Citations: Power Laws: Minutes from an Infinite Paradise

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M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York, 1991.

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Computationally Measurable Temporal Differences between Speech.. - Gerhard (2003) (Correct)

....with a low power fundamental, it is possible to mistake an upper harmonic for the fundamental. Humans do this as well, and it is a result more of the signal itself than of the recognition algorithm. A period k signal can become a period 2k signal through a process called period doubling [65, 29]. At the transition point, it is unclear whether it is appropriate to count the period as k or 2k. This transition point is unstable, so it is uncommon to hear signals of ambiguous pitch in nature. However, it does indicate that period doubling errors may be a di#cult problem to overcome. ....

Manfred R. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H.Freeman, New York, 1991.

A Stochastic Model for the Evolution of the Web - Levene, Fenner, Loizou, Wheeldon (2002) (5 citations) (Correct)

....for example Zipf s law [Rap82] which states that relative frequency of words in a text is inversely proportional to their rank, and Lotka s law [Nic89] which is an inverse square law stating that the number of authors making n contributions is proportional to n 2 . We refer the reader to [Sch91] for more examples of power law distributions. Recently several researchers have detected power law distributions in the Internet [FFF99] and World Wide Web [BKM 00, DKM 01] topologies. In order to understand how these power law distributions emerge and how the Web has evolved and is ....

M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New York, NY, 1991.

A Stochastic Model for the Evolution of the Web - Levene, Fenner, Loizou, Wheeldon (2002) (5 citations) (Correct)

Simulation-Based Analysis of Cascading Failure for.. - Parrilo, Lall.. (Correct)

....These laws are of the form y = Cx fi , and therefore, when plotted on a log log scale, they are represented by straight lines. There has been a flurry of activity in the last decade centered around the appearance 6 Figure 4: Binary tree of power laws in many natural phenomena (see for example [12]) The exact causes and consequences of these power law distributions are still not understood, but see [2] for an investigation of the role of design in this question. 4.1 Why power laws An important question raised by the experimental results is related to the existence of intrinsic reasons ....

M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, 1991.

Indexing Multimedia Databases - Faloutsos (Correct)

.... 2 (X 2 GammaY 2 ) 2 (X 1 GammaY 1 ) 2 (X 2 GammaY 2 ) 2 (X 3 GammaY 3 ) 2 : 54 Colored noises ffl brown noise (1=f 2 energy spectrum) j random walk (stock price movements, currency exchange rates) Mandelbrot] Man77] ffl pink noise (1=f energy spectrum) works of art [Sch91] ffl black noise (1=f b b 2) water level of rivers [Sch91] 55 Examples of colored noises: 1 2 5 10. 20. 50.100. 200. loglog DFT of White Noise 0.05 0.1 0.2 0.5 1 1 2 5 10. 20. 50.100. 200. log log DFT of Brown Noise 0.1 1 10. 50 100 150 200 250 DFT of White Noise 0.2 0.4 ....

.... ) 2 (X 3 GammaY 3 ) 2 : 54 Colored noises ffl brown noise (1=f 2 energy spectrum) j random walk (stock price movements, currency exchange rates) Mandelbrot] Man77] ffl pink noise (1=f energy spectrum) works of art [Sch91] ffl black noise (1=f b b 2) water level of rivers [Sch91] 55 Examples of colored noises: 1 2 5 10. 20. 50.100. 200. loglog DFT of White Noise 0.05 0.1 0.2 0.5 1 1 2 5 10. 20. 50.100. 200. log log DFT of Brown Noise 0.1 1 10. 50 100 150 200 250 DFT of White Noise 0.2 0.4 0.6 0.8 1 1.2 1.4 50 100 150 200 250 DFT of Brown Noise ....

Manfred Schroeder. Fractals, Chaos, Power Laws: Minutes From an Infinite Paradise. W.H. Freeman and Company, New York, 1991.

Using the Fractal Dimension to Cluster Datasets - Barbara, Chen (1999) (6 citations) (Correct)

....to find self similarity, i.e. an invariance with respect to the scale used. The structures that appear as a consequence of self similarity are known as fractals [17] 2 Fractals have been used in numerous disciplines (for a good coverage of the topic of fractals and their applications see [22]) In the database arena, fractals have been successfully used to analyze R trees [7] Quadtrees [6] model distributions of data [8] and selectivity estimation [2] Fractal sets are characterized by their fractal dimension. In truth, there exists an infinite family of fractal dimensions. By ....

M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New York, 1991.

Emergence and Levels of Abstraction - Damper (Correct)

....emergence as a principle seems opposed to reductionism a widely accepted doctrine which has proved enormously fruitful in science. Accordingly, the relation between the two is explored before concluding. 2 Emergence: Basic Ideas To introduce the topic of emergence, let us consider (to cite Schroeder 1991, p. 35) one of the most surprising instances of a power law in the humanities , namely Zipf s (1949) law according to which the frequency of occurrence, f , of words is (approximately) inversely proportional to their rank 1 , r , for many natural languages. That is: f # 1 r and log f ....

....intelligence was, however, demolished by Mandelbrot (1961) who was able to show that a language composed by randomly striking typewriter keys also obeyed Zipf s law. Today, this law is generally recognised as just one of a number of scaling or power laws occurring widely in the natural world (Schroeder 1991; Gell Mann 1994; Bak 1996; Casti 1997) For many authors, Zipf s law is a prime example of what today we would call an emergent property. For instance, Casti (1997, p. 128 9) writes that this law: is not a pattern that can be seen in the individual words . but rather emerges from ....

Schroeder, M. (1991). Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York, NY: W. H. Freeman.

On Generalized Entropies and Scale-Space - Sporring, Weickert (1997) (3 citations) (Correct)

....rate of replication with scale, the so called Lyaponov exponent. In the theory of multi fractals there is not assumed to be one global scaling exponent but an entire spectrum of exponents each associated with an area in the image . In the following will be given a summary of multi fractal theory [11, 23, 24]. Given an density function ae, a discretization can be performed as, p i(l) Z x2 Omega i(l) ae(x) dx; where Omega i(l) is the i th box in the grid of boxes of width i covering the domain of x. The point i(l) then has a fractal behavior with scaling exponent ff 2 IR if it converges ....

Manfred Schroeder. Fractals, Chaos, Power Laws -- Minutes from an infinite paradise. W. H. Freeman and Company, 1991.

Density Biased Sampling: An Improved Method for Data Mining.. - Palmer, Faloutsos (2000) (11 citations) (Correct)

.... they have been found in the frequency distribution of vocabulary words in text (English and Latin works of literature [24] the Bible [6] the distribution of city populations [24] distribution of first and last names of people [5] sales patterns [6] income distributions (the Pareto law [20]) and distribution of website hits [13] The main contribution of this paper is to introduce a new sampling technique and an e#cient algorithm that improves on uniform sampling when cluster sizes are skewed. The rest of the paper is organized as follows. First, we present density biased sampling ....

Manfred Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York, 1991.

Deflating the Dimensionality Curse using Multiple Fractal.. - Pagel, Korn, Faloutsos (2000) (6 citations) (Correct)

....U.S. Bureau of Census (E=2, N= 27,282, D 0 = 1.719 and D 2 = 1.518 ) ffl LBcty road intersections in Long Beach County from TIGER census data (E=2, N= 36,548, D 0 = 1.728, and D 2 = 1. 732 ) ffl Sierpinski a synthetic, non uniform fractal depicted in Fig 1, known as Sierpinski s triangle [30]. It was generated by embedding a Sierpinski triangle in E = 10 dimensional space, with D 0 = D 2 = 1.585. The Sierpinski triangle was on a two dimensional plane, which was deliberately not aligned with any of the axes; ffl plane uniformly generated points for a range of sizes N (from 1 500K) ....

Manfred Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York, 1991.

Fourier-Transform Based Techniques in Efficient Retrieval of.. - Rafiei (1999) (1 citation) (Correct)

....Distance based Retrieval of Similar Time Sequences 19 coefficients, i.e. they have a skewed energy spectrum of the form O(F Gamma2b ) for b 0:5 where F denotes the frequency. For example, classical music and jazz fall in the class of pink noise whose energy spectrum is O(F Gamma1 ) WS90, Sch91] stock prices and exchange rates fall in the class of brown noise whose energy spectrum is O(F Gamma2 ) Man83, Cha84] and the water level of rivers falls in the class of black noise for which b 1 ( Man83, Sch91] To retrieve similar time sequences stored in the index we may invoke ....

.... in the class of pink noise whose energy spectrum is O(F Gamma1 ) WS90, Sch91] stock prices and exchange rates fall in the class of brown noise whose energy spectrum is O(F Gamma2 ) Man83, Cha84] and the water level of rivers falls in the class of black noise for which b 1 ( Man83, Sch91] To retrieve similar time sequences stored in the index we may invoke one of the following spatial queries: ffl Proximity Query: Given a query point Q and a threshold ffl, find all points X such that the Euclidean distance D( X ; Q) ffl. ffl Nearest Neighbor Query: Given a ....

Manfred Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New York, 1991.

Analysis of Performance of Symmetric Second-Order Line.. - Pronzato, Wynn.. (1999) (Correct)

....n Gamma1 [ med(p=q; p 0 =q 0 ) where the union is taken over all adjacent pairs fp=q; p 0 =q 0 g in F n Gamma1 . For example, F 1 = f0; 1 2 ; 1g ; F 2 = f0; 1 3 ; 1 2 ; 2 3 ; 1g ; F 3 = f0; 1 4 ; 1 3 ; 2 5 ; 1 2 ; 3 5 ; 2 3 ; 3 4 ; 1g : One can easily check (see [12], p. 337 and [2] that there are exactly 2 n Gamma1 Farey fractions of level n, that is, elements in F n n F n Gamma1 , and all of them have a finite continued fraction representation p=q = a 1 ; a 2 ; a k ] satisfying a 1 : a k = n. Also, the number of elements in F n is jF n j ....

....of the Farey tree is that the maximum value of the denominator q among all the points p=q with g.c.d(p; q) 1 and continued fraction expansion p=q = a 1 ; a 2 ; a k ] such that P k i=1 a i N , is achieved when p=q = FN =FN 1 . This property can easily be proved by induction, see also [12], p. 339. 5.3 Asymptotic performance Using (9) we can easily evaluate the performance of any symmetric algorithm in terms of the asymptotic convergence rate R defined as follows: R(e 1 ) lim sup N 1 [LN 1 (e 1 ) 1 N = lim sup N 1 [L 1 N Y n=1 r n (e 1 ) 1 N : 12) 12 Since L ....

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M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York, 1991.

Chaotic Mining: Knowledge Discovery Using the Fractal Dimension.. - Barbar� (Correct)

....possible to find self similarity, i.e. an invariance with respect to the scale used. The structures that appear as a consequence of self similarity are known as fractals [12] Fractals have been used in numerous disciplines (for a good coverage of the topic of fractals and their applications see [14]) In the database arena, fractals have been sucessfully used to analyze R trees [6] Quadtrees [5] model distributions of data [7] and selectivity estimation [3] Fractal sets are characterized by their fractal dimension. In truth, there exists an infinite family of fractal dimensions. By ....

M. Schroeder. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman, New York, 1991.

Graphs over Time: Densification Laws, Shrinking.. - Leskovec, Kleinberg, .. (Correct)