.uk 37
INFOCOM Dataset: What Happens if We Reshuffle the Sequence? h = 1, t min =12 am, t max = 12pm, no runs = 50 mirco.musolesi@cl.cam.ac.uk 38
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