broadcasting on random networks
play

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

Jan 20, 2024 •329 likes •1.42k views

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

  1. Motivation: Broadcasting on Trees Intuition: In tree T , layers grow exponentially with rate br( T ) and information contracts with rate (1 − 2 δ ) 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

  2. 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

  3. Formal Model: Broadcasting on Bounded Indegree DAGs Fix infinite DAG with single source node. X k , j ∈ { 0 , 1 } – node random variable at j th position in level k level �,� �,� level �,� �,� level �,� �,� �,� �,� level �,� �,� �,� � �� �,� � �� A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 7 / 26

  4. Formal Model: Broadcasting on Bounded Indegree DAGs Fix infinite DAG with single source node. X k , j ∈ { 0 , 1 } – node random variable at j th position in level k L k – 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

  5. Formal Model: Broadcasting on Bounded Indegree DAGs Fix infinite DAG with single source node. X k , j ∈ { 0 , 1 } – node random variable at j th position in level k L k – 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

  6. Formal Model: Broadcasting on Bounded Indegree DAGs Fix infinite DAG with single source node. X k , j ∈ { 0 , 1 } – node random variable at j th position in level k L k – number of nodes at level k d – indegree of each node � 1 � X 0 , 0 ∼ Bernoulli level �,� � 2 Every edge is independent �,� BSC with crossover level �,� �,� � 0 , 1 � � probability δ ∈ . 2 level � �,� �,� �,� �,� level � vertices �,� �,� �,� � �� �,� � �� A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 7 / 26

  7. Formal Model: Broadcasting on Bounded Indegree DAGs Fix infinite DAG with single source node. X k , j ∈ { 0 , 1 } – node random variable at j th position in level k L k – number of nodes at level k d – indegree of each node � 1 � X 0 , 0 ∼ Bernoulli level �,� � 2 Every edge is independent �,� BSC with crossover level �,� �,� � 0 , 1 � � probability δ ∈ . 2 level � Nodes combine inputs with �,� �,� �,� �,� d -ary Boolean functions. This defines joint distribution of { X k , j } . level � vertices �,� �,� �,� � �� �,� � �� A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 7 / 26

  8. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? level �,� � �,� level �,� �,� � level � �,� �,� �,� �,� level � vertices �,� �,� �,� � �� �,� � �� A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 8 / 26

  9. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? Binary Hypothesis Testing: Let ˆ X k ML ( X k ) ∈ { 0 , 1 } be maximum likelihood (ML) decoder with probability of error: � � P ( k ) X k ˆ ML � P ML ( X k ) � = X 0 , 0 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 8 / 26

  10. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? Binary Hypothesis Testing: Let ˆ X k ML ( X k ) ∈ { 0 , 1 } be maximum likelihood (ML) decoder with probability of error: = 1 � � � � P ( k ) X k ˆ ML � P � � ML ( X k ) � = X 0 , 0 1 − � P X k | X 0 =1 − P X k | X 0 =0 . � TV 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 8 / 26

  11. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? Binary Hypothesis Testing: Let ˆ X k ML ( X k ) ∈ { 0 , 1 } be maximum likelihood (ML) decoder with probability of error: = 1 � � � � P ( k ) X k ˆ ML � P � � ML ( X k ) � = X 0 , 0 1 − � P X k | X 0 =1 − P X k | X 0 =0 . � TV 2 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

  12. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? Binary Hypothesis Testing: Let ˆ X k ML ( X k ) ∈ { 0 , 1 } be maximum likelihood (ML) decoder with probability of error: = 1 � � � � P ( k ) X k ˆ ML � P � � ML ( X k ) � = X 0 , 0 1 − � P X k | X 0 =1 − P X k | X 0 =0 . � TV 2 By data processing inequality, TV distance contracts as k increases. Broadcasting/Reconstruction possible if: ML < 1 k →∞ P ( k ) � � lim ⇔ lim � P X k | X 0 =1 − P X k | X 0 =0 TV > 0 � 2 k →∞ and Broadcasting/Reconstruction impossible if: ML = 1 k →∞ P ( k ) � � lim ⇔ lim � P X k | X 0 =1 − P X k | X 0 =0 TV = 0 . � 2 k →∞ A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 8 / 26

  13. Broadcasting Problem Let X k � ( X k , 0 , . . . , X k , L k − 1 ). Can we decode X 0 from X k as k → ∞ ? Binary Hypothesis Testing: Let ˆ X k ML ( X k ) ∈ { 0 , 1 } be maximum likelihood (ML) decoder with probability of error: = 1 � � � � P ( k ) X k ˆ ML � P � � ML ( X k ) � = X 0 , 0 1 − � P X k | X 0 =1 − P X k | X 0 =0 . � TV 2 By data processing inequality, TV distance contracts as k increases. Broadcasting/Reconstruction possible iff: ML < 1 k →∞ P ( k ) � � lim ⇔ lim � P X k | X 0 =1 − P X k | X 0 =0 TV > 0 . � 2 k →∞ For which δ , d , { L k } , and Boolean processing functions is reconstruction possible? A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 8 / 26

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. Outline Introduction 1 Results on Random DAGs 2 Phase Transition for Majority Processing Impossibility Results for Broadcasting Phase Transition for NAND Processing Deterministic Broadcasting DAGs 3 Conclusion 4 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 10 / 26

  20. �,� level � �,� level �,� �,� � level � �,� �,� �,� �,� level � vertices �,� �,� �,� � �� �,� � �� Random DAG Model Fix { L k } and d > 1. A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 11 / 26

  21. Random DAG Model Fix { L k } and d > 1. For each node X k , 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

  22. Random DAG Model Fix { L k } and d > 1. For each node X k , 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

  23. Random DAG Model Fix { L k } and d > 1. For each node X k , 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 � L k − 1 1 σ k � j =0 X k , j – sufficient statistic of X k for σ 0 = X 0 , 0 L k 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

  24. ✶ Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 12 / 26

  25. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if L k ≥ C ( δ, d ) log( k ), then reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 S k � ✶ � � is majority decoder. 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 12 / 26

  26. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if L k ≥ C ( δ, d ) log( k ), then reconstruction possible: < 1 � � � � P ( k ) ˆ lim ML ( G ) ≤ lim sup S k � = X 0 , 0 k →∞ E P 2 k →∞ where ˆ σ k ≥ 1 S k � ✶ � � is majority decoder. 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 12 / 26

  27. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if L k ≥ C ( δ, d ) log( k ), then reconstruction possible: < 1 � � � � P ( k ) ˆ lim ML ( G ) ≤ lim sup S k � = X 0 , 0 k →∞ E P 2 k →∞ where ˆ σ k ≥ 1 S k � ✶ � � is majority decoder. 2 δ maj , 1 � � Suppose δ ∈ . Then, there exists D ( δ, d ) > 1 such that if 2 � D ( δ, d ) k � L k = o , then reconstruction impossible: ML ( G ) = 1 k →∞ P ( k ) lim G - a . s . 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 12 / 26

  28. Proof Intuition Suppose d = 3 and δ maj = 1 6 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  29. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], level k has i.i.d. random bits i.i.d. X k , j ∼ 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

  30. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], level k has i.i.d. random bits i.i.d. X k , j ∼ majority(Bernoulli( σ ∗ δ ) , Bernoulli( σ ∗ δ ) , Bernoulli( σ ∗ δ )) where σ ∗ δ = σ (1 − δ ) + δ (1 − σ ), and L k − 1 � L k σ k = X k , j ∼ binomial( L k , E [ σ k | σ k − 1 = σ ] ) . j =0 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  31. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], level k has i.i.d. random bits i.i.d. X k , j ∼ majority(Bernoulli( σ ∗ δ ) , Bernoulli( σ ∗ δ ) , Bernoulli( σ ∗ δ )) where σ ∗ δ = σ (1 − δ ) + δ (1 − σ ), and L k − 1 � L k σ k = X k , j ∼ binomial( L k , g δ ( σ )) . j =0 Define the cubic polynomial: g δ ( σ ) � E [ σ k | σ k − 1 = σ ] = P ( X k , j = 1 | σ k − 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

  32. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  33. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . Fixed Point Analysis: Case δ < δ maj : 1 � 1 2 0 0 1 1 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  34. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . Fixed Point Analysis: σ k “concentrates” at fixed point near X 0 , 0 Case δ < δ maj : 3 fixed points 1 � 1 2 0 0 1 1 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  35. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . Fixed Point Analysis: Case δ < δ maj : 3 fixed points Case δ > δ maj : 1 1 � � 1 1 2 2 0 0 0 1 1 0 1 1 2 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  36. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . Fixed Point Analysis: σ k → 1 2 a . s . if δ > δ maj and L k = ω (log( k )) Case δ < δ maj : 3 fixed points Case δ > δ maj : 1 fixed point 1 1 � � 1 1 2 2 0 0 0 1 1 0 1 1 2 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  37. Proof Intuition Suppose d = 3 and δ maj = 1 6 . Conditioned on σ k − 1 = σ ∈ [0 , 1], L k σ k ∼ binomial( L k , g δ ( σ )). Define the cubic polynomial g δ ( σ ) � ( σ ∗ δ ) 3 + 3( σ ∗ δ ) 2 (1 − σ ∗ δ ). Concentration: For large k , σ k ≈ g δ ( σ k − 1 ) given σ k − 1 . Converse uses key property : Lip( g δ ) ≤ 1 ⇔ g δ has unique fixed point. Case δ < δ maj : 3 fixed points Case δ > δ maj : 1 fixed point 1 1 � � 1 1 2 2 0 0 0 1 1 0 1 1 2 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 13 / 26

  38. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if � � P ( k ) < 1 L k ≥ C ( δ, d ) log( k ), then lim ML ( G ) 2 . k →∞ E δ maj , 1 � � Suppose δ ∈ . Then, there exists D ( δ, d ) > 1 such that if 2 k →∞ P ( k ) D ( δ, d ) k � ML ( G ) = 1 � L k = o , then lim 2 G - a . s . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 14 / 26

  39. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if � � P ( k ) < 1 L k ≥ C ( δ, d ) log( k ), then lim ML ( G ) 2 . k →∞ E δ maj , 1 � � Suppose δ ∈ . Then, there exists D ( δ, d ) > 1 such that if 2 k →∞ P ( k ) D ( δ, d ) k � ML ( G ) = 1 � L k = o , then lim 2 G - a . s . Remarks: δ maj = 1 6 for d = 3 appears in reliable computation [vNe56, HW91]. δ maj for odd d ≥ 3 also relevant in reliable computation [ES03]. δ maj 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

  40. Random DAG with Majority Processing Theorem (Phase Transition for d ≥ 3) Consider random DAG model with d ≥ 3 and majority processing (with 2 d − 2 ties broken randomly). Let δ maj � 1 2 − ⌈ d / 2 ⌉ ). ⌈ d / 2 ⌉ ( d Suppose δ ∈ (0 , δ maj ). Then, there exists C ( δ, d ) > 0 such that if � � P ( k ) < 1 L k ≥ C ( δ, d ) log( k ), then lim ML ( G ) 2 . k →∞ E δ maj , 1 � � Suppose δ ∈ . Then, there exists D ( δ, d ) > 1 such that if 2 k →∞ P ( k ) D ( δ, d ) k � ML ( G ) = 1 � L k = o , then lim 2 G - a . s . Questions: Broadcasting possible with sub-logarithmic L k ? Broadcasting possible when δ > δ maj with other processing functions? What about d = 2? A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 14 / 26

  41. Optimality of Logarithmic Layer Size Growth Broadcasting possible with sub-logarithmic L k ? Proposition (Layer Size Impossibility Result) For any deterministic DAG, if: log( k ) � , L k ≤ � 1 d log 2 δ then reconstruction impossible for all processing functions: ML = 1 k →∞ P ( k ) lim 2 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 15 / 26

  42. Optimality of Logarithmic Layer Size Growth Broadcasting possible with sub-logarithmic L k ? Proposition (Layer Size Impossibility Result) For any deterministic DAG, if: log( k ) � , L k ≤ � 1 d log 2 δ then reconstruction impossible for all processing functions: ML = 1 k →∞ P ( k ) lim 2 . No, broadcasting impossible with sub-logarithmic L k ! A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 15 / 26

  43. Partial Converse Results Broadcasting possible when δ > δ maj with other processing functions? Proposition (Single Vertex Reconstruction) Consider random DAG model with d ≥ 3. If δ ∈ (0 , δ maj ), L k ≥ C ( δ, d ) log( k ), and processing functions are majority, then single vertex reconstruction possible: P ( X k , 0 � = X 0 , 0 ) < 1 lim sup 2 . k →∞ A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 16 / 26

  44. Partial Converse Results Broadcasting possible when δ > δ maj with other processing functions? Proposition (Single Vertex Reconstruction) Consider random DAG model with d ≥ 3. If δ ∈ (0 , δ maj ), L k ≥ C ( δ, d ) log( k ), and processing functions are majority, then single vertex reconstruction possible: P ( X k , 0 � = X 0 , 0 ) < 1 lim sup 2 . k →∞ δ maj , 1 � � � d 2 k � If δ ∈ , d is odd, lim k →∞ L k = ∞ , and inf n ≥ k L n = O , then 2 single vertex reconstruction impossible for all processing functions (which may be graph dependent): �� � � lim � P X k , 0 | G , X 0 , 0 =1 − P X k , 0 | G , X 0 , 0 =0 = 0 . k →∞ E � � � TV Remark: Converse uses reliable computation results [HW91, ES03]. A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 16 / 26

  45. Partial Converse Results Broadcasting possible when δ > δ maj with other processing functions? Proposition (Information Percolation [ES99, PW17]) For any deterministic DAG, if: � � δ > 1 1 1 √ 2 − and L k = o ((1 − 2 δ ) 2 d ) k 2 d then reconstruction impossible for all processing functions: ML = 1 k →∞ P ( k ) lim 2 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 16 / 26

  46. Partial Converse Results Broadcasting possible when δ > δ maj with other processing functions? Proposition (Information Percolation [ES99, PW17]) For any deterministic DAG, if: � � δ > 1 1 1 √ 2 − > δ maj and L k = o ((1 − 2 δ ) 2 d ) k 2 d then reconstruction impossible for all processing functions: ML = 1 k →∞ P ( k ) lim 2 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 16 / 26

  47. ✶ 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 δ nand � 3 − 7 . 4 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 17 / 26

  48. 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 δ nand � 3 − 7 . 4 Suppose δ ∈ (0 , δ nand ). Then, there exist C ( δ ) > 0 and t ( δ ) ∈ (0 , 1) such that if L k ≥ C ( δ ) log( k ), then reconstruction possible: < 1 � P ( k ) � � � ˆ lim ML ( G ) ≤ lim sup T 2 k � = X 0 , 0 k →∞ E P 2 k →∞ where ˆ T k � ✶ { σ k ≥ t ( δ ) } is thresholding decoder. A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 17 / 26

  49. 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 δ nand � 3 − 7 . 4 Suppose δ ∈ (0 , δ nand ). Then, there exist C ( δ ) > 0 and t ( δ ) ∈ (0 , 1) such that if L k ≥ C ( δ ) log( k ), then reconstruction possible: < 1 � P ( k ) � � � ˆ lim ML ( G ) ≤ lim sup T 2 k � = X 0 , 0 k →∞ E P 2 k →∞ where ˆ T k � ✶ { σ k ≥ t ( δ ) } is thresholding decoder. δ nand , 1 � � Suppose δ ∈ . Then, there exist D ( δ ) , E ( δ ) > 1 such that if 2 � D ( δ ) k � L k = o and lim inf k →∞ L k > E ( δ ), then reconstruction impossible: ML ( G ) = 1 k →∞ P ( k ) lim G - a . s . 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 17 / 26

  50. 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 δ nand � 3 − 7 . 4 Suppose δ ∈ (0 , δ nand ). Then, there exist C ( δ ) > 0 and t ( δ ) ∈ (0 , 1) such that if L k ≥ C ( δ ) log( k ), then reconstruction possible: < 1 � P ( k ) � � � ˆ lim ML ( G ) ≤ lim sup T 2 k � = X 0 , 0 k →∞ E P 2 k →∞ where ˆ T k � ✶ { σ k ≥ t ( δ ) } is thresholding decoder. δ nand , 1 � � Suppose δ ∈ . Then, there exist D ( δ ) , E ( δ ) > 1 such that if 2 � D ( δ ) k � L k = o and lim inf k →∞ L k > E ( δ ), then reconstruction impossible: ML ( G ) = 1 k →∞ P ( k ) lim G - a . s . 2 Remark: δ nand appears in reliable computation [EP98, Ung07]. A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 17 / 26

  51. Outline Introduction 1 Results on Random DAGs 2 Deterministic Broadcasting DAGs 3 Existence of DAGs where Broadcasting is Possible Construction of DAGs where Broadcasting is Possible Conclusion 4 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 18 / 26

  52. 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

  53. 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 , δ maj ), and L k ≥ C ( δ, d ) log( k ), there exists DAG with majority processing functions such that reconstruction possible: ML < 1 k →∞ P ( k ) lim 2 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 19 / 26

  54. 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 , δ maj ), and L k ≥ C ( δ, d ) log( k ), there exists DAG with majority processing functions such that reconstruction possible: ML < 1 k →∞ P ( k ) lim 2 . 0 , 1 � � Can we construct such DAGs for any δ ∈ ? 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 19 / 26

  55. � � 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 B n = ( U n , V n , E n ) with disjoint vertex sets U n , V n of cardinality | U n | = | V n | = n , edge multiset E n , and the lossless expansion property: � 2 � n ∀ S ⊆ U n , | S | = ⇒ | Γ( S ) | ≥ 1 − d | S | d 6 / 5 d 1 / 5 where Γ( S ) � { v ∈ V n : ∃ u ∈ S , ( u , v ) ∈ E n } is neighborhood of S . vertices � � A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 20 / 26

  56. 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 B n = ( U n , V n , E n ) with disjoint vertex sets U n , V n of cardinality | U n | = | V n | = n , edge multiset E n , and the lossless expansion property: � 2 � n ∀ S ⊆ U n , | S | = ⇒ | Γ( S ) | ≥ 1 − d | S | d 6 / 5 d 1 / 5 where Γ( S ) � { v ∈ V n : ∃ u ∈ S , ( u , v ) ∈ E n } is neighborhood of S . � � A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 20 / 26

  57. 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 B n = ( U n , V n , E n ) with disjoint vertex sets U n , V n of cardinality | U n | = | V n | = n , edge multiset E n , and the lossless expansion property: � 2 � n ∀ S ⊆ U n , | S | = ⇒ | Γ( S ) | ≥ 1 − d | S | d 6 / 5 d 1 / 5 where Γ( S ) � { v ∈ V n : ∃ u ∈ S , ( u , v ) ∈ E n } 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

  58. Construction of DAGs where Broadcasting is Possible 0 , 1 � � Fix any δ ∈ and any sufficiently large odd d = d ( δ ). 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 21 / 26

  59. Construction of DAGs where Broadcasting is Possible 0 , 1 � � Fix any δ ∈ and any sufficiently large odd d = d ( δ ). 2 Fix L 0 = 1, L k = N for k ∈ { 1 , . . . , ⌊ M ⌋} where N = N ( δ ) sufficiently N / (4 d 12 / 5 ) � � large and M = exp , and ∀ r ≥ 1 , M 2 r − 1 < k ≤ M 2 r , L k = 2 r N such that L k = Θ(log( k )). A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 21 / 26

  60. Construction of DAGs where Broadcasting is Possible 0 , 1 � � Fix any δ ∈ and any sufficiently large odd d = d ( δ ). 2 Fix L 0 = 1, L k = N for k ∈ { 1 , . . . , ⌊ M ⌋} where N = N ( δ ) sufficiently N / (4 d 12 / 5 ) � � large and M = exp , and ∀ r ≥ 1 , M 2 r − 1 < k ≤ M 2 r , L k = 2 r N such that L k = Θ(log( k )). Construct bounded degree deterministic “expander DAG”: Each X 1 , j has one edge from X 0 , 0 . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 21 / 26

  61. Construction of DAGs where Broadcasting is Possible 0 , 1 � � Fix any δ ∈ and any sufficiently large odd d = d ( δ ). 2 Fix L 0 = 1, L k = N for k ∈ { 1 , . . . , ⌊ M ⌋} where N = N ( δ ) sufficiently N / (4 d 12 / 5 ) � � large and M = exp , and ∀ r ≥ 1 , M 2 r − 1 < k ≤ M 2 r , L k = 2 r N such that L k = Θ(log( k )). Construct bounded degree deterministic “expander DAG”: Each X 1 , j has one edge from X 0 , 0 . Case L k +1 = L k : Edge multiset X k → X k +1 given by expander B L k . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 21 / 26

  62. Construction of DAGs where Broadcasting is Possible 0 , 1 � � Fix any δ ∈ and any sufficiently large odd d = d ( δ ). 2 Fix L 0 = 1, L k = N for k ∈ { 1 , . . . , ⌊ M ⌋} where N = N ( δ ) sufficiently N / (4 d 12 / 5 ) � � large and M = exp , and ∀ r ≥ 1 , M 2 r − 1 < k ≤ M 2 r , L k = 2 r N such that L k = Θ(log( k )). Construct bounded degree deterministic “expander DAG”: Each X 1 , j has one edge from X 0 , 0 . Case L k +1 = L k : Edge multiset X k → X k +1 given by expander B L k . Case L k +1 = 2 L k : Both edge multisets X k → ( X k +1 , 0 , . . . , X k +1 , L k − 1 ) and X k → ( X k +1 , L k , . . . , X k +1 , L k +1 − 1 ) given by expander B L k . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 21 / 26

  63. 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

  64. 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

  65. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  66. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 level 𝑁 + 1 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  67. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 𝐶 � level 𝑁 + 1 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  68. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 𝐶 � 𝐶 � level 𝑁 + 1 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  69. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 𝐶 � 𝐶 � level 𝑁 + 1 𝐶 �� level 𝑁 + 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  70. Construction of DAGs where Broadcasting is Possible Illustration of “Expander DAG”: level 0 level 1 𝐶 � level 2 level 𝑁 ‐ 1 𝐶 � level 𝑁 𝐶 � 𝐶 � level 𝑁 + 1 𝐶 �� level 𝑁 + 2 level 𝑁 � ‐ 1 𝐶 �� level 𝑁 � A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 22 / 26

  71. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  72. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  73. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . Let S k � { nodes equal to 1 at level k } . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  74. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . Let S k � { nodes equal to 1 at level k } . Call node at level k + 1 “bad” if it is connected to ≥ 1 + d 4 nodes in S k . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  75. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . Let S k � { nodes equal to 1 at level k } . Call node at level k + 1 “bad” if it is connected to ≥ 1 + d 4 nodes in S k . Expansion Property: If |S k | ≤ d − 6 / 5 N , then we have ≤ 8 d − 7 / 5 N “bad” nodes. A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  76. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . Let S k � { nodes equal to 1 at level k } . Call node at level k + 1 “bad” if it is connected to ≥ 1 + d 4 nodes in S k . Expansion Property: If |S k | ≤ d − 6 / 5 N , then we have ≤ 8 d − 7 / 5 N “bad” nodes. Main Lemma: Given |S k | ≤ d − 6 / 5 N , we have |S k +1 | ≤ d − 6 / 5 N 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

  77. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proof Sketch: Suppose edges from level k to k + 1 given by expander B N . Let S k � { nodes equal to 1 at level k } . Call node at level k + 1 “bad” if it is connected to ≥ 1 + d 4 nodes in S k . Expansion Property: If |S k | ≤ d − 6 / 5 N , then we have ≤ 8 d − 7 / 5 N “bad” nodes. Main Lemma: Given |S k | ≤ d − 6 / 5 N , we have |S k +1 | ≤ d − 6 / 5 N with high probability, as “good” nodes have low probability of becoming 1. If X 0 , 0 = 0, then |S k | likely to remain small as k → ∞ . A. Makur, E. Mossel, Y. Polyanskiy (MIT) Broadcasting on Random Networks 10 July 2019 23 / 26

  78. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proposition (Computational Complexity of DAG Construction) 0 , 1 � � For any δ ∈ , the d -regular bipartite expander graphs for levels 2 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

  79. Construction of DAGs where Broadcasting is Possible Theorem (Broadcasting in Expander DAG) For “expander DAG” with majority processing, reconstruction possible: < 1 � � ˆ lim sup S k � = X 0 , 0 P 2 k →∞ where ˆ σ k ≥ 1 � � S k = ✶ is majority decoder. 2 Proposition (Computational Complexity of DAG Construction) 0 , 1 � � For any δ ∈ , the d -regular bipartite expander graphs for levels 2 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

Recommend

REQUEST FOR BID / PROPOSAL Middle East Broadcasting Networks, Inc. (MBN) Springfield, VA

REQUEST FOR BID / PROPOSAL Middle East Broadcasting Networks, Inc. (MBN) Springfield, VA

REQUEST FOR BID / PROPOSAL Middle East Broadcasting Networks, Inc. (MBN) Springfield, VA Broadcast Graphics Presentation System for Alhurra TV Background Middle East Broadcasting Networks, Inc. (MBN) is a 501(c)(3) non-profit corporation

228 views • 9 slides
Performance Evaluation and Improvement of Broadcasting Algorithms in Ad Hoc Networks Hao Zhang

Performance Evaluation and Improvement of Broadcasting Algorithms in Ad Hoc Networks Hao Zhang

Performance Evaluation and Improvement of Broadcasting Algorithms in Ad Hoc Networks Hao Zhang and Zhong Ping Jiang ECE Department Polytechnic University, New York 1 Ad Hoc Networks No base station, self-organized Energy and bandwidth

364 views • 21 slides
Minimum-Energy Broadcasting in Multi-hop Wireless Networks Using a Single Broadcast Tree * IOANNIS

Minimum-Energy Broadcasting in Multi-hop Wireless Networks Using a Single Broadcast Tree * IOANNIS

Mobile Networks and Applications 11, 361375, 2006 2006 Springer Science + Business Media, LLC. Manufactured in The Netherlands. C DOI: 10.1007/s11036-006-5189-6 Minimum-Energy Broadcasting in Multi-hop Wireless Networks Using a Single

419 views • 15 slides
Applications of Random Networks Analysis of real networks How to build revisited Complex

Applications of Random Networks Analysis of real networks How to build revisited Complex

Applications of Random Networks Applications of Random Networks Analysis of real networks How to build revisited Complex Networks, Course 295A, Spring, 2008 Motifs References Prof. Peter Dodds Department of Mathematics &amp; Statistics

514 views • 15 slides
Random Access Networks Random Access networks are characterised by the absence of a channel-

Random Access Networks Random Access networks are characterised by the absence of a channel-

Chapter 3 Random Access Networks Random Access networks are characterised by the absence of a channel- controlled access mechanism. To some extent, a station is free to broadcast its packets on the network at a time determined by the station

309 views • 15 slides
BROADCASTING IN WIRELESS NETWORKS WITH NETWORK CODING Pouya Ostovari, Jie Wu, and Abdallah

BROADCASTING IN WIRELESS NETWORKS WITH NETWORK CODING Pouya Ostovari, Jie Wu, and Abdallah

DEADLINE-AWARE BROADCASTING IN WIRELESS NETWORKS WITH NETWORK CODING Pouya Ostovari, Jie Wu, and Abdallah Khreishah Computer &amp; Information Sciences Department, Temple University, USA Center for Networked Computing Agenda

1k views • 69 slides
Outline Applications of Random Networks Random Networks Applications of Random Networks

Outline Applications of Random Networks Random Networks Applications of Random Networks

Applications of Outline Applications of Random Networks Random Networks Applications of Random Networks Analysis of real Analysis of real networks networks How to build revisited How to build revisited Complex Networks, Course 295A,

367 views • 4 slides
Networks and Governance in Broadcasting Stephen Pratten Department of Management Kings

Networks and Governance in Broadcasting Stephen Pratten Department of Management Kings

Networks and Governance in Broadcasting Stephen Pratten Department of Management Kings College London CBR Summit: 29-30 March 2006 Innovation and Governance Motivation Network forms of organisation have been identified within the

350 views • 19 slides
The Trend of Digital Broadcasting in Japan February 12 th 2004 Broadcasting Technology Division

The Trend of Digital Broadcasting in Japan February 12 th 2004 Broadcasting Technology Division

The Trend of Digital Broadcasting in Japan February 12 th 2004 Broadcasting Technology Division Information and Communications Policy Bureau MPHPT Digital Terrestrial Television Broadcasting Topics now Dec. 1st 2003 Start of Digital

694 views • 35 slides
Popular Communication Operations - One-one sending/one-all broadcasting - All-all broadcasting P

Popular Communication Operations - One-one sending/one-all broadcasting - All-all broadcasting P

CS140, 2014 II-1 Popular Communication Operations - One-one sending/one-all broadcasting - All-all broadcasting P 0 P 1 P 2 P 0 P 2 P 1 - Accumulation (or gather) P 0 . . . P 2 P n P 1 - Reduction I nitially : V 0 V 1 . . . V p

385 views • 6 slides
Random Graphs Random Graphs 1 / 19 Generative Models Could hope to understand networks in the

Random Graphs Random Graphs 1 / 19 Generative Models Could hope to understand networks in the

Random Graphs Random Graphs 1 / 19 Generative Models Could hope to understand networks in the real world if we had descriptions of the processes that generate them. Describe such processes via random algorithms. The likelihood of generating a

472 views • 19 slides
Core Models of Complex Networks Principles of Complex Systems Generalized random networks

Core Models of Complex Networks Principles of Complex Systems Generalized random networks

Core Models of Complex Networks Core Models of Complex Networks Principles of Complex Systems Generalized random networks CSYS/MATH 300, Spring, 2013 | #SpringPoCS2013 Small-world networks Main story Generalized affiliation networks

2.16k views • 99 slides
Peer-to-Peer Networks 09 Random Graphs for Peer-to-Peer-Networks Christian Ortolf Technical

Peer-to-Peer Networks 09 Random Graphs for Peer-to-Peer-Networks Christian Ortolf Technical

Peer-to-Peer Networks 09 Random Graphs for Peer-to-Peer-Networks Christian Ortolf Technical Faculty Computer-Networks and Telematics University of Freiburg Peer-to-Peer Networking Facts Hostile environment - Legal situation - Egoistic

882 views • 57 slides
Outline Markov networks (a.k.a. Markov random fields) Markov Networks Reading: Michael

Outline Markov networks (a.k.a. Markov random fields) Markov Networks Reading: Michael

Outline Markov networks (a.k.a. Markov random fields) Markov Networks Reading: Michael Jordan, Graphical Models , Statistical Science (Special November 12, 2009 Issue on Bayesian Statistics), 19, 140- CS 486/686 155, 2004. University

325 views • 4 slides
Random walks on finite networks Andr e Schumacher &lt;schumach@tcs.hut.fi&gt; February 06, 2006

Random walks on finite networks Andr e Schumacher &lt;schumach@tcs.hut.fi&gt; February 06, 2006

Random walks on finite networks Andr e Schumacher &lt;schumach@tcs.hut.fi&gt; February 06, 2006 Doyle &amp; Snell 1.3.1-1.3.4 Random walks on finite networks [1] Overview Short review of recent electric network models Model of

403 views • 14 slides
Outline Random Networks Basics Basics Basics Definitions Definitions How to build

Outline Random Networks Basics Basics Basics Definitions Definitions How to build

Random Networks Random Networks Outline Random Networks Basics Basics Basics Definitions Definitions How to build Definitions How to build Complex Networks, Course 295A, Spring, 2008 Some visual examples Some visual examples How to

1.27k views • 16 slides
Telecommunication and Broadcasting Regulation in Europe - A need for convergent Regulation?

Telecommunication and Broadcasting Regulation in Europe - A need for convergent Regulation?

Telecommunication , Broadcasting and Tajikistans WTO Commitments Conference Organized by the Ministry of Economic Development and Trade and the OSCE Office in Tajikistan (OiT) Telecommunication and Broadcasting Regulation in Europe - A need

411 views • 12 slides
Good Morning! MCS1450/ BMS1301 Introduction to Broadcasting Ulrich Werner History and

Good Morning! MCS1450/ BMS1301 Introduction to Broadcasting Ulrich Werner History and

Good Morning! MCS1450/ BMS1301 Introduction to Broadcasting Ulrich Werner History and development of broadcasting, including the influence of broadcasting media in the democratic society. Course description in the curriculum.

853 views • 31 slides
Maintenance of random logical networks Romaric Duvignau DCS seminar, Chalmers October 4, 2017

Maintenance of random logical networks Romaric Duvignau DCS seminar, Chalmers October 4, 2017

Maintenance of random logical networks Romaric Duvignau DCS seminar, Chalmers October 4, 2017 Plan 1 A Quick Example of P2P Networks 2 A New Model of Evolution 3 A Concrete Example: Uniform k -out Random Graphs 4 A More General Question Romaric

976 views • 87 slides
Outline Scheidegger Networks Networks Scheidegger NetworksA Bonus First return First

Outline Scheidegger Networks Networks Scheidegger NetworksA Bonus First return First

Scheidegger Outline Scheidegger Networks Networks Scheidegger NetworksA Bonus First return First return random walk random walk Calculation References References Complex Networks, Course 295A, Spring, 2008 First return random walk

468 views • 3 slides
Bayesian networks (1) Lirong Xia Random variables and joint distributions A random variable is

Bayesian networks (1) Lirong Xia Random variables and joint distributions A random variable is

Bayesian networks (1) Lirong Xia Random variables and joint distributions A random variable is a variable with a domain Random variables: capital letters, e.g. W, D, L values: small letters, e.g. w, d, l A joint distribution over

898 views • 32 slides
Networks out of Control: Models and Methods for Random Networks Matthias Grossglauser Patrick

Networks out of Control: Models and Methods for Random Networks Matthias Grossglauser Patrick

Networks out of Control: Models and Methods for Random Networks Matthias Grossglauser Patrick Thiran School of Computer and Communication Sciences Ecole Polytechnique F ed erale de Lausanne (EPFL) M. Grossglauser &amp; P. Thiran, 2005

1.77k views • 149 slides
Educational Programming Jim Princivalle The Corporation for Public Broadcasting The Public

Educational Programming Jim Princivalle The Corporation for Public Broadcasting The Public

The Evolution of Educational Programming Jim Princivalle The Corporation for Public Broadcasting The Public Broadcasting Act of 1967 Signed into law by Lyndon B. Johnson Established the Corporation for Public Broadcasting (CPB)

347 views • 12 slides
Broadcasting &amp; ICT in the MENA Countries -Prospects &amp; Investment Opportunities-

Broadcasting &amp; ICT in the MENA Countries -Prospects &amp; Investment Opportunities-

Broadcasting &amp; ICT in the MENA Countries -Prospects &amp; Investment Opportunities- Chairmans Introduction Dr. Fares Lubbadeh Statu tus s of TV Broadcasting adcasting in the Arab ab World rld In accordance with ASBU latest

329 views • 11 slides

More recommend