Broadcasting on Random Networks Anuran Makur, Elchanan Mossel, and - - PowerPoint PPT Presentation

Broadcasting on Random Networks

Anuran Makur, Elchanan Mossel, and Yury Polyanskiy

EECS and Mathematics Departments Massachusetts Institute of Technology

ISIT 2019

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 1 / 26

Outline

1

Introduction Motivation Formal Model and Broadcasting Problem Related Models in the Literature

2

Results on Random DAGs

3

Deterministic Broadcasting DAGs

4

Conclusion

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 2 / 26

Motivation: Information Propagation in 2D Grid

How does information spread in time?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 3 / 26

Motivation: Information Propagation in 2D Grid

How does information spread in time?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 3 / 26

Motivation: Information Propagation in 2D Grid

How does information spread in time?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 3 / 26

Motivation: Information Propagation in 2D Grid

How does information spread in time?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 3 / 26

Motivation: Information Propagation in 2D Grid

How does information spread in time? Can we invent relay functions so that far boundary contains non-trivial information about the original bit?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 3 / 26

Motivation: Broadcasting on Trees

Fix infinite tree T with branching number br(T).

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 4 / 26

Motivation: Broadcasting on Trees

Fix infinite tree T with branching number br(T).

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 4 / 26

Motivation: Broadcasting on Trees

Fix infinite tree T with branching number br(T). Root X0,0 ∼ Bernoulli 1

2

• ,

, , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 4 / 26

Motivation: Broadcasting on Trees

Fix infinite tree T with branching number br(T). Root X0,0 ∼ Bernoulli 1

2

• Edges are independent BSCs with crossover probability δ ∈
• 0, 1

2

• .

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 4 / 26

Motivation: Broadcasting on Trees

Fix infinite tree T with branching number br(T). Root X0,0 ∼ Bernoulli 1

2

• Edges are independent BSCs with crossover probability δ ∈
• 0, 1

2

• .

Let P(k)

ML = P

ˆ X k

ML(Xk) = X0,0

• , where Xk =
• Xk,0, . . . , Xk,br(T)k−1
• .

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 4 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If δ < 1

2 − 1 2√ br(T), then reconstruction possible:

lim

k→∞P(k) ML < 1 2.

If δ > 1

2 − 1 2√ br(T), then reconstruction impossible:

lim

k→∞P(k) ML = 1 2.

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If (1 − 2δ)2 br(T) > 1, then reconstruction possible: lim

k→∞P(k) ML < 1 2.

If (1 − 2δ)2 br(T) < 1, then reconstruction impossible: lim

k→∞P(k) ML = 1 2.

Proof Idea: Strong data processing inequality [AG76, ES99]

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If (1 − 2δ)2 br(T) > 1, then reconstruction possible: lim

k→∞P(k) ML < 1 2.

If (1 − 2δ)2 br(T) < 1, then reconstruction impossible: lim

k→∞P(k) ML = 1 2.

Proof Idea: Strong data processing inequality [AG76, ES99] If PY |X = BSC(δ), then for any U → X → Y : I(U; Y ) ≤ (1 − 2δ)2I(U; X).

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If (1 − 2δ)2 br(T) > 1, then reconstruction possible: lim

k→∞P(k) ML < 1 2.

If (1 − 2δ)2 br(T) < 1, then reconstruction impossible: lim

k→∞P(k) ML = 1 2.

Proof Idea: Strong data processing inequality [AG76, ES99] If PY |X = BSC(δ), then for any U → X → Y : I(U; Y ) ≤ (1 − 2δ)2I(U; X). For any 0 ≤ j < br(T)k, I(X0,0; Xk,j) ≤ (1 − 2δ)2k.

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If (1 − 2δ)2 br(T) > 1, then reconstruction possible: lim

k→∞P(k) ML < 1 2.

If (1 − 2δ)2 br(T) < 1, then reconstruction impossible: lim

k→∞P(k) ML = 1 2.

Proof Idea: Strong data processing inequality [AG76, ES99] If PY |X = BSC(δ), then for any U → X → Y : I(U; Y ) ≤ (1 − 2δ)2I(U; X). For any 0 ≤ j < br(T)k, I(X0,0; Xk,j) ≤ (1 − 2δ)2k. br(T)k paths from X0 to Xk: I(X0; Xk)≤

• br(T)(1 − 2δ)2

k

.

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Theorem (Phase Transition for Trees [KS66, BRZ95, EKPS00])

If (1 − 2δ)2 br(T) > 1, then reconstruction possible: lim

k→∞P(k) ML < 1 2.

If (1 − 2δ)2 br(T) < 1, then reconstruction impossible: lim

k→∞P(k) ML = 1 2.

Proof Idea: Strong data processing inequality [AG76, ES99] Layers grow by br(T) and information contracts by (1 − 2δ)2. So, whichever effect wins determines reconstruction.

, , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 5 / 26

Motivation: Broadcasting on Trees

Intuition: In tree T, layers grow exponentially with rate br(T) and information contracts with rate (1 − 2δ)2. So, whichever effect wins determines reconstruction.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 6 / 26

Motivation: Broadcasting on Trees

Intuition: In tree T, layers grow exponentially with rate br(T) and information contracts with rate (1 − 2δ)2. So, whichever effect wins determines reconstruction. If intuition correct, then broadcasting impossible on finite-dimensional grids, because layers grow polynomially.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 6 / 26

Motivation: Broadcasting on Trees

Intuition: In tree T, layers grow exponentially with rate br(T) and information contracts with rate (1 − 2δ)2. So, whichever effect wins determines reconstruction. If intuition correct, then broadcasting impossible on finite-dimensional grids, because layers grow polynomially. Can there be any graph with sub-exponentially growing layer sizes such that reconstruction possible?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 6 / 26

Motivation: Broadcasting on Trees

Intuition: In tree T, layers grow exponentially with rate br(T) and information contracts with rate (1 − 2δ)2. So, whichever effect wins determines reconstruction. If intuition correct, then broadcasting impossible on finite-dimensional grids, because layers grow polynomially. Can there be any graph with sub-exponentially growing layer sizes such that reconstruction possible? Surprise: Yes, and in fact, even logarithmic growth suffices (doubly-exponential reduction compared to trees (!)). But need nice loops to aggregate information.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 6 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite directed acyclic graph (DAG) with single source node.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite DAG with single source node. Xk,j ∈ {0, 1} – node random variable at jth position in level k

, , , , , , , , , , , ,

level level level level

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite DAG with single source node. Xk,j ∈ {0, 1} – node random variable at jth position in level k Lk – number of nodes at level k

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite DAG with single source node. Xk,j ∈ {0, 1} – node random variable at jth position in level k Lk – number of nodes at level k d – indegree of each node

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite DAG with single source node. Xk,j ∈ {0, 1} – node random variable at jth position in level k Lk – number of nodes at level k d – indegree of each node

, , , , , , , , , , , ,

level level level level

• vertices

X0,0 ∼ Bernoulli 1

2

• Every edge is independent

BSC with crossover probability δ ∈

• 0, 1

2

• .
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Formal Model: Broadcasting on Bounded Indegree DAGs

Fix infinite DAG with single source node. Xk,j ∈ {0, 1} – node random variable at jth position in level k Lk – number of nodes at level k d – indegree of each node

, , , , , , , , , , , ,

level level level level

• vertices

X0,0 ∼ Bernoulli 1

2

• Every edge is independent

BSC with crossover probability δ ∈

• 0, 1

2

• .

Nodes combine inputs with d-ary Boolean functions. This defines joint distribution of {Xk,j}.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 7 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞?

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞? Binary Hypothesis Testing: Let ˆ X k

ML(Xk) ∈ {0, 1} be maximum

likelihood (ML) decoder with probability of error: P(k)

ML P

• ˆ

X k

ML(Xk) = X0,0

• .
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞? Binary Hypothesis Testing: Let ˆ X k

ML(Xk) ∈ {0, 1} be maximum

likelihood (ML) decoder with probability of error: P(k)

ML P

• ˆ

X k

ML(Xk) = X0,0

• = 1

2

• 1 −
• PXk|X0=1 − PXk|X0=0
• TV
• .
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞? Binary Hypothesis Testing: Let ˆ X k

ML(Xk) ∈ {0, 1} be maximum

likelihood (ML) decoder with probability of error: P(k)

ML P

• ˆ

X k

ML(Xk) = X0,0

• = 1

2

• 1 −
• PXk|X0=1 − PXk|X0=0
• TV
• .

By data processing inequality, TV distance contracts as k increases.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞? Binary Hypothesis Testing: Let ˆ X k

ML(Xk) ∈ {0, 1} be maximum

likelihood (ML) decoder with probability of error: P(k)

ML P

• ˆ

X k

ML(Xk) = X0,0

• = 1

2

• 1 −
• PXk|X0=1 − PXk|X0=0
• TV
• .

By data processing inequality, TV distance contracts as k increases. Broadcasting/Reconstruction possible if: lim

k→∞ P(k) ML < 1

2 ⇔ lim

k→∞

• PXk|X0=1 − PXk|X0=0
• TV > 0

and Broadcasting/Reconstruction impossible if: lim

k→∞ P(k) ML = 1

2 ⇔ lim

k→∞

• PXk|X0=1 − PXk|X0=0
• TV = 0 .
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Broadcasting Problem

Let Xk (Xk,0, . . . , Xk,Lk−1). Can we decode X0 from Xk as k → ∞? Binary Hypothesis Testing: Let ˆ X k

ML(Xk) ∈ {0, 1} be maximum

likelihood (ML) decoder with probability of error: P(k)

ML P

• ˆ

X k

ML(Xk) = X0,0

• = 1

2

• 1 −
• PXk|X0=1 − PXk|X0=0
• TV
• .

By data processing inequality, TV distance contracts as k increases. Broadcasting/Reconstruction possible iff: lim

k→∞ P(k) ML < 1

2 ⇔ lim

k→∞

• PXk|X0=1 − PXk|X0=0
• TV > 0 .

For which δ, d, {Lk}, and Boolean processing functions is reconstruction possible?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 8 / 26

Related Models in the Literature

Communication Networks: Sender broadcasts single bit through network.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 9 / 26

Related Models in the Literature

Communication Networks: Sender broadcasts single bit through network. Reliable Computation and Storage: [vNe56, HW91, ES03, Ung07] Broadcasting model is noisy circuit to remember a bit using perfect gates and faulty wires.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 9 / 26

Related Models in the Literature

Communication Networks: Sender broadcasts single bit through network. Reliable Computation and Storage: Broadcasting model is noisy circuit to remember a bit using perfect gates and faulty wires. Probabilistic Cellular Automata: Impossibility of broadcasting on 2D regular grid parallels ergodicity of 1D probabilistic cellular automata.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 9 / 26

Related Models in the Literature

Communication Networks: Sender broadcasts single bit through network. Reliable Computation and Storage: Broadcasting model is noisy circuit to remember a bit using perfect gates and faulty wires. Probabilistic Cellular Automata: Broadcasting on 2D regular grid parallels 1D probabilistic cellular automata. Ancestral Data Reconstruction: Reconstruction on trees ⇔ Infer trait of ancestor from observed population.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 9 / 26

Related Models in the Literature

Communication Networks: Sender broadcasts single bit through network. Reliable Computation and Storage: Broadcasting model is noisy circuit to remember a bit using perfect gates and faulty wires. Probabilistic Cellular Automata: Broadcasting on 2D regular grid parallels 1D probabilistic cellular automata. Ancestral Data Reconstruction: Reconstruction on trees ⇔ Infer trait of ancestor from observed population. Ferromagnetic Ising Models: [BRZ95, EKPS00] Reconstruction impossible on tree ⇔ Free boundary Gibbs state of Ising model on tree is extremal.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 9 / 26

Outline

1

Introduction

2

Results on Random DAGs Phase Transition for Majority Processing Impossibility Results for Broadcasting Phase Transition for NAND Processing

3

Deterministic Broadcasting DAGs

4

Conclusion

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 10 / 26

Random DAG Model

Fix {Lk} and d > 1.

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 11 / 26

Random DAG Model

Fix {Lk} and d > 1. For each node Xk,j, randomly and independently select d parents from level k − 1 (with repetition). This defines random DAG G.

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 11 / 26

Random DAG Model

Fix {Lk} and d > 1. For each node Xk,j, randomly and independently select d parents from level k − 1 (with repetition). This defines random DAG G. P(k)

ML(G) – ML decoding probability of error for DAG G

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 11 / 26

Random DAG Model

Fix {Lk} and d > 1. For each node Xk,j, randomly and independently select d parents from level k − 1 (with repetition). This defines random DAG G. P(k)

ML(G) – ML decoding probability of error for DAG G

σk

1 Lk

Lk−1

j=0 Xk,j – sufficient statistic of Xk for σ0 = X0,0

in the absence of knowledge of G

, , , , , , , , , , , ,

level level level level

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 11 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

✶

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 12 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk ✶

• σk ≥ 1

2

• is majority decoder.
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 12 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then reconstruction possible: lim

k→∞ E

• P(k)

ML(G)

• ≤ lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk ✶

• σk ≥ 1

2

• is majority decoder.
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 12 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then reconstruction possible: lim

k→∞ E

• P(k)

ML(G)

• ≤ lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk ✶

• σk ≥ 1

2

• is majority decoder.

Suppose δ ∈

• δmaj, 1

2

• . Then, there exists D(δ, d) > 1 such that if

Lk = o

• D(δ, d)k

, then reconstruction impossible: lim

k→∞ P(k) ML(G) = 1

2 G-a.s.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 12 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], level k has i.i.d. random bits Xk,j

i.i.d.

∼ majority(Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ)) where σ ∗ δ = σ(1 − δ) + δ(1 − σ)

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], level k has i.i.d. random bits Xk,j

i.i.d.

∼ majority(Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ)) where σ ∗ δ = σ(1 − δ) + δ(1 − σ), and Lkσk =

Lk−1

• j=0

Xk,j ∼ binomial(Lk, E[σk|σk−1 = σ] ) .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], level k has i.i.d. random bits Xk,j

i.i.d.

∼ majority(Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ), Bernoulli(σ ∗ δ)) where σ ∗ δ = σ(1 − δ) + δ(1 − σ), and Lkσk =

Lk−1

• j=0

Xk,j ∼ binomial(Lk, gδ(σ)) . Define the cubic polynomial: gδ(σ) E[σk|σk−1 = σ] = P(Xk,j = 1|σk−1 = σ) = (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ) .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1. Fixed Point Analysis: Case δ < δmaj:

1 2 1 1 2 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1. Fixed Point Analysis: σk “concentrates” at fixed point near X0,0 Case δ < δmaj: 3 fixed points

1 2 1 1 2 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1. Fixed Point Analysis: Case δ < δmaj: 3 fixed points

1 2 1 1 2 1

• Case δ > δmaj:

1 2 1 1 2 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1. Fixed Point Analysis: σk → 1

2 a.s. if δ > δmaj and Lk = ω(log(k))

Case δ < δmaj: 3 fixed points

1 2 1 1 2 1

• Case δ > δmaj: 1 fixed point

1 2 1 1 2 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Proof Intuition

Suppose d = 3 and δmaj = 1

6.

Conditioned on σk−1 = σ ∈ [0, 1], Lkσk ∼ binomial(Lk, gδ(σ)). Define the cubic polynomial gδ(σ) (σ ∗ δ)3 + 3(σ ∗ δ)2(1 − σ ∗ δ). Concentration: For large k, σk ≈ gδ(σk−1) given σk−1. Converse uses key property: Lip(gδ) ≤ 1 ⇔ gδ has unique fixed point. Case δ < δmaj: 3 fixed points

1 2 1 1 2 1

• Case δ > δmaj: 1 fixed point

1 2 1 1 2 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 13 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then lim

k→∞E

• P(k)

ML(G)

• < 1

2.

Suppose δ ∈

• δmaj, 1

2

• . Then, there exists D(δ, d) > 1 such that if

Lk = o

• D(δ, d)k

, then lim

k→∞P(k) ML(G) = 1 2 G-a.s.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 14 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then lim

k→∞E

• P(k)

ML(G)

• < 1

2.

Suppose δ ∈

• δmaj, 1

2

• . Then, there exists D(δ, d) > 1 such that if

Lk = o

• D(δ, d)k

, then lim

k→∞P(k) ML(G) = 1 2 G-a.s.

Remarks: δmaj = 1

6 for d = 3 appears in reliable computation [vNe56, HW91].

δmaj for odd d ≥ 3 also relevant in reliable computation [ES03]. δmaj for d ≥ 3 relevant in recursive reconstruction on trees [Mos98].

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 14 / 26

Random DAG with Majority Processing

Theorem (Phase Transition for d ≥ 3)

Consider random DAG model with d ≥ 3 and majority processing (with ties broken randomly). Let δmaj 1

2 −

2d−2

⌈d/2⌉(

d ⌈d/2⌉).

Suppose δ ∈ (0, δmaj). Then, there exists C(δ, d) > 0 such that if Lk ≥ C(δ, d) log(k), then lim

k→∞E

• P(k)

ML(G)

• < 1

2.

Suppose δ ∈

• δmaj, 1

2

• . Then, there exists D(δ, d) > 1 such that if

Lk = o

• D(δ, d)k

, then lim

k→∞P(k) ML(G) = 1 2 G-a.s.

Questions: Broadcasting possible with sub-logarithmic Lk? Broadcasting possible when δ > δmaj with other processing functions? What about d = 2?

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 14 / 26

Optimality of Logarithmic Layer Size Growth

Broadcasting possible with sub-logarithmic Lk?

Proposition (Layer Size Impossibility Result)

For any deterministic DAG, if: Lk ≤ log(k) d log 1

2δ

, then reconstruction impossible for all processing functions: lim

k→∞ P(k) ML = 1

2 .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 15 / 26

Optimality of Logarithmic Layer Size Growth

Broadcasting possible with sub-logarithmic Lk?

Proposition (Layer Size Impossibility Result)

For any deterministic DAG, if: Lk ≤ log(k) d log 1

2δ

, then reconstruction impossible for all processing functions: lim

k→∞ P(k) ML = 1

2 . No, broadcasting impossible with sub-logarithmic Lk!

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 15 / 26

Partial Converse Results

Broadcasting possible when δ > δmaj with other processing functions?

Proposition (Single Vertex Reconstruction)

Consider random DAG model with d ≥ 3. If δ ∈ (0, δmaj), Lk ≥ C(δ, d) log(k), and processing functions are majority, then single vertex reconstruction possible: lim sup

k→∞

P(Xk,0 = X0,0) < 1 2 .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 16 / 26

Partial Converse Results

Broadcasting possible when δ > δmaj with other processing functions?

Proposition (Single Vertex Reconstruction)

Consider random DAG model with d ≥ 3. If δ ∈ (0, δmaj), Lk ≥ C(δ, d) log(k), and processing functions are majority, then single vertex reconstruction possible: lim sup

k→∞

P(Xk,0 = X0,0) < 1 2 . If δ ∈

• δmaj, 1

2

• , d is odd, lim

k→∞Lk = ∞, and inf n≥kLn = O

• d2k

, then single vertex reconstruction impossible for all processing functions (which may be graph dependent): lim

k→∞ E

• PXk,0|G,X0,0=1 − PXk,0|G,X0,0=0
• TV
• = 0 .

Remark: Converse uses reliable computation results [HW91, ES03].

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 16 / 26

Partial Converse Results

Broadcasting possible when δ > δmaj with other processing functions?

Proposition (Information Percolation [ES99, PW17])

For any deterministic DAG, if: δ > 1 2 − 1 2 √ d and Lk = o

• 1

((1 − 2δ)2d)k

• then reconstruction impossible for all processing functions:

lim

k→∞ P(k) ML = 1

2 .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 16 / 26

Partial Converse Results

Broadcasting possible when δ > δmaj with other processing functions?

Proposition (Information Percolation [ES99, PW17])

For any deterministic DAG, if: δ > 1 2 − 1 2 √ d > δmaj and Lk = o

• 1

((1 − 2δ)2d)k

• then reconstruction impossible for all processing functions:

lim

k→∞ P(k) ML = 1

2 .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 16 / 26

Random DAG with NAND Processing

What about d = 2?

Theorem (Phase Transition for d = 2)

Consider random DAG model with d = 2 and NAND processing functions. Let δnand 3−

√ 7 4

. ✶

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 17 / 26

Random DAG with NAND Processing

What about d = 2?

Theorem (Phase Transition for d = 2)

Consider random DAG model with d = 2 and NAND processing functions. Let δnand 3−

√ 7 4

. Suppose δ ∈ (0, δnand). Then, there exist C(δ) > 0 and t(δ) ∈ (0, 1) such that if Lk ≥ C(δ) log(k), then reconstruction possible: lim

k→∞ E

• P(k)

ML(G)

• ≤ lim sup

k→∞

P

• ˆ

T2k = X0,0

• < 1

2 where ˆ Tk ✶{σk ≥ t(δ)} is thresholding decoder.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 17 / 26

Random DAG with NAND Processing

What about d = 2?

Theorem (Phase Transition for d = 2)

Consider random DAG model with d = 2 and NAND processing functions. Let δnand 3−

√ 7 4

. Suppose δ ∈ (0, δnand). Then, there exist C(δ) > 0 and t(δ) ∈ (0, 1) such that if Lk ≥ C(δ) log(k), then reconstruction possible: lim

k→∞ E

• P(k)

ML(G)

• ≤ lim sup

k→∞

P

• ˆ

T2k = X0,0

• < 1

2 where ˆ Tk ✶{σk ≥ t(δ)} is thresholding decoder. Suppose δ ∈

• δnand, 1

2

• . Then, there exist D(δ), E(δ) > 1 such that if

Lk = o

• D(δ)k

and lim inf

k→∞ Lk > E(δ), then reconstruction impossible:

lim

k→∞ P(k) ML(G) = 1

2 G-a.s.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 17 / 26

Random DAG with NAND Processing

What about d = 2?

Theorem (Phase Transition for d = 2)

Consider random DAG model with d = 2 and NAND processing functions. Let δnand 3−

√ 7 4

. Suppose δ ∈ (0, δnand). Then, there exist C(δ) > 0 and t(δ) ∈ (0, 1) such that if Lk ≥ C(δ) log(k), then reconstruction possible: lim

k→∞ E

• P(k)

ML(G)

• ≤ lim sup

k→∞

P

• ˆ

T2k = X0,0

• < 1

2 where ˆ Tk ✶{σk ≥ t(δ)} is thresholding decoder. Suppose δ ∈

• δnand, 1

2

• . Then, there exist D(δ), E(δ) > 1 such that if

Lk = o

• D(δ)k

and lim inf

k→∞ Lk > E(δ), then reconstruction impossible:

lim

k→∞ P(k) ML(G) = 1

2 G-a.s. Remark: δnand appears in reliable computation [EP98, Ung07].

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 17 / 26

Outline

1

Introduction

2

Results on Random DAGs

3

Deterministic Broadcasting DAGs Existence of DAGs where Broadcasting is Possible Construction of DAGs where Broadcasting is Possible

4

Conclusion

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 18 / 26

Existence of DAGs where Broadcasting is Possible

Probabilistic Method: Random DAG broadcasting ⇒ DAG where reconstruction possible exists.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 19 / 26

Existence of DAGs where Broadcasting is Possible

Probabilistic Method: Random DAG broadcasting ⇒ DAG where reconstruction possible exists. For example:

Corollary (Existence of Deterministic Broadcasting DAGs)

For every d ≥ 3, δ ∈ (0, δmaj), and Lk ≥ C(δ, d) log(k), there exists DAG with majority processing functions such that reconstruction possible: lim

k→∞ P(k) ML < 1

2 .

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 19 / 26

Existence of DAGs where Broadcasting is Possible

Probabilistic Method: Random DAG broadcasting ⇒ DAG where reconstruction possible exists. For example:

Corollary (Existence of Deterministic Broadcasting DAGs)

For every d ≥ 3, δ ∈ (0, δmaj), and Lk ≥ C(δ, d) log(k), there exists DAG with majority processing functions such that reconstruction possible: lim

k→∞ P(k) ML < 1

2 . Can we construct such DAGs for any δ ∈

• 0, 1

2

• ?
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 19 / 26

Regular Bipartite Expander Graphs

Proposition (Existence of Expander Graphs [Pin73, SS96])

For all (large) d and all sufficiently large n, there exists d-regular bipartite graph Bn = (Un, Vn, En) with disjoint vertex sets Un, Vn of cardinality |Un| = |Vn| = n, edge multiset En, and the lossless expansion property: ∀S ⊆ Un, |S| = n d6/5 ⇒ |Γ(S)| ≥

• 1 −

2 d1/5

• d|S|

where Γ(S) {v ∈ Vn : ∃u ∈ S, (u, v) ∈ En} is neighborhood of S.

• vertices
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 20 / 26

Regular Bipartite Expander Graphs

Proposition (Existence of Expander Graphs [Pin73, SS96])

For all (large) d and all sufficiently large n, there exists d-regular bipartite graph Bn = (Un, Vn, En) with disjoint vertex sets Un, Vn of cardinality |Un| = |Vn| = n, edge multiset En, and the lossless expansion property: ∀S ⊆ Un, |S| = n d6/5 ⇒ |Γ(S)| ≥

• 1 −

2 d1/5

• d|S|

where Γ(S) {v ∈ Vn : ∃u ∈ S, (u, v) ∈ En} is neighborhood of S.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 20 / 26

Regular Bipartite Expander Graphs

Proposition (Existence of Expander Graphs [Pin73, SS96])

For all (large) d and all sufficiently large n, there exists d-regular bipartite graph Bn = (Un, Vn, En) with disjoint vertex sets Un, Vn of cardinality |Un| = |Vn| = n, edge multiset En, and the lossless expansion property: ∀S ⊆ Un, |S| = n d6/5 ⇒ |Γ(S)| ≥

• 1 −

2 d1/5

• d|S|

where Γ(S) {v ∈ Vn : ∃u ∈ S, (u, v) ∈ En} is neighborhood of S. Intuition: Expander graphs are sparse, but have high connectivity.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 20 / 26

Construction of DAGs where Broadcasting is Possible

Fix any δ ∈

• 0, 1

2

• and any sufficiently large odd d = d(δ).
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 21 / 26

Construction of DAGs where Broadcasting is Possible

Fix any δ ∈

• 0, 1

2

• and any sufficiently large odd d = d(δ).

Fix L0 = 1, Lk = N for k ∈ {1, . . . , ⌊M⌋} where N = N(δ) sufficiently large and M = exp

• N/(4d12/5)
• , and

∀ r ≥ 1, M2r−1 < k ≤ M2r , Lk = 2rN such that Lk = Θ(log(k)).

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 21 / 26

Construction of DAGs where Broadcasting is Possible

Fix any δ ∈

• 0, 1

2

• and any sufficiently large odd d = d(δ).

Fix L0 = 1, Lk = N for k ∈ {1, . . . , ⌊M⌋} where N = N(δ) sufficiently large and M = exp

• N/(4d12/5)
• , and

∀ r ≥ 1, M2r−1 < k ≤ M2r , Lk = 2rN such that Lk = Θ(log(k)). Construct bounded degree deterministic “expander DAG”: Each X1,j has one edge from X0,0.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 21 / 26

Construction of DAGs where Broadcasting is Possible

Fix any δ ∈

• 0, 1

2

• and any sufficiently large odd d = d(δ).

Fix L0 = 1, Lk = N for k ∈ {1, . . . , ⌊M⌋} where N = N(δ) sufficiently large and M = exp

• N/(4d12/5)
• , and

∀ r ≥ 1, M2r−1 < k ≤ M2r , Lk = 2rN such that Lk = Θ(log(k)). Construct bounded degree deterministic “expander DAG”: Each X1,j has one edge from X0,0. Case Lk+1 = Lk: Edge multiset Xk → Xk+1 given by expander BLk.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 21 / 26

Construction of DAGs where Broadcasting is Possible

Fix any δ ∈

• 0, 1

2

• and any sufficiently large odd d = d(δ).

Fix L0 = 1, Lk = N for k ∈ {1, . . . , ⌊M⌋} where N = N(δ) sufficiently large and M = exp

• N/(4d12/5)
• , and

∀ r ≥ 1, M2r−1 < k ≤ M2r , Lk = 2rN such that Lk = Θ(log(k)). Construct bounded degree deterministic “expander DAG”: Each X1,j has one edge from X0,0. Case Lk+1 = Lk: Edge multiset Xk → Xk+1 given by expander BLk. Case Lk+1 = 2Lk: Both edge multisets Xk → (Xk+1,0, . . . , Xk+1,Lk−1) and Xk → (Xk+1,Lk, . . . , Xk+1,Lk+1−1) given by expander BLk.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 21 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 level 𝑁

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 level 𝑁 + 1 level 𝑁

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 𝐶 level 𝑁 + 1 level 𝑁

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 𝐶 level 𝑁 + 1 𝐶 level 𝑁

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 𝐶 level 𝑁 + 1 𝐶 level 𝑁 𝐶 level 𝑁 + 2

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Illustration of “Expander DAG”:

level 0 level 1 𝐶 level 2 𝐶 level 𝑁 ‐ 1 𝐶 level 𝑁 + 1 𝐶 level 𝑁 𝐶 level 𝑁 + 2 𝐶 level 𝑁 level 𝑁 ‐ 1

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 22 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.
• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN. Let Sk {nodes equal to 1 at level k}.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN. Let Sk {nodes equal to 1 at level k}. Call node at level k + 1 “bad” if it is connected to ≥ 1 + d

4 nodes in Sk.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN. Let Sk {nodes equal to 1 at level k}. Call node at level k + 1 “bad” if it is connected to ≥ 1 + d

4 nodes in Sk.

Expansion Property: If |Sk| ≤ d−6/5N, then we have ≤ 8d−7/5N “bad” nodes.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN. Let Sk {nodes equal to 1 at level k}. Call node at level k + 1 “bad” if it is connected to ≥ 1 + d

4 nodes in Sk.

Expansion Property: If |Sk| ≤ d−6/5N, then we have ≤ 8d−7/5N “bad” nodes. Main Lemma: Given |Sk| ≤ d−6/5N, we have |Sk+1| ≤ d−6/5N with high probability, as “good” nodes have low probability of becoming 1.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proof Sketch: Suppose edges from level k to k + 1 given by expander BN. Let Sk {nodes equal to 1 at level k}. Call node at level k + 1 “bad” if it is connected to ≥ 1 + d

4 nodes in Sk.

Expansion Property: If |Sk| ≤ d−6/5N, then we have ≤ 8d−7/5N “bad” nodes. Main Lemma: Given |Sk| ≤ d−6/5N, we have |Sk+1| ≤ d−6/5N with high probability, as “good” nodes have low probability of becoming 1. If X0,0 = 0, then |Sk| likely to remain small as k → ∞.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proposition (Computational Complexity of DAG Construction)

For any δ ∈

• 0, 1

2

• , the d-regular bipartite expander graphs for levels

0, . . . , k of “expander DAG” can be constructed in: deterministic quasi-polynomial time O( exp( Θ(log(k) log log(k)) ) ), Remark: Enumerate all d-regular bipartite graphs and test expansion.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Construction of DAGs where Broadcasting is Possible

Theorem (Broadcasting in Expander DAG)

For “expander DAG” with majority processing, reconstruction possible: lim sup

k→∞

P

• ˆ

Sk = X0,0

• < 1

2 where ˆ Sk = ✶

• σk ≥ 1

2

• is majority decoder.

Proposition (Computational Complexity of DAG Construction)

For any δ ∈

• 0, 1

2

• , the d-regular bipartite expander graphs for levels

0, . . . , k of “expander DAG” can be constructed in: deterministic quasi-polynomial time O( exp( Θ(log(k) log log(k)) ) ), randomized polylogarithmic time O( log(k) log log(k) ) with positive success probability (which depends on δ but not k). Remark: Generate uniform random d-regular bipartite graphs.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 23 / 26

Outline

1

Introduction

2

Results on Random DAGs

3

Deterministic Broadcasting DAGs

4

Conclusion

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 24 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing Broadcasting in random DAGs with d = 2 and NAND processing

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing Broadcasting in random DAGs with d = 2 and NAND processing Broadcasting in “expander DAG” construction

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing Broadcasting in random DAGs with d = 2 and NAND processing Broadcasting in “expander DAG” construction Future Directions: Prove conjecture that for random DAG with odd d ≥ 3 (or d = 2), reconstruction impossible for all processing functions when δ ≥ δmaj (or δ ≥ δnand).

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing Broadcasting in random DAGs with d = 2 and NAND processing Broadcasting in “expander DAG” construction Future Directions: Prove conjecture that for random DAG with odd d ≥ 3 (or d = 2), reconstruction impossible for all processing functions when δ ≥ δmaj (or δ ≥ δnand). Find polynomial time construction of DAGs with sufficiently large d given some δ such that broadcasting possible.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Conclusion

Main Contributions: Broadcasting in random DAGs with d ≥ 3 and majority processing Broadcasting in random DAGs with d = 2 and NAND processing Broadcasting in “expander DAG” construction Future Directions: Prove conjecture that for random DAG with odd d ≥ 3 (or d = 2), reconstruction impossible for all processing functions when δ ≥ δmaj (or δ ≥ δnand). Find polynomial time construction of DAGs with sufficiently large d given some δ such that broadcasting possible. Construct DAGs with arbitrary d ≥ 3 and δ < δmaj, or d = 2 and δ < δnand, such that broadcasting possible.

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 25 / 26

Thank You!

• A. Makur, E. Mossel, Y. Polyanskiy (MIT)

Broadcasting on Random Networks 10 July 2019 26 / 26