Scaling-law relations characterize an immense number of natural processes, prominently in the form of

- scaling-law distributions,
- scale-free networks,
- cumulative relations of stochastic processes.

*f*(

*x*) depends on a power of

*x*. Two properties of such laws can easily be shown:

- a logarithmic mapping yields a linear relationship,
- scaling the function’s argument
*x*preserves the shape of the function*f*(*x*), called scale invariance.

### Scaling-Law Distributions

Scaling-law distributions have been observed in an extraordinary wide range of natural phenomena: from physics, biology, earth and planetary sciences, economics and finance, computer science and demography to the social sciences; see (Newman, 2004). It is truly amazing, that such diverse topics as- the size of earthquakes, moon craters, solar flares, computer files, sand particle, wars and price moves in financial markets,
- the number of scientific papers written, citations received by publications, hits on webpages and species in biological taxa,
- the sales of music, books and other commodities,
- the population of cities,
- the income of people,
- the frequency of words used in human languages and of occurrences of personal names,
- the areas burnt in forest fires,

### Scale-Free Networks

Another modern research field marked by the ubiquitous appearance of scaling-law relations is the study of complex networks. Many different phenomena in the physical (e.g., computer networks, transportation networks, power grids, spontaneous synchronization of systems of lasers), biological (e.g., neural networks, epidemiology, food webs, gene regulation), and social (e.g., trade networks, diffusion of innovation, trust networks, research collaborations, social affiliation) worlds can be understood as network based. In essence, the links and nodes are abstractions describing the system under study via the interactions of the elements comprising it. In graph theory, the*degree*of a node (or vertex),

*k*, describes the number of links (or edges) the node has to other nodes. The degree distribution gives the probability distribution of degrees in a network. For scale-free networks, one finds that the probability that a node in the network connects with

*k*other nodes follows a scaling law. Again, this power law is characterized by the existence of highly connected hubs, whereas most nodes have small degrees. Scale-free networks are

- characterized by high robustness against random failure of nodes, but susceptible to coordinated attacks on the hubs, and
- thought to arise from a dynamical growth process, called preferential attachment, in which new nodes favor linking to existing nodes with high degrees.

*small-world*property, is separate from a scale-free degree distribution, although scale-free networks are also small-world networks; (Strogatz and Watts, 1998). For small-world networks, although most nodes are not neighbors of one another, most nodes can be reached from every other by a surprisingly small number of hops or steps. Most real-world complex networks - such as those listed at the beginning of this section - show both scale-free and small-world characteristics. Some general references include (Barabasi, 2002), (Albert and Barabasi, 2001), and (Newman, 2003). Emergence of scale-free networks in the preferential attachment model (Albert and Barabasi, 1999). An alternative explanation to preferential attachment, introducing non-topological values (called fitness) to the vertices, is given in (Caldarelli

*et al*., 2002).

### Cumulative Scaling-Law Relations

Next to distributions of random variables, scaling laws also appear in collections of random variables, called stochastic processes. Prominent empirical examples are financial time-series, where one finds empirical scaling laws governing the relationship between various observed quantities. See (Guillaume*et al.*, 1997), (Dacorogna

*et al.*, 2001) and (Glattfelder

*et al.*, 2011).

###
*References*

**Albert R. and Barabasi A.-L.**, 1999, Emergence of Scaling in Random Networks, http://www.arXiv.org/abs/cond-mat/9910332.

**Albert R. and Barabasi A.-L.**, 2001, Statistical Mechanics of Complex Networks, http://www.arXiv.org/abs/cond-mat/0106096.

**Barabasi A.-L.**, 2002, Linked — The New Science of Networks, Perseus Publishing, Cambridge, Massachusetts.

**Caldarelli G., Capoccio A.**, Rios P. D. L., and Munoz M. A., 2002, Scale- free Networks without Growth or Preferential Attachment: Good get Richer, http://www.arXiv.org/abs/cond-mat/0207366.

**Dacorogna M. M., Gencay R., Müller U. A., Olsen R. B., and Pictet O. V.**, 2001, An Introduction to High-Frequency Finance, Academic Press, San Diego, CA.

**Glattfelder J. B., Dupuis A., and Olsen R. B.**, 2011, Patterns in high-frequency FX data: Discovery of 12 empirical scaling laws

**,**Quantitative Finance, 11(4), 599 - 614.

**Guillaume D. M., Dacorogna M. M., Dave R. D., Müller U. A., Olsen R. B., and Pictet O. V.**, 1997, From the Bird’s Eye to the Microscope: A Survey of New Stylized Facts of the Intra-Daily Foreign Exchange Markets, Finance and Stochastics, 1, 95–129.

**Mandelbrot B. B.**, 1963, The variation of certain speculative prices, Journal of Business, 36, 394–419.

**Newman M. E. J.**, 2003, The Structure and Function of Complex Networks, http://www.arXiv.org/abs/cond-mat/0303516.

**Newman M. E. J.**, 2004, Power Laws, Pareto Distributions and Zipf ’s Law, http://www.arXiv.org/abs/cond-mat/0412004.

**Sornette D.**, 2006, Critical Phenomena in Natural Sciences, Series in Synergetics. Springer, Berlin, 2nd edition.

**Strogatz S. H. and Watts D. J.**, 1998, Collective Dynamics of ‘Small-World’ Networks, Nature, 393, 440–442.

See more in Appendix C of Ownership Networks and Corporate Control: Mapping Economic Power in a Globalized World, J.B. Glattfelder, 2010

*[This post was originally posted on my now obsolete tech blog in September 2007 http://blogs.olsen.ch/jbg/2007/09/10/scaling-laws/]*

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