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

The physical Internet:

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

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.

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/N unsatisfiable, 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

Cost of WALKSAT

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

SIDecimation truncates WALKSAT cost

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.

Propagating surveys or beliefs in a “cavity”

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.

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

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

Optimal solutions to the selfish network

Either a complete graph, A < 2

Or a star configuration if A > 2

Selfish solutions with finite information horizon

Open star

10 < A < 20

Tree

N < A

2-core (sites not in tree)

3-core

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

And on the Avogadro scale?

As presently conceived, bothBiological 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.