the links several times during the study period whereas the bulk dry carrier (Fig. 5b) passes almost every link exactly once. The oil tanker (Fig. 5c) commutes a few times between some ports, but by and large also serves most links only once. 

We can express these trends in terms of a “regularity index” p that quantifies how much the frequency with which each link is used deviates from a random network. Consider the trajectory of a ship calling S times at N distinct ports and travelling on L distinct links. We compare the mean number of journeys per link f$_{real}$ = S/L to the average link usage f$_{ran}$ in an ensemble of randomized trajectories with the same number of nodes N and port calls S . To quantify the difference between real and random trajectories we calculate the Z score p = ( f$_{real}$ − f$_{ran}$ ) /σ (where σ is the standard deviation of f in the random ensemble). If p = 0, the real trajectory is indistinguishable from a random walk, whereas larger values of p indicate that the movement is more regular. Figures 5d-f present the distributions of the regularity index p for the different fleets. For container ships, p is distributed broadly around p ≈ 2, thus supporting our earlier observation that most container ships provide regular services between ports along their way. Trajectories of bulk dry carriers and oil tankers, on the other hand, appear essentially random with the vast majority of ships near p = 0. 

## VI. APPROXIMATING TRAFFIC FLOWS USING THE GRAVITY MODEL 

In this article, we view global ship movements as a network based on detailed arrival and departure records. Until recently, surveys of seaborne trade had to rely on far less data: only the total number of arrivals at some major ports were publicly accessible, but not the ships’ actual paths (Zachcial & Heideloff 2001). Missing information about the frequency of journeys, thus, had to be replaced by plausible assumptions, the gravity model being the most popular choice. It posits that trips are, in general, more likely between nearby ports than between ports far apart. If d$_{ij}$ is the distance between ports i and j , the decline in mutual interaction is expressed in terms of a distance deterrence function f ( d$_{ij}$ ). The number of journeys from i to j then takes the form F$_{ij}$ = a$_{i}$b$_{j}$O$_{i}$I$_{j}$f ( d$_{ij}$ ), where O$_{i}$ is the total number of departures from port i and I$_{j}$ the number of arrivals at j (Haynes & Fotheringham 1984). The coefficients a$_{i}$ and b$_{j}$ are needed to ensure ∑$_{j}$$_{F}$$_{ij}$ = O$_{i}$ and ∑$_{i}$$_{F}$$_{ij}$ = I$_{j}$ . 

How well can the gravity model approximate real ship traffic? We choose a truncated 

