Semester projects - Principles of Complex Systems CSYS/MATH 300, Fall, 2010

5 of 49 topics: with r g~5:8 km, br5 1.656 0.15 and k5 350km (Fig. 1d, see Supplementary Information for statistical validation). Lévy flights are characterized by a high degree of intrinsic heterogeneity, raising the possibility that equation (2) could emerge from an ensemble of identical agents, each following a Lévy trajectory. Th refore, we determined P(rg) for an ensemble of agents following a random walk (RW), Lévy flight (LF) or truncated Lévy flight (TLF) (Fig. 1d). We found that an ensemble of Lévy agents display a significant degree of heterogeneity in rg; however, this was not sufficient to explain the truncated power-law distribution P(rg) exhibited by the mobile phone users. Taken together, Fig. 1c and d suggest that the difference in the range of typical mobility patterns of individuals (rg) has a strong impact on the truncated Lévy behaviour seen in equation (1), ruling out hypothesis A. If individual trajectories are described by an LF or TLF, then the radius of gyration should increase with time as rg(t), t b) (ref. 21), whereas, for an RW, rg(t), t; that is, the longer we observe a user, the higher the chance that she/he will travel to areas not visited before. To check the validity of these predictions, we measured the time dependence of the radius of gyration for users whose gyration radius would be considered small (rg(T)# 3 km), medium (20, rg(T)# 30 km) or large (rg(T). 100 km) at the end of our observation period (T5 6months). The results indicate that the time dependence of the average radius of gyration of mobile phone users is better approximated by a logarithmic increase, not only a manifestly slower dependence than the one predicted by a power law but also one that may appear similar to a saturation process (Fig. 2a and Supplementary Fig. 4). In Fig. 2b, we chose users with similar asymptotic rg(T) after T5 6months, and measured the jump size distribution P(Drjrg) for each group. As the inset of Fig. 2b shows, users with small rg travel mostly over small distances, whereas those with large rg tend to display a combination of many small and a few larger jump sizes. Once we rescaled the distributions with rg (Fig. 2b), we found that the data collapsed into a single curve, suggesting that a single jump size distribution characterizes all users, independent of their rg. This indicates that P Dr rg !! " *r{a g F Dr $ rg " # , where a< 1.26 0.1 and F(x) is an rg-independent function with asymptotic behaviour, that is, F(x), x for x, 1 and F(x) rapidly decreases for x? 1. Therefore, the travel patterns of individual users may be approximated by a Lévy flight up to a distance characterized by rg. Most important, however, is the fact that the individual trajectories are bounded beyond rg; thus, large displacements, which are the source of the distinct and anomalous nature of Lévy flights, are statistically absent. To understand the relationship between the different exponents, we note that themeasured probability distributions are related