Unfolding Accessibility of Temporal Networks
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This document introduces methods for representing and analyzing the accessibility and path structure of temporal networks. It begins by contrasting static and temporal network representations, with temporal networks accounting for the chronological sequence of edges. Accessibility matrices and graphs are described as ways to measure the number of paths between nodes. Examples are provided analyzing accessibility in static and temporal networks using empirical data on livestock trade and human contact networks. The analysis finds temporal representations better capture causal paths. A software tool for calculating temporal network accessibility is also introduced.
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Unfolding Accessibility of Temporal Networks
- 1. unfolding accessibility of temporal networks Hartmut "Hardy" Lentz Thomas Selhorst Igor Sokolov
- 2. static network representation representation: Adjacency matrix A 1 4 2 3
- 3. static network representation representation: Adjacency matrix A 1 4 2 3 the big picture
- 6. chronologic sequence of edges fundamental difference to static: causal paths Mon Tue Wed causalpath causal paths
- 7. accessibility matrices & accessibility graphs adjacency matrix A: number of paths of length 1 square of adjacency matrix A2: number of paths of length 2 Example (static):
- 8. accessibility matrices & accessibility graphs initial graph its accessibility-graph
- 9. accessibility matrices of static networks example: Random DiGraph 1000 nodes 2000 edges
- 10. accessibility matrices of static networks example: Random DiGraph 1000 nodes 2000 edges static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20
- 11. accessibility matrices of static networks example: Random DiGraph 1000 nodes 2000 edges static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20
- 14. recap: the idea Network unfold accessibility ... static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20
- 20. accessibility matrices of temporal networks
- 21. accessibility matrices of temporal networks easy implementation!
- 22. normalization. actual number of paths not relevant
- 23. normalization. actual number of paths not relevant for convenience: Boolean matrices
- 24. application.
- 27. characteristic time scales results I static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20 static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20 static random network
- 28. characteristic time scales results I static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20 static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20 static random network Probability 0 0.002 0.004 0.006 Pathdensity 0 0.05 0.10 Shortest path duration (days) 0 100 200 300 400 Pig trade network
- 29. characteristic time scales results I breeding slaughter fattening piglet production T=180 days Probability 0 0.002 0.004 0.006 Pathdensity 0 0.05 0.10 Shortest path duration (days) 0 100 200 300 400 Pig trade network
- 30. characteristic time scales results I breeding slaughter fattening piglet production T=180 days Probability 0 0.002 0.004 0.006 Pathdensity 0 0.05 0.10 Shortest path duration (days) 0 100 200 300 400 Pig trade network Pork production chain
- 32. causal fidelity results II comparison: static vs. temporal accessibility-graph
- 33. causal fidelity results II comparison: static vs. temporal accessibility-graph number of paths: static representation: 1.4 billion temporal representation: 1.0 billion
- 34. causal fidelity results II comparison: static vs. temporal accessibility-graph number of paths: static representation: 1.4 billion temporal representation: 1.0 billion 28 % of all paths in the aggregated network do actually not exist!
- 35. comparison: static vs. temporal accessibility-graph causal fidelity results II Definition: causal fidelity
- 36. comparison: static vs. temporal accessibility-graph causal fidelity results II static path density temporal (real) path densityDefinition: causal fidelity
- 37. comparison: static vs. temporal accessibility-graph causal fidelity results II static path density temporal (real) path densityDefinition: causal fidelity pig trade net
- 38. further datasets. sexual contact network face-to-face conference contacts Rocha et al., PNAS 2010 sociopatterns.org
- 39. sexual contact network results IIIProbability 0 2 4×10 −3 Pathdensity 0 0.1 0.2 0.3 Shortest path duration (years) 0 1 2 3 4 5 6 Sexual contacts
- 40. sexual contact network results IIIProbability 0 2 4×10 −3 Pathdensity 0 0.1 0.2 0.3 Shortest path duration (years) 0 1 2 3 4 5 6 Sexual contacts causal fidelity
- 41. conference contacts results IIIProbability 0 5 10×10 −3 Pathdensity 0 0.5 1.0 Shortest path duration (hours) 0 10 20 30 40 50 60 Conference contacts
- 42. conference contacts results IIIProbability 0 5 10×10 −3 Pathdensity 0 0.5 1.0 Shortest path duration (hours) 0 10 20 30 40 50 60 Conference contacts causal fidelity
- 44. network perspective: accessibility path-lengths new static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20 static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20 recap.
- 45. network perspective: accessibility path-lengths new static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20 static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20 new: temporal accessibility Pig trade Probability 0 0.002 0.004 0.006 Pathdensity 0 0.05 0.10 Shortest path duration (days) 0 100 200 300 400 Pig trade network recap.
- 46. network perspective: accessibility path-lengths new static Densityρ(P) 0 0.2 0.4 0.6 0.8 Power n 0 5 10 15 20 static probability 0 0.05 0.10 0.15 0.20 Shortest path length 0 5 10 15 20 new: temporal accessibility Pig trade Probability 0 0.002 0.004 0.006 Pathdensity 0 0.05 0.10 Shortest path duration (days) 0 100 200 300 400 Pig trade network recap. new: causal fidelity
- 47. unfolding accessibility software-tool easy-usage object based tool github repository: hartmutlentz/TemporalNetworkAccessibility Download Language Uses
- 48. example usage
- 51. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools
- 52. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools At = AdjMatrixSequence("Edges.txt",directed=True)
- 53. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools At = AdjMatrixSequence("Edges.txt",directed=True) c = At.unfold_accessibility()
- 54. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools At = AdjMatrixSequence("Edges.txt",directed=True) c = At.unfold_accessibility() h = Tools.cdf2histogram(c)
- 55. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools At = AdjMatrixSequence("Edges.txt",directed=True) c = At.unfold_accessibility() h = Tools.cdf2histogram(c) Tools.dict2file(c,"Path-density.txt")
- 56. example usage # File: Unfold_Accessibility.py from MatrixList import AdjMatrixSequence, Tools At = AdjMatrixSequence("Edges.txt",directed=True) c = At.unfold_accessibility() h = Tools.cdf2histogram(c) Tools.dict2file(c,"Path-density.txt") Tools.dict2file(h,"Histogram.txt")
- 57. more features
- 58. more features stepwise aggregation temporal clustering coefficient
- 59. more features talk: TDN tue, 15:00 stepwise aggregation temporal clustering coefficient
- 60. more features talk: TDN tue, 15:00 additional class: temporal edge-lists randomization techniques stepwise aggregation temporal clustering coefficient
- 61. more features talk: TDN tue, 15:00 talk: NetSci wed, 17:20 additional class: temporal edge-lists randomization techniques stepwise aggregation temporal clustering coefficient
- 62. paper. Unfolding Accessibility provides a Macroscopic Approach to Temporal Networks Hartmut H K Lentz, Thomas Selhorst and Igor M Sokolov Phys. Rev. Lett, 2013
- 63. “ ” No man ever steps into the same river twice. Except Chuck Norris! - Heraclitus