1. Iterative Byzantine Vector Consensus in Incomplete Graphs Nitin Vaidya University of Illinois at Urbana-Champaign ICDCN presentation by Srikanth Sastry Google

2. Byzantine Fault Model  Faulty nodes may misbehave arbitrarily, including sending inconsistent messages to neighbors 2

3. This Paper 3  Which directed graphs can solve iterative Byzantine vector consensus?

4. Vector Consensus  Input: d-dimensional vector of reals  Output: Vector in convex hull of inputs at fault-free nodes 4

5. Exact Vector Consensus in Complete Graphs Necessary and sufficient condition for a complete graph of N nodes tolerating f faults  Synchronous: N ≥ max(3f, (d+1)f+1)  Asynchronous: N ≥ (d+2)f+1 [STOC13: Mendes-Herlihy PODC13: Vaidya-Garg] 5

6. This Paper  Incomplete directed graphs  Iterative approximate consensus  Synchronous system … similar results can be obtained for asynchronous 6

7. Iterative Structure  Each node maintains a state: initial state = input vector  Each iteration … communicate with neighbors … update state  Output = state at termination

8. Correctness Conditions  Termination after finite number of iterations  Validity: State of a fault-free node always in convex hull of input at fault-free nodes (“Minimal state”)  ε-Agreement: Corresponding elements of output vectors at fault-free nodes within ε at termination 8

9. Necessary Condition on Network Graph 9

10. Notation A B if there exists a node in B with c+1 incoming edges from A 10 c

11. A Necessary Condition 11

12. A Necessary Condition  Proof by contradiction … omitted  Do not know if this condition is tight 12

13. A Sufficient Condition 13

14. A Sufficient Condition 14 Partition nodes into 4 sets F L C R ≥ df+1 i j

15. A Sufficient Condition potential fault set ≥ df+1 i j L, R non-empty

16. A Sufficient Condition 16 ≥ df+1 i j i or j exists

17. Sufficiency Proof  Algorithm & its correctness under the sufficient condition 17

18. Iterative Algorithm  Obtain current state of all neighbors  Find Tverberg point for each subset of (d+1)f+1 states (own & neighbors)  New state = average of all Tverberg points Terminate after number of rounds that depends on ε and the input domain 18

19. Tverberg Point for a Multiset S  Partition S into f+1 non-empty subsets such that the intersection of their convex hull is non-empty  Any point in the intersection is a Tverberg point  Intersection is non-empty if |S| ≥ (d+1)f+1 [Tverberg Theorem] 19

20. Proof of Correctness  V i [t] = state of fault-free node i at the end of iteration t  Think of the vector state as a point in Euclidean space  V[t] = Vector of states of fault-free nodes 20

21. Proof of Correctness  For a given execution, always possible to find a row stochastic matrix M[t] such that V[t] = M[t] V[t-1]  Row stochastic: All elements non-negative Each row adds to 1 21

22. Proof of Correctness  Rest follows from the previous matrix equation … proof structure used in our prior work too … borrowed from work on non-fault-tolerant consensus (decentralized consensus [Tsitsiklis])  Our main contribution is to prove that the matrix form holds with Byzantine faults … proof omitted 22

23. Summary  Vector consensus  Using simple iterative structure  Minimal state  Sufficient condition ≠ Necessary condition 23

24. Open Problem  Finding conditions such that Sufficient condition = Necessary condition 24

25. Thanks ! 25