PageRank Centrality of dynamic graph structures

(PageRank) Centrality
of dynamic graph
structures
David F. Gleich!
Computer Science"
Purdue University
1
David Gleich · Purdue
AN14 · MS59

Models and algorithms for high performance !
matrix and network computations
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2
1
error
1
std
0
2
(b) Std, s = 0.39 cm
10
error
0
0
10
std
0
20
(d) Std, s = 1.95 cm
model compared to the prediction standard de-
bble locations at the ﬁnal time for two values of
= 1.95 cm. (Colors are visible in the electronic
approximately twenty minutes to construct using
s.
ta involved a few pre- and post-processing steps:
m Aria, globally transpose the data, compute the
nd errors. The preprocessing steps took approx-
recise timing information, but we do not report
Tensor eigenvalues"
and a power method

FIGURE 6 – Previous work
from the PI tackled net-
work alignment with ma-
trix methods for edge
overlap:
i
j j0
i0
OverlapOverlap
A L B
This proposal is for match-
ing triangles using tensor
methods:
j
i
k
j0
i0
k0
TriangleTriangle
A L B
t
r
o
s.
g
n.
o
n
s
s-
g
maximize
P
ijk Tijk xi xj xk
subject to kxk2 = 1
where ! ensures the 2-norm
[x(next)
]i = ⇢ · (
X
jk
Tijk xj xk + xi )
SSHOPM method due to "
Kolda and Mayo
Big data methods
SIMAX ‘09, SISC ‘11,MapReduce ‘11, ICASSP ’12
Network alignment
ICDM ‘09, SC ‘11, TKDE ‘13
Fast & Scalable"
Network centrality
SC ‘05, WAW ‘07, SISC ‘10, WWW ’10, …
Data clustering
WSDM ‘12, KDD ‘12, CIKM ’13 …
Ax = b
min kAx bk
Ax = x
Massive matrix "
computations
on multi-threaded
and distributed
architectures

I hope to add power-grid networks soon!
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Centrality measures
“relative importance in a
network” –Wikipedia
“it’s a guess about what
might be important” -Me
They tell us something
about a network
considering it’s topology.
They need to be deployed
with extreme care!
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From Wikipedia

Centrality measures of
dynamic graphs
Something about my network is changing, what
should I do?

1.  Recompute at each change
2.  Batch up changes, and periodically recompute
3.  Efﬁciently update (i.e. recompute smartly!)
4.  Approximately update/compute
5.  Do something else.
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What else to do???
“If the optimization is hard, you should be
solving a different optimization problem” "
–Cris Moore
1.  Des Higham et al. "
Adopt the fundamentals to discrete time
2.  Use dynamical system generalizations,
Gleich and Rossi 2012/2014; and "
Des Higham et al. 2014
3.  Likely more too…
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Smart centrality for the "
smart grid?
You need to adapt your centrality measure for
your application! (Or try to get lucky!)
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Application to the power grid
Prior work
•  Kim, Obah, 2007; Jin et al., 2010; Adolf et al., 2011; Halappanavar et
al., 2012
has found that graph properties have important
correlations with power-grid vulnerabilities and
contingency analysis
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1.  Perspectives on PageRank
2.  PageRank as a dynamical system and
time-dependent teleportation
3.  Predicting using PageRank
4.  Applications to the power-grid?
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The random surfer model!
At a node …
1.  follow edges with prob α
2.  do something else with prob (1-α)
Google’s PageRank is one
possible answer
PageRank by Google
1
2
3
4
5
6
The Model
1. follow edges uniformly with
probability , and
2. randomly jump with probability
1 , we’ll assume everywhere is
equally likely
The places we ﬁnd the
surfer most often are im-
portant pages.
The important pages are the
places we are most likely to ﬁnd
the random surfer
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My preferred version "
of PageRank
A PageRank vector x is the solution of the linear system:
(I – αP) x = (1 – α) v
where P is a column stochastic matrix, 0 ≤ α < 1, and v is a
probability vector.
tails
!
2
6
6
4
1/6 1/2 0 0 0 0
1/6 0 0 1/3 0 0
1/6 1/2 0 1/3 0 0
1/6 0 1/2 0 0 0
1/6 0 1/2 1/3 0 1
1/6 0 0 0 1 0
3
7
7
5
| {z }
P
P j 0
eT P=eT
Just three ingredients!
vi 0, eT
v = 1
↵ usually 0.5 to 0.99
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This deﬁnition applies to a
remarkable variety of problems
1.  GeneRank
2.  ProteinRank
3.  FoodRank
4.  SportsRank
5.  HostRank
6.  TrustRank
7.  BadRank
8.  ObjectRank
9.  ItemRank
10.  ArticleRank
11.  BookRank
12.  FutureRank
13.  TimedPageRank
14.  SocialPageRank
15.  DiffusionRank
16.  ImpressionRank
17.  TweetRank
18.  TwitterRank
19.  ReversePageRank
20.  PageTrust
21.  PopRank
22.  CiteRank
23.  FactRank
24.  InvestorRank
25.  ImageRank
26.  VisualRank
27.  QueryRank
28.  BookmarkRank
29.  StoryRank
30.  PerturbationRank
31.  ChemicalRank
32.  RoadRank
33.  PaperRank
34.  Etc…
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The teleportation distribution v
models where surfers “restart”

What if this changes with time?
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Let’s look at how PageRank
evolves with iterations
x(k)
= x(k+1)
x(k)
= ↵Px(k)
+ (1 ↵)v x(k)
= (1 ↵)v (I ↵P)x(k)
x0
(t) = (1 ↵)v (I ↵P)x(t)
PageRank is the steady-state solution of the ODE
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A dynamical system for "
time-dependent teleportation
+ Easy to integrate
+ Easy to understand
+ Possible to treat analytically!
– Need to “model time” (not dimensionless)
– Still useful to have a data assimilation model
x0
(t) = (1 ↵)v(t) (I ↵P)x(t)
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Need a symplectic integrator
(or self-correcting…)
We use a standard RK integrator "
(ode45 in Matlab)
We used the formulation

to maintain x(t) as a probability distribution

x0
(t) = (1 ↵)v(t) ( I ↵P)x(t)
= (1 ↵)eT
v(t) + ↵eT
x(t)
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Where is this model realistic?
On Wikipedia, we have
hourly visit data that provides
a coarse measure of outside
interest
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Now PageRank values are
time-series, not static scores
1 MainPage 2 FrancisMag 3
11 501(c) 12 Searching 1
Earthquake
Australian
Earthquake
occurs!
Main page
Time
Time
Importance
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Some quick theory
x(t) = exp[ (I ↵P)t]x(0)
+ (1 ↵)
Z t
0
exp[ (I ↵P)(t ⌧)]v(⌧) d⌧.
x0
(t) = (1 ↵)v(t) (I ↵P)x(t)
Z t
0
exp[ (I ↵P)(t ⌧)]v(⌧) d⌧
= (I ↵P) 1
v exp[ (I ↵P)t](I ↵P) 1
v
x(t) = exp[ (I ↵P)t](x(0) x) + x
For
general
v(t)
For
static
v(t) = v
The original "
PageRank vector
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Thus we recover "
the original PageRank vector "
if interest stops changing.
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Modeling cyclical behavior
Cyclically switch between teleportation vectors vj
v(t) =
1
k
kX
j=1
vj
⇣
cos(t + (j 1)2⇡
k ) + 1
⌘
0 20 40 60 80
0
0.05
0.1
0.15
0.2
time
Time−dependentteleportation
Page 1
Page 2
Page 3
Page 4
v1
v2
v1
v2
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0 5 10 15 20
0.1
0.2
0.3
0.4
0.5
time
DynamicPageRank
Page 1
Page 2
Page 3
Page 4
Cyclical behavior in the time-
dependent PageRank scores
1
2
3
4
0 20 40 60 80
0
0.05
0.1
0.15
0.2
time
Time−dependentteleportation
Page 1
Page 2
Page 3
Page 4
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Modeling cyclical behavior
Cyclically switch between teleportation vectors vj
v(t) =
1
k
kX
j=1
vj
⇣
cos(t + (j 1)2⇡
k ) + 1
⌘
x(t) = x + Re {s exp(ıt)}
Then the eventual solution is
(I ↵P)x = (1 ↵)
1
k
Ve
(I ↵
1+ı P)s
= (1 ↵) 1
k(1+ı) V exp(ıf)
PageRank vector with average teleportation
PageRank with
complex teleportation
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Summary
If you have cyclical interest on a node, we have
a NEW centrality measure that provides the
magnitude of the oscillation based on PageRank
with complex valued “teleportation.”
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Thus we can determine "
the size of the oscillation "
for the case of cyclical
teleportation
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Is it useful? Let’s try and
predict retweets on Twitter
We crawled Twitter and gathered "
a graph of who follows who and "
how active each user is in a month
This yields a graph and 6 vectors v!
!
Our goal is to predict how many tweets you’ll
send next month based on the current month!
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… and then there are details I can go into …
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The results
Dataset Type ✓ Error Ratio
s (timescale)
1 2 6 1
TWITTER stationary 0.01 0.635 0.929 0.913 0.996
0.50 0.636 0.735 0.854 0.939
1.00 0.522 0.562 0.710 0.963
non-stationary 0.01 0.461 0.841 1.001 0.992
0.50 0.261 0.608 0.585 0.929
1.00 0.137 0.605 0.617 0.918
Err Ratio = SMAPE of tweets + Time-dependent PR / SMAPE of tweets only
If this ratio < 1, then using Time-dependent PR helps
Stationary nodes are those with small maximum change in scores
Non-stationary nodes are those with large maximum change in scores
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Using Granger Causality to study link
relationships on Wikipedia
51 Greygoo 52 pageprotec 53 R
61 Science 62 Gackt 63 T
71 Madonna(en 72 Richtermag 73 T
81 Livingpeop 82 Mathematic 83 S
91 Categories 92 Germany 93 M
ogy 20 Geography
atic 30 Biography
en(f 40 Earthquake
io 50 Raceandeth
60 Football(s
Earthquake
Richter Mag.
Causes?
Of course! We build this into the model.

But, the question is, which of these are
preserved after incorporating the effects
of page view data?

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To the power grid …
Line failures in the grid
can be anticipated via
linearized DC
dynamics

Hines el al.?
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c = diag(B (L)+
BT
)

The PageRank problem & "
the Laplacian
Combinatorial "
Laplacian
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1. (I ↵AD 1
)x = (1 ↵)v;
2. (I ↵A)y = (1 ↵)D 1/2
v,
where A = D 1/2
AD 1/2
and x = D1/2
y; and
3. [ D + L]z = v where ↵ = 1/(1 + ) and x = Dz.
Let x(↵) solve PageRank and
let vT
e = 0.
Then lim↵!1 x(↵) ! SL+
v
where S is a scaling matrix.

Some potential applications
1.  PageRank can be thought of as a type of
regularization; often helps improve on simple
centrality baselines
2.  Limits of PageRank interpolate between centrality
and spectral clustering [Mahoney, Orecchia, and
Vishnoi]
3.  Time dependent teleportation models; adaptations
to node dropouts possible.
4.  Use PageRank on the line graph?
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Results on the power grid
… pending …
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Questions, Conclusions, and
References!
Questions!
How to validate some of these
ideas?
Too simplistic?
Other power-grid problems
where similar ideas may be
able to help?
Collaborators?????

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AN14 · MS59
Dear David, Please
remember to repeat
the question!
Paper Gleich & Rossi, Internet Mathematics, 2014
Code https://www.cs.purdue.edu/homes/dgleich/codes/dynsyspr-im
Conclusions!
Centrality is more
complicated than just
one method.
It’s possible to tune
centrality measures to
different structures and
this makes it a ﬂexible
setup."

PageRank Centrality of dynamic graph structures

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