Discrete Probability on Graphs: Estimation, Reconstruction of

1
Discrete Probability on Graphs Estimation,
Reconstruction of Optimization on Networks

• Elchanan Mossel
• UC Berkeley
• At IPAM Mar 2007

2
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?

3
Part 0Two Types of Network Problems
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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.

5
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

6
part IEstimation in Markov Random Fields
7
Gibbs Measures / Graphical Models

• 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 AV 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.

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Message Passing Algorithms / The Replica Method

• 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
• 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

G
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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?

G
T
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Message Passing Algorithms in Coding

• In coding
• BP is used to decode Low Density Parity Check
  Codes (LDPC) Gallager62
• Proved to be efficient Luby-Mitzenmacher-Shokroll
  ahi-Spielman-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.

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Message Passing Algorithms -Random 3-SAT
n
x1
x2
x3
x4
x5
x6
x7
x8
m ?n

WalkSAT

Survey propagation
Not satisfiable
Satisfiable
Belief propagation
Not satisfiable
Satisfiable
Myopic
PLR
?
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Message Passing Algorithms for Random 3-SAT

• 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

M. Talagrand
G. Parisi
B. Selman
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Decay of correlation for 3-SAT extended MRF
Partial assignments
0, 1n assignments
01??1?0?
01101???
stars
?10?11??
01101???
????????
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part IIReconstructing Stochastic Network from
observations
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Main Problem

• How to reconstruct the network topology from
  observations at a (sub)-set of the nodes?
• The Example Reconstructing Trees.

16
Two Tree Inference Problems

• In Evolution
• Given a tree of species / mothers, can we infer
  ancestral sequence at the root from contemporary
  samples?
• Phase Transition
• Trade-off between noise and duplication?
• Reconstructing Evolution
• Is it possible to reconstruct evolutionary
  history from genetic sequences?

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Defn Markov Model on a Tree
001100011101000011000100
0
s(r)

• Ising/BSC/CFN Model
• Tree T (V,E)
• Node states
• Mutation probabilities
• Number of leaves n

pra
prc
0
s(a)
pab
pa3
1
0
s(c)
s(b)
pc4
pc5
pb1
pb2
1
0
0
0
1
0 Purines (A,G) 1 Pyrimidines (C,T)
s(1)
s(4)
s(5)
s(3)
s(2)
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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)
  Felsenstein04
• TCS (Learning) Ambainis-Desper-Farach-Kannan97
  , Farach-Kannan96, Cryan-Goldberg-Goldberg02
  
• M-Roch
• Combinatorial Phylogeny Erdos-Steel-Szekely-Warn
  ow97, 98, M07

s(1)
s(4)
s(5)
s(3)
s(2)
1
1
1
0
0
0
1
1
0
0
1
0
0
1
1
0
1
1
1
0
1
1
1
0
0
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Phase Transition for the Ising model
LOW Temp
HIGH Temp
bias
no bias
typical boundary
typical boundary
2 ?2 gt 1
2?2 lt 1
The transition at 2 ?2 1 was proved
by Bleher-Ruiz-Zagrebnov95,
Ioffe96,Evans-Kenyon-Peres-Schulman00,
Kenyon-Mossel-Peres01,Martinelli-Sinclair-Wei
tz04, Borgs-Chayes-M-Roch06. Also,
spin-glass case studied by Chayes-Chayes-Sethna
-Thouless86. Solvability for 2 ?2 gt 1 was
first proved by Higuchi77 (and
Kesten-Stigum66).
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Steels Favorite Conjecture
n of leaves k of samples
Reconstruction Problem
Phylogeny
conj
N
k n?(1)
conj
k ?(log n)
Y
proof
N
k n?(1)
M03 (J. Comp. Biol.)
proof
k ?(log n)
Y
Random Cluster Model M-Steel04 (Math.
Biosciences.)
CFN Model M-04 (Transaction of AMS),
Daskalakis-M-Roch (STOC06)
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Polynomial Lower Bound at High Mutations
Conditional Independence Data Processing Lemma

• Proof

• In fact
• M06 (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)

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Reconstruction from short sequences

• Th Daskalakis-M-Roch (STOC06) If T is a tree
  on n leaves s.t.
• For all e, ?min lt ?(e)lt ?max and 2?2min gt 1, ?max
  lt 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).

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Proof Distance Methods

• Associate to each edge e the weight ln (1- 2pe)
• 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
• Estimate pi,j from sequences
• Deduce the topology of the tree
• Problem Need exp. long sequences
• ESSW log n radius neighborhoods determine the
  tree ) poly(n) sequence length suffices.

Back
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Four-Point Method
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Balanced Trees

• Two-Step Algorithm M, 2004
• 1) Reconstruct one (or a few) level(s)
• 2) Infer sequences at roots
• 3) Start over

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General Trees Daskalakis, M, Roch, 2006
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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

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Blindfolded Cherry Picking I Edge Disjointness
True Tree
Non Edge-Disjoint Reconstruction
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Blindfolded Cherry Picking II Weight Estimation
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Blindfolded Cherry Picking III Collisions
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Tree Reconstruction in a Nutshell

• Similar Techniques apply to other tree networks
  for example
• Reconstructing Multicast Networks (Liang-M-Yu,
  Bhamidi-Rajagopal-Roch)

32
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 IIIOptimization over Stochastic Networks
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Motivating Problem

• Problem
• Optimization over stochastic models defined on
  networks.
• Examples
• 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.

35
models of collective behavior

• examples
• joining a riot
• adopting a product
• going to a movie
• model features
• binary decision
• cascade effect
• network structure

36
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-Richardson02
• collaborative filtering system
• use MRF to compute influence of each customer

37
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
1.0
0.25
0.5
0.5
0.5
0.5
1.0
0.75
0.5
0.5
0.5
0.25
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generalized models

• graph G(V,E) initial activated set S0
• generalized threshold model Kempe-Kleinberg-Tardo
  s03,05
• 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 KKT03,05
• 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 KKT03 - the two models are equivalent

39
influence maximization

• definition - the influence ?(S) given the initial
  seed S is the expected size of the infected set
  at termination
• definition - in the influence maximization
  problem (IMP), we want to find the seed S of
  fixed size k that maximizes the influence
• theorem KKT03 - the IMP is NP-hard
• reduction from Set Cover ground set U
  u1,,un and collection of cover subsets S1,,Sm
  

u1
S1
u2
S2
independent cascade model
u3
S3


un
Sm
40
submodularity

• definition - a set function f V -gt R is
  submodular if for all A, B in V
• example f(S) g(S) where g is concave
• interpretation discrete concavity or
  diminishing returns, indeed submodularity
  equivalent to
• threshold models
• it is natural to assume that the activation
  functions have diminishing returns
• supported by observations of Leskovec-Adamic-Hube
  rman06 in the context of viral marketing

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main result

• theorem M-Roch06 first conjectured in KKT03
  - in the generalized threshold model, if all
  activation functions are monotone and submodular,
  then the influence is also submodular
• corollary M-Roch06 - IMP admits a (1 - e-1 -
  ?)-approximation algorithm (for all ? gt 0)
• this follows from a general result on the
  approximation of submodular functions
  Nemhauser-Wolsey-Fisher78
• known special cases KKT03,05
• linear threshold model, independent cascade model
• decreasing cascade model, normalized submodular
  threshold model

42
related work

• sociology
• threshold models Granovetter78, Morris00
• cascades Watts02
• data mining
• viral marketing KKT03,05, Domingos-Richardso
  n02
• recommendation networks Leskovec-Singh-Kleinberg
  05, Leskovec-Adamic-Huberman06
• economics
• game-theoretic point of view Ellison93,
  Young02
• probability theory
• Markov random fields, Glauber dynamics
• percolation
• interacting particle systems voter model,
  contact process

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proof sketch
44
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
• indeed, at termination
• (note this works with . replaced with any w
  monotone, submodular)

45
proof ideas

• our goal
• antisense coupling
• obvious way to couple use same ?us 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 ?us at beginning
• can unveil only what we need to know

46
piecemeal growth

• process started at S (St)
• partition of S S(1),,S(K)
• consider the process (Tt)
• pick ?us
• 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

47
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 ?us
• run piecemeal process with seeds S(1),,S(K)
  until termination
• add T and continue with threshold values
• lemma - the sets S(K1)n-1 and T(K1)n-1 have the
  same distribution

48
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

simulation 1
simulation 2
49
proof I
ANTI
50
proof II
ANTI
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proof III

• new processes have correct final distribution
• up to time 2n-1, Bt Ct and At Dt so that
• for time 2n, note that
• so by monotonicity and submodularity
• then proceed by induction

52
general result

• we have proved
• theorem Mossel-R06 - in the generalized
  threshold model, if all activation functions are
  submodular, then for any monotone, submodular
  function w, the generalized influence
• is submodular
• Note A closure property for sub-modular
  functions!

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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?