A Random Walk Around The Block Johan Ugander Stanford University - - PowerPoint PPT Presentation

A Random Walk Around The Block

Johan Ugander Stanford University Joint work with: Isabel Kloumann (Facebook) 
 & Jon Kleinberg (Cornell) Google Mountain View August 17, 2016

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T.

• ●
• ●
• ●
• ●
• target set T

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T.

• ●
• ●
• ●
• ●
• target set T

seed set S

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T.

• ●
• ●
• ●
• ●
• seed set S

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T. seed set S

• ●
• ●
• ●
• ●
• Scored by


Personalized PageRank

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T.

• Applications:
• Broadly: ranking on graphs, recommendation systems
• Spam filtering (Wu & Chellapilla ’07)
• Community detection (Weber et al. ’13)
• Missing data inference (Mislove et al. ’14)
• Common methods:
• Semi-supervised learning (Zhu et al. ’03)
• Diffusion-based classification 


(Jeh & Widom ’03, Kloster & Gleich ’14)

• Outwardness, modularity and more 


(Bagrow ’08, Kloumann & Kleinberg ’14)

• ●
• ●
• ●
• ●

S e e d s e t e x p a n s i

• n
• Given a graph G=(V, E), goal is to accurately identify


a target set T ⊂ V from a smaller seed set S ⊂ T.

• Applications:
• Broadly: ranking on graphs, recommendation systems
• Spam filtering (Wu & Chellapilla ’07)
• Community detection (Weber et al. ’13)
• Missing data inference (Mislove et al. ’14)
• Common methods:
• Semi-supervised learning (Zhu et al. ’03)
• Diffusion-based classification 


(Jeh & Widom ’03, Kloster & Gleich ’14)

• Outwardness, modularity and more 


(Bagrow ’08, Kloumann & Kleinberg ’14)

• ●
• ●
• ●
• ●

R e c a l l c u r v e s f

• r

s e e d s e t e x p a n s i

• n
• Recall curve: true positive rate, as a function of the number 

• f items returned based on small uniformly random seed set.
• Kloumann & Kleinberg ’14 tested many different methods 

• n data, broadly found Personalized PageRank to be best.

Kloumann & Kleinberg ‘14

R e c a l l c u r v e s f

• r

s e e d s e t e x p a n s i

• n
• Recall curve: true positive rate, as a function of the number 

• f items returned based on small uniformly random seed set.
• Kloumann & Kleinberg ’14 tested many different methods 

• n data, broadly found Personalized PageRank to be best.
• Truncated PPR (first K steps) comparable to PPR from K=4.
• Heat Kernel later found comparable to PPR.

Kloumann & Kleinberg ‘14

• Classification based on random walk landing probabilities
• , probability that a random walk starting in S is at v after k steps.
• , truncated vector of landing probabilities.
• Personalized PageRank and Heat Kernel ranking:
• General diffusion score function:

D i f f u s i

• n
• b

a s e d n

• d

e c l a s s i fi c a t i

• n

score(v) =

∞

X

k=1

wkrv

k

rv

k

PPR(v) ∝

∞

X

k=1

(αk)rv

k

HK(v) ∝

∞

X

k=1

✓tk k! ◆ rv

k

(rv

1, rv 2, ..., rv K)

• ●
• ●
• ●
• ●

• Personalized PageRank and Heat Kernel


= two parametric families of linear weights

• Question in this work: 


What weights are “optimal” for diffusion-based classification?

D i f f u s i

• n
• b

a s e d n

• d

e c l a s s i fi c a t i

• n

20 40 60 80 100 10

−5

10 t=1 t=5 t=15 α=0.85 α=0.99 Weight Length

(Kloster & Gleich, ’14)

PPR HK wk = αk wk = tk/k!

score(v) =

K

X

k=1

wkrv

k

T h e s t

• c

h a s t i c b l

• c

k m

• d

e l

• C blocks
• Focus on C=2 blocks: 1=“Target”, 2=“Other”
• n1, n2 nodes in blocks
• Independent edge probabilities:
• Edge probability within a block = pin
• Edge probability across blocks = pout
• (Results for C>2 as well, see paper)
• Model with many names:
• Stochastic Block Model (Holland et al. ’83)
• Affiliation Model (Frank-Harary ’82)
• Planted Partition Model (Dyer-Frieze ’89)

pin pin pout pout

T h e S B M r e s

• l

u t i

• n

l i m i t

• Find true partition in poly(n) time w.h.p. as n→∞ :
• Dyer-Frieze ’89:

If pin - pout = O(1)

• Condon-Karp ’01: If pin - pout ≥ Ω(n-1/2)
• McSherry ’01:

If pin - pout ≥ Ω((pout(log n)/n)-1/2)

• pin

pin pout pout

T h e S B M r e s

• l

u t i

• n

l i m i t

• Find true partition in poly(n) time w.h.p. as n→∞ :
• Dyer-Frieze ’89:

If pin - pout = O(1)

• Condon-Karp ’01: If pin - pout ≥ Ω(n-1/2)
• McSherry ’01:

If pin - pout ≥ Ω((pout(log n)/n)-1/2)

• Find partition positively correlated with true partition:
• Coja-Oghlan ’06: If pin - pout ≥ Ω((pout/n)-1/2),
• pin

pin pout pout

T h e S B M r e s

• l

u t i

• n

l i m i t

• Find true partition in poly(n) time w.h.p. as n→∞ :
• Dyer-Frieze ’89:

If pin - pout = O(1)

• Condon-Karp ’01: If pin - pout ≥ Ω(n-1/2)
• McSherry ’01:

If pin - pout ≥ Ω((pout(log n)/n)-1/2)

• Find partition positively correlated with true partition:
• Coja-Oghlan ’06: If pin - pout ≥ Ω((pout/n)-1/2),
• If and only if (a-b)2 > 2(a+b) (pin = a/n, pout = b/n):
• Decelle et al ’11: Conjecture and belief propagation numerics
• Mossel et al ’12,’13, Massoulié ’13, Abbe et al. ’14: Proven
• Recent extensions:
• More than two blocks (e.g. Neeman-Netrapalli ’14)
• Unequal block sizes (e.g. Zhang et al. ’16)
• pin

pin pout pout

T h e S B M r e s

• l

u t i

• n

l i m i t

• Is block recovery/classification over? No!
• Unsupervised vs. semi-supervised
• Empirical graphs != SBMs
• Optimal algorithms not practical
• Beyond asymptotic limits, what are decay rates?
• Rather than being “problem down” (SBM classification), this talk will

be “method up”: how to tune diffusion weights to find seed sets?

• Possible variations: Diffusion weights for seed set expansion in 


core-periphery models? Latent space models (Hoff et al. 2002)? Etc.

score(v) =

K

X

k=1

wkrv

k

pin pin pout pout

D i f f u s i

• n
• b

a s e d c l a s s i fi c a t i

• n

i n S B M s

• SBMs present a natural binary classification problem.
• Recall notation:
• , probability that a random walk starting in S is at v after k steps.
• , truncated vector of landing probabilities.
• Choices of define sweep directions through space.
• Optimistically:

rv

k

(rv

1, rv 2, ..., rv K)

ri rj

Target block nodes Other block nodes (w1, ..., wK)

T h e s p a c e

• f

l a n d i n g p r

• b

a b i l i t i e s

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05
• SBM: 2000 nodes, Target & Other blocks, pin = 0.2, pout = 0.05
• One seed node (uniformly at random from Target set)

T h e s p a c e

• f

l a n d i n g p r

• b

a b i l i t i e s

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05
• SBM: 2000 nodes, Target & Other blocks, pin = 0.2, pout = 0.05
• One seed node (uniformly at random from Target set)

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05

T h e s p a c e

• f

l a n d i n g p r

• b

a b i l i t i e s

• Geometric discriminant function: sweeps through the space

• f landing probabilities following vector from b to a.

a b

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05

T h e s p a c e

• f

l a n d i n g p r

• b

a b i l i t i e s

• Fisher discriminant functions: Clearly exist better linear 


and quadratic functions. Forward pointer, will return.

T h e s p a c e

• f

l a n d i n g p r

• b

a b i l i t i e s

• 0.000

0.002 0.004 0.006 0e+00 4e−04 8e−04 1−step Landing prob 2−step Landing prob

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• ●
• ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 3−step Landing prob 4−step Landing prob

• Focus on deriving optimal Geometric discriminant function first.


G e

• m

e t r i c d i s c r i m i n a n t f u n c t i

• n

s

• Let be the landing probabilities of a node
• Let be the Target class centroid
• Let be the Other class centroid
• Then is the geometric discriminant function.
• Notice: increases when r moves in direction of a - b.
• Can classify nodes based on thresholds of .

r = (r1, . . . , rK) a = (a1, . . . , aK) b = (b1, . . . , bK) f(r) = (a − b)T r f(r) f(r)

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05

a b

P e r s

• n

a l i z e d P a g e R a n k i s “

• p

t i m a l ”

• Main Theorem (informal version). 


For 2-block SBM with equal sized blocks and edge densities , : 
 



 and the optimal geometric classifier is therefore: . 


which is PPR(!) with .
 ak − bk = ✓pin − pout pin + pout ◆k pin pout α∗ = ✓pin − pout pin + pout ◆

K

X

k=1

(α∗)krk ,

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05

a b

P e r s

• n

a l i z e d P a g e R a n k i s “

• p

t i m a l ”

• Main Theorem (informal version). 


For 2-block SBM with equal sized blocks and edge densities , : 
 



 and the optimal geometric classifier is therefore: . 


which is PPR(!) with .
 ak − bk = ✓pin − pout pin + pout ◆k pin pout α∗ = ✓pin − pout pin + pout ◆

K

X

k=1

(α∗)krk ,

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05

a b

• Two main parts:
• 1. Centroids a, b concentrate on quantities


determined by the solution to a linear 
 recurrence relation.

• 2. That linear recurrence relation can be 


solved and yields PPR.

P P R i s “

• p

t i m a l ” : P r

• f

i d e a

• Part 1: Concentration of landing probabilities

Lemma 1. For any ✏, > 0, there is an n sufficiently large such that the random landing probabilities (ˆ a1, ...., ˆ aK) and (ˆ b1, ...,ˆ bK) for a uniform random walk on Gn starting in the seed block satisfy the following conditions with probability at least 1 − for all k > 0: Nˆ ak ∈  (1 − ✏) Ak Ak + Bk , (1 + ✏) Ak Ak + Bk

• and

(1) Nˆ bk ∈  (1 − ✏) Bk Ak + Bk , (1 + ✏) Bk Ak + Bk

• ,

(2) where Ak, Bk are the solutions to the matrix recurrence relation ( Ak = N(pinAk−1 + poutBk−1) Bk = N(poutAk−1 + pinBk−1), with A0 = 1, B0 = 0.

P P R i s “

• p

t i m a l ” : P r

• f

i d e a

• Part 1: Concentration of landing probabilities
• interpretable as length-k walk count to nodes in block 1 vs. 2.
• For large n, block walk counts increase by factors of ~E[degree].

Ak, Bk

Lemma 1. For any ✏, > 0, there is an n sufficiently large such that the random landing probabilities (ˆ a1, ...., ˆ aK) and (ˆ b1, ...,ˆ bK) for a uniform random walk on Gn starting in the seed block satisfy the following conditions with probability at least 1 − for all k > 0: Nˆ ak ∈  (1 − ✏) Ak Ak + Bk , (1 + ✏) Ak Ak + Bk

• and

(1) Nˆ bk ∈  (1 − ✏) Bk Ak + Bk , (1 + ✏) Bk Ak + Bk

• ,

(2) where Ak, Bk are the solutions to the matrix recurrence relation ( Ak = N(pinAk−1 + poutBk−1) Bk = N(poutAk−1 + pinBk−1), with A0 = 1, B0 = 0.

M

• r

e g e n e r a l S B M s

• For SBMs with C>2 blocks and/or with arbitrary P:
• Seed set expansion asks: identify nodes in a target block set.
• With conditions on equal expected degrees, PPR(!).
• Without conditions, still:
• Asymptotically optimal weights for geometric classification


still obtainable from solutions to a matrix recurrence relation.

block 1 block 2 block 3 block 4 block 1 block 2 block 3 block 4

E m p i r i c a l v s . t h e

• r

e t i c a l c e n t r

• i

d s

• 2048-node, 4-block SBM, empirical class centroids vs. theory:

1e−3 2e−3 3e−3

rk

empirical centroids predicted centroids

1 2 3 4 5 6 7 8 9 k 1e−6 1e−5 | ˆ wk − Ψk|

Error

• a, Target blocks
• b, Other blocks

block 1 block 2 block 3 block 4 block 1 block 2 block 3 block 4

1e−3 2e−3 3e−3

rk

empirical centroids predicted centroids

1 2 3 4 5 6 7 8 9 k 1e−6 1e−5 | ˆ wk − Ψk|

Error

• a, Target blocks
• b, Other blocks

block 1 block 2 block 3 block 4 block 1 block 2 block 3 block 4

E m p i r i c a l v s . t h e

• r

e t i c a l c e n t r

• i

d s

• 2048-node, 4-block SBM, empirical class centroids vs. theory:

From matrix 
 recurrence relation

T h e

• r

i e s

• f

g r a p h d i f f u s i

• n
• Other motivations for PPR:
• Random Surfer Model (Brin-Page ’98)
• Cheeger inequalities for PPR, HK (Andersen et al ’06, Chung ’09)
• Local spectral algorithm with regularization (Mahoney et al. ’12)
• Our work shows PPR can be derived as “optimal” geometric classifier.
• Also motivates how to choose PPR , as = .

α

= ✓pin − pout pin + pout ◆

α

• ●
• ●
• ●
• ●

T h e

• r

i e s

• f

g r a p h d i f f u s i

• n
• Other motivations for PPR:
• Random Surfer Model (Brin-Page ’98)
• Cheeger inequalities for PPR, HK (Andersen et al ’06, Chung ’09)
• Local spectral algorithm with regularization (Mahoney et al. ’12)
• Our work shows PPR can be derived as “optimal” geometric classifier.
• Also motivates how to choose PPR , as = .
• Most importantly: also opens door to methods beyond PPR.
• ●
• ●
• ●
• ●
• α

= ✓pin − pout pin + pout ◆

α

P P R i s “

• p

t i m a l ” i n a n a r r

• w

s e n s e

0.006

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• p_in=0.2, p_out=0.05
• Discriminant functions that model higher moments of point clouds?

a b

F i s h e r d i s c r i m i n a n t f u n c t i

• n

s

0.005

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• Discriminant functions that model higher moments of point clouds.

(a, Σa) (b, Σb)

• Let z be the latent class of each node.
• Capture (mean, variance) of class point clouds:
• Log-likelihood ratio as discriminant function:

F i s h e r d i s c r i m i n a n t f u n c t i

• n

s

0.005

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• Pr(r|z = 1) ∝ |Σa|− 1

2 exp

✓ −1 2(r − a)T Σ−1

a (r − a)

◆ Pr(r|z = 0) ∝ |Σb|− 1

2 exp

✓ −1 2(r − b)T Σ−1

b (r − b)

◆ g(r) = log Pr(r|z = 1) Pr(z = 1) Pr(r|z = 0) Pr(z = 0)

• Three approaches:
• We call the first two methods


QuadSBMRank, LinSBMRank.

• Perhaps reasonable to assume


equal covariances; effective.

• PPR follows from an assumption of 


uniform variance, no covariance.

F i s h e r d i s c r i m i n a n t f u n c t i

• n

s

General : g2(r) ∝

• Σ−1

a a − Σ−1 b b

T r + 1

2rT

Σ−1

b

− Σ−1

a

• r

Assume Σa = Σb = Σ : g1(r) ∝ Σ−1(a − b)T r Assume Σa = Σb = I : g0(r) ∝ (a − b)T r

0.005

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• g2(r)

g1(r) g0(r)

• Three approaches:
• We call the first two methods


QuadSBMRank, LinSBMRank.

• Perhaps reasonable to assume


equal covariances; effective.

• PPR follows from an assumption of 


uniform variance, no covariance.

• Open challenge: Possible to 


show asymptotic normality and
 characterize covariance matrices?

F i s h e r d i s c r i m i n a n t f u n c t i

• n

s

General : g2(r) ∝

• Σ−1

a a − Σ−1 b b

T r + 1

2rT

Σ−1

b

− Σ−1

a

• r

Assume Σa = Σb = Σ : g1(r) ∝ Σ−1(a − b)T r Assume Σa = Σb = I : g0(r) ∝ (a − b)T r

0.005

• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ●
• ● ●
• ●
• 0e+00

4e−04 8e−04 0e+00 4e−04 8e−04 2−step Landing prob 3−step Landing prob

• g2(r)

g1(r) g0(r)

E v a l u a t i

• n

: r e c a l l c u r v e s

• SBM with 2 blocks, 64 nodes/block, 1 seed node.
• Recall that Belief Propagation reaches resolution limit.
• Easy instance (pin >> pout):
• Everything does well.

E v a l u a t i

• n

: r e c a l l c u r v e s

• SBM with 2 blocks, 64 nodes/block, 1 seed node.
• Recall that Belief Propagation reaches resolution limit.
• Hard instance…
• PPR/HK lost all recall, LinSBMRank and QuadSBMRank near BP.

E v a l u a t i

• n

: r e c a l l c u r v e s

• SBM with 2 blocks, 64 nodes/block, 1 seed node.
• Recall that Belief Propagation reaches resolution limit.
• Even harder instance…
• LinSBMRank and QuadSBMRank outperforming BP by a hair…?

E v a l u a t i

• n

: r e c a l l c u r v e s

• SBM with 2 blocks, 64 nodes/block, 1 seed node.
• Recall that Belief Propagation reaches resolution limit.
• Impossible (pin = pout):
• Nothing works.

E v a l u a t i

• n

: r e s

• l

u t i

• n

l i m i t

• Pearson correlation r between true partition and inferred partition.
• Empirically, we see LinSBMRank and QuadSBMRank get very close 


to resolution limit (dotted line), with slower decay rate. PPR, HK, LinSBMRank, QuadSBMRank, BP

C

• n

c l u s i

• n

s

• Personalized PageRank with = is optimal geometric 


discriminant function for balanced 2-block SBM.

• Geometric discriminant functions for more general block models 


follow from recurrence relation.

• Landing probabilities are correlated; correcting for higher moments in

the space of landing probabilities greatly improves classification.

• In practice: fit GMMs in space of landing probs.
• A new perspective on diffusion-based ranking


that can hopefully open new doors.

• ●
• ●
• ●
• ●
• =

✓pin − pout pin + pout ◆

α

• Pre-print:


Isabel Kloumann, Johan Ugander, Jon Kleinberg
 “Block Models and Personalized PageRank”
 arXiv:1607.03483

O p e n d i r e c t i

• n

s

• ●
• ●
• ●
• ●
• Model covariance of landing probabilities?
• Currently requires at least ~logarithmic degrees (we think);


possible to derive weights for bounded degree SBMs?

• Better classifiers in the space of landing probabilities for 

• ther random walks? (Non-backtracking, etc.)
• Not just SBM? Optimal weights for dcSBM, core-periphery, 


Hoff latent space model, etc, etc.

• Slow decay beyond resolution limit?
• Pre-print:


Isabel Kloumann, Johan Ugander, Jon Kleinberg
 “Block Models and Personalized PageRank”
 arXiv:1607.03483