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Importance

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Presentation slides for:

T. Takaguchi, N. Sato, K. Yano, N. Masuda.
Importance of individual events in temporal networks.
Preprint: arXiv:1205.4808

Published in: Technology, Business
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Importance

  1. 1. Importance of individual eventsin temporal networksTaro Takaguchi1, Nobuo Sato2, Kazuo Yano2, and Naoki Masuda11 Department of Mathematical Informatics, The University of Tokyo2 Central Research Laboratory, Hitachi, Ltd.
  2. 2. Interests: patterns in human communication behavior By garryknight By opacity twitter.com/#!/duncanjwatts By infomatique photos from flickr 2
  3. 3. More extensive data, more detailed analysis • Huge populations (~millions) • High temporal resolution (~minute) • Additional information (e.g., locations, history of purchases) Cell-phone calling network Business Microscope system (Onnela et al., NJP 2007) (Hitachi, Ltd., Japan) Name tag with an infrared module http://www.hitachi-hitec.com/jyouhou/business-microscope/ 3
  4. 4. Temporal networksReviewed by Holme and Saramäki, Phys. Rep. 2012Represented by sequences of events with time stamps 1 static (aggregated) network 1 2 2 3 3 4 4 ✓ node 1 → node 4 (temporal path) time - node 4 → node 1 4
  5. 5. Impact of interevent intervals Different temporal paths from node 2 to node 3 may have different impacts on epidemics, information propagation, etc. 1 1 1 1 1 2 2 2 3 3 3 time 5
  6. 6. Question: which events are important? Evaluate the importance of each event • time-dependent centrality of links 1 2 1 2 1 2 3 4 3 4 3 4 time 6
  7. 7. Importance of events Defined by the amount of new information about others Note: “information” ≠ contents of conversation 1 2 3 4 time 7
  8. 8. Importance of events Defined by the amount of new information about others Note: “information” ≠ contents of conversation Before the event: 1 2 3 latest information 4 time 8
  9. 9. Importance of events Defined by the amount of new information about others Note: “information” ≠ contents of conversation Before the event: 1 2 3 latest information 4 time 9
  10. 10. Importance of events Defined by the amount of new information about others Note: “information” ≠ contents of conversation After the event: 1 2 3 latest information 4 time 10
  11. 11. Importance of events Defined by the amount of new information about others Note: “information” ≠ contents of conversation After the event: 1 2 3 latest information 4 time 11
  12. 12. Concept (1): vector clock and latencyLamport, Commun. ACM 1978; Mattern, 1988 Vector clock of node At time , has the latest information about at time Example: time 12
  13. 13. Concept (2): advance of eventKossinets et al., Proc. 14th ACM SIGKDD 2008 Advance for owing to an event between and⇒ time⇒ 13
  14. 14. Calculation of importance Assumption: • Individuals can be involved in multiple events in a single snapshot. • Information can spread up to hops within a snapshot. (called “horizon” in Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009)Read the given event sequence in the chronological order.1. Update every ‘s information about .2. Calculate and for all the events at .3. Importance = symmetrized advance 14
  15. 15. Calculation of importance Assumption: • Individuals can be involved in multiple events in a single snapshot. • Information can spread up to hops within a snapshot. (called “horizon” in Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009)Read the given event sequence in the chronological order.1. Update every ‘s information about .2. Calculate and for all the events at .3. Importance = symmetrized advance 15
  16. 16. Calculation of advance (1) Source node (defined for each ) h-neighbors having the latest information about & being at the shortest distance from Snapshot at : source node 16
  17. 17. Calculation of advance (2) Contributing neighbors ‘s neighbors that are on a shortest path from a nearest source node (about ) to and contribute . Snapshot at : source node : contributing neighbor 17
  18. 18. Case 1: multiple source nodes with different distances Assumption: Only the closest ones convey the information. is not a contributing neighbor. Snapshot at : source node : contributing neighbor 18
  19. 19. Case 2: multiple source nodes with the same distance Assumption: Contributing neighbors equally contribute regardless of the number of shortest paths they bridge. and contribute . Snapshot at : source node : contributing neighbor 19
  20. 20. Application to real data 20
  21. 21. Research questions 1. How is the importance distributed? Broadly? 2. Is the advance asymmetric? (i → j versus j → i) 3. Is the importance “valid”? Data set Situation Company office in Japan Participants 163 Period / resolution 73 days / 1 min Total events 118,546 Data was collected by World Signal Center, Hitachi, Ltd. 21
  22. 22. Parameter We set . Information can spread to all nodes in the connected component within a snapshot. 22
  23. 23. 1,2. Importance is broadly distributed & asymmetric frequency of events max = min on the diagonal 23
  24. 24. 3. Is the importance of event “valid”? Event removal test Hypothesis: Removal of events with large importance values 1. makes “temporal distance” longer. 2. makes node pairs disconnected. time 24
  25. 25. Two measures to characterize the connectivityReachability ratio (Holme, PRE 2005) with at least one temporal path from to disconnected fully connectedNetwork efficiency (Tang et al., Proc. 2nd ACM SIGCOMM WOSN 2009) : time average of latency disconnected fully connected or large latency with small latency 25
  26. 26. Time average of latencyPan & Saramäki, PRE 2011 Problem: is not defined for Solution: a periodic boundary condition sum of Time average 26
  27. 27. Five schemes of event removal • ascending/descending orders of the importance • ascending/descending orders of the link weight # events on the link • random order Fraction of connected pairs Shortness of temporal paths ? fraction of removed events 27
  28. 28. Ascending/descending orders of the link weight 1. Choose a link with the smallest/largest weight. # events on the link 2. Remove an event on the link at random. Decrease the weight of the link by one. Static (aggregated) network 28
  29. 29. Event removal tests based on the importance 1. Removal of 80% unimportant events influences little (Robustness). 2. Removal of 20% important events considerably decreases connectivity. 1.0 1.0 0.8 0.8 netwrok efficiencyreachability ratio 0.6 0.6 ascending I ij descending I ij 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 29
  30. 30. Comparison with the results based on the link weight Event removals based on temporal/static information are similar but different. 1.0 1.0 0.8 0.8 netwrok efficiencyreachability ratio 0.6 0.6 ascending I ij descending I ij 0.4 ascending weight 0.4 descending weight 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 30
  31. 31. Removal of weak links fragments static network “Strength of weak ties” property (Granovetter, AJS 1973; Onnela et al., PNAS 2007) Weak links connect different communities mainly composed of strong links. Takaguchi et al., PRX 2011 31
  32. 32. Do we need to consider the importance? A criticism Ascending-link-weight removal efficiently cuts off temporal paths. Information about the importance is not necessary. YES, we do need consider the importance, because: 1. Events on weak links are necessary but NOT sufficient for connecting efficient temporal paths. 2. Events with large importance are necessary and sufficient for connecting efficient temporal paths. 32
  33. 33. Correlates of the importance value Spearman’s rank correlation coefficient between the importance value and Length of the # total events # total events # partners of IEI involving i or j i or j 0.819 0.701 0.701 0.630 IEI: interevent interval time 33
  34. 34. Latest IEI approximates the importance 1.0 1.0 (a) (b) 0.8 0.8 network efficiencyreachability ratio 0.6 0.6 0.4 0.4 ascending I ij 0.2 descending I ij 0.2 ascending IEI descending IEI 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 34
  35. 35. Origin of the robustness Bursty activity patterns (Barabási, Nature 2005) time of a typical individual (Takaguchi et al., PRX 2011) 35
  36. 36. Exploration of the effect of burstinessCarry out the event removal tests for the temporal networks generated by (i) Shuffled IEIs (interevent intervals) For each pair, time (ii) Poissonized IEIs Reassign random time to each event. Events follow Poisson process. 36
  37. 37. Characteristics conserved / lost by the randomizations Poissonized Original Shuffled IEIs IEIs Weighted network structure ✓ ✓ ✓ Burstiness ✓ ✓ - Temporal correlations, etc. ✓ - - 37
  38. 38. 1. Temporal correlation is not necessary Results for Shuffled IEIs Results for the original data 1.0 1.0 (a) (a) 0.8 0.8 network efficiencyreachability ratio 0.6 0.6 ascending I ij descending I ij 0.4 ascending weight 0.4 descending weight random order 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 38
  39. 39. 2. Burstiness (long-tailed IEIs) is essential Results for Poissonized IEIs ≠ Results for the original data & Shuffled IEIs Removal of unimportant events rapidly spoils network efficiency. 1.0 1.0 (b) (b) 0.8 0.8 network efficiencyreachability ratio 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 39
  40. 40. Effect of the weighted network structure (iii) Rewiring 1. Make an Erdös-Rényi random graph with the same number of nodes and links as the original data. 2. Put the event sequences on the original links onto links in the random graph. time time original network rewired network 40
  41. 41. Characteristics conserved / lost by the randomizations Poissonized Original Shuffled IEIs Rewiring IEIs Weighted network ✓ ✓ ✓ - structure Burstiness ✓ ✓ - ✓ Temporal correlation, ✓ - - △ etc. link weight distribution ✓ ✓ ✓ ✓ 41
  42. 42. 3. Heterogeneity in link weights is sufficient Results for Rewiring Results for the original data Skewed degree dist., community, structure-weight corr., etc. are irrelevant. 1.0 1.0 (c) (c) 0.8 0.8 network efficiencyreachability ratio 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 42
  43. 43. Effect of network structure Can bustiness explain the heterogeneity in the importance even without the heterogeneity in the link weight? Regular random graph IEI distributions power-law + cutoff exponential (Poisson process) i.i.d. time 60 events on each link 43
  44. 44. Burstiness is a main cause of the robustness Power-law IEIs on the RRG Exponential IEIs on the RRG 1.0 1.0 (a) (b) 0.8network efficiency 0.8 network efficiency 0.6 0.6 0.4 0.4 0.2 ascending I ij 0.2 descending I ij random order 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 fraction of removed events fraction of removed events 44
  45. 45. Summary • Importance of events in temporal networks - Based on advance of vector clocks in an event • Heterogeneity in the importance - Long-tailed distribution and strong asymmetry • Robustness of empirical temporal networks - Connectivity conserved after removing 80% unimportant events • Origin of the robustness - Bursty activity patterns (i.e., long-tailed IEIs) - Heterogeneity in the link weight Reference Taro Takaguchi, Nobuo Sato, Kazuo Yano, and Naoki Masuda, “Importance of individual events in temporal networks”, New Journal of Physics 14, 093003 (2012). [Open Access] 45

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