Time walkers and spatial dynamics of ageing information

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

Philosophy

**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:

- R1. At every time step, a new time walker is born at the source point on the lattice and tagged with its birth time which it carries throughout its life.
- R2. At every time step, all time walkers make an attempt to step to a random neighboring lattice point and one of the events (a)-(c) occurs:
- (a) If a time walker (TW
_{j}) steps to a lattice point previously visited by another time walker (TW_{i}) which carried newer information than TW_{j}carries, then TW_{j}is eliminated from the model. - (b) If TW
_{j}steps to a lattice point previously visited by TW_{i}which carried older information than TW_{j}, then TW_{j}completes the step and updates the age and the arrow at the lattice point (moving the arrow to point toward TW_{j}'s previous position). - (c) If a TW visits the same lattice point more than once, the TW completes the step but does not update the arrow. We included this rule to ensure that all visited points connect back to the information source without closed loops.

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:

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