Temporal Reachability Graphs - Semantic Scholar
´s - lip6 thales & upmc sorbonne universite
Temporal Reachability Graphs John Whitbeck, Marcelo Dias de Amorim, Vania Conan and Jean-Loup Guillaume
August 25th, 2012
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces Time 0 0 0 1 1 1 1 2 ···
Node 1 a d c a d c b a
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
Node 2 b e e b e e d b
Event UP UP UP DOWN DOWN DOWN UP UP
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces Time 0 0 0 1 1 1 1 2 ···
Node 1 a d c a d c b a
Node 2 b e e b e e d b
Event UP UP UP DOWN DOWN DOWN UP 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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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 Tolerance (δ) δ = 1s
TRG at t = 0s a e b c
d
TRG at t = 1s a e b
δ = 2s
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 5/16
b c
a
e d
c
d
a
e d
c
d
a
TRG at t = 2s a e b
b c
e d
b c
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
“Adding” reachability graphs
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
´s - lip6 thales & upmc sorbonne universite
“Adding” reachability graphs
“Rδ ⊕ Rµ = Rδ+µ ”
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Messages are created out of sync with the contact trace’s granularity (t 6= kη) • Good upper (too many arcs) and lower (a few missed arcs)
approximations • Bounds are equal for all t = kη
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Density
Upper bound 0.48 Lower bound
0.5 0.4
0.44
0.3
0.4
0.2
0.36
0.1
Dominating set size from lower and upper bounds 1350
1360
1370
1380
Dominating set size
0.6
0.52
0 1390
1400
Time (s)
Example taken from the Rollernet trace with τ = 5s and δ = 1min
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
1 0.8 0.6 0.4 0.2 0
1 0.8 0.6 0.4 0.2 0
δ = 10s
20
30
40
50
1 0.8 0.6 0.4 0.2 0
60
70
80 1 0.8 0.6 0.4 0.2 0
δ = 1min 20
30
40
1 0.8 0.6 0.4 0.2 0
50
60
70
80 1 0.8 0.6 0.4 0.2 0
δ = 3min
20
30
40
50
Time (min) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 12/16
60
70
80
Dominating set size
Proportion of connected pairs
Example 1 : Rollernet (τ = 5s)
´s - lip6 thales & upmc sorbonne universite
1 0.8 0.6 0.4 0.2 0
1 0.8 0.6 0.4 0.2 0
δ = 20min
4
5
6
1 0.8 0.6 0.4 0.2 0
7
8
9
10
11 1 0.8 0.6 0.4 0.2 0
δ = 1h 4
5
6
1 0.8 0.6 0.4 0.2 0
7
8
9
10
11 1 0.8 0.6 0.4 0.2 0
δ = 2h 4
5
6
7
8
Time (h) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 13/16
9
10
11
Dominating set size
Proportion of connected pairs
Example 2 : Stanford High (τ = 1s)
´s - lip6 thales & upmc sorbonne universite
Avg. density
Bounds 1 0 0.8 2 0.6 5 0.4 20 0.2 10 0.25 0 0 60 120 Delay δ (s)
180
Rollernet : Average density vs. maximum delay δ for different edge traversal times τ Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 14/16
´s - lip6 thales & upmc sorbonne universite
Avg. dom. set size
Bounds 1 0.8 0.6 0.4 0.2 0
from top to bottom: τ = 20, 10, 5, 2, 0.25, 0
0
60 120 Delay δ (s)
180
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
´s - lip6 thales & upmc sorbonne universite
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
Temporal Reachability Graphs John Whitbeck, Marcelo Dias de Amorim, Vania Conan and Jean-Loup Guillaume
August 25th, 2012
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces Time 0 0 0 1 1 1 1 2 ···
Node 1 a d c a d c b a
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 2/16
Node 2 b e e b e e d b
Event UP UP UP DOWN DOWN DOWN UP UP
´s - lip6 thales & upmc sorbonne universite
Intro : Contact Traces Time 0 0 0 1 1 1 1 2 ···
Node 1 a d c a d c b a
Node 2 b e e b e e d b
Event UP UP UP DOWN DOWN DOWN UP 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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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 Tolerance (δ) δ = 1s
TRG at t = 0s a e b c
d
TRG at t = 1s a e b
δ = 2s
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 5/16
b c
a
e d
c
d
a
e d
c
d
a
TRG at t = 2s a e b
b c
e d
b c
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
“Adding” reachability graphs
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
´s - lip6 thales & upmc sorbonne universite
“Adding” reachability graphs
“Rδ ⊕ Rµ = Rδ+µ ”
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 9/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Messages are created out of sync with the contact trace’s granularity (t 6= kη) • Good upper (too many arcs) and lower (a few missed arcs)
approximations • Bounds are equal for all t = kη
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
But not quite so easy...
Density
Upper bound 0.48 Lower bound
0.5 0.4
0.44
0.3
0.4
0.2
0.36
0.1
Dominating set size from lower and upper bounds 1350
1360
1370
1380
Dominating set size
0.6
0.52
0 1390
1400
Time (s)
Example taken from the Rollernet trace with τ = 5s and δ = 1min
Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 10/16
´s - lip6 thales & upmc sorbonne universite
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
´s - lip6 thales & upmc sorbonne universite
1 0.8 0.6 0.4 0.2 0
1 0.8 0.6 0.4 0.2 0
δ = 10s
20
30
40
50
1 0.8 0.6 0.4 0.2 0
60
70
80 1 0.8 0.6 0.4 0.2 0
δ = 1min 20
30
40
1 0.8 0.6 0.4 0.2 0
50
60
70
80 1 0.8 0.6 0.4 0.2 0
δ = 3min
20
30
40
50
Time (min) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 12/16
60
70
80
Dominating set size
Proportion of connected pairs
Example 1 : Rollernet (τ = 5s)
´s - lip6 thales & upmc sorbonne universite
1 0.8 0.6 0.4 0.2 0
1 0.8 0.6 0.4 0.2 0
δ = 20min
4
5
6
1 0.8 0.6 0.4 0.2 0
7
8
9
10
11 1 0.8 0.6 0.4 0.2 0
δ = 1h 4
5
6
1 0.8 0.6 0.4 0.2 0
7
8
9
10
11 1 0.8 0.6 0.4 0.2 0
δ = 2h 4
5
6
7
8
Time (h) Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 13/16
9
10
11
Dominating set size
Proportion of connected pairs
Example 2 : Stanford High (τ = 1s)
´s - lip6 thales & upmc sorbonne universite
Avg. density
Bounds 1 0 0.8 2 0.6 5 0.4 20 0.2 10 0.25 0 0 60 120 Delay δ (s)
180
Rollernet : Average density vs. maximum delay δ for different edge traversal times τ Whitbeck et al. — Temporal Reachability Graphs — August 25th, 2012 14/16
´s - lip6 thales & upmc sorbonne universite
Avg. dom. set size
Bounds 1 0.8 0.6 0.4 0.2 0
from top to bottom: τ = 20, 10, 5, 2, 0.25, 0
0
60 120 Delay δ (s)
180
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
´s - lip6 thales & upmc sorbonne universite
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
