Animated Time Lines:  Coordination of Spatial and Temporal Information
Sandra L. Arlinghaus, Michael Batty, and John D. Nystuen
with input from Naru Shiode*

    Animation is important because it permits the portrayal of spatial information as a rapid sequence of snapshots.  Thus, it integrates time with space.  Many of us think of cartoons of cute animals bouncing around on a movie screen when we think of "animation."  As is the case with most enduring ideas, this concept, too, has its amusing side as well as its scholarly side. Several aspects of the scholarly side have been explored in previous articles in this journal (see list of links, below), which by its internet transmission alone, lends itself as a fine medium in which to embed animations. In this paper we suggest the power of animation not only to simplify complexity, but also to coordinate sequences of information portrayed graphically.
     Havlin [4] used data plots to measure differences between ranks [8] over time.  He captured the differences in ranks, as a "Havlin score" on the vertical axis and the difference in time on the horizontal axis [2].  Vilensky [7] used Havlin plots to compare texts on the basis of word frequencies; books by the same author showed more in common on this factor than did books by different authors.  The idea of measuring such differences is one that applies to a whole range of topics:  from phase-shift diagrams, to word frequencies and authorship [7], to oil recovery [5],  to agricultural applications [6], and beyond.  Recently, Batty and Shiode [2] plotted populations for 459 British municipalities in Wales, Scotland, and England in 1901 by their rank differences every 10 years (Figure 1).  Casual knowledge [2] suggests a British urban spatial system that is stable in form; most large cities entered the urban system by 1901.  The single Havlin plot (Figure 1, Shiode created this original Havlin plot) displays a remarkably complex data set at a single glance.  It also suggests the underlying stability of the dynamics of this urban system through similarity of successive pattern as one moves from left to right:  color, only, serves as a guide to tracking the pattern.

Figure 1.  Static Havlin plot for rank differences in British population, 1901-1991 [2].

     To gain a more focused look at this pattern, we animate the Havlin plot: a dynamic visual serves as a model for the underlying dynamic system (Figure 2).  When the plot is animated, the time lines come into the picture in sequence;  the corresponding sequence of dates for each time line is coordinated, using color, to enter the picture along with the curve.  Coordination of sequences within a single image can underscore elements of that image's content.  Figure 2 emphasizes the time at which a particular curve enters the system so that one can easily track the dynamics of the British urban spatial system through a complex data set.  Note in particular that it is far easier to distinguish the 1971 curve from the 1981 curve in the animated figure than it is in the static figure.

Figure 2.  Animated Havlin plot coordinated with time legend.

The Havlin plot is an effective means of portraying complex urban dynamics in the British system because new cities do not enter the picture in any significant manner.  If one wishes, instead, to look at a corresponding plot for United States cities, the problem of cities entering and leaving the system obscures meaning to the pattern.  Thus, Batty [1] focuses instead on this entry/exit of cities from the broad urban system and expresses it in terms of half-life:  the extent to which new cities enter the list of top 100 cities, at each time slice, and the extent to which cities already in this list leave the list. A plot of these values [1], tallied in a matrix, results in a somewhat inverted Havlin plot (Figure 3).

Figure 3.  The Lives of US Cities from 1790 to 2000 [1].

Again, animation offers a dynamic view of a dynamic system.  Time lines in Figure 4a portray an animated sequence of Figure 3:  the moving blue line simplifies the otherwise complex visual pattern.  The interested reader, of course, will ask about the derivation of these time lines.  In conventional format, that reader might be referred to a matrix and left to run an eye up and down columns to understand the derivation of the plotted curves.  Figure 4b imitates that eye movement with an animated matrix.  The blue color of the curves in Figure 4a coordinates with the blue color of the matrix column outlines in Figure 4b to show clearly which column corresponds to which plotted curve:  coordination of animations clarifies mathematical process.  Coordination of images, each of which is itself an animated image, presents a sort of "meta" animation:  an animation of animations.  Thus, the transformation from the numerical to the geometric is emphasized.

Figure 4a.  Animated sequence of time lines.

Figure 4b.  Animated sequence of matrix columns.

Figure 4.  Animated sequence of time lines (4a) coordinated with animated sequence of matrix columns (4b) to show derivation of plotted curves.

     In the current Internet climate, this technique of image coordination can be effective only if the targeted audience is known to have high speed connections to the Internet or has the capability to download and view the image on a modern computer. Otherwise, the timing between successive images quickly goes out of kilter; fortunately, however, modern computing capability is rapidly becoming more affordable and more widespread.  Figure 4, viewed as a single whole, displays this sort of synthesis of animations.  Dynamic images of all sorts, including coordinated sets of images that are themselves dynamic, serve as models for dynamic systems.

*Naru Shiode (SUNY-Buffalo) produced the data for the Havlin plots (see )

  1. Batty, Michael.  The Emergence of Cities:  Complexity and Urban Dynamics, in press, 2003.
  2. Batty, Michael and Shiode, Naru.  Population Growth Dynamics in Cities, Countries, and Communication Systems, in press, 2003.
  3. Bunde, A. and Havlin, Shlomo [eds.]  Fractals and Disordered Systems, 2nd ed.,  Springer, New York, 1996.
  4. Havlin, Shlomo.  The distance between Zipf plots.  Physica A 216 (1): 148-150, 1995.
  5. Lee, Youngki; Andrade, Jr., Jose S.;  Buldyrev, Sergey V.; Dokholyan, Nikolay V.; Havlin, Shlomo; King, Peter R.; Paul, Gerald; and, Stanley, H. Eugene.  Traveling time and Traveling Length in Critical Percolation Clusters, Physical Review E, Volume 60, Number 3, September 1999.
  6. Stucki, J. W. and Lee, K.  Improving Soil Tests for Potassium: Fundamental Considerations for Partitioning Between Fixed and Exchangeable Forms and Redox Effects, Illinois Fertilizer Conference Proceedings, January 25-27, 1999.
  7. Vilensky, B. Can analysis of word frequency distinguish between writings of different authors?  Physica A 231:  705-711, 1996.
  8. Zipf, G. K.  Human Behavior and The Principle of Least Effort, Cambridge, MA, Addison-Wesley, 1949.

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