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Local information transfer as a spatiotemporal filter for complex systems

Joseph T. Lizier, Mikhail Prokopenko, Albert Y. Zomaya

arXiv:0809.3275v1nlin.CGnlin.AOnlin.PS

TL;DR

The paper addresses the lack of quantitative, spatially and temporally resolved measures linking information transfer to coherent structures in complex systems. It derives local transfer entropy from transfer entropy and applies it to cellular automata, where the resulting profiles filter coherent structure and quantitatively support particles as dominant information-transfer agents.

  • Problem

    Quantitative studies connecting information transfer to local coherent structures are comparatively absent, despite information transfer’s recognized role in complex systems.

  • Method

    The paper derives local transfer entropy from averaged transfer entropy to profile transfer into each spatiotemporal point, including different conditioning parameters and metric forms.

  • Results

    For cellular automata, local transfer entropy provides a coherent-structure filter and quantitatively supports particles, including gliders and domain walls, as information-transfer agents.

  • Takeaways & Limitations

    Local transfer entropy offers spatiotemporal profiles that reveal transfer patterns unavailable from averaged transfer entropy alone and can investigate analogous structures beyond cellular automata.

Abstract

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We present a measure of local information transfer, derived from an existing averaged information-theoretical measure, namely transfer entropy. Local transfer entropy is used to produce profiles of the information transfer into each spatiotemporal point in a complex system. These spatiotemporal profiles are useful not only as an analytical tool, but also allow explicit investigation of different parameter settings and forms of the transfer entropy metric itself. As an example, local transfer entropy is applied to cellular automata, where it is demonstrated to be a novel method of filtering for coherent structure. More importantly, local transfer entropy provides the first quantitative evidence for the long-held conjecture that the emergent traveling coherent structures known as particles (both gliders and domain walls, which have analogues in many physical processes) are the dominant information transfer agents in cellular automata.

I. INTRODUCTION

Information transfer is central to complex spatiotemporal behavior, but quantitative studies remain comparatively scarce. The paper develops local transfer entropy to examine transfer at each spatiotemporal point and applies it to cellular automata.

  • I. INTRODUCTION: Information transfer is considered vital to complex nonlinear behavior across cellular automata and several physical and biological systems.Examples include CA particles, solitons, wave fragments, crystal phase transitions, intelligent agents, and neural structure.
  • I. INTRODUCTION: Quantitative studies of information transfer are comparatively absent despite extensive complexity measures and debate over where transfer is maximized.The introduction contrasts proposed maxima in complex dynamics with intermediate levels associated with chaos.
  • I. INTRODUCTION: Local transfer entropy derives spatiotemporal information-transfer profiles from averaged transfer entropy, enabling analysis of hotspots and metric parameters.It supports examination of destination-history conditioning and additional information sources beyond what averaged transfer entropy provides.
  • I. INTRODUCTION: Cellular automata offer a model-system basis for interpreting quantitative transfer because their emergent particles and domains have been qualitatively studied extensively.Existing CA filters provide comparison methods for the proposed continuous spatiotemporal profiles.
  • I. INTRODUCTION: The study applies local transfer entropy to elementary cellular automata to test whether particles are dominant information-transfer agents.The particle framework includes gliders and domain walls against background domains.

III. LOCAL INFORMATION TRANSFER

The paper introduces transfer entropy and then derives local transfer entropy, while also distinguishing apparent and complete forms and discussing relevant parameters.

  • III. LOCAL INFORMATION TRANSFER: The paper derives local transfer entropy from transfer entropy and develops apparent and complete forms for local analysis.It also examines transfer-entropy parameters and self-information transfer.

A. Transfer Entropy

Transfer entropy addresses the lack of directionality and dynamical specificity in mutual information by measuring source information about a destination’s next state conditional on its past. The paper adopts this formulation as the basis for a local measure.

  • A. Transfer Entropy: Mutual information lacks inherent directionality and measures static shared information, problems not resolved by simple time-lagged variants.The criticism applies to equivalent non-information-theoretical definitions as well.
  • A. Transfer Entropy: Transfer entropy measures the deviation from independence of a destination’s state transition given the previous state of a source.It is presented as a truly dynamic measure and can be expressed as conditional mutual information.
  • A. Transfer Entropy: Transfer entropy quantifies average information from the source about the destination’s next state conditioned on the destination’s past.The measure is defined in bits and is applicable to deterministic and stochastic systems.
  • A. Transfer Entropy: Transfer entropy has been applied to sensorimotor networks and information closure, while alternative perturbation-based and cellular-automaton measures remain comparison points.The paper proposes future comparison with a causality-oriented information-flow measure.
  • A. Transfer Entropy: The paper identifies a gap between transfer entropy’s quantitative definition and local information-transfer instances such as cellular-automaton particles.Those instances are local in space and time and therefore require a local measure for investigation.

B. Local Transfer Entropy

Local transfer entropy extracts an observation-level information-transfer contribution from averaged transfer entropy and assigns it to each destination, time, and source direction. In lattice systems, the resulting profiles expose transfer channels while destination-history conditioning removes non-traveling self-influence.

  • B. Local Transfer Entropy: The derivation rewrites the averaged transfer entropy as a probability-weighted sum over observed state-transition tuples.Finite observations estimate tuple probabilities from their counts.
  • B. Local Transfer Entropy: Moving the logarithm inside the sum converts the averaged metric into a sum over actual observations, identifying a local transfer-entropy contribution at each observation.The local measure is therefore the quantity whose expectation recovers global transfer entropy.
  • B. Local Transfer Entropy: Local transfer entropy is defined for every destination element, time, and causal source, producing a spatiotemporal profile for each information channel or direction.For cellular automata, sensible source offsets satisfy |j| ≤ r, and j denotes spatial displacement per time step.
  • B. Local Transfer Entropy: The local cellular-automaton quantity measures transfer from a source-cell block to a destination cell conditioned on the destination’s k-state history.The source block has size l and the destination is evaluated at time n + 1.
  • B. Local Transfer Entropy: Conditioning on the destination’s history removes non-traveling self-influence that could otherwise appear as independent source-to-destination transfer.This issue arises in systems with bidirectional information transfer, including cellular automata.
  • B. Local Transfer Entropy: The asymptotic destination-history length k →∞ is considered most correct for non-Markovian dynamics, although finite-k estimates are retained because the limit is generally infeasible.The notation drops l when the default source-history length is l = 1.

C. Complete and Apparent Transfer Entropy

The paper distinguishes apparent transfer entropy from complete transfer entropy by conditioning on other causal information sources. For cellular automata, complete conditioning removes misleading source effects and yields a nonnegative local measure in deterministic systems.

  • Local transfer entropy is negative when the source-conditioned probability of the destination’s actual next state is lower than its source-independent probability.Such negativity indicates that the source is misleading about the destination transition in the given context.
  • Apparent transfer entropy measures source information without conditioning on other possible contributors, so those contributors may be mistaken for source influence.The paper denotes conditioning on no other information contributors as apparent transfer entropy.
  • Complete transfer entropy conditions on the destination’s past and all other causal information contributors, isolating information supplied by the source.For ECAs, the other contributors are the destination’s neighboring cells excluding the measured source and the destination’s previous value.
  • In deterministic systems such as ECAs, complete conditioning ensures local complete transfer entropy is nonnegative because all other causal sources are included.The averaged complete transfer entropy can be constructed for any system by conditioning out all causal contributors apart from the source under consideration.

D. Summed Information Transfer Profiles

The summed information transfer profiles combine transfer entropies across source positions, including a distinct self-information-transfer component for the destination’s own past.

  • Self-information transfer uses the destination’s immediate past as the source and measures information about its next state beyond the prior history.The paper interprets this as traveling information with instantaneous velocity zero.
  • Although self-information transfer is not especially useful alone, it completes summed profiles that also include transfer entropies from spatially distinct sources.These summed profiles are defined separately for apparent and complete transfer entropy.

IV. RESULTS AND DISCUSSION

The experiments evaluate local transfer-entropy profiles across selected elementary cellular-automaton rules, parameters, and metric forms. They focus on whether the profiles reveal regular and irregular particle structures while using repeated long-lattice simulations.

  • Each cellular-automaton instance used 10 000 randomized cells, discarded 30 settling steps, and recorded 600 further time steps.Results were confirmed by at least 10 runs with different randomized initial states, using periodic boundary conditions.
  • The study varied history length k, compared elementary-cellular-automaton types, and contrasted apparent with complete local transfer-entropy profiles.The source-history parameter l was fixed at 1.
  • Probability distributions for cellular automata were estimated from all spatiotemporal observations of the corresponding channel rather than only the measured source-destination pair.This choice is described as appropriate for spatially ordered systems with homogeneous agents such as cellular automata.
  • The analysis concentrated on rule 110 with regular gliders and rule 18 with irregular domain walls to test whether local information-transfer profiles highlight both particle types.The selected rules also enable comparison with other filtering techniques.

A. Base comparison cases

For ECA Rule 110, default local mutual information and transfer-entropy profiles fail to distinguish gliders from background, whereas larger history lengths reveal coherent information-transfer patterns. The profiles show gliders as dominant, but not exclusive, information-transfer agents.

  • Base comparison cases: Default k = 1 local mutual information and apparent or complete transfer entropy do not distinguish gliders from the background in ECA Rule 110.The same basic metrics were also unsuccessful for other displacements and cellular-automaton rules.
  • Gliders as dominant information transfer agents: For k = 6, summed local complete transfer entropy highlights gliders, with transfer aligned to their macroscopic direction of motion.The summed profile also resembles filtering obtained with other techniques, unlike averaged transfer entropy alone.
  • Gliders as dominant information transfer agents: A close-up glider profile reveals two consecutive rightward information transfers followed by a pause, while some visually marked points are predictable from the domain history.At one transfer point, the local complete transfer entropy is 4.7 bits.
  • Gliders as dominant information transfer agents: Using k = 16 better approximates k →∞, concentrating larger transfer values near leading glider edges and largely removing orthogonal or vertically predictable structure.The authors state that k →∞ is more correct but not computationally feasible, so k = 16 provides a practical estimate.
  • Gliders as dominant information transfer agents: Small non-zero transfer can occur inside the periodic domain because neighboring sources add information about whether a glider is incoming.For one such point, the source-conditioned probability is 0.96 and local complete transfer entropy is 0.057 bits.
  • Gliders as dominant information transfer agents: Because orthogonal transfer remains weaker and less coherent, the authors describe gliders as dominant rather than the only information-transfer agents.The conclusion is supported for ECA Rule 110 and has also been verified for ECA Rule 54.

C. Domain walls as dominant information transfer agents

In ECA Rule 18, local complete transfer entropy highlights domain walls as dominant information-transfer agents, while also revealing structured transfer within domains.

  • Local complete transfer entropy highlights Rule 18 domain walls as strong information-transfer regions in each channel.The summed profile quantitatively confirms domain walls as dominant information-transfer agents against the domain.
  • Similar local-transfer results were observed for ECA Rule 146.
  • Rule 18 domains exhibit a spatial-temporal period-2 transfer pattern corresponding closely to a period-2 spatial epsilon-machine description.
  • Domain-wall motion produces high transfer entropy in its direction because the intruding domain informs the destination’s next state.
  • Rule 22 shows substantial transfer at many points but no coherent structure, consistent with its lack of known domain walls.

D. Apparent transfer entropy

Apparent transfer entropy largely identifies gliders and domain walls like the complete metric, but it can expose misleading sources and differs in domains.

  • Local apparent transfer entropy highlights gliders and domain walls as dominant agents in their direction of motion.
  • For Rule 110, positive apparent-transfer profiles closely resemble complete-transfer profiles, while orthogonal channels can contain negative values.
  • A Rule 110 glider example yields 2.65 bits of positive apparent transfer in its motion direction and −2.05 bits orthogonally.The orthogonal source is misleading, whereas the complete metric measures 0.00 bits for that channel at the same point.
  • In Rule 18, apparent transfer is positive along domain-wall motion but vanishes throughout the domain, unlike the periodic complete-transfer pattern.The complete metric detects 1 bit at alternate domain sites because it conditions on the rest of the neighborhood.
  • The apparent and complete metrics can produce different insights, so both viewpoints remain valuable.

E. Averaged transfer entropies

Averaged transfer entropies provide limited structural insight: complete values decrease with conditioning length, while apparent values may increase and neither converges over the measured range.

  • Average complete transfer entropies decrease as conditioning length k increases, whereas average apparent transfer entropy can increase.The difference reflects whether other information sources are conditioned out.
  • Figure 7 plots average complete and apparent transfer entropies against k for channels j = 1 and −1 in ECA Rule 110.
  • Neither metric reaches a limiting value over the measured k range, supporting k →∞ as the most correct choice in principle.In practice, k is limited by sample size; previous sections used k = 16 to retain enough observations for probability estimation.
  • Averages reveal little about glider particles and obscure distinctions between apparent and complete metrics that local profiles expose.

V. CONCLUSION

The paper introduces local transfer entropy as a spatiotemporal analytic and filtering tool for cellular automata. It distinguishes metric forms and conditioning choices, and provides quantitative support that gliders and domain walls transfer information.

  • Local transfer entropy characterizes information transfer into each spatiotemporal point and provides profiles unavailable from averaged transfer entropy alone.
  • Applied to cellular automata, local transfer entropy acts as a continuous coherent-structure filter with multiple directional views and a combined profile.
  • The method need not use a new filter for every cellular automaton, but its probability distribution functions must be recalculated for each one.
  • Local transfer entropy provided the first quantitative support for particles—gliders and domain walls—as information-transfer agents in cellular automata.The result is connected to analogous coherent structures and hypothesized information-transfer agents in physical systems.
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