Transcript Document

Discrete Probability on Graphs:
Estimation, Reconstruction of &
Optimization on
Networks
Elchanan Mossel
UC Berkeley
At: IPAM Mar 2007
Outline: Stochastic Models on Networks
• Disclaimer: Big field ; Biased choice of examples … - an
applied view.
• Part 0: Two types of Network problems.
• Part I: Estimation of statistical quantities in Gibbs-measures
/ Markov Random Fields
• part II: Reconstruction of Stochastic Networks from
observations.
–
-
Tree Networks.
Directed Acyclic graphs.
• part III: Optimization over stochastic models defined on
networks
- Which functions of stochastic models can be
(approximately) optimized efficiently?
January 15, 2007
2/31
Part 0:
Two Types of Network
Problems
Two types of Network problems
• Type 1: Structural Network problems.
• Type 2: Distributional Network problems.
• This talk: Mostly Distributional network problems.
• Examples of Structural Network problems:
• “Clustering”: Partition a graph G = (V,E) to V = V1,…,Vk
such that each Vi is “big” and there is a small number of
edges between Vi and Vj for i  j.
• “Ranking”: Given a random walk on a finite set, find the
stationary distribution.
• Spectral Techniques are applicable for both problems.
January 15, 2007
4/31
A hard Structural Network Problem
• The “Graph Isomorphism Problem”
Given two graphs (G,E) and (H,F) is there an “isomorphism”,
f : G ! H one to one s.t. (v1,v2) 2 E iff (f(v1),f(v2)) 2 F.
• Clear: if two graphs isomorphic, then they have same
spectral structure, but this is not enough …
• Other open problems exits in this area …
• Example of recent work:
January 15, 2007
5/31
part I:
Estimation in Markov
Random Fields
Gibbs Measures / Graphical Models


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A Gibbs Measure on a (finite) graph G=(V,E) is given by
Node potentials (v : v 2 V) and
Edge Potentials (e : e 2 E)
The probability of  = ((v) : v 2 V) 2 A|V| is given by
P[] = Z-1
£
v 2 V v[(v)] £
e=(v,u) 2 Ee[(v),(u)]
G



Gibbs measures introduced in Statistical Physics.
Essential in Machine Learning.
Also known as Markov Random Fields, Graphical Models etc.
January 15, 2007
Diffusion of Influence in Social Networks
7
Message Passing Algorithms / The Replica Method

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
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Statistical Problem: Given a Gibbs measure: estimate
P[(0) = a]
Equivalent to many other Inference Problems.
Computational View: Problem can be NP hard (to approximate)
even in very simple cases.
Statistical Physics view: Find Dynamics / Markov Chains that
have P as stationary measure.
Statistical Physics Insight:



G
Rapid Convergence of Dynamics  spatial correlation decay.
A very active area of research ; Fascinating Challenges.
Artificial Intelligence / Neuroscience / Replica view:
Solve problem by “Message Passing”
January 15, 2007
Diffusion of Influence in Social Networks
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Message Passing Algorithms / The Replica Method






Message Passing Algorithms are used to estimate
probabilities on graphical models.
Examples: Warning Propagation, Sum-Product, Belief
Propagation etc.
All of these algorithms do exact calculation for an associated
computation tree.
Example: Belief Propagation (BP) is a popular method in
AI/Coding for estimating marginal probabilities P[(0) = a] for
a Gibbs measure G.
It is equivalent [TatikondaJordan02] to calculating marginal
probabilities P[(0) = a] on the computation tree T(G).
Question: How come message passing algorithms work in
practice?
January 15, 2007
Diffusion of Influence in Social Networks
G
T
9
Message Passing Algorithms in Coding

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In coding:
BP is used to decode Low Density Parity Check Codes (LDPC)
[Gallager62]
Proved to be efficient [Luby-Mitzenmacher-ShokrollahiSpielman-98, Richardson-Urbanke-01]
Message passing algorithms work “because:”
LDPC factor graphs are locally “tree-like” &
Individual constraints “push” toward the correct code
word.
Actual analysis uses recursion of random variables on the
tree.
January 15, 2007
Diffusion of Influence in Social Networks
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Message Passing Algorithms -Random 3-SAT
x1
x2
x3
x4
x5
x6
x7
n
x8
m = n
WalkSAT
Survey propagation
Satisfiable
Belief propagation
Satisfiable
Myopic
Not
satisfiable
Not
satisfiable
PLR
0 January 15, 2007
1.63

3.52
3.95
Diffusion of Influence in Social Networks
4.27
4.51
11
Message Passing Algorithms for Random 3-SAT


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Message passing algorithms work because:
Random-SAT graphs are locally “tree-like”
Far away variables are uncorrelated:
Speculation 1: For Belief Propagation: Variables are un-correlated in
a “standard sense” when  · 3.95
Thm: (Maneva-M-Wainwright-05): Survey Propagation is just Belief
Propagation on an extended Markov Random Field.
Speculation 2: For Survey Propagation: Variables are un-correlated
in the extended Markov Random Field for all .
Speculations 1 & 2 are under heated discussions between Physicists,
Computer Scientists and Mathematicians …
15, 2007
M. January
Talagrand
in Social Networks
G. ParisiDiffusion of Influence
B. Selman
12
Decay of correlation for 3-SAT extended MRF
{0, 1}n assignments
Partial assignments
0110
1011
01101
# stars
01101

January 15, 2007
Diffusion of Influence in Social Networks
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part II:
Reconstructing Stochastic
Network from observations
Main Problem:
• How to reconstruct the network topology from
observations at a (sub)-set of the nodes?
• The Example: Reconstructing Trees.
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15
Two Tree Inference Problems

In Evolution:


Phase Transition:


Given a tree of species /
mothers, can we infer
ancestral sequence at the root
from contemporary samples?
Trade-off between noise and
duplication?
Reconstructing Evolution:

Is it possible to reconstruct
evolutionary history from
genetic sequences?
January 15, 2007
16
Defn: Markov Model on a Tree
…001100011101000011000100…

s(r)
Ising/BSC/CFN Model:


Tree: T = (V,E)
Node states:
pra
s(v) 0,1: v V 


s(a)
Number of leaves: n
0: Purines (A,G)
1: Pyrimidines (C,T)
January 15, 2007
prc
0
pab
Mutation probabilities:
0  pe  1 2 : e  E
0
s(b)
pb1
0
s(1)
pa3
0
pb2
0 1
s(2)
s(3)
1
s(c)
pc4
pc5
0
1
s(4) s(5)
17
Defn: Phylogenetic Reconstruction Problem
Phylogenetic Reconstruction:



Given: k i.i.d. samples at the n leaves
Task: fully reconstruct the model, i.e.
find tree and mutation probabilities
(and, if possible, do so efficiently)
Studied in:
Biology (dozens of books, 1000s of
papers) [Felsenstein’04]

TCS (Learning): [Ambainis-DesperFarach-Kannan’97], [FarachKannan’96], [Cryan-GoldbergGoldberg’02]
[M-Roch ]

Combinatorial Phylogeny: [Erdos‘98], [M’07]
January Steel-Szekely-Warnow’97,
15, 2007
s(1)
s(2)
0
0
1
0
1
0
0
1
0
0
s(3)
1
0
0
1
0
s(4) s(5)
1
1
1
1
0
1
1
1
1
1

+
prc
pc5

18
Phase Transition for the Ising model
LOW
Temp
“typical”
boundary
bias
2 2 > 1
HIGH
Temp
“typical”
boundary
no bias
22 < 1
The transition at 2 2 = 1 was proved by:
[Bleher-Ruiz-Zagrebnov’95], [Ioffe’96],[Evans-Kenyon-Peres-Schulman’00],
[Kenyon-Mossel-Peres’01],[Martinelli-Sinclair-Weitz’04], [Borgs-Chayes-M-Roch’06].
Also, “spin-glass” case studied by [Chayes-Chayes-Sethna-Thouless’86]. Solvability for
2 2 > 1January
was first
proved by [Higuchi’77] (and [Kesten-Stigum’66]).
15, 2007
  2 (M )
19
n = # of leaves
k = # of samples
Steel’s Favorite Conjecture
Reconstruction Problem
Phylogeny
N
conj
Y
conj
N
proof
k = n(1)
k =(log n)
k = n(1)
[M’03 (J. Comp. Biol.)]
Y
proof
k =(log n)
Random Cluster Model: [M-Steel’04 (Math. Biosciences.)]
CFN Model: [M-04’ (Transaction of AMS)],
January 15, 2007
[Daskalakis-M-Roch’ (STOC06)]
20
Polynomial Lower Bound at High Mutations
Proof:

Conditional Independence + Data
Processing Lemma
X=T
L
Known
q-L

?
?
*k
Known
*k
In fact:


[M’06: (IEEE. Comp. Bio. & BioInfo)]: “Shallow Part” of the tree can
be efficiently reconstructed when k = O(log n) for all mutation rates.
Also in practice [Daskalakis-Hill-Jaffe-Mihaescu-M-Rao (Recomb06)]
January 15, 2007
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Reconstruction from short sequences

Th [Daskalakis-M-Roch (STOC’06)]: If T is a tree on n
leaves s.t.


For all e, min < (e)< max and 22min > 1, max < 1.
Then there exists a polynomial time algorithm that uses
sequences of length k = O(log n – log ) to reconstruct the
topology with probability 1- in polynomial time where the
constant depends on (min, max).
January 15, 2007
22
Proof: Distance Methods


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Associate to each edge e the weight ln (12pe)
For any two leaves i and j:
ln(1 – 2 pi,j) =  ln (1 – 2 pe)
where the sum is over all e in the path
connecting a to b.
Reconstruction Algorithm:




r
(ra)
a
(ab)
Estimate pi,j from sequences
b
Deduce the topology of the tree
(b1)
Problem: Need exp. long sequences
ESSW: “log n” radius neighborhoods
determine the tree ) poly(n) sequence length 1
suffices.
January 15, 2007
(rc)
(a3)
(b2)
2
c
(c4)
3
4
(c5)
5
23
Back
Four-Point Method
  D(a, c)  D(b, d )  D(a, b)  D(c, d )
a
c
a
b
b
d
c
d
January 15, 2007
0
0
a
b
d
c
0
24
Balanced Trees

Two-Step Algorithm [M, 2004]:



1) Reconstruct one (or a few) level(s)
2) Infer sequences at roots
3) Start over
January 15, 2007
25
General Trees [Daskalakis, M, Roch, 2006]
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26
Blindfolded Cherry Picking

Need “only” one extra step in the algorithm

Main Loop:




1) Distance estimation
2) Identify cherries from the next level
3) Sequence reconstruction
4) Detect “fake cherries”
January 15, 2007
27
Blindfolded Cherry Picking I: Edge Disjointness
Non Edge-Disjoint Reconstruction
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True Tree
28
Blindfolded Cherry Picking II: Weight Estimation
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29
Blindfolded Cherry Picking III: Collisions
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30
Tree Reconstruction in a Nutshell
Tree reconstruction can be
solved from very short sequences

There exists a good estimator
for root reconstruction
Similar Techniques apply to other tree networks – for example

Reconstructing Multicast Networks (Liang-M-Yu, BhamidiRajagopal-Roch)
January 15, 2007
31
Back to General Problem:
• How to reconstruct the network topology from
observations at a (sub)-set of the nodes?
• Example 3: Reconstructing Markov Random Fields from
observations at a subset of the nodes ???
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part III:
Optimization over Stochastic Networks
Motivating Problem
•
Problem:
–
•
Examples:
–
–
–
–
–
•
•
Optimization over stochastic models defined on networks.
Which Genes to knock out in order to kill a cancer cell?
Which computers to immune in order make a networks robust?
Which computers to attack in order to fail the network?
Which individuals to immune to stop a disease from spreading.
Viral Marketing: Which individuals to expose to a product so as to
maximize its distribution?
One case Study: Influence in Social Networks
Joint work with Sebastien Roch.
January 15, 2007
34/31
models of collective behavior
•
examples:
–
–
–
•
joining a riot
adopting a product
going to a movie
model features:
–
–
–
binary decision
cascade effect
network structure
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viral marketing
• referrals, word-of-mouth can be very effective
–
ex.: Hotmail
• viral marketing
–
–
goal: mining the network value of potential customers
how: target a small set of trendsetters, seeds
• example [Domingos-Richardson’02]
–
–
collaborative filtering system
use MRF to compute “influence” of each customer
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independent cascade model
• when a node is activated
–
–
it gets one chance to activate each neighbour
probability of success from u to v is pu,v
0.5
0.33
0.25
0.5
1.0
0.5
0.5
0.5
1.0
0.75
0.5
0.5
January 15, 2007
0.25
0.5
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generalized models
• graph G=(V,E); initial activated set S0
•
generalized threshold model [Kempe-Kleinberg-Tardos’03,’05]
–
–
–
•
activation functions: fu(S) where S is set of activated nodes
threshold value: u uniform in [0,1]
dynamics: at time t,set St to St-1 and add all nodes with fu(St-1)  u
(note the process stops after (at most) n-1 steps)
generalized cascade model [KKT’03,’05]
–
when node u is activated:
• gets one chance to activate each neighbours
• probability of success from u to v: pu(v,S) where S is set of nodes who have
already tried (and failed) to activate u
–
•
assumption: the pu(v,.)’s are “order-independent”
theorem [KKT’03] - the two models are equivalent
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influence maximization
• definition - the influence (S) given the initial seed S is the
expected size of the infected set at termination
 (S)  E S Sn1 
• definition - in the influence maximization problem (IMP), we want
to find the seed S of fixed size k that maximizes the influence

S*  argmax (S) : S  V, S  k
• theorem [KKT’03] - the IMP is NP-hard
–
reduction from Set Cover: ground set U = {u1,…,un} and collection of cover
S ,…,S
subsets
1
m
u1
u2
u3
un

ui  S j
…
…
independent
cascade
model
S1
S2
S3
(ui ,S j )  E
Sm

January 15, 2007

S, S  k,  (S)  n  k ?
39/31
submodularity
• definition - a set function f : V -> R is submodular if for all A, B in V
f (A)  f (B)  f (A  B)  f (A  B)
• example: f(S) = g(|S|) where g is concave

• interpretation: “discrete concavity” or “diminishing returns”, indeed
submodularity equivalent to
S  T,v  V,
f (T {v})  f (T)  f (S {v})  f (S)
• threshold models:

–
–
it is natural to assume that the activation functions have diminishing
returns
supported by observations of [Leskovec-Adamic-Huberman’06] in the
context of viral marketing
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main result
• theorem [M-Roch’06; first conjectured in KKT’03] - in the generalized
threshold model, if all activation functions are monotone and
submodular, then the influence is also submodular
• corollary [M-Roch’06] - IMP admits a (1 - e-1 - )-approximation
algorithm (for all  > 0)
–
this follows from a general result on the approximation of submodular
functions [Nemhauser-Wolsey-Fisher’78]
• known special cases [KKT’03,’05]:
–
–
linear threshold model, independent cascade model
decreasing cascade model, “normalized” submodular threshold model
S  T, pu (v,S)  pu (v,T) or equiv.

January 15, 2007
f u (S {v})  f u (S) f u (T {v})  f u (T)

1 f u (S)
1 f u (T)
41/31
related work
•
sociology
–
–
threshold models: [Granovetter’78], [Morris’00]
cascades: [Watts’02]
• data mining
–
–
viral marketing: [KKT’03,’05], [Domingos-Richardson’02]
recommendation networks: [Leskovec-Singh-Kleinberg’05], [LeskovecAdamic-Huberman’06]
• economics
–
game-theoretic point of view: [Ellison’93], [Young’02]
• probability theory
–
–
–
Markov random fields, Glauber dynamics
percolation
interacting particle systems: voter model, contact process
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proof sketch
coupling
• we use the generalized threshold model
• arbitrary sets A, B; consider 4 processes:
–
–
–
–
(At) started at A
(Bt) started at B
(Ct) started at AB
(Dt) started at AB
• it suffices to couple the 4 processes in such a way that for all t
C t  At  Bt
Dt  At  Bt
• indeed, at termination
An1  Bn1
  An1  Bn1  An1  Bn1  Cn1  Dn1
(note this works with |.| replaced with any w monotone, submodular)

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proof ideas
• our goal:
Ct  At  Bt
(1)
Dt  At  Bt
(2)
• antisense coupling
–
–
–
–
obvious way to couple: use same u’s for all 4 processes

satisfies
(1) but not (2)
“antisense”: using u for (At) and (1-u) for (Bt) “maximizes union”
we combine both couplings
• piecemeal growth
–
–
seed sets can be introduced in stages
we add AB then A\B and finally B\A
• need-to-know
–
–
not necessary to pick all u’s at beginning
can unveil only what we need to know:
v  f v St2 , f v St1?
January 15, 2007

45/31
piecemeal growth
• process started at S: (St)
• partition of S: S(1),…,S(K)
• consider the process (Tt):
–
–
–
–
pick u’s
run the process with seed S(1) until termination
add S(2) and continue until termination
add S(3) and so on
• lemma - the sets Sn-1 and TKn-1 have the same distribution
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antisense coupling
• disjoint sets: S, T
• partition of S: S(1),…,S(K)
• piecemeal process with seeds S(1),…,S(K),T: (St)
• consider the process (Tt):
–
–
–
pick u’s
run piecemeal process with seeds S(1),…,S(K) until termination
add T and continue with threshold values
v '1 v  f v TKn1
• lemma - the sets S(K+1)n-1 and T(K+1)n-1 have the same distribution

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need-to-know
• proof of lemma
–
–
–
run the first K stages identically in both processes
note that for all v not in SKn-1 = TKn-1, v is uniformly distributed in
[fv(TKn-1),1]
but v’ = 1 - v + fv(TKn-1) has the same distribution
v  f v St2 , f v St1?

simulation 1
January 15, 2007
simulation 2
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proof I
ANTI
January 15, 2007
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proof II
ANTI
January 15, 2007
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proof III
• new processes have correct final distribution
• up to time 2n-1, Bt = Ct and At = Dt so that
Ct  At  Bt
Dt  At  Bt
• for time 2n, note that
B2n1  D2n1
B2n  B2n1  (T \ S)
D2n  D2n1  (T \ S)
• so by monotonicity and submodularity
f v (B2n )  f v (B2n1)  f v (D2n )  f v (D2n1 )

• then proceed by induction

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general result
• we have proved:
theorem [Mossel-R’06] - in the generalized threshold model, if all
activation functions are submodular, then for any monotone, submodular
function w, the generalized influence
 w (S)  E S [w(Sn1 )]
is submodular
• Note: A closure property for sub-modular functions!

January 15, 2007
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Future Research Directions
• Study optimization problems for other stochastic models defined on
networks.
• And another annoying problem where discrete probability may help:
• Are there (easily computable? Probabilistic?) invariants of unlabelled
graphs that uniquely determine them?
• Motivation: Can one efficiently check if two graphs are isomorphic?
January 15, 2007
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