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Computation in Complex Systems

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The Local Information Dynamics of Distributed Computation in Complex Systems

Part of the book series: Springer Theses ((Springer Theses))

Abstract

Complex systems science is the study of large collections of (generally simple) entities, where the global behaviour is a non-trivial result of the local interactions of the individual elements [1]. This approach seeks a fundamental understanding of how such collective behaviour results from these interactions between simple individuals. In particular, it seeks to gain and apply this understanding across many different disciplines, examining both natural and man-made systems as apparently diverse as insect colonies, the brain, the immune system, economies and the world wide web [2]. Complex behaviour is often described as incorporating elements of both order and disorder (or chaos), and these elements can be seen in all of the above, e.g. path-following (order) versus exploration (disorder) in ant foraging.

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Notes

  1. 1.

    For introductions to the beginnings of complex systems science, see [2, 11, 12, 14].

  2. 2.

    In the sense of chaotic behaviour.

  3. 3.

    Particularly those with less well-defined or more complex transitions.

  4. 4.

    \(H_{\mu X} (k)\) here is equivalent to \(h_\mu (L-1)\) in [39]. This means the sum in Eq. (2.18) starts from \(k=0\) as equivalent to \(L=1\).

  5. 5.

    The excess entropy was labelled the “effective measure complexity” by Grassberger in [42].

  6. 6.

    Local information-theoretic measures are known as point-wise measures elsewhere [44].

  7. 7.

    Appendix A describes two different approaches that have been presented to quantifying partial localisations of the mutual information \(I(x_n;Y)\) [48]. The partial localisation \(I(x_n;Y)\) considers how much information \(I(x_n;Y)\) a specific value \(x_n\) at time step \(n\) gives about what value \(Y\) might take. We note this is distinct from the full localisations \(i(x_n;y_n)\) that we consider here; this quantifies the amount of information conveyed by a specific value \(x_n\) about the specific value \(y_n\) that \(Y\) actually takes at time step \(n\) (or vice-versa). Our interest lies in these full localisations \(i(x_n;y_n)\), as they quantify the specific amount of information involved or manipulated in the dynamics of the computation at time step \(n\) with the given realisation \(\left\{ x_n,y_n \right\} \). Appendix A demonstrates that there is only one approach to quantifying full localisations \(i(x_n;y_n)\) that fulfils both additivity and symmetry properties.

    We also note that a similar approach to “localising” information-theoretic values is by using sliding windows of observations (e.g. [49]). While this does provide a more local measure than averaging over all available observations, it is not local in the same sense as the term is used here (i.e. it does not look at the information involved in the computation at a single specific time step).

  8. 8.

    The transfer entropy is arguably the most important measure used in this thesis. As such, recommendations on approaches to compute it will be followed for other measures also for consistency. The transfer entropy will be introduced in Chap. 4

  9. 9.

    An attractor is a single global state \(\mathbf x _n = \{ \ldots , x_{i-1,n}, x_{i,n}, x_{i+1,n}, \ldots \}\) or periodic sequence of global states that a CA (generally considering fixed sizes) can reach after a finite number of time steps but then never leave (unless some stochasticity is introduced into the dynamics).

  10. 10.

    A transient is a path of global states that a CA could traverse before reaching an attractor. For deterministic CAs, two or more transient paths can converge on the same next state, but a given state cannot diverge via multiple transient paths to more than one next state. A CA of finite size \(C\) cells must reach an attractor after a finite number of time steps (since there are a finite number \(b^C\) of possible global states, where \(b\) is the base or number of possible discrete states for each cell).

  11. 11.

    Of course, blinkers can be considered to be vertical or non-moving gliders, while both are particles.

  12. 12.

    However as discussed in Chap. 4 this is a symmetric measure not capturing directional transfer.

  13. 13.

    Langton’s use of mutual information in [15] is a symmetric measure not capturing directional transfer. As we will describe in Sect. 3.1, Grassberger inferred the excess entropy to be infinite in some CAs under certain circumstances by studying trends of the entropy rate [42, 114]. The study did not make any direct measurements of the excess entropy apart from these inferences, and focussed on collective excess entropy. Crutchfield and Feldman measure the excess entropy in spatially-extended blocks in various CAs in an ensemble study in [19], however this is not an information storage since the measurement is not made temporally.

  14. 14.

    Measured using the CimulA package [100] over 600 time steps of 100,000 cells, with light cone depths of 3 time steps.

  15. 15.

    Lattice systems are those with a regular spatial ordering for their agents or variables, typically being placed on a one or two-dimensional array.

  16. 16.

    The work in [69] is part of an interesting trend towards the use of information-theoretic measures to study network topology, e.g. see also [70]. Though information-theoretic, since these measures study topology rather than time-series dynamics they remain out of scope here.

  17. 17.

    Though, of course either of these can arise at random.

  18. 18.

    There has been some debate about the best updating scheme to model GRNs [143], and variations on the synchronous CRBN model are known to produce different behaviours. However, the relevant phase transitions are known to exist in all updating schemes, and their properties depend more on the network size than on the updating scheme [144]. As such, the use of CRBNs is justified for ensemble studies such as ours [128].

  19. 19.

    The justification or otherwise of the suggestion that natural evolution is driven by the intrinsic forces of information processing is irrelevant to whether information-driven design can be used as a successful tool for artificial systems.

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Lizier, J.T. (2013). Computation in Complex Systems. In: The Local Information Dynamics of Distributed Computation in Complex Systems. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32952-4_2

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