Introduction - The distribution of information is essential for living system's ability to coordinate and adapt. Random walkers are often used to model this distribution process and, in doing so, one effectively assumes that information maintains its relevance over time. But the value of information in social and biological systems often decays with time and must continuously be updated. To capture the spatial dynamics of aging information, we introduce the time walkers. A time walker moves just like a random walker, but interacts with traces left by other walkers, some representing older information, some newer. The traces form a navigable information landscape which has self-similarity features of river networks of flowing water. We found that search processes using the age landscape is far superior to a random search and has scaling properties of loop-erased random walks.
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 (TWj) steps to a lattice point previously visited by another time walker (TWi) which carried newer information than TWj carries, then TWj is eliminated from the model.
- (b) If TWj steps to a lattice point previously visited by TWi which carried older information than TWj, then TWj completes the step and updates the age and the arrow at the lattice point (moving the arrow to point toward TWj'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: