computing parameters of sequence based dynamic graphs
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Computing Parameters of Sequence-based Dynamic Graphs Ralf Klasing LaBRI, CNRS, University of Bordeaux, France **This is a joint work with Arnaud Casteigts, Yessin M. Neggaz, and Joseph G. Peters. Dynamic Networks Highly dynamic networks? 1

  1. Transitive Closures Computation Minimization version (find the temporal diameter d ) ... G 2 G 1 Temporal Connectivity 9

  2. Transitive Closures Computation Minimization version (find the temporal diameter d ) ... G 2 G 1 Temporal Connectivity 9

  3. Transitive Closures Computation Minimization version (find the temporal diameter d ) ... G 2 × G 1 Temporal Connectivity 9

  4. Transitive Closures Computation Minimization version (find the temporal diameter d ) ... G 2 × × G 1 Temporal Connectivity 9

  5. Transitive Closures Computation Minimization version (find the temporal diameter d ) Strategy: ascending walk ... × G 2 × × G 1 Temporal Connectivity 9

  6. Transitive Closures Computation Minimization version (find the temporal diameter d ) Strategy: ascending walk ... × G 2 × × G 1 Temporal Connectivity 9

  7. Transitive Closures Computation Minimization version (find the temporal diameter d ) Strategy: ascending walk ... × G 2 × × G 1 Temporal Connectivity 9

  8. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × ×× ... × G 2 × × G 1 Temporal Connectivity 9

  9. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × The total length of the ×× ladders is O ( δ ) ... × At most O ( δ ) binary G 2 × concatenation and × G 1 completeness tests Temporal Connectivity 9

  10. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × The total length of the ×× ladders is O ( δ ) ... × At most O ( δ ) binary G 2 × concatenation and × G 1 completeness tests Disjointness property: cat ( G ( i , j ) , G ( i ′ , j ′ ) ) = G ( i , j ′ ) Temporal Connectivity 9

  11. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × The total length of the ×× ladders is O ( δ ) ... × At most O ( δ ) binary G 2 × concatenation and × G 1 completeness tests Disjointness property: cat ( G ( i , j ) , G ( i ′ , j ′ ) ) = G ( i , j ′ ) Temporal Connectivity 9

  12. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × The total length of the ×× ladders is O ( δ ) ... × At most O ( δ ) binary G 2 × concatenation and × G 1 completeness tests Disjointness property: cat ( G ( i , j ) , G ( i ′ , j ′ ) ) = G ( i , j ′ ) If G ( i , j ) is complete, then G ( i ′ , j ′ ) is complete, for all i ′ ≤ i and j ′ ≥ j Temporal Connectivity 9

  13. Transitive Closures Computation Minimization version (find the temporal diameter d ) × Strategy: ascending walk × The total length of the ×× ladders is O ( δ ) ... × At most O ( δ ) binary G 2 × concatenation and × G 1 completeness tests Disjointness property: cat ( G ( i , j ) , G ( i ′ , j ′ ) ) = G ( i , j ′ ) If G ( i , j ) is complete, then G ( i ′ , j ′ ) is complete, for all i ′ ≤ i and j ′ ≥ j Temporal-Diameter is solvable with O ( δ ) elementary operations Temporal Connectivity 9

  14. Online Algorithms Temporal Connectivity 10

  15. Online Algorithms The optimal algorithms can be adapted to an online setting The sequence of graphs G 1 , G 2 , G 3 , ... of G is processed in the order of reception Amortized cost of O ( 1 ) elementary operations per graph received Dynamic version: consider only the recent history Temporal Connectivity 11

  16. Generic Framework Temporal Connectivity 12

  17. A Generic Framework Solve other problems using the same framework Temporal Connectivity 13

  18. A Generic Framework Solve other problems using the same framework × × ×× ... × G 2 × × G 1 Framework generalization Transitive closures concatenation Completeness test Transitive closure Temporal Connectivity 13

  19. A Generic Framework Solve other problems using the same framework × × ×× ... × G 2 × × G 1 Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 13

  20. A Generic Framework Solve other problems using the same framework × × ×× ... × G 2 × × G 1 Minimization problems Find the smallest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 13

  21. A Generic Framework Solve other problems using the same framework × × ×× ... × G 2 × × G 1 Minimization problems Maximization problems V.S Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 13

  22. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  23. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  24. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  25. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  26. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  27. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  28. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  29. A Generic Framework Solve other problems using the same framework × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  30. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  31. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  32. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  33. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  34. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  35. A Generic Framework Solve other problems using the same framework × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  36. A Generic Framework Solve other problems using the same framework × × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  37. A Generic Framework Solve other problems using the same framework × × × × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  38. A Generic Framework Solve other problems using the same framework × × ×× × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  39. A Generic Framework Solve other problems using the same framework × × ×× × ×× ... × × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  40. A Generic Framework Solve other problems using the same framework × × ×× × × ×× × ... × G T × G 2 × G 1 Minimization problems V.S Maximization problems Find the smallest value Find the largest value Framework generalization Transitive closures concatenation → Composition operation Completeness test → Test operation Transitive closure → Super node Temporal Connectivity 14

  41. A Generic Framework Solve other problems using the same framework × × ×× × × ×× × ... × G T G 2 × × G 1 Minimization problems Maximization problems V.S Find the smallest value Find the largest value Temporal Connectivity 14

  42. A Generic Framework Solve other problems using the same framework × × ×× × × ×× × ... × G T G 2 × × G 1 Minimization problems Maximization problems V.S Find the smallest value Find the largest value Requirements test ( G ( i , j ) ) = true ⇔ { G i , G i + 1 , . . . , G j } satisfies the property P The composition operation is associative Only minimization: If test ( G ( i , j ) ) = true then test ( G ( i ′ , j ′ ) ) = true , ∀ i ′ ≤ i , j ′ ≥ j Only maximization: If test ( G ( i , j ) ) = true then test ( G ( i ′ , j ′ ) ) = true , ∀ i ′ ≥ i , j ′ ≤ j Temporal Connectivity 14

  43. Round-trip Temporal Connectivity Round-trip Temporal Connectivity A dynamic graph G is round-trip temporal connected if and only if a back-and-forth journey exists from any node to all other nodes. Temporal Connectivity 15

  44. Round-trip Temporal Connectivity Round-trip Temporal Connectivity A dynamic graph G is round-trip temporal connected if and only if a back-and-forth journey exists from any node to all other nodes. Round-Trip-Temporal-Diameter (minimization) Finding the smallest duration in which there exists a back-and-forth journey from any node to all other nodes. Temporal Connectivity 15

  45. Round-trip Temporal Connectivity Round-trip Temporal Connectivity A dynamic graph G is round-trip temporal connectivity if and only if a back-and-forth journey exists from any node to all other nodes. Round-Trip-Temporal-Diameter (minimization) Finding the smallest duration in which there exists a back-and-forth journey from any node to all other nodes. Round-trip transitive closure Super node: Round-trip transitive closure concatenation Composition operation: 2 , 4 7 , 6 7 , 6 2 , 4 2 , 4 7 , 6 5 7 6 , 7 2 , 7 7 , 4 7 , 4 2 , 7 7 = 7 = 3 , 3 , rtcat ∪ ⟲ 7 6 7 3 2 6 6 , , 2 , 4 7 , 7 7 , 7 6 , 5 , 6 , 5 7 , 7 , 7 7 , 7 , 6 , 4 4 6 , 6 , 5 6 , 6 6 , 6 6 , 5 6 , 6 G ( 1 , 5 ) ∪ ⟲ G ( 6 , 7 ) G ( 1 , 5 ) G ( 6 , 7 ) G ( 1 , 5 ) → ( 6 , 7 ) G ( 1 , 7 ) Test operation: Round-trip completeness Temporal Connectivity 16

  46. Bounded Realization of the footprint Time-bounded edge reappearance A dynamic graph G has a time-bounded edge reappearance with a bound b if the time between two appearances of the same edge is at most b . Temporal Connectivity 17

  47. Bounded Realization of the footprint Time-bounded edge reappearance A dynamic graph G has a time-bounded edge reappearance with a bound b if the time between two appearances of the same edge is at most b . Bounded-Realization-of-the-Footprint (minimization) Finding the smallest b such that in every subsequence of length b in the sequence G , all the edges of the footprint appear at least once. Temporal Connectivity 17

  48. Bounded Realization of the footprint Time-bounded edge reappearance A dynamic graph G has a time-bounded edge reappearance with a bound b if the time between two appearances of the same edge is at most b . Bounded-Realization-of-the-Footprint (minimization) Finding the smallest b such that in every subsequence of length b in the sequence G , all the edges of the footprint appear at least once. Union graphs Super node: Union Composition operation: Equality to the footprint Test operation: Temporal Connectivity 18

  49. T-interval Connectivity Definition: T -interval connectivity A dynamic graph G is T -interval connected if and only if every T length sequence of graphs has a common connected spanning sub-graph. Temporal Connectivity 19

  50. T-interval Connectivity Definition: T -interval connectivity A dynamic graph G is T -interval connected if and only if every T length sequence of graphs has a common connected spanning sub-graph. Temporal Connectivity 20

  51. T-interval Connectivity Definition: T -interval connectivity A dynamic graph G is T -interval connected if and only if every T length sequence of graphs has a common connected spanning sub-graph. T-Interval-Connectivity (maximization) Finding the largest T for which the graph is T -interval connected. G Temporal Connectivity 20

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