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From the scale-free Web to Avogadro-scale Engineering Complexity Science Symposium, London, March 2004 Scott Kirkpatrick, Hebrew University, Jerusalem With thanks to: Byung-Gon Chun, Johannes J Schneider, Uri Gordon, Erik Aurell

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The physical Internet:

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Outline of this talk Information networks: vast in scale and scale-free Physical Internet – 10^9 hosts, 10^4 AS’s, accelerating growth Web of online information – 10^14 B on surface alone Genetic expression data – 10^9 Designed objects also reach 10^9 units Microprocessors and systems on a chip Logistics for airlines, modern armies New methods for optimal design on these scales Methods for distributed management of growth

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VLSE: very large scale engineering. Engineering has become a study of open systems “Avogadro scale” a serious possibility Physical methods and insights relevant System properties known to be “good” or “not good enough” at the extremes of their parameter space What happens in between these phases? We have limited tools for understanding such transitions. Physics of disordered materials: Sharp vs. smeared – Harris criterion Glass transitions Combinatorics on large scales Sharp property crossovers – Friedgut’s theorem We should care because computing is HARD at phase boundaries.

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SAT and 3-SAT: classic test cases Parameters: N variables, M constraining clauses, M/N constant For 3-SAT, phase diagram is known: M/N< 3.9“easy,” satisfiable (probably P) WALKSAT gives linear cost in easy regime 3.9 < M/N < 4.27“hard,” but still satisfiable 4.15 < M/N < 4.27 1-RSB stable, implies solutions cluster 4.27… < M/Nunsatisfiable, exponential cost Recent advances, applying message-passing, push boundary of solubility in the “hard-SAT” region from N = 300 (exact methods) to N = 10^7(using survey propagation as heuristic). General technique – soften discrete variables into beliefs or surveys

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Cost of WALKSAT Random walk search is linear in N, diverges in hard-SAT region

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SIDecimation truncates WALKSAT cost

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Surveys and Beliefs for the SAT problem Beliefs – probabilities that the spin is up or down Avg. over satisfying configurations, as estimated by the local tree Surveys – probabilities that the spin is up or down ( In all sat configurations) This leaves a third possibility – spins that do both at different times. Equations for beliefs and for surveys are nearly identical. We can define hybrid methods by interpolating. They prove useful. Use beliefs or surveys to guide decimation – this solves problem.

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Propagating surveys or beliefs in a “cavity”

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Self-organizing Networks, e.g. the Internet Model introduced by Papadimitriou Who pays for a network to come into existence? What happens when only selfish incentives are acted on? This is the domain of game theory The “price of anarchy” is the ratio of best to worst case Nash equilibria (Trivia question: of what utility function is TCP/IP a Nash equilibrium?) Recently studied by Fabrikant et al, others. Can compare selfish optimization (Nash equilibria) to “social” equilibria (global optimization). Available results characterize best and bound worst case Both selfish and global optimization are in fact sharply distributed.

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A Model for Network Formation Each link is purchased at a cost of A. Once purchased, it is available to all. Each site minimizes cost of their links plus sum of all distances in hops to all other sites. Network must be connected. Can improve this to model real asynchronous decision processes. Sum of the costs to all sites is the “social” cost. Ratio of the social cost of worst-case Nash Equilibrium to the social optimum is the price of anarchy. This is 4/3 if A < 2, and bounded between 3 and A for A large. A < 2 equilibria range from clique to star A > 2 equilibria include star, complex stars, trees A > N trees stable

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Experiments to compare selfish and social optimization(BG Chun, UCB Oceanstore group ; Johannes J Schneider, Mainz) 100 “overlay network” sites Distance metric in data presented – hops In more realistic studies, built a realistic underlay network, used estimated delays Selfish optima found by allowing each site to make asynchronous single greedy moves until Nash equilibrium is reached Social optima found by annealing Neither worst nor best case seen in practice for large A Social optimization uses more links, gives more robust networks

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Optimal solutions to the selfish network Either a complete graph, A < 2 Or a star configuration if A > 2

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Selfish solutions with finite information horizon Open star 10 < A < 20 Tree N < A

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2-core (sites not in tree)

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3-core

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Implications for design of P2P networks Routing solutions for P2P delivery of stored material take two forms: unstructured, structured Structured (Chord, Butterfly, …) use rules for assigning all links, with some local optimization Unstructured (Gnutella, E-Donkey, all present deployments) discover their links. There appears to be significant potential for optimization of either selfish or social cost. Start with always sharing the cost of a link between 2 ends. Doing this within unstructured networks is easiest

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And on the Avogadro scale? As presently conceived, both Biological nanoscale fabrication, and Quantum Computing may involve Avogadro’s number (10^21 molecules), but are not scale-free Typically a unique product is manufactured Designers unwilling to accept risk of randomized control When flexible manufacturing reaches nanoscale, all this must change, requiring distributed scale-free management, and soft measures of quality to replace the rigid agent policy approach now prevalent.

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