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| A '''Lévy flight''' is a [[random walk]] in which the step-lengths have a [[probability distribution]] that is [[heavy-tailed distribution|heavy-tailed]]. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. The "Lévy" in "Lévy flight" is a reference to the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]].
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| The term "Lévy flight" was coined by [[Benoît Mandelbrot]],<ref name=D1>Mandelbrot (1982, p.289)</ref> who used this for one specific definition of the distribution of step sizes. He used the term '''Cauchy flight''' for the case where the distribution of step sizes is a [[Cauchy distribution]],<ref>Mandelbrot (1982, p.290)</ref> and '''Rayleigh flight''' for when the distribution is a [[normal distribution]]<ref>Mandelbrot (1982, p.288)</ref> (which is not an example of a heavy-tailed probability distribution).
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| Later researchers have extended the use of the term "Lévy flight" to include cases where the random walk takes place on a discrete grid rather than on a continuous space.<ref name=Kleinberg/><ref name=Li/>
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| A Lévy flight is a [[random walk]] in which the steps are defined in terms of the step-lengths, which have a certain [[probability distribution]], with the directions of the steps being isotropic and random.
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| The particular case for which Mandelbrot used the term "Lévy flight"<ref name=D1/> is defined by the [[survivor function]] (commonly known as the survival function) of the distribution of step-sizes, ''U'', being<ref>Mandelbrot (1982, p.294)</ref>
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| :<math>\operatorname{Pr}(U>u) = \begin{cases}
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| 1 &:\ u < 1,\\
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| u^{-D} &:\ u \ge 1.
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| \end{cases}</math>
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| Here ''D'' is a parameter related to the [[fractal dimension]] and the distribution is a particular case of the [[Pareto distribution]]. Later researchers allow the distribution of step sizes to be any distribution for which the [[survival function]] has a power-like tail{{citation needed|date=August 2011}}
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| :<math>\operatorname{Pr}(U>u) = O(u^{-k}),</math>
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| for some ''k'' satisfying 1 < k < 3. (Here the notation ''O'' is the [[Big O notation]].) Such distributions have an infinite [[variance]]. Typical examples are the symmetric [[stable distribution]]s.
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| ==Properties==
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| Lévy flights are, by construction, [[Markov property|Markov processes]]. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a [[stable distribution]].{{citation needed|date=December 2010}}
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| The exponential scaling of the step lengths gives Lévy flights a [[scale invariance|scale invariant]] property,{{citation needed|date=December 2010}} and they are used to model data that exhibits clustering.{{citation needed|date=December 2010}}
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| [[Image:LevyFlight.svg|thumb|left|256px|Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. [[stable distribution|stable]]) distribution with ''α'' = 1 and ''β'' = 0 which is a [[Cauchy distribution]]. Note the presence of large jumps in location compared to the Brownian motion illustrated in Figure 2.]]
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| [[Image:BrownianMotion.svg|thumb|right|256px|Figure 2. An example of 1000 steps of an approximation to a Brownian motion type of Lévy flight in two dimensions. The origin of the motion is at [0, 0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. [[stable distribution|stable]]) distribution with ''α'' = 2 and ''β'' = 0 (''i.e.,'' a [[normal distribution]]).]]
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| ==Applications==
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| The definition of a Lévy flight stems from the mathematics related to [[chaos theory]] and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include [[earthquake]] data analysis, [[financial mathematics]], [[cryptography]], signals analysis as well as many applications in [[astronomy]], [[biology]], and [[physics]].
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| Another application is the [[Lévy flight foraging hypothesis]]. When sharks and other ocean predators can’t find food, they abandon [[Brownian motion]], the random motion seen in swirling gas molecules, for Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 radio-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including [[silky shark]]s, [[yellowfin tuna]], blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns.<ref>{{cite web |last=Witze |first=Alexandra |title=Sharks Have Math Skills |url=http://news.discovery.com/animals/sharks/sharks-math-hunt.htm |work=discovery.com |accessdate=22 February 2013}}</ref><ref>{{cite web |last=Dacey |first=James |title=Sharks hunt via Lévy flights |url=http://physicsworld.com/cws/article/news/2010/jun/11/sharks-hunt-via-levy-flights |work=physicsworld.com |accessdate=22 February 2013}}</ref> Birds and other animals<ref>{{cite journal |title=Optimizing the success of random searches |doi=10.1038/44831 |journal=Nature |year=1999 |last1=Viswanathan |first1=G. M. |last2=Buldyrev |first2=S. V. |last3=Havlin |first3=Shlomo |authorlink3=Shlomo Havlin |last4=da Luz |first4=M. G. E. |last5=Raposo |first5=E. P. |last6=Stanley |first6=H. E. |authorlink6=H. Eugene Stanley |volume=401 |issue=6756 |page=911|bibcode = 1999Natur.401..911V }}</ref> (including humans)<ref>{{cite web|url=http://well.blogs.nytimes.com/2014/01/01/navigating-our-world-like-birds-and-bees/ |date=January 1, 2014 |title=Navigating Our World Like Birds and Bees |first=Gretchen |last=Reynolds |work=[[The New York Times]]}}</ref> follow Lévy paths when searching for food.
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| Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha.<ref name=Kleinberg/><ref name=Li/>
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| ==See also==
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| *[[Fat-tailed distribution]]
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| *[[Heavy-tailed distribution]]
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| *[[Lévy process]]
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| *[[Lévy alpha-stable distribution]]
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| *[[Lévy flight foraging hypothesis]]
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| == Notes ==
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| {{reflist|refs=
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| <ref name=Kleinberg>{{cite journal|title=Navigation in a small world|doi=10.1038/35022643 |journal=Nature|year=2000|volume=406|pages=845|issue=6798|pmid=10972276|author=J. M. Kleinberg|bibcode = 2000Natur.406..845K }}</ref>
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| <ref name=Li>{{cite journal|title=Towards Design Principles for Optimal Transport Networks|journal=PRL|year=2010|volume=104|pages=018701|
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| url=http://havlin.biu.ac.il/Publications.php?keyword=Towards+Design+Principles+for+Optimal+Transport+Networks&year=*&match=all|author=G. Li, S. D. S. Reis, A. A. Moreira, [[Shlomo Havlin|S. Havlin]], [[H. Eugene Stanley|H. E. Stanley]], and J. S. Andrade, Jr|bibcode=2010PhRvL.104a8701L|last2=Reis|last3=Moreira|last4=Havlin|last5=Stanley|last6=Andrade|doi=10.1103/PhysRevLett.104.018701}}</ref>
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| }}
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| ==References==
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| *{{cite book |last=Mandelbrot |first=Benoit B. |authorlink=Benoît Mandelbrot |title=The Fractal Geometry of Nature |year=1982 |publisher=W. H. Freeman |location=New York |isbn=0-7167-1186-9 |edition=Updated and augm. |oclc=7876824}}
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| ==Further reading==
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| *{{cite doi|10.1016/S0378-4371(02)01157-3}}
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| *{{cite doi|10.1016/S0378-4371(00)00071-6}}
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| *{{cite doi|10.1063/1.527644}}
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| *{{Cite journal |last1=Shlesinger |first1=Michael F. |last2=Klafter |first2=Joseph |last3=Zumofen |first3=Gert |title=Above, below and beyond Brownian motion |doi=10.1119/1.19112 |journal=American Journal of Physics |volume=67 |issue=12 |pages=1253–1259 |date=December 1999 |url=http://caos.fs.usb.ve/~srojas/Teaching/USB/MC_Intro/MC_readings_a/MC_a4_brownian_1.pdf|bibcode = 1999AmJPh..67.1253S }}
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| ==External links==
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| *[http://plus.maths.org/issue11/features/physics_world/ A comparison of the paintings of Jackson Pollock to a Lévy flight model]
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| {{Fractals}}
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| {{DEFAULTSORT:Levy Flight}}
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| [[Category:Fractals]]
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| [[Category:Stochastic processes]]
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While this dream of yours could very well become a reality in the next several years, you might want to consider a few things before becoming a full-time work from home employee. Working from home might seem like a dream - and since many companies are going the virtual route, more employees are now expecting flexible work options - such as telecommuting and working virtually or from home. According to a survey commissioned by Mom Corps, a national flexible staffing firm, 51 percent of working adults (ages 18 to 44) plan to look for a new job within the next three years with an employer that offers flexible work options.
For most workers, this means a computer, reliable Internet access and a telephone. Companies often opt to allow workers to stay at home to save costs; so don't expect that these types of things will be provided to you as they normally are in an office setting. Do you have the necessary tools to get work done? If something else is required to successfully complete your daily tasks, add that to the list as well.
When working from home, your internal motivation must be strong enough to overcome your desire to sleep in, do laundry or talk to your friend on the phone for an hour. No one will be around to remind you to get that project done or to tell you to stop playing online games. Can you motivate yourself?
Working from home might seem like a dream - and since many companies are going the virtual route, more employees are now expecting flexible work options - such as telecommuting and working virtually or from home. According to a survey commissioned by Mom Corps, a national flexible staffing firm, 51 percent of working adults (ages 18 to 44) plan to look for a new job within the next three years with an employer that offers flexible work options. While this dream of yours could very well become a reality in the next several years, you might want to consider a few things before becoming a full-time work from home employee.
Companies often opt to allow workers to stay at home to save costs; so don't expect that these types of things will be provided to you as they normally are in an office setting. For most workers, this means a computer, reliable Internet access and a telephone. If something else is required to successfully complete your daily tasks, add that to the list as well. Do you have the necessary tools to get work done?
Can you motivate yourself? No one will be around to remind you to get that project done or to tell you to stop playing online games. When working from home, your internal motivation must be strong enough to overcome your desire to sleep in, do laundry or talk to your friend on the phone for an hour.
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