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

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:

- The relationship between the length l of the longest river stream (equivalent to the largest number of tracking steps) within a drainage area and the size of the drainage area a, is is lah with h=0.63. This is referred to as the Hack's law and h≈0.57-0.60 which for river networks of water.

- Histograms of river lengths and drainage areas are scale-free, p(l)∼l and p(a)∼a with γ=1.40 and τ=1.37. Corresponding exponents for water river networks are γ=1.8±0.1 and τ=1.43±0.02 which means that the information rivers are in general longer, of more serpentine shape, and tend to have larger and wider drainage areas than real river networks.

Tutorial

The Java applet illustrates three basic features the time walkers. These are all controlled by the buttons located in the 'Display mode' panel. The function of each button is the following:

- Time walkers. By clicking on the 'Time walkers' button the motion of individual time walkers can be followed as they move on the lattice after being released from the source (black circle). At birth each time walker is given a colour which fill the lattice points that the time walker visits. If the time walker enters a node previously visited by another younger time walker it will die which is indicated with a yellow flash (see dynamical rules in the 'Philosophy' section in the left coloumn). An additional source can be added by first clicking on the 'Add source' button in the 'Action button' panel and then clicking on the desired location on the lattice. There is also a possibility to add an absorbing boundary killing those time walkers that touch it. The absorbing wall is created by clicking the 'Add' button under 'Add absorbing wall' inside the "Action button" panel. The wall is removed when clicking the 'Remove' button. Moving the source and adding boundaries work in all display modes.

- Age landscape. After the release of a large number of time walkers, each lattice site is marked with an age and the whole lattice can be visualised as an age landscape. Colouring towards dark brown indicate old age which eventually fade a way into the white background. This symbolises the decay of the value of information as time progresses. The colour palette is inspired from Salvador Dali's famous painting The persistence of memory. The painting shows a dream where time has no significance; events occur simultaneously without any obvious causality. This contrasts our time walkers picture where time is everything.

- River landscape. As time walkers move on the lattice they mark each lattice point with two pieces of information: the age of the youngest time walker to visit that point and the direction from which it came. By clicking the 'River landscape' button the latter is explored. The arrows form a navigable information landscape which can be used to, for instance, locate the time walker source from any given point in the lattice. If releasing trackers that follow the arrows in the landscape and keep count of the number of trackers that go through every lattice point on their way to the source, which is analogous to river depth, an information river landscape is built up. White to blue colouring shallow to deep river. The dark brown line shows the main river from the chosen lattice point indicated by the red circle, to the source. This point can change by clicking on 'Change' in the 'Simulation stats' box. Additional statistics of the selected lattice point can be viewed also be viewed in the 'Simulation stats' box. For instance Node is the node number, Age is the age of the latest visited time walker and Arrow indicate the direction of the stored information arrow.