Networks: Lecture 66.207/14.15: Networks Lecture 6: Growing Random Networks and Power Laws Daron Acemoglu and Asu Ozdaglar MIT September 28, 2009 1Networks: Lecture 6 Outline Growing random networks Power-law degree distributions: Rich-Get-Richer effects Models: Uniform attachment model Preferential attachment model Reading: EK, Chapter 18. Jackson, Chapter 5, Sections 5.1-5.2. 2Networks: Lecture 6 Growing Random Networks So far, we have focused on static random graph models in which edges among “fixed” n nodes are formed via random rules in a static manner. Erd¨os-Renyi model has small distances, but low clustering and a rapidly falling degree distribution. Configuration model generates arbitrary degree distributions. Small-world model provides a tractable model that has small distances and high clustering. Most networks form dynamically whereby new nodes are born over time and form attachments to existing nodes when they are born. Example: Consider the creation of web pages. When a new web page is designed, it includes links to existing web pages. Over time, an existing page will be linked to by new web pages. The same phenomenon true in many other networks: Networks of friendships, citations, professional relationships. Evolution over time introduces a natural heterogeneity to nodes based on their age in a growing network. 3Networks: Lecture 6 Emergence of Degree Distributions These considerations motivate dynamic or generative models of networks. These models also provide foundations for the emergence of natural linkage structures or degree distributions. What degree distributions are observed in real-world networks? In social networks, degree distributions can be viewed as a measure of “popularity” of the nodes. Popularity is a phenomenon characterized by extreme imbalances: while almost everyone goes through life known only to people in their immediate social circles, a few people achieve wide visibility. Let us focus on the concrete example of World Wide Web (WWW), i.e., network of web pages. In studies over many different Web snapshots taken at different points in time, it has been observed that the degree distribution obeys a power law distribution, i.e., the fraction of web pages with k in-links (or out-links) is approximately proportional to k−2.1 (or k−2.7). 4Networks: Lecture 6 Power Law Distribution—1 Many social and biological phenomena also governed by power laws. Population sizes of cities observed to follow a power law distribution. Number of copies of a gene in a genome follows a power law distribution. Some physicists think these correspond to some “universal laws”, as illustrated by the following quote from Barabasi that appeared in the April 2002 issue of the Scientist: “What do proteins in our bodies, the Internet, a cool collection of atoms, and sexual networks have in common? One man thinks he has the answer and it is going to transform the way we view the world.” A nonnegative random variable X is said to have a power law distribution if P(X ≥ x) ∼ cx−α , for constants c > 0 and α > 0. (Here f (x) ∼ g (x) represents that the limit of the ratios goes to 1 as x grows large.) Roughly speaking, in a power law distribution, asymptotically, the tails fall of polynomially with power α. 5Networks: Lecture 6 Power Law Distribution—2 Such a distribution leads to much heavier tails than other common models, such as Gaussian and exponential distributions. In the context of the WWW, this implies that pages with large numbers of in-links are much more common than we’d expect in a Gaussian distribution. This accords well with our intuitive notion of popularity exhibiting extreme imbalances. One specific commonly used power law distribution is the Pareto distribution, which satisfies P(X ≥ x) = � x �−α , t for some α > 0 and t > 0. The Pareto distribution requires X ≥ t. The density function for the Pareto distribution is f (x ) = αtαx−α−1 . For a power law distribution, usually α falls in the range 0 < α ≤ 2, in which case X has infinite variance. If α ≤ 1, then X also has infinite mean. 6Networks: Lecture 6 Examples A simple method for providing a quick test for whether a data-set exhibits a power-law distribution is to plot the (complementary) cumulative distribution function or the density function on a log-log scale. Courtesy of Society for Industrial and Applied Mathematics. Used with permission. 14 The structure and function of complex networks1 10 10010-410-21001 10 100 100010-410-210010010210410610-810-610-410-21001 10 100 100010-410-310-210-11000 10 2010-310-210-11001 1010-310-210-1100(a) collaborationsin mathematics(b) citations(c) World Wide Web(d) Internet(e) power grid(f) proteininteractionsFIG. 6 Cumulative degree distributions for six different networks. The horizontal axis for each panel is vertex degree k (or in-degree for the citation and Web networks, which are directed) and the vertical axis is the cumulative probability distribution ofdegrees, i.e., the fraction of vertices that have degree greater than or equal to k. The networks shown are: (a) the collaborationnetwork of mathematicians [182]; (b) citations between 1981 and 1997 to all papers cataloged by the Institute for ScientificInformation [351]; (c) a 300 million vertex subset of the World Wide Web, circa 1999 [74]; (d) the Internet at the level ofautonomous systems, April 1999 [86]; (e) the power grid of the western United States [416]; (f) the interaction network ofproteins in the metabolism of the yeast S. Cerevisiae [212]. Of these networks, three of them, (c), (d) and (f), appear to havepower-law degree distributions, as indicated by their approximately straight-line forms on the doubly logarithmic scales, andone (b) has a power-law tail but deviates markedly from power-law behavior for small degree. Network (e) has an exponentialdegree distribution (note the log-linear scales used in this panel) and network (a) appears to have a truncated power-law degreedistribution of some type, or possibly two separate power-law regimes with different exponents.within domains [338].2. Maximum degreeThe maximum degree kmaxof a vertex in a networkwill in general depend on the size of the network. Forsome calculations on networks the value of this maxi-mum degree matters (see, for example, Sec. VIII.C.2).In work on scale-free networks, Aiello et al. [8] assumedthat the maximum degree was approximately the