Localized Self- healing using Expanders Gopal Pandurangan Nanyang Technological University, Singapore Amitabh Trehan Technion - Israel Institute of Technology, Haifa, IL TTI-C heal

PODC’11 G. Pandurangan, A. Trehan Epic Fail Adsense Mar 2010, Google, May 15, 2009 Twitter, August 6, 2009 Facebook, August 6, 2009 Skype, August 15, 2007

PODC’11 G. Pandurangan, A. Trehan How to self-heal? Brain: component fails, brain rewires and does without it Computer networks: components fail, network fails until components fixed.

An autonomic system Self-managing: Self-configuring Self-healing Self-optimizing Self-protecting PhD Dissertation’10 Amitabh Trehan

PODC’11 G. Pandurangan, A. Trehan Autonomic Computing IBM’s autonomic computing initiative Self-CHOP

PODC’11 G. Pandurangan, A. Trehan Self-healing A self-healing system, starting from a correct state, under attack from an adversary, goes only temporarily out of a correct state. Our work: Under attack from powerful adversary, maintain certain topological properties within acceptable bounds.

PODC’11 G. Pandurangan, A. Trehan Ensuring Robustness Want to ensure that our network is robust to node failures. Idea: build some redundancy into the network? Example: Connectivity Use k-connected graph. Price: degree must be at least k.

PODC’11 G. Pandurangan, A. Trehan Ensuring Robustness Want to ensure that our network is robust to node failures. Idea: build some redundancy into the network? Example: Connectivity Use k-connected graph. Price: degree must be at least k. Expensive!

PODC’11 G. Pandurangan, A. Trehan Model Start: a network G. An adversary inserts or deletes nodes. After each node addition/deletion, we can add and/or drop some edges between pairs of nearby nodes, to “heal” the network.

PODC’11 G. Pandurangan, A. Trehan Challenge 1: properties conflict Low degree increase => high diameter/stretch/ poorer expansion?

PODC’11 G. Pandurangan, A. Trehan Challenge 2: local fixing of global properties Low diameter => high degree increase? ✴ Limited global Information with nodes ✴ Limited resources and time constraints

PODC’11 G. Pandurangan, A. Trehan Our Self-healing Goals Healing should be fast, local and distributed. Certain topological properties should be maintained within bounds: - Connectivity - Degree - Stretch - Spectral properties (~Expansion/Conductance)

PODC’11 G. Pandurangan, A. Trehan A series of unfortunate events

PODC’11 G. Pandurangan, A. Trehan Xheal Goals Maintain connectivity. Edge Expansion of graph not much worse than ‘original’ graph. Distance between any two nodes shouldn’t increase by too much (low stretch). If vertex v starts with degree d, then its degree should never be much more than d. Healing should be fast and localized.

Comparing results G: healed network G’: graph of only insertions and original nodes

PODC’11 G. Pandurangan, A. Trehan Main Result A distributed algorithm, Xheal such that: Degree increase: Degree of node in G ≤  times degree in G’ G’ 3 G 5 vv

PODC’11 G. Pandurangan, A. Trehan Main Result (Contd..) Stretch: Distance between any two nodes in G = O(log n) times their distance in G’ G u v d(u,v) = 5 G’ u v d(u,v) = 3

PODC’11 G. Pandurangan, A. Trehan Main Result (Contd..) Spectral properties:  h(G t ) ≥ min(, h(G′ t )), for constant ≥ 1 (If G′ t is a (bounded degree) expander, so is G t )  (Bounded 2 nd smallest eigen value): Put equation here?

PODC’11 G. Pandurangan, A. Trehan Main Result (Contd..) Costs: - Deletions (by Law-Siu implementation): ‣ O(log n) rounds per deletion. ‣ Amortized O(k.(log n)A(p) * ) messages for healing by k-degree expander. * A(p) is average degree of deleted nodes over p deletions i.e (put in formula).

PODC’11 G. Pandurangan, A. Trehan Xheal: Outline Node inserted without restrictions. On node deletion, its neighbors reattach to form a primary expander cloud (k- degree expander). Over further deletions.... Multiple primaries joined by secondary expander clouds using ‘free’ nodes*. If no ‘free’ nodes (happens over a large number of deletions), clouds merged into new primary expander cloud. *`Free’ nodes: nodes not participating in secondary clouds.

PODC’11 G. Pandurangan, A. Trehan Xheal: Outline (Contd..) Each node of degree d part of at most d primary clouds and one secondary cloud. All clouds maintained as expanders. Efficient distributed implementation dependant on distributed expander construction (Using Law-Siu construction in this paper).

PODC’11 G. Pandurangan, A. Trehan Healing by expanders Lemma: At end of any timestep t, h(G t ) min(c’,h(G’ t )), c’ 1, a fixed constant generalization of base case: Assume deletion at timestep t =1, h(G 1 ) min(c’,h(G’ 1 )), c’ 1, a fixed constant

PODC’11 G. Pandurangan, A. Trehan Proof ๏ h(G 1 ) min(c’,h(G 0 )) - h(G) = E S,S’ (G) / S(G), S(G) ≤ n/2 I : k-reg expander subgraph replacing deleted node

PODC’11 G. Pandurangan, A. Trehan Proof (contd) E(I) intersection ES,S’(G_1) is null : latex equations here

PODC’11 G. Pandurangan, A. Trehan Proof (contd..) E(I) intersection ES,S’(G_1) not null : latex equations here, 2 parts

PODC’11 G. Pandurangan, A. Trehan Future Directions Improving distributed construction of expander graphs (will enhance Xheal): - Deterministic, or improved randomized. Self-healing routing Load-balanced self-healing: Chord like structures? Small world models? Extend model and algorithms: Byzantine faults, multiple failures, sensor networks, social networks, self-*.

PODC’11 G. Pandurangan, A. Trehan Summary Efficient, distributed algorithm Xheal for self- healing spectral properties (expansion), stretch, degree and connectivity. Xheal ensures maintainance of good expansion, stretch of at most log n, with constant degree increase, low latency and messages. Distributed implementation using distributed expander construction techniques; better techniques can improve implementation.

PODC’11 G. Pandurangan, A. Trehan Thank You

PODC’11 G. Pandurangan, A. Trehan More future directions Behavioral self-healing in social networks Self-* problems Network evolution and group formation Byzantine agreement: byzantine faults

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