1. Power-law performance ranking relationship in exponentially growing populations Chunhui Cai 1, Da-Liang Li 2, Qi Ouyang 2, Lei-Han Tang 1,3, Yuhai Tu 2,4 1 Hong Kong Baptist University 2 CQB-Peking University 3 Beijing Computational Science Research Center 4 IBM TJ Watson Research Center 6 th KIAS Conference on StatPhys: NCPCS 2014, 8-11 July, 2014, Seoul Korea

2. Outline TOP500: just another power-law? Phenomenology: The constant growth and insertion hypothesis A possible microscopic mechanism -The pick-and-improve-retire (PAIR) model -Mapping to randomly branching tree and extremal value statistics -Velocity selection -Front profile and power-law distribution -Rank-performance fluctuations Analysis of the SC500 data: comparison with theory Push or pull fronts? Summary and conclusions

3. Hong Kong is currently not on the list! November 2007 release CountriesCountShare %Rmax Sum (GF)Rpeak Sum (GF)Processor Sum United States28356.60%416431262139391024339 United Kingdom489.60%512400818366115244 Germany316.20%536464753361143392 Japan204.00%29111840563464218 France173.40%22255534298854548 Taiwan112.20%10296616282114024 China102.00%8717615585622836 India91.80%19452430365134932 Spain91.80%13745221222524332 Russia71.40%8261512275611796 Sweden71.40%18617627873327720 Switzerland71.40%7456810241820368

4. Proposed configuration Slope = 0.75 Doubles each year

5. Engineering

6. Business

7.  Seen in many big engineering projects, economic and social data (known as Pareto distribution or Zipf’s law)  Exponents vary over a broad range  Are there common mechanisms?

8. Phenomenology: The constant growth and insertion hypothesis Li Daliang ln(RMax) ln(rank) ln(b) ln(1+a) insertion growth

9. Li Daliang Phenomenology: The constant growth and insertion hypothesis ln(RMax) ln(rank) ln(b) ln(1+a) ln[X(R,t+1)] ln[X(R,t)] Solution:

10. Outline TOP500: just another power-law? Phenomenology: The constant growth and insertion hypothesis A possible microscopic mechanism -The pick-and-improve-retire (PAIR) model -Mapping to randomly branching tree and extremal value statistics -Velocity selection -Front profile and power-law distribution -Rank-performance fluctuations Analysis of the SC500 data: comparison with theory Push or pull fronts? Summary and conclusions

11. The punch line When embarking on an expensive project, decisions are made by following a local leader in the sector. Performance of the newly introduced node is a certain percentage better than the existing one.

12. The pick-and-improve-retire (PAIR) model s = rate a given node acquires a follower. r = rate a given node retires (< s). Performance X of nodes measured on logarithmic scale x = log X. ρ(y) = distribution of the increment in performance (log scale) for the follower, taken here to be a Gaussian function, simulation

13. Mapping to the randomly branching tree (a)Ensemble description n(x,t) = number density of nodes (b)Front propagation, distribution of x max (t). (Directed polymer on Cayley tree, extremal statistics) t x y n(x,t)n(x,t)

14. Ensemble description t x y dt Look for traveling wave with exponential front,

15. Velocity selection λ O Solution in parameter space Increasing performance for the population even when offsprings on average perform worse than parents! λcλc

16. Performance-rank distribution Number of nodes with performance better than x Exponential growth with power-law rank distribution Agrees perfectly with simulations!

17. Rank-performance fluctuations

18. Heuristic argument for independent branching Fluctuations δN in the number of insertion events N in a given performance and time interval is proportional to N 1/2. Hence Confirmed by tour de force analytic calculation of the two-point distribution function for the branching process

19. Two-point distribution function PAIR model: Solved using Fourier transform (translational symmetry) Nodes of F(k) on the complex k-plane

20. Outline TOP500: just another power-law? Phenomenology: The constant growth and insertion hypothesis A possible microscopic mechanism -The pick-and-improve-retire (PAIR) model -Mapping to randomly branching tree and extremal value statistics -Velocity selection -Front profile and power-law distribution -Rank-performance fluctuations Analysis of the SC500 data: comparison with theory Push or pull fronts? Summary and conclusions

21. 43 rd release

22. The lists (42 analyzed) allow one to trace the identity of individual nodes (8,696 computers) Performance (Rmax) Entry/exit time (release No.) Rank (in each release) Expansion rate:

23. Rank-resolved insertion and retirement rates Weak dependence on rank

24. The parent-offspring association Each new node in a new release belongs to one of the following 4 categories: 1.Same location and same computer (2596 cases) “copier” 2.Same location but a new computer (3781 cases) “upgrader” 3.New location and same computer (1826 cases) “copier” 4.New location and new computer (1931 cases) “intruder”, about 19% Used in the simulation of the PAIR model

25. Comparison between SC500 data and simulations SC500 data shows stronger fluctuation with ζ=0.8.

26. Push or pull fronts? Push front: growth driven by nodes behind Pull front: growth driven by the leader, stronger fluctuation

27. Summary Pareto distribution and Zipf’s law arises under the “constant growth and expansion rate” hypothesis A microscopic PAIR model is mapped to the randomly branching tree problem The microscopic model not only yields the observed power-law rank-size distribution, but also exhibits a fluctuation phenomenon with 1/R scaling. Detailed analysis of the SC500 data confirms hypothesis and allows determination of the model parameters. Somewhat stronger fluctuations observed in the real data. Alternation of push and pull fronts?

28. Thank you for your attention!