A Parsimonious Statistical Protocol for Generating Power-Law Networks Shilpa Ghadge 1 Timothy Killingback 2 Bala Sundaram 3 Duc A. Tran 1 1 Department of Computer Science 2 Department of Mathematics 3 Department of Physics University of Massachusetts, Boston, MA 02125 Email: {shilpa.ghadge001, timothy.killingback, bala.sundaram, duc.tran}@umb.edu Abstract—We propose a new mechanism for generating net- works with a wide variety of degree distributions. The idea is a modification of the well-studied preferential attachment scheme in which the degree of each node is used to determine its evolving connectivity. Modifications to this base protocol to include features other than connectivity have been considered in building the network. However, schemes based on preferential attachment in any form require substantial information on the entire network. We propose instead a protocol based only on a single statistical feature which results from the reasonable assumption that the effect of various attributes, which determine the ability of each node to attract other nodes, is multiplicative. This composite attribute or fitness is lognormally distributed and is used in forming the complex network. We show that, by varying the parameters of the lognormal distribution, we can recover both exponential and power-law degree distributions. The exponents for the power-law case are in the correct range seen in real- world networks such as the World Wide Web and the Internet. Further, as power-law networks with exponents in the same range are a crucial ingredient of efficient search algorithms in peer-to- peer networks, we believe our network construct may serve as a basis for new protocols that will enable peer-to-peer networks to efficiently establish a topology conducive to optimized search procedures. Index Terms—Power-law networks, growing random networks, lognormal distribution, peer-to-peer networks, search in power- law networks I. I NTRODUCTION In the last decade there has been much interest in studying complex real-world networks and attempting to find theoretical models that elucidate their structure. Although empirical net- works have been studied for some time; the recent surge in activity is often seen as having started with Watts and Strogatz’s paper on “small world networks” [1]. More recently, the major focus of research has moved from small-world networks to “scale-free” networks, which are characterized by having power-law degree distributions [2]. Empirical studies have shown that in many large networks, like the one shown in Fig. 1 and including the World-Wide-Web [3], the Internet [4], metabolic networks [5], protein networks [6], co-authorship networks [7], and sexual contact networks [8], the degree distribution exhibits a power-law tail: that is, if p(k) is the fraction of nodes in the network having degree k (i.e. having k connections to other nodes) then (for suitably large k) p(k)= ck -λ , (1) Fig. 1. Diagram in the figure is of Internet connections, showing the major Metropolitan Area Exchanges (MAE), by K.C. Claffy, republished on Albert-L´ aszl´ o Barab´ asi’s Self-Organized Networks Gallery web page ”http://www.nd.edu/ networks/Image Gallery/gallery.htm”. The colors reflect the volume of network traffic. where c =(λ - 1)m λ-1 is a normalization factor and m is the minimal degree in the network. One of the earliest theoretical models of a complex network, that of a random graph, was proposed and studied in detail by Erd´ os and R´ enyi [9]–[11] in a famous series of papers in the 1950s and 1960s. The Erd´ os-R´ enyi random graph model consists of n nodes (or vertices) joined by links (or edges), where each possible edge between two vertices is present independently with probability p and absent with probability 1 - p. The degree distribution of the Erd´ os-R´ enyi random graph model is easy to determine. The probability p(k) that a vertex in a random graph has exactly degree k is given by the binomial distribution p(k)=  n - 1 k  p k (1 - p) n-k-1 . (2) In the limit when n ≫ kz, where z =(n - 1)p is the mean degree, the degree distribution becomes the Poisson 978-1-4244-4581-3/09/$25.00 ©2009 IEEE