Intentional Walks on Scale Free Small Worlds

We present a novel algorithm that generates scale free small world graphs such as those found in the World Wide Web,social and metabolic networks. We use the generated graphs to study the dynamics of a realistic search strategy on the graphs, and find that they can be navigated in a very short number of steps. Small world and scale free graphs, which are at the heart of systems as diverse as the World Wide Web [1], the call logs of telephone networks [2], social and professional acquaintances [12,23,19,11], power grids [23] and metabolic networks [7,21], have attracted a lot of attention from the physics community in recent years (see the reviews [3,11]). One reason for the interest has been the number of dynamical processes such as percolation [4,10,22,20], epidemic spreading [15,10,23], random walks [14] and message-passing [8] which are of fundamental importance to statistical physics and have numerous applications in areas such as robustness of the Internet and the power grid, the spread of epidemics in societies, the spread of computer viruses in computer networks, routing in large computer networks [16] and in measuring the efficiency of online algorithms which utilize (www) network topology [18]. We present a novel algorithm that generates scale free small world graphs such as those found in the WorldWide Web,social networks and metabolic networks. We use the generated graphs to study the dynamics of a realistic search strategy on the graphs, and find that they can be navigated in a very short number of steps. Watts and Strogatz [23] first developed a procedure for generating graphs which have both short path lengths and clustering. This was an improvement over traditional Erdos-Renyi random graphs [17]. The Watts-Strogatz procedure however, lacks an important property exhibited by social and other networks, i.e. their approximate power-law distribution in the number of a node’s links. This distribution amounts to stating that a few nodes or people or sites in the web have very many links whereas most have a few. Whereas there are some small world graphs that are not power-law like (e.g. the electric power grid), many are scale free, such as the call graph of largescale telephone use, the Web, the Internet backbone, and metabolic networks. Recently, Barabasi et al [5], described a procedure for producing random graphs with a power-law distribution while failing to produce graphs that also have the clustering property of small worlds. While this work generates networks analogous to the power grid, it fails at generating the clustering property known to exist in the link structure of the World Wide Web [1], metabolic networks [21] or social networks [12]. The issue of navigation also received a partial answer in a paper by Kleinberg [8], who used a 2-D lattice substrate and a regular distribution of links. Motivated by real experiments with social networks, Kleinberg was concerned with how, given the fact that short paths existed, one could find them without complete global information. The treatment given in [8] had an elegant result, but the underlying graph model did not reflect all of the important features real world problems. An important shortcoming is its particular assumption of an inverse square correlation that implies that a majority of ones contacts lie in close geographical proximity. What happens if a large fraction of people know as many people outside of their city or state as inside? Would it become impossible to pass messages efficiently? What happens if the graph representing the social network, cannot be embedded on a two-dimensional lattice? Is it possible to devise an optimal strategy to navigate these networks?