t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Temporal Reachability Graphs John Whitbeck, Marcelo Dias de Amorim, Vania Conan and Jean-Loup Guillaume August 25th, 2012
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Intro : Contact Traces Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Intro : Contact Traces Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Intro : Contact Traces Time Node 1 Node 2 Event 0 a b UP 0 d e UP 0 c e UP 1 a b DOWN 1 d e DOWN 1 c e DOWN 1 b d UP 2 a b UP · · · Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Intro : Contact Traces Time Node 1 Node 2 Event 0 a b UP 0 d e UP 0 c e UP 1 a b DOWN 1 d e DOWN 1 c e DOWN 1 b d UP 2 a b UP · · · Contact trace : symmetric single-hop information Opportunistic routing : asymmetric multi-hop connectivity over time Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Outline 1 From time-varying connectivity graphs to reachability graphs 2 Efficient calculation of reachability graphs 3 Results : bounds on communication capabilities Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 3/16
From time-varying connectivity graphs to reachability graphs
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Temporal reachability graphs TRG Definition In a ( τ, δ )-reachability graph, an arc exists from node A to B at time t if a space-time path exists from A to B leaving A at time t and arriving at B before t + δ given that each single-hop takes time τ . Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 5/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Temporal reachability graphs TRG Definition In a ( τ, δ )-reachability graph, an arc exists from node A to B at time t if a space-time path exists from A to B leaving A at time t and arriving at B before t + δ given that each single-hop takes time τ . Delay TRG TRG TRG Tolerance ( δ ) at t = 1s at t = 0s at t = 2s a a a e e e b b b δ = 1s c c c d d d a a a e e e b b b δ = 2s c c c d d d Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 5/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Time-varying dominating set TVDS Definition A time-varying dominating set (TVDS) of a temporal reachability graph, is a time-varying set of nodes such at at all times t , the node in the TVDS are a regular dominating set of the directed reachability graph at time t . Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 6/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Why reachability graphs ? • On reachability graphs, certain routing performance questions become easy • Upper-bound on average delivery ratio at time t (e.g., point-to-point, broadcast) • Size of the“temporal dominating set”at time t (for offloading) • New analysis angles on connectivity graphs • Asymmetric / Symmetric connectivity phases • Good receivers = large incoming node degree • Good senders = large outgoing node degree Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 7/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Why reachability graphs ? • On reachability graphs, certain routing performance questions become easy • Upper-bound on average delivery ratio at time t (e.g., point-to-point, broadcast) • Size of the“temporal dominating set”at time t (for offloading) • New analysis angles on connectivity graphs • Asymmetric / Symmetric connectivity phases • Good receivers = large incoming node degree • Good senders = large outgoing node degree The real challenge is calculating a reachability graph from a regular time-varying graph ! Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 7/16
Efficient calculation of reachability graphs
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 “Adding” reachability graphs Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 “Adding” reachability graphs “ R δ ⊕ R µ = R δ + µ ” Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 But not quite so easy... Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 But not quite so easy... Messages are created out of sync with the contact trace’s granularity ( t � = k η ) • Good upper (too many arcs) and lower (a few missed arcs) approximations • Bounds are equal for all t = k η Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 But not quite so easy... 0.52 0.6 Upper bound Dominating set size 0.5 Lower bound 0.48 0.4 Density 0.44 0.3 0.2 0.4 0.1 Dominating set size from 0.36 lower and upper bounds 0 1350 1360 1370 1380 1390 1400 Time (s) Example taken from the Rollernet trace with τ = 5 s and δ = 1 min Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 But not quite so easy... Graph can evolve faster than the transmission time ( η < τ ) • Composition over families of TRGs with close δ values • Parallel computation of families Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
Results : bounds on communication capabilities
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Example 1 : Rollernet ( τ = 5 s ) 1 1 0.8 0.8 δ = 10s 0.6 0.6 Proportion of connected pairs 0.4 0.4 0.2 0.2 0 0 Dominating set size 20 30 40 50 60 70 80 1 1 0.8 0.8 0.6 0.6 0.4 0.4 δ = 1min 0.2 0.2 0 0 20 30 40 50 60 70 80 1 1 0.8 0.8 0.6 δ = 3min 0.6 0.4 0.4 0.2 0.2 0 0 20 30 40 50 60 70 80 Time (min) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 12/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Example 2 : Stanford High ( τ = 1 s ) 1 1 δ = 20min 0.8 0.8 0.6 0.6 Proportion of connected pairs 0.4 0.4 0.2 0.2 0 0 Dominating set size 4 5 6 7 8 9 10 11 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 δ = 1h 0.2 0 0 4 5 6 7 8 9 10 11 1 1 0.8 0.8 0.6 0.6 0.4 δ = 2h 0.4 0.2 0.2 0 0 4 5 6 7 8 9 10 11 Time (h) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 13/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Bounds 1 0 Avg. density 0.8 2 0.6 5 0.4 20 0.2 10 0.25 0 0 60 120 180 Delay δ (s) Rollernet : Average density vs. maximum delay δ for different edge traversal times τ Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 14/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Bounds Avg. dom. set size 1 0.8 from top to bottom: τ = 20, 10, 5, 0.6 2, 0.25, 0 0.4 0.2 0 0 60 120 180 Delay δ (s) Rollernet : Average dominating set size vs. maximum delay δ for different values of τ (in seconds). Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 14/16
t h a l e s & u p m c s o r b o n n e u n i v e r s i t ´ e s - l i p 6 Conclusions Contributions • Formalization of temporal reachability graphs (TRG) • Fast implementation of the conversion from regular time-varying graphs • Powerful tool for analyzing performance bounds in opportunistic networks (e.g., asymmetry, max delivery ratio) • Opens up many new perspectives (modeling, community detection) Lessons for opportunistic networks • Point to point communications with acceptable delays are very hard • However usually possible to reach everyone in the network through a small dominating set (Offloading) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 15/16
Thank You ! More info at : http://www-npa.lip6.fr/~whitbeck Calculation & visualization code : http://github.com/neush/ditl
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