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                                                Time




    Dynamic PageRank using Evolving
    Teleportation
    Ryan A. Rossi
    David F. Gleich




                      Tunisia   Egypt   Libya        Tunisia   Egypt   Libya   Tunisia   Egypt   Libya   Tunisia   Egypt   Libya
Problem: Importance of nodes is NOT static

                                               (static PageRank)

                      Evolving in reality!




Ryan Rossi (Purdue)                   Dynamic PageRank
Problem: Importance of nodes is NOT static

                                                          Formulate PageRank
                                                          as Dynamical System!
                      Evolving in reality!
                                                         Importance of 100 nodes
                                                            changing over time


 Dynamic Generalization
 of PageRank

 Helps in prediction!

Ryan Rossi (Purdue)                   Dynamic PageRank
Detecting dynamic anomalies
Dynamic Ranks
                                                                   Australian                      Spike!
                                                                   Earthquake
                                                                     occurs!


                                                                               Earthquake
                                                   Prediction
                                      observed                            future
                                      values                              values
                TV shows/
              “American Idol”                                                               Earthquake


                                                                   Modeling
                                                                                                  Causes?
     1 TimAllen        2 TheOffice(     3 DrivingMis          human dynamics
                                                          4 JoannaPaci 5 AmericanId              6 BloodDrive


    11 KrisAllen      12 KatharineM    13 AmericanId      14 DavidFoste     15 ListofTheO        16 TheOffice

           Clustering nodes with similar
   21 TheOffice(  22 TheLastHou 23 AmericanId            24 TheLastHou      25 JasonKay         26 CamillaBel
           time-series patterns                                                             Richter Mag.
   31 AsherRoth       32 DwightSchr     33 B.J.Novak      34 PromNight(     35 JennaFisch       36 RashidaJon
Ryan Rossi (Purdue)                          Dynamic PageRank
1           2                 Static PageRank Model. At a node, a
                                            random surfer can:
                      3       4             1. follow edges uniformly with probability
                                               α, and
                      5
                                             2. randomly jump with probability 1 − α
                                                 (for now, assume vi = 1/n)
                                                     The nodes that are visited most
                                                          often are important!



                                            Induces a Markov chain model
                          (random walk)

                                            Or the linear system
where




Ryan Rossi (Purdue)                       Dynamic PageRank
1           2                   Static PageRank Model. At a node, a
                                              random surfer can:
                      3       4               1. follow edges uniformly with probability
                                                 α, and
                      5
                                               2. randomly jump with probability 1 − α
                                                   (for now, assume vi = 1/n)

                                    Too simplistic! that important! most
                                            The nodes
                                                 often are
                                                           are visited

                                  Graph & attributes evolve!
                          Importance continuously changes!
                                        Induces a Markov chain model
                          (random walk)

                                              Or the linear system
where




Ryan Rossi (Purdue)                         Dynamic PageRank
Majority of work focuses on static networks!


   Combine PageRank with crawling process
   S. Abiteboul, M. Preda, & G. Cobena:
   Adaptive on-line page importance computation

   Walks on dynamic graphs
   P. Grindrod, D. Higham, M. Parsons, & E. Estrada:
   Communicability Across Evolving Networks

   Other work:
   J. O’Madadhain & P. Smyth,
   EventRank: A framework for ranking time-varying networks




Ryan Rossi (Purdue)                   Dynamic PageRank
All of these techniques are not
                         placed in the context of a
                              dynamical system



              We want to gain additional flexibility
                by adapting these problems as
                 continuous dynamical systems


Ryan Rossi (Purdue)              Dynamic PageRank
Evolving teleportation
                      96
                                                                    (e.g. pageviews)
             105
                           281

                                 42
              11
                      27




   ⋯
                                 time



                                                           Importance continuously changes
                                                           as the external influence evolves!

   Dynamic PageRank

                                                    ⋯                                           ⋯
Ryan Rossi (Purdue)                     Dynamic PageRank
Evolving teleportation
                      96                               113
                                                                                 (e.g. pageviews)
             105                             139
                           281                               397

                                 42                                64
              11                              16
                      27                               21




   ⋯
                                      time                         time


                                                                        Importance continuously changes
                                                                        as the external influence evolves!

   Dynamic PageRank

                                                                   ⋯                                         ⋯
Ryan Rossi (Purdue)                                Dynamic PageRank
96                        113                                   103
             105                      139                                       125
                           281                        397                                   331

                                 42                         64                                    53
              11                       16                                        12
                      27                        21                                    39




   ⋯                                                                  ⋯                                ⋯
                                                                                                  time



                                                                 Importance continuously changes
                                                                 as the external influence evolves!

   Dynamic PageRank

                                                            ⋯                                            ⋯
Ryan Rossi (Purdue)                         Dynamic PageRank
Changes in PageRank
                                             values evolve



                                          Dynamical System


                                         Dynamic Teleportation

Ryan Rossi (Purdue)   Dynamic PageRank
Dynamic Teleportation Model




    Generalization of static PageRank. If v(t) = v stops
    changing, then we recover the original PageRank vector x as the
    steady-state solution:



Ryan Rossi (Purdue)           Dynamic PageRank
 A principled dynamical system framework for studying these
         problems
     Flexibility to choose our algorithm to solve it
     Determines the effective length scale
     Seamlessly generalizes PageRank for dynamics
     We can easily and naturally incorporate the complete set of
         dynamic components




Ryan Rossi (Purdue)            Dynamic PageRank
Evolve the dynamical system,




                      Select any standard method!

      forward Euler
                                 Family of
                                Runge-Kutta                  …
                                 methods
                                                                 Many others!
                  Classical methods       Adaptive methods
                   RK2,…,RK4,…
Ryan Rossi (Purdue)                   Dynamic PageRank
Evolve the dynamical system,



                            Forward Euler




Ryan Rossi (Purdue)         Dynamic PageRank
How we map updates to v into the dynamical system time
   determines the effective length-scale that we are looking at
                                                   time-scale of dynamical system
         Relationship?
                                                   time-scale of application
          x(1)?  1 sec, 1 min,...?




Ryan Rossi (Purdue)                   Dynamic PageRank
How we map updates to v into the dynamical system time
   determines the effective length-scale that we are looking at
                                                          Equivalent to running the
                                                       time-scale of dynamical system
         Relationship?                                       power-method until
                                                       time-scale of application
                                                           convergence each hour!
          x(1)?  1 sec, 1 min,...?
                                           time in application
                                h=1
                                 t=1
                                                                    60 iterations
                      time-scale = 1 (1 min)
                      (application)                              between each hour

                                h=1
                                 t=1
                                                                  3 iterations after
                      time-scale = 1 (20 min)
                      (application)                              each hourly change
Ryan Rossi (Purdue)                       Dynamic PageRank
v(t) changes at fixed intervals
                        Better idea might be to smooth out these “jumps”!
                                         Feature of the new model!
                       0.2

                      0.18
                                            Utilize this informationh=1
                      0.16
                                               from the evolution 1 (12 min)
                                                                    t=
                                                                                           time-scale = 1 hour
Convergence Measure




                      0.14
                                                                                (application)
                      0.12

                       0.1

                      0.08

                      0.06

                      0.04                                                                        5 iterations after
                      0.02
                                                                                                 each hourly change
                        0
                             0   5   10   15   20      25       30   35    40     45   50
                                                    Iteration


Ryan Rossi (Purdue)                                                       Dynamic PageRank
 Transient
      — Instantaneous values of

    Summary & Cumulative
      — Any summary function s(⋅) of the time-series:
            integral, min, max, variance



    Difference Rank



   Among many others...

Ryan Rossi (Purdue)                  Dynamic PageRank
 Wikipedia
        — Hyperlink graph
        — Hourly pageviews


     Twitter
        — Who-follows-whom
        — Tweet rates (monthly)




          Dataset     Nodes     Edges             tmax Period    Average pi Max pi
          Wikipedia   4,143,840 72,718,664          20 hours     1.3225     334,650
          Twitter     465,022   835,424               6 months   0.5569     1056


Ryan Rossi (Purdue)                     Dynamic PageRank
Nope, pageviews and degree uncorrelated!
                  8
                          correlation=0.02
                  7

                           High degree,
In Degree (Log)




                  6
                           Low pageviews
                  5


                  4


                  3


                  2

                                                                                         High pageviews,
                  1
                                                                                         Low degree
                  0
                      0      1       2       3       4    5        6     7       8   9
                                                 Total Pageviews
                                                      (Log)
Ryan Rossi (Purdue)                                           Dynamic PageRank
Main Finding: Combing the external
                        influence with the graph, produces
                        something new, that is not captured
                        by the other methods




Ryan Rossi (Purdue)   Dynamic PageRank
Learn model as



                                              (Exponential moving avg)


Predicts p(t+1) as




Evaluate models (total errors) as



Ryan Rossi (Purdue)        Dynamic PageRank
Base Model. Only pageviews (or tweet-rates)
   Dynamic PageRank. Pageviews and Dynamic PageRank time-series


           Dataset     Forecasting       Dynamic PageRank   Base Model
                       Non-stationary             0.4349      0.5028
           Wikipedia
                       Stationary                 0.3672      0.4373
                       Non-stationary             0.4852      1.2333
           Twitter
                       Stationary                 0.6690      0.9180


                       Main Finding. Dynamic PageRank time-series
                       provides valuable information for forecasting
                       future pageviews (or tweet-rates)

Ryan Rossi (Purdue)                  Dynamic PageRank
Many applications such as
   Base Model. Only pageviews (or tweet-rates) systems
         • Actively adapting caches in large DB
   Dynamic PageRank. Pageviews and Dynamic PageRank time-series
         • Dynamically recommending pages


           Dataset     Forecasting       Dynamic PageRank   Base Model
                       Non-stationary             0.4349      0.5028
           Wikipedia
                       Stationary                 0.3672      0.4373
                       Non-stationary             0.4852      1.2333
           Twitter
                       Stationary                 0.6690      0.9180




Ryan Rossi (Purdue)                  Dynamic PageRank
Top 100 pages that fluctuate the most!



                      Dynamic PageRank identifies interesting pages
                      that pertain to recent external interest.




Ryan Rossi (Purdue)                   Dynamic PageRank
Top 100 pages that fluctuate the most!


                      Pages related to a recent
                      Australian earthquake!




Ryan Rossi (Purdue)                 Dynamic PageRank
Top 100 pages that fluctuate the most!




                      Just released movie
                         “Watchmen”




Ryan Rossi (Purdue)                  Dynamic PageRank
Top 100 pages that fluctuate the most!




                         Famous co-
                      host/musician that
                            died




Ryan Rossi (Purdue)                 Dynamic PageRank
Top 100 pages that fluctuate the most!




        Recent “American
          Idol” gossip

Ryan Rossi (Purdue)         Dynamic PageRank
Top 100 pages that fluctuate the most!




                                                A remembrance of Eve
                                               Carson from a contestant
                                                  on “American Idol”



        Recent “American
          Idol” gossip

Ryan Rossi (Purdue)         Dynamic PageRank
Top 100 pages that fluctuate the most!

                      Main Finding. These examples reveal
                      the ability of our Dynamic PageRank
                      to mesh the network structure with
                      changes in external interest!




Ryan Rossi (Purdue)          Dynamic PageRank
 Clustering PageRank trends
     Granger Causality
     Better algorithms (RK4,…)
     Put more theoretical teeth behind these results




Ryan Rossi (Purdue)          Dynamic PageRank
0.25
                                                                                        Well-separated and unique!
                                              Temporal Pattern1
                                              Temporal Pattern2
Normalized Dynamic PageRank




                                              Temporal Pattern3
                                              Temporal Pattern4
                               0.2            Temporal Pattern5


                                         Centroids!
                              0.15



                                                                                                                Most nodes stationary!
                               0.1




                              0.05




                                0
                                     0    2         4       6     8    10    12   14   16   18    20
                                                                      Time

              Ryan Rossi (Purdue)                                                      Dynamic PageRank
Non-stationary nodes (and clusters)
                                                                 Potential Anomalies: Large-scale disasters, breaking news
                              0.25
                                              Temporal Pattern1
                                              Temporal Pattern2
Normalized Dynamic PageRank




                                              Temporal Pattern3
                                              Temporal Pattern4
                               0.2            Temporal Pattern5


                                         Centroids!
                              0.15



                                                                                                          Most nodes stationary!
                               0.1




                              0.05




                                0
                                     0    2         4       6     8    10    12   14   16   18    20
                                                                      Time

              Ryan Rossi (Purdue)                                                      Dynamic PageRank
1 TimAllen       2 TheOffice(     3 DrivingMis    4 Jo


                                                                                                            11 KrisAllen    12 KatharineM    13 AmericanId    14 D


                                         Allows us identify nodes that become                              21 TheOffice(    22 TheLastHou    23 AmericanId    24 T


                                         important around similar times (nodes                             31 AsherRoth     32 DwightSchr    33 B.J.Novak     34 P

                                         w/ similar trends of importance may be                            41 TheOffice(    42 SeanHannit    43 Drake(ente    44 P

                                         related)                                                          51 SaraPaxton    52 BobbyBrown       53 Sting        54


                                                                                                           61 CelticWoma    62 PaulWalker    63 TheHauntin      64
                              0.25
                                               Temporal Pattern1                                           71 TracyMorga    72 YouSpinMeR    73 AnnCoulter     74
                                               Temporal Pattern2
Normalized Dynamic PageRank




                                               Temporal Pattern3
                                               Temporal Pattern4                                           81 JoBethWill    82 AHaunting     83 Octopussy      84
                               0.2             Temporal Pattern5

                                                                                                           91 MarcoPierr     92 Rebirth(Li   93 LietoMe(TV    94 T
                                          Centroids!
                              0.15
                                                                                                               1 Chile       2 WorldWarII        3 Iraq       4 An

                                                                                                               11 Jew          12 Brazil     13 Frenchlang    14 S
                               0.1
                                                                                                            21 Caribbean      22 Judaism     23 RomanCatho       2

                                                                                                              31 Rome       32 NaziGerman       33 2007          3
                              0.05

                                                                                                               41 2005       42 Christiani    43 Christian       4


                                0                                                                              51 2004          52 Gold         53 2008        54
                                     0     2         4       6     8    10    12   14   16   18    20
                                                                       Time                                    61 God        62 Wiktionary    63 Mammal        64

              Ryan Rossi (Purdue)                                                       Dynamic PageRank    71 LatinAmeri    72 Disappeare    73 Yearofbirt   74 Y
Question: Does an earthquake at
                                    time t cause people to visit Richter
                                    magnitude page at t+1?

                                   Causes?
                      Earthquake                        Richter Mag.


             Statement on Granger Causality (Stronger version)
             1. cause must occur before the effect
             2. cause contains information about the effect
             3. cause and effect must be linked in the graph

Ryan Rossi (Purdue)                  Dynamic PageRank
Multivariate regression                 lag




                                                                   vector of errors
   vector of response variables
                                   regression coefficients to estimate


    Granger Causality exists if the error by using the time-series x
    in the forecast model is smaller than without considering x:




       Significance of the difference in error is measured using the F-test

Ryan Rossi (Purdue)                 Dynamic PageRank
0.000406***
                                      Significant!
                      Earthquake                                Richter Mag.




             Caused by Earthquake in Australia            p-value
             Earthquake preparedness                      0.000607***
             Aftershock                                   0.009619**
             Asperity                                     0.001601**
             Stick-slip phenomenon                        0.002312**
             Landslide dam                                0.004820**
                                            pval < 0.5 (*), 0.01 (**), 0.001 (***)

Ryan Rossi (Purdue)                    Dynamic PageRank
0.000406***
                                      Significant!
                              Main Finding. Allows us to identify the
                      Earthquake                        Richter Mag.
                              pages that influence the others with
                              regards to how users find information

             Caused by Earthquake in Australia            p-value
             Earthquake preparedness                      0.000607***
             Aftershock                                   0.009619**
             Asperity                                     0.001601**
             Stick-slip phenomenon                        0.002312**
             Landslide dam                                0.004820**
                                            pval < 0.5 (*), 0.01 (**), 0.001 (***)

Ryan Rossi (Purdue)                    Dynamic PageRank
 Introduced dynamical system framework for PageRank
     Stated a dynamic Generalization of PageRank
     Dynamic PageRank can help in prediction
     Useful for many other applications




Ryan Rossi (Purdue)          Dynamic PageRank
Thanks!


                                Questions?
                               rrossi@purdue.edu
                      http://www.cs.purdue.edu/homes/rrossi


Ryan Rossi (Purdue)                  Dynamic PageRank
Ryan Rossi (Purdue)   Dynamic PageRank
Hourly
                                                 Pageviews
   Earthquake
   Preparedness

                      Earthquake      132       172




                                                      time
                       Richter
                                       35       31
                        Mag.



                                                      Charles
                                                      Richter

Ryan Rossi (Purdue)          Dynamic PageRank
Earthquake
   Preparedness

                      Earthquake      132       172     764


                                                  Spike in the number of pageviews
                                                          for that given hour!


                                                              time
                       Richter
                                       35       31      56
                        Mag.



                                                      Charles
                                                      Richter

Ryan Rossi (Purdue)          Dynamic PageRank
ΔPR importance
                       substantially increases!
   Earthquake
   Preparedness

                      Earthquake      132       172     764


                                                  Spike in the number of pageviews
                                                          for that given hour!


                                                              time
                       Richter
                                       35       31      56
                        Mag.



                                                      Charles
                                                      Richter

Ryan Rossi (Purdue)          Dynamic PageRank
ΔPR importance
                            substantially increases!
   Earthquake
   Preparedness

                           Earthquake      132       172     764

After a few iterations,
importance diffuses                                    Spike in the number of pageviews
from Earthquake to                                             for that given hour!
Richter Mag!
Direct result of meshing                                           time
graph with pageviews!       Richter
                                            35       31      56
                             Mag.



                                                           Charles
                                                           Richter

Ryan Rossi (Purdue)               Dynamic PageRank
ΔPR importance
                            substantially increases!
   Earthquake
   Preparedness

                           Earthquake      132       172   764

After a few iterations,
importance diffuses                                    Spike in the number of pageviews
from Earthquake to                                             for that given hour!
Richter Mag!
Direct result of meshing                                         time
graph with pageviews!       Richter
                                            35       31    56           becomes important
                             Mag.
                                                                            at this time


                            Hence, Richter magnitude receives a high dynamic
                            PageRank score, becoming increasingly important at this
                                                      Charles
                            time, while its pageviews are not significantly increasing.
                                                      Richter

Ryan Rossi (Purdue)               Dynamic PageRank
Earthquake
   Preparedness

                      Earthquake      132       172   764    3406




                                                                  time
                       Richter
                                       35       31    56     1447
                        Mag.



                                                 In the next hour, we find that
                                                    Charles
                                                 the pageviews of Richter spike!
                                                     Richter
                                                 Reinforcing the importance!

Ryan Rossi (Purdue)          Dynamic PageRank
Earthquake
   Preparedness

                                Earthquake      132       172   764    3406

                          Dynamic PageRank is
                        predictive (by definition)!
                       Importance of Richter magnitude captured by
                      dynamic PageRank an hour earlier than when it time
                      actually became important (spike in pageviews)
                                   Richter
                                                 35       31    56     1447
                                  Mag.



                                                           In the next hour, we find that
                                                              Charles
                                                           the pageviews of Richter spike!
                                                               Richter
                                                           Reinforcing the importance!

Ryan Rossi (Purdue)                    Dynamic PageRank
 Real-world networks are naturally dynamic
        — Information Networks (e.g., Wikipedia: article-links-article)
        — Social Networks (e.g., Twitter: who-follows-whom)
        — Biological Networks
        …                            ⇒
                                                        Importance changes!




  Static methods fail to capture the temporal flow of information
         Lead to misleading or simply incorrect conclusions




Ryan Rossi (Purdue)                  Dynamic PageRank
Graph
                      dynamic networks




 ⋯                                              ⋯       ⋯

                                                        time



Ryan Rossi (Purdue)          Dynamic PageRank
Graph
                                      dynamic networks                 Attributes    ✓
                             External Influence (e.g., pageviews)

                      96                              113                            103
                                             139                               125
                           281                              397                            331

                                 42                               64                             53
              11                              16                                12
                      27                               21                            39




 ⋯                                                                     ⋯                              ⋯

                                                                                                      time



Ryan Rossi (Purdue)                                Dynamic PageRank

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Dynamic PageRank using Evolving Teleportation

  • 1. ⋯ ⋯ Time Dynamic PageRank using Evolving Teleportation Ryan A. Rossi David F. Gleich Tunisia Egypt Libya Tunisia Egypt Libya Tunisia Egypt Libya Tunisia Egypt Libya
  • 2. Problem: Importance of nodes is NOT static (static PageRank) Evolving in reality! Ryan Rossi (Purdue) Dynamic PageRank
  • 3. Problem: Importance of nodes is NOT static Formulate PageRank as Dynamical System! Evolving in reality! Importance of 100 nodes changing over time Dynamic Generalization of PageRank Helps in prediction! Ryan Rossi (Purdue) Dynamic PageRank
  • 4. Detecting dynamic anomalies Dynamic Ranks Australian Spike! Earthquake occurs! Earthquake Prediction observed future values values TV shows/ “American Idol” Earthquake Modeling Causes? 1 TimAllen 2 TheOffice( 3 DrivingMis human dynamics 4 JoannaPaci 5 AmericanId 6 BloodDrive 11 KrisAllen 12 KatharineM 13 AmericanId 14 DavidFoste 15 ListofTheO 16 TheOffice Clustering nodes with similar 21 TheOffice( 22 TheLastHou 23 AmericanId 24 TheLastHou 25 JasonKay 26 CamillaBel time-series patterns Richter Mag. 31 AsherRoth 32 DwightSchr 33 B.J.Novak 34 PromNight( 35 JennaFisch 36 RashidaJon Ryan Rossi (Purdue) Dynamic PageRank
  • 5. 1 2 Static PageRank Model. At a node, a random surfer can: 3 4 1. follow edges uniformly with probability α, and 5 2. randomly jump with probability 1 − α (for now, assume vi = 1/n) The nodes that are visited most often are important! Induces a Markov chain model (random walk) Or the linear system where Ryan Rossi (Purdue) Dynamic PageRank
  • 6. 1 2 Static PageRank Model. At a node, a random surfer can: 3 4 1. follow edges uniformly with probability α, and 5 2. randomly jump with probability 1 − α (for now, assume vi = 1/n) Too simplistic! that important! most The nodes often are are visited Graph & attributes evolve! Importance continuously changes! Induces a Markov chain model (random walk) Or the linear system where Ryan Rossi (Purdue) Dynamic PageRank
  • 7. Majority of work focuses on static networks! Combine PageRank with crawling process S. Abiteboul, M. Preda, & G. Cobena: Adaptive on-line page importance computation Walks on dynamic graphs P. Grindrod, D. Higham, M. Parsons, & E. Estrada: Communicability Across Evolving Networks Other work: J. O’Madadhain & P. Smyth, EventRank: A framework for ranking time-varying networks Ryan Rossi (Purdue) Dynamic PageRank
  • 8. All of these techniques are not placed in the context of a dynamical system We want to gain additional flexibility by adapting these problems as continuous dynamical systems Ryan Rossi (Purdue) Dynamic PageRank
  • 9. Evolving teleportation 96 (e.g. pageviews) 105 281 42 11 27 ⋯ time Importance continuously changes as the external influence evolves! Dynamic PageRank ⋯ ⋯ Ryan Rossi (Purdue) Dynamic PageRank
  • 10. Evolving teleportation 96 113 (e.g. pageviews) 105 139 281 397 42 64 11 16 27 21 ⋯ time time Importance continuously changes as the external influence evolves! Dynamic PageRank ⋯ ⋯ Ryan Rossi (Purdue) Dynamic PageRank
  • 11. 96 113 103 105 139 125 281 397 331 42 64 53 11 16 12 27 21 39 ⋯ ⋯ ⋯ time Importance continuously changes as the external influence evolves! Dynamic PageRank ⋯ ⋯ Ryan Rossi (Purdue) Dynamic PageRank
  • 12. Changes in PageRank values evolve Dynamical System Dynamic Teleportation Ryan Rossi (Purdue) Dynamic PageRank
  • 13. Dynamic Teleportation Model Generalization of static PageRank. If v(t) = v stops changing, then we recover the original PageRank vector x as the steady-state solution: Ryan Rossi (Purdue) Dynamic PageRank
  • 14.  A principled dynamical system framework for studying these problems  Flexibility to choose our algorithm to solve it  Determines the effective length scale  Seamlessly generalizes PageRank for dynamics  We can easily and naturally incorporate the complete set of dynamic components Ryan Rossi (Purdue) Dynamic PageRank
  • 15. Evolve the dynamical system, Select any standard method! forward Euler Family of Runge-Kutta … methods Many others! Classical methods Adaptive methods RK2,…,RK4,… Ryan Rossi (Purdue) Dynamic PageRank
  • 16. Evolve the dynamical system, Forward Euler Ryan Rossi (Purdue) Dynamic PageRank
  • 17. How we map updates to v into the dynamical system time determines the effective length-scale that we are looking at time-scale of dynamical system Relationship? time-scale of application x(1)?  1 sec, 1 min,...? Ryan Rossi (Purdue) Dynamic PageRank
  • 18. How we map updates to v into the dynamical system time determines the effective length-scale that we are looking at Equivalent to running the time-scale of dynamical system Relationship? power-method until time-scale of application convergence each hour! x(1)?  1 sec, 1 min,...? time in application h=1 t=1 60 iterations time-scale = 1 (1 min) (application) between each hour h=1 t=1 3 iterations after time-scale = 1 (20 min) (application) each hourly change Ryan Rossi (Purdue) Dynamic PageRank
  • 19. v(t) changes at fixed intervals Better idea might be to smooth out these “jumps”! Feature of the new model! 0.2 0.18 Utilize this informationh=1 0.16 from the evolution 1 (12 min) t= time-scale = 1 hour Convergence Measure 0.14 (application) 0.12 0.1 0.08 0.06 0.04 5 iterations after 0.02 each hourly change 0 0 5 10 15 20 25 30 35 40 45 50 Iteration Ryan Rossi (Purdue) Dynamic PageRank
  • 20.  Transient — Instantaneous values of  Summary & Cumulative — Any summary function s(⋅) of the time-series: integral, min, max, variance  Difference Rank Among many others... Ryan Rossi (Purdue) Dynamic PageRank
  • 21.  Wikipedia — Hyperlink graph — Hourly pageviews  Twitter — Who-follows-whom — Tweet rates (monthly) Dataset Nodes Edges tmax Period Average pi Max pi Wikipedia 4,143,840 72,718,664 20 hours 1.3225 334,650 Twitter 465,022 835,424 6 months 0.5569 1056 Ryan Rossi (Purdue) Dynamic PageRank
  • 22. Nope, pageviews and degree uncorrelated! 8 correlation=0.02 7 High degree, In Degree (Log) 6 Low pageviews 5 4 3 2 High pageviews, 1 Low degree 0 0 1 2 3 4 5 6 7 8 9 Total Pageviews (Log) Ryan Rossi (Purdue) Dynamic PageRank
  • 23. Main Finding: Combing the external influence with the graph, produces something new, that is not captured by the other methods Ryan Rossi (Purdue) Dynamic PageRank
  • 24. Learn model as (Exponential moving avg) Predicts p(t+1) as Evaluate models (total errors) as Ryan Rossi (Purdue) Dynamic PageRank
  • 25. Base Model. Only pageviews (or tweet-rates) Dynamic PageRank. Pageviews and Dynamic PageRank time-series Dataset Forecasting Dynamic PageRank Base Model Non-stationary 0.4349 0.5028 Wikipedia Stationary 0.3672 0.4373 Non-stationary 0.4852 1.2333 Twitter Stationary 0.6690 0.9180 Main Finding. Dynamic PageRank time-series provides valuable information for forecasting future pageviews (or tweet-rates) Ryan Rossi (Purdue) Dynamic PageRank
  • 26. Many applications such as Base Model. Only pageviews (or tweet-rates) systems • Actively adapting caches in large DB Dynamic PageRank. Pageviews and Dynamic PageRank time-series • Dynamically recommending pages Dataset Forecasting Dynamic PageRank Base Model Non-stationary 0.4349 0.5028 Wikipedia Stationary 0.3672 0.4373 Non-stationary 0.4852 1.2333 Twitter Stationary 0.6690 0.9180 Ryan Rossi (Purdue) Dynamic PageRank
  • 27. Top 100 pages that fluctuate the most! Dynamic PageRank identifies interesting pages that pertain to recent external interest. Ryan Rossi (Purdue) Dynamic PageRank
  • 28. Top 100 pages that fluctuate the most! Pages related to a recent Australian earthquake! Ryan Rossi (Purdue) Dynamic PageRank
  • 29. Top 100 pages that fluctuate the most! Just released movie “Watchmen” Ryan Rossi (Purdue) Dynamic PageRank
  • 30. Top 100 pages that fluctuate the most! Famous co- host/musician that died Ryan Rossi (Purdue) Dynamic PageRank
  • 31. Top 100 pages that fluctuate the most! Recent “American Idol” gossip Ryan Rossi (Purdue) Dynamic PageRank
  • 32. Top 100 pages that fluctuate the most! A remembrance of Eve Carson from a contestant on “American Idol” Recent “American Idol” gossip Ryan Rossi (Purdue) Dynamic PageRank
  • 33. Top 100 pages that fluctuate the most! Main Finding. These examples reveal the ability of our Dynamic PageRank to mesh the network structure with changes in external interest! Ryan Rossi (Purdue) Dynamic PageRank
  • 34.  Clustering PageRank trends  Granger Causality  Better algorithms (RK4,…)  Put more theoretical teeth behind these results Ryan Rossi (Purdue) Dynamic PageRank
  • 35. 0.25 Well-separated and unique! Temporal Pattern1 Temporal Pattern2 Normalized Dynamic PageRank Temporal Pattern3 Temporal Pattern4 0.2 Temporal Pattern5 Centroids! 0.15 Most nodes stationary! 0.1 0.05 0 0 2 4 6 8 10 12 14 16 18 20 Time Ryan Rossi (Purdue) Dynamic PageRank
  • 36. Non-stationary nodes (and clusters) Potential Anomalies: Large-scale disasters, breaking news 0.25 Temporal Pattern1 Temporal Pattern2 Normalized Dynamic PageRank Temporal Pattern3 Temporal Pattern4 0.2 Temporal Pattern5 Centroids! 0.15 Most nodes stationary! 0.1 0.05 0 0 2 4 6 8 10 12 14 16 18 20 Time Ryan Rossi (Purdue) Dynamic PageRank
  • 37. 1 TimAllen 2 TheOffice( 3 DrivingMis 4 Jo 11 KrisAllen 12 KatharineM 13 AmericanId 14 D Allows us identify nodes that become 21 TheOffice( 22 TheLastHou 23 AmericanId 24 T important around similar times (nodes 31 AsherRoth 32 DwightSchr 33 B.J.Novak 34 P w/ similar trends of importance may be 41 TheOffice( 42 SeanHannit 43 Drake(ente 44 P related) 51 SaraPaxton 52 BobbyBrown 53 Sting 54 61 CelticWoma 62 PaulWalker 63 TheHauntin 64 0.25 Temporal Pattern1 71 TracyMorga 72 YouSpinMeR 73 AnnCoulter 74 Temporal Pattern2 Normalized Dynamic PageRank Temporal Pattern3 Temporal Pattern4 81 JoBethWill 82 AHaunting 83 Octopussy 84 0.2 Temporal Pattern5 91 MarcoPierr 92 Rebirth(Li 93 LietoMe(TV 94 T Centroids! 0.15 1 Chile 2 WorldWarII 3 Iraq 4 An 11 Jew 12 Brazil 13 Frenchlang 14 S 0.1 21 Caribbean 22 Judaism 23 RomanCatho 2 31 Rome 32 NaziGerman 33 2007 3 0.05 41 2005 42 Christiani 43 Christian 4 0 51 2004 52 Gold 53 2008 54 0 2 4 6 8 10 12 14 16 18 20 Time 61 God 62 Wiktionary 63 Mammal 64 Ryan Rossi (Purdue) Dynamic PageRank 71 LatinAmeri 72 Disappeare 73 Yearofbirt 74 Y
  • 38. Question: Does an earthquake at time t cause people to visit Richter magnitude page at t+1? Causes? Earthquake Richter Mag. Statement on Granger Causality (Stronger version) 1. cause must occur before the effect 2. cause contains information about the effect 3. cause and effect must be linked in the graph Ryan Rossi (Purdue) Dynamic PageRank
  • 39. Multivariate regression lag vector of errors vector of response variables regression coefficients to estimate Granger Causality exists if the error by using the time-series x in the forecast model is smaller than without considering x: Significance of the difference in error is measured using the F-test Ryan Rossi (Purdue) Dynamic PageRank
  • 40. 0.000406*** Significant! Earthquake Richter Mag. Caused by Earthquake in Australia p-value Earthquake preparedness 0.000607*** Aftershock 0.009619** Asperity 0.001601** Stick-slip phenomenon 0.002312** Landslide dam 0.004820** pval < 0.5 (*), 0.01 (**), 0.001 (***) Ryan Rossi (Purdue) Dynamic PageRank
  • 41. 0.000406*** Significant! Main Finding. Allows us to identify the Earthquake Richter Mag. pages that influence the others with regards to how users find information Caused by Earthquake in Australia p-value Earthquake preparedness 0.000607*** Aftershock 0.009619** Asperity 0.001601** Stick-slip phenomenon 0.002312** Landslide dam 0.004820** pval < 0.5 (*), 0.01 (**), 0.001 (***) Ryan Rossi (Purdue) Dynamic PageRank
  • 42.  Introduced dynamical system framework for PageRank  Stated a dynamic Generalization of PageRank  Dynamic PageRank can help in prediction  Useful for many other applications Ryan Rossi (Purdue) Dynamic PageRank
  • 43. Thanks! Questions? rrossi@purdue.edu http://www.cs.purdue.edu/homes/rrossi Ryan Rossi (Purdue) Dynamic PageRank
  • 44. Ryan Rossi (Purdue) Dynamic PageRank
  • 45. Hourly Pageviews Earthquake Preparedness Earthquake 132 172 time Richter 35 31 Mag. Charles Richter Ryan Rossi (Purdue) Dynamic PageRank
  • 46. Earthquake Preparedness Earthquake 132 172 764 Spike in the number of pageviews for that given hour! time Richter 35 31 56 Mag. Charles Richter Ryan Rossi (Purdue) Dynamic PageRank
  • 47. ΔPR importance substantially increases! Earthquake Preparedness Earthquake 132 172 764 Spike in the number of pageviews for that given hour! time Richter 35 31 56 Mag. Charles Richter Ryan Rossi (Purdue) Dynamic PageRank
  • 48. ΔPR importance substantially increases! Earthquake Preparedness Earthquake 132 172 764 After a few iterations, importance diffuses Spike in the number of pageviews from Earthquake to for that given hour! Richter Mag! Direct result of meshing time graph with pageviews! Richter 35 31 56 Mag. Charles Richter Ryan Rossi (Purdue) Dynamic PageRank
  • 49. ΔPR importance substantially increases! Earthquake Preparedness Earthquake 132 172 764 After a few iterations, importance diffuses Spike in the number of pageviews from Earthquake to for that given hour! Richter Mag! Direct result of meshing time graph with pageviews! Richter 35 31 56 becomes important Mag. at this time Hence, Richter magnitude receives a high dynamic PageRank score, becoming increasingly important at this Charles time, while its pageviews are not significantly increasing. Richter Ryan Rossi (Purdue) Dynamic PageRank
  • 50. Earthquake Preparedness Earthquake 132 172 764 3406 time Richter 35 31 56 1447 Mag. In the next hour, we find that Charles the pageviews of Richter spike! Richter Reinforcing the importance! Ryan Rossi (Purdue) Dynamic PageRank
  • 51. Earthquake Preparedness Earthquake 132 172 764 3406 Dynamic PageRank is predictive (by definition)! Importance of Richter magnitude captured by dynamic PageRank an hour earlier than when it time actually became important (spike in pageviews) Richter 35 31 56 1447 Mag. In the next hour, we find that Charles the pageviews of Richter spike! Richter Reinforcing the importance! Ryan Rossi (Purdue) Dynamic PageRank
  • 52.  Real-world networks are naturally dynamic — Information Networks (e.g., Wikipedia: article-links-article) — Social Networks (e.g., Twitter: who-follows-whom) — Biological Networks … ⇒ Importance changes! Static methods fail to capture the temporal flow of information Lead to misleading or simply incorrect conclusions Ryan Rossi (Purdue) Dynamic PageRank
  • 53. Graph dynamic networks ⋯ ⋯ ⋯ time Ryan Rossi (Purdue) Dynamic PageRank
  • 54. Graph dynamic networks Attributes ✓ External Influence (e.g., pageviews) 96 113 103 139 125 281 397 331 42 64 53 11 16 12 27 21 39 ⋯ ⋯ ⋯ time Ryan Rossi (Purdue) Dynamic PageRank