Cs599: Structure and Dynamics of Networked Information (spring 2005) 03/09/2005: Power Laws via Optimization Scribes: Ranjit Raveendran and Farnoush Banaei-kashani

In the past lecture, we investigated the preferential attachment model for the generation of power law degree distributions. While “rich-get-richer” models may be considered close to the truth for WWW-like graphs, they do not seem appropriate for graphs such as the Internet, as there appears no natural reason why nodes would choose to attach to high-degree nodes (or be more likely to find out about them). Hence, this class of models is not very useful for explaining the power laws observed in the Internet at the AS level [4]. Fabrikant et al. [3] argue that the reason power laws evolve in the Internet is a heuristic optimization performed by the owners of machines. Indeed, optimization as a cause for power laws has been investigated before. Mandelbrot [5] and Zipf [7] argue that power laws in word lengths are caused by the “Principle of Least Effort”: languages evolve so as to optimize the average information transmitted per character or phoneme. (Notice, however, that the same result about word lengths can be obtained by assuming that characters, including the space bar, are pressed completely randomly [6].) Carlson and Doyle [2] extend the argument for file sizes and other parameters, and Fabrikant et al. [3] apply it to Internet-like graphs. They posit the following graph growth model. A communication tree is built as nodes arrive uniformly at random in a unit square. Let O be the first node that arrives, and assume that it arrives at the center of the square. Each node i arriving subsequently connects to a node j that had arrived earlier. The issue is which node j should i connect to. [3] argues that nodes want to be central in the network, i.e., few hops from O. At the same time, they want to have low “last mile” cost for their connection to j. The tradeoff is accomplished by considering the objective function dij + βhj where dij is the Euclidean distance between nodes i and j, and hj is the number of hops from j to O in the communication tree, and β is a given constant. That is, node i connects to the node j minimizing dij + βhj , and consequently has hi = hj + 1. Depending on the value of the parameter β, the graph evolves in very different ways.