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Power Law Statistics for Earthquakes, Forest fires, Networks, and Other Complex Systems

Power Law Statistics for Earthquakes, Forest fires, Networks, and Other Complex Systems. Kan Chen Department of Computational Science, NUS. Outlines: I Earthquake statistics and the Gutenberg-Richter law II Sandpile model and self-organized criticality

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Power Law Statistics for Earthquakes, Forest fires, Networks, and Other Complex Systems

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  1. Power Law Statistics for Earthquakes, Forest fires, Networks, and Other Complex Systems Kan Chen Department of Computational Science, NUS Outlines: I Earthquake statistics and the Gutenberg-Richter law II Sandpile model and self-organized criticality III Other possible examples of self-organized criticality IV Organization and dynamics of complex networks

  2. Earthquake Phenomenology It is generally believed that earthquakes result from a stick-slip dynamics involving the Earth’s crust sliding along faults.  San-Andreas Fault marks the contact between the Pacific and North American Plates San-Andreas fault, where the great 1906 San Francisco earthquake occurred

  3. Earthquake Phenomenology The recent earthquake in Taiwan (Sept. 21, 1999) also causes tremendous damages. The size of an earthquake is often given in term of magnitude (Richter scale), which is roughly the energy in logarithmic scale. An increase in magnitude by 1 leads to about 32-fold increase in the energy release (10-fold increase in amplitude of seismic shocks). An earthquake of magnitudes of 7 or higher typically causes tremendousdamages.  Main shock and aftershocks

  4. What can we understand about earthquakes? Can we predict earthquake?             ---  it is as difficult as predicting the individual ball movement in a giant Galton Board For Gaton's board even though we cannot predict the movement of an individual ball, but we can still give a probabilistic description of the movement of the ball. The probability P(x) of the horizontal position x of the ball can be described by a Gaussian distribution (Bell curve). Similarly, even though we cannot predict precisely the size of the next earthquake, we can calculate the probability that an earthquake of a given size will occur.

  5. Gutenberg Richter Law • Despite of the apparently complexity involved in the dynamics of earthquake, the probability distribution of earthquakes follows a glaringly simple distribution function known as the Gutenberg-Richter law: • The probability of having an earthquake of energy E, • P(E) = c 1/E1+B, • where the exponent B is a universal exponent in the sense that it does not depend on particular geographic area. • When plotted on a log-log scale, such power laws appear as straight lines, where the slope is -B:  log (P(E)) = log (c) – (1+B) log(E) Earthquake Statistics (1995)

  6. Gutenberg-Richter Law There are more small earthquakes than large ones. But there is no apparent cut-off in the possible size of an earthquake; earthquakes of all sizes are possible. • If the probability distribution were of the form exp(-E/E0), then there would be a well defined cut-off at the energy E0 such that the probability for an earthquake of size E much greater than E0 is exponentially small.

  7. Earthquake Size Distribution • M Energy(TNT) Descriptions Frequency (num/year) • 1~2 30lb ~ 1 ton Blast at a construction site • 2 1/2  4.6 tons Smallest felt earthquakes; • Daisy Cutter >10,0000 • 3 29 tons 50,000 • 4 1000 tons Atomic bomb;  sleep awaken 6000 • 5 32,000 tons Smallest damaging shocks; • difficult to walk 800 • 1 million tons  Hydrogen bomb 120 • 32 million tons 18 • 1 billion tons 1 • 32 billion tons 1~2 in a century • Chile (1960), Alaska(1964), Aceh(2004) MM+1  E  32 E The mechanism for small and large earthquakes is the same (the statistics is related) P(E) = c  E-1-B P(E1)/P(E2) = (E1/E2)-1-B P(M = 4) / P(M = 8) = 6000 In Singapore, P(M=4) = 1 per year  P(M=8)=1/6000 per year. In Taiwan, P(M=8)=1/100 per year  P(M=4)=60 per year

  8. Sub, Super, and Critical States Chain reactions of dominoes In the critical system (modelled by placing dominoes randomly on about half of the segments in a diamond grid) many sizes of chain reactions (avalanches) were produced when the dominoes in the bottom row were tipped over.

  9. Sub, Super, and Critical States The sub-critical system (where the density of the dominoes is less than the critical value) produced only small chain reactions The supercritical system (where the density of dominoes is greater than the critical value) exploded with activity. In our earthquake model, if the initial average stress is high, we start with a supercritical system; if the initial average stress is low, then we start with a sub-critical system. Eventually the system evolves to a critical state, which is signified by the power-law distribution of the sizes of earthquakes. Thus the critical state is a very special one that is characterized by chain reactions of all sizes (just like earthquakes we generated in our earthquake model).

  10. Percolation and criticality A simple percolation model (applet). You may view the lattice as a forest: a given cell is either empty (with probability 1 - p) or is occupied by a tree (with probability p); p is the density of the trees in the forest. We ask the question, if we start a fire on the left-edge of the forest, what is the probability that the fire will reach (percolate to) the right-edge? How does the percolation probabilitydepend on the density of the forest p.

  11. Sand Pile and Self-organized Criticality Why and how do complex systems, such as the earth’s crust (where earthquakes occur), evolve to such critical states? To answer this question, it is better to study much simpler systems, then generalize our understanding of simple model systems to realistic complex systems --- This is the approach we adopt in our study of science of complexity. We hope to be able to understand universal properties such the Gutenberg-Richter laws exhibited in earthquake statistics.    A deceptively simple system serves as a paradigm for self-organized criticality: a pile of sand.  Artwork by Elaine Wiesenfeld (from Bak, How Nature Works)

  12. A Rice Pile Experiment confirms Self-organized Criticality

  13. Sand Pile Model The model is not a realistic model of sand pile. But it captures the basic features of avalanches, and it provides a good illustration of self-organized criticality. • A variable (local height difference) z is defined at each cell • Addition of sand: pick a random cell, increase z by one. z --> z + 1 • Toppling: if z > 4, then z-> z - 4 and  z -> z + 1 for all its neighbors.

  14. Sand Pile Model Office version of the model (Suggested by Peter Grassberger

  15. Butterfly Effect A small perturbation can have drastic consequences. Chaos scientists call this the butterfly effect:  A butterfly moving its wings in South American will affect the weather in the United States. What is the “butterfly effect” in SOC systems such as a sand pile? In a pile of sand, the majority of perturbations have little effect (leading to no or small avalanches). But there are perturbations that cause big avalanches. But it is difficult to predict which are "more important" sand grains. A possible example of butterfly effect:     Butterfly ballot at Miami Dade County  US presidential election Second Gulf War The effect of any perturbation will affect the entire system if we wait long enough.  In a lifetime, everybody will have his 15 minutes of fame (Andy Warhol).   Comparison of two simulations of the earthquake model (starting from small differences in the stress distribution) [applet]

  16. Other Possible Examples of SOC --- Forest Fires Source: Bruce D. Malamud, Gleb Morein, Donald L. Turcotte, Science, Vol 281, page 1840 (1998). how to manage forest?

  17. The World Wide Web The probability distribution of web sites vs. the number of visitors (in a given day) is a power law (source:Lada A. Adamic)

  18. Human Society Wars and conflicts, migration, city distribution Zipf's law describes the relationship between the size of the city and its rank (Rank-Size distribution)  References on Zipf's law        Zipf's law is  named after the Harvard linguistic professor George Kingsley Zipf (1902-1950).         S  ~  c / Rg , g = 1 in the original Zipf's law  Zipf law also describes income distribution.

  19. Distribution of City Sizes Data for US cities (g ~ 0.8) [Source: Oxford University Press City          Pop. (thousands)     Rank New York             7333             1Los Angeles           3449            2Chicago                 2732            3Houston                 1702            4Philadelphia           1524             5San Diego              1152             6Phoenix                  1049            7Dallas                   1023              8San Antonio           999              9Detroit                 992              10

  20. Distribution of City Sizes • How about Chinese cities? • (g ~ 0.5) • Cities            Pop. (thousands)     Rank •     Shanghai        8206                 1    BEIJING      7362                  2    Hong Kong      6190                  3    Tianjin           5804                  4    Qingdo           5125                  5    Shenyang       4655                  6    Guangzhou      3918                  7    Wuhan            3833                 8    Tai'an            3825                  9     Harbin           3597                 10 Why is the Zipf exponent for the size distribution of the Chinese cities different from that of the US cities? What social-economic dynamics gives rise to the difference?

  21. Word                Frequency            Rank         the                     0.0653                 1        of                       0.0282                 2        to                       0.0262                 3        and                     0.0243                 4        a                        0.0242                 5        in                       0.0195                  6        is                        0.0107                 7        for                     0.0102                 8        that                   8.9320e-3             9        on                      7.8680e-3            10he                      5.9770e-3            16        not                    3.9910e-3             26        or                      2.5540e-3            39        you                    2.3230e-3            45she                     1.5920e-3            63    australian               1.5220e-3            68her                     1.4970e-3            69    million                    1.3400e-3            78him                     9.4000e-4            98    market                   7.3500e-4            130      think                    5.3300e-4            171    computer                1.6300e-4            693 • Does the brain operates at the edge of chaos? Genius or a mad man? • Zipf's law for the Rank - Word Count distribution • Sydney Morning Herald Word Database  (Source: Simon Dennis, University of Queensland) The Brain

  22. Economy and Financial Market Boom and bust Can one manage the economy by manipulating a few global parameters? Can the economy be described using equilibrium theory used by many main-stream economists Probability of Stock Price Change over an interval, P(DS)  = c /DS4

  23. Punctuated Equilibrium and Evolution Biological extinction and punctuated equilibrium --- a controversial subject It is estimated that among four billion species which have existed on the Earth since life first appeared less than 50 million (about 1 percent) are still alive. The pattern of extinction events from the fossil history, as recorded by J. J. Sepkoski. Extinction events recorded over 600 million years. The curve shows the estimated percentage of species that became extinct during consecutive periods of 5 million years.

  24. Punctuated Equilibrium From the pattern of extinction events, one can see that There are periods of relatively little activity (stasis) interrupted by narrow intervals, or bursts, with large activity in the history of biological evolution. This "punctuated equilibrium" was first noted by Gould and Eldredge. The biological system only appears to be in equilibrium within a certain time scale. From the data collected by Sepkoski, it is also found that the size distribution of extinction events can roughly be interpreted as a power law, just like the Gutenberg-Richter law for earthquakes. Does biological system also operate in a critical state?

  25. Evolution in a test tube Lenski's experiment on E-coli Evolution in a test tube (Lenski's experiment on E-coli) Fossil Evidence (Jackson and Cheetham) on Punctuated Equilibrium Gradualism vs. Punctuated Equilbirum

  26. Bak-Sneppen Model It is a very simple toy model of an ecology of evolving species. The underlying picture is the one where species interact with each other. The random mutation and then selection of the fitter variants affect the fitness of other species in the global ecology. In the Bak-Sneppen model, species are placed on a one-dimensional ring, for simplicity and bookkeeping. This can be thought of as a food chain (a food web is more realistic). Initially, the species are assigned random numbers, {fi}, between 0 and 1, where fi represents the fitness of species i. Each species only interacts with its nearest neighbors. 

  27. Scale Free Network • A scale-free network is a specific kind of network in which the distribution of connectivity is extremely uneven (it follows a power law) • This kind of connectedness dramatically influences the way the network operates, including how it responds to catastrophic events. The Internet, WWW, many other large-scale networks have been shown to be scale-free networks. • Physicists Barabasi and Albert constructed a simple model of growing network, which is shown to be scale-free. • Start with a node • A new node is connected to one of the existing nodes. But the probability of choosing the existing node to connect to Prob ~ The number of connections it already has • The more connections a node has, the more likely it will get connected, because the number of agents a node has is equal to how many nodes it connects to.

  28. Dynamics of Networks • Kauffman’s NK Boolean network • Random network • Each node has K inputs. s(t) = f({si(t-1)}, i=1,..,K) • K = 1  fixed point • K = 2  periodic (the distribution of the periods, P(T), is a power law) • K > 3  chaotic In real biological network, K is typically much greater than 3; but the network is stable

  29. Dynamics of Networks • Growing Directed Network • Network is grown by adding nodes one by one (starting with an initial cluster) • Each node has K inputs. s(t) = f({si(t-1)}, i=1,..,K) But the new node cannot influence the existing nodes (like a citation network) • Stable for all K (the maximum period is 2K+1) • Dynamics insensitive to preferential attachment. Size distribution of descendant clusters is a power law. • Growing Directed Network with a fraction of link reversals • Appears to be critical (the distribution of the periods is a power law) Could the growing directed network with a fraction of link reversals serves as a better model for biological networks?

  30. Conclusions • Power-law statistics often means the underlying dynamics is critical. Examples: Earthquakes, forest fires. • Such critical states emerge naturally without fine tuning  Self-organized Criticality • Criticality  Universality. It is possible to understand universal properties using simple models • Many interesting networks also exhibit power law statistics. • Network organization and dynamics are closely related. • Emergence of power laws in network dynamics

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