Dynamic PageRank using Evolving Teleportation

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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!


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

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


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


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


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


Evolving teleportation
96
(e.g. pageviews)
105
281

42
11
27

⋯
time

Importance continuously changes
as the external influence evolves!

Dynamic PageRank

⋯ ⋯

Evolving teleportation
96 113
(e.g. pageviews)
105 139
281 397

42 64
11 16
27 21

⋯
time time


Dynamic PageRank

⋯ ⋯

96 113 103
105 139 125
281 397 331

42 64 53
11 16 12
27 21 39

⋯ ⋯ ⋯
time


Dynamic PageRank

⋯ ⋯

Changes in PageRank
values evolve

Dynamical System

Dynamic Teleportation


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:


 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


Evolve the dynamical system,

Select any standard method!

forward Euler
Family of
Runge-Kutta …
methods
Many others!
Classical methods Adaptive methods
RK2,…,RK4,…

Evolve the dynamical system,

Forward Euler


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,...?


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

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


 Transient
— Instantaneous values of

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

 Difference Rank

Among many others...


 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


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)

Main Finding: Combing the external
influence with the graph, produces
something new, that is not captured
by the other methods


Learn model as

(Exponential moving avg)

Predicts p(t+1) as

Evaluate models (total errors) as


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
Twitter

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


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
Wikipedia
Twitter


Top 100 pages that fluctuate the most!

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



Pages related to a recent
Australian earthquake!



Just released movie
“Watchmen”



Famous co-
host/musician that
died



Recent “American
Idol” gossip



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

Recent “American
Idol” gossip



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


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


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


Non-stationary nodes (and clusters)
Potential Anomalies: Large-scale disasters, breaking news
0.25
Temporal Pattern1
Temporal Pattern2

Temporal Pattern3
Temporal Pattern4

Centroids!
0.15

Most nodes stationary!
0.1

0.05

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


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

Temporal Pattern3
Temporal Pattern4 81 JoBethWill 82 AHaunting 83 Octopussy 84

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


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


0.000406***
Significant!

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 (***)


0.000406***
Significant!
Main Finding. Allows us to identify the
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 (***)


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


Thanks!

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


Hourly
Pageviews
Earthquake
Preparedness

Earthquake 132 172

time
Richter
35 31
Mag.

Charles
Richter


Earthquake
Preparedness

Earthquake 132 172 764

Spike in the number of pageviews
for that given hour!

time
Richter
35 31 56
Mag.

Charles
Richter


ΔPR importance
substantially increases!
Earthquake
Preparedness


Spike in the number of pageviews
for that given hour!

time
Richter
35 31 56
Mag.

Charles
Richter


ΔPR importance
Earthquake
Preparedness


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


ΔPR importance
Earthquake
Preparedness


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


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!


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!


 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


Graph
dynamic networks

⋯ ⋯ ⋯

time


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


Dynamic PageRank using Evolving Teleportation

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