This applet visualises a model of spatial distribution of ageing information. Detailed information as well as instructions on how to use the applet is given below the applet window. In order to run the applet you need to have Java Runtime Environment installed
Time walker dynamics - We emit our time walkers from a single source (black circle), from which new information is created, on a two-dimensional lattice. To model the interactions between time walkers, we store two pieces of information at each lattice point: (i) the age of the youngest time walker to visit that point, and (ii) an arrow pointing to the lattice point from which the youngest time walker arrived. The time walkers obey the following rules:
Figure illustrating the dynamics of time walkers. (a) Illustration of rules R1 and R2 applied to three walkers released at different times: old (brown), intermediate (red), and young (yellow). The yellow TW can cross the trace of the brown TW (and the red TW), but the red TW cannot cross the trace of the yellow one and is eliminated from the model. (b) Snapshot of a simulation of TWs moving on a two-dimensional lattice. We use lighter colours to indicate younger walkers. (c) Mean square distance (MSD) of TWs as a function of their age: Blue circles are for distances reached by living TWs at a given age and red dots are for distances reached and age obtained when eliminated (fitted by MSD &sim age for old TWs). (d) Distribution of TW lifetimes fitted by age-2.
The age landscape - By continuously releasing time walkers onto the lattice an age landscape is built up. In general, since fewer steps are required to reach the central points than the remote corners of the lattice, points close to the information source have younger information than points farther way. However, the landscape can be highly intermittent, with plateaus of nearly constant age demonstrating the near space-filling property of random walkers in two dimensions, separated by sharp edges created by new time walkers entering a region left untouched for a long time, leading to a huge age variance.
Figure showing a three dimensional image and properties of the age landscape. (a) Information landscape created by continually releasing TWs from a fixed point source (blue sink in foreground). Hills and crevasses represent old and young information. The age of each lattice point is equal to the age of the last TW that visited. Blue streaks show a few traces of tracking walkers, released close to the edge of the lattice, that follow the arrows pointing toward the source in the landscape, creating an information river network. Panels (b)-(d) contain statistics of the information river network: (a) is the relation between upstream drainage area a and the length l of the longest upstream river (Hack's law) sampled over all lattice points. Panels (c) and (d) are histograms of a and l, showing that they are scale-free. Dashed lines are linear fits.
Navigation in the age landscape - As the time walkers move over the lattice, they mark each lattice point with updated ages and arrows providing directions toward the time walkers' source, which together effectively define the information landscape. To quantify the value of the information left by the time walkers, as measured by its age, we release passive tracking walkers on the lattice and study the pathways they take from their release points back to the source point, as they follow the arrows stored at each lattice point. By releasing a rain of trackers over the whole lattice, the myriad streams that correspond to the trajectories of the various trackers form an information river network . We define a stream's depth at a given coordinate P as the number of trackers that pass through P if released upstream of P. In this way, the stream's depth is analogous to the upstream drainage area of a river of flowing water. By evaluating the trajectories from a large number of tracking walkers, we can use the well-known scaling relations of actual rivers to characterize our information river network. We found that: