Say a complex system C has an attribute A. Can I assert that a change of A is always – to some degree – dependent upon A’s past?

My reasoning is the following:

  • Any attribute of C is influenced by C.
  • Therefore A’s current value is always a result of C.
  • C’s (i.e. the set of actors/nodes) interaction with A depends on the value of A.
  • A changes from the interaction of C with A

Hence, there always exists a dependence.

Note that I am not claiming that this dependence implies predictability.

up vote 1 down vote accepted

There are at least some types of systems for which your assertion:

C’s (i.e. the set of actors/nodes) interaction with A depends on the value of A.

is not always correct:

  • The most simple example would be if A is a “write-only-memory”, whose value is overwritten with the value of another node irrespective of its current value and never read. In such a situation, A’s current value only depends on the past of the rest of C, but not on its own past.

  • A more realistic example are networks of pulse-coupled neurons. Here you can have parts of the network that only react to external pulses, do not feed back into the rest of the network, and are constructed such that the effect of each pulse ceases completely before the next one.

Note that the exceptional property is rather that A’s state does not influence the future of the system (and thus also itself) rather than not being dependent on itself – though the two are technically equivalent.

However, for many types of complex systems, unidirectional couplings and in particular memoryless states are not possible. For example, ODE systems are defined through changes to their states and thus the past state always has an influence.

  • Thank you for the answer Wrzlprmft. So if it is that, in complex systems, the value of an attribute does not always depend on it's past, then is there a term to classify these 2 states of being dependent and independent of the past? – cardycakes Jun 12 at 19:12
  • I am not aware of a term for this and wouldn’t be surprised if none existed. Properties that are completely forgotten are rather rare and not particularly interesting. (Also please see my edit.) – Wrzlprmft Jun 12 at 19:21
  • Also, let me muse for a second. With the "write-only-memory" system, surely the wider system that interacted with any memory component could introduce some form of dependence? The memory component need no awareness of it's previous state(s) to have its state dependent (to some degree) on the actors or nodes that will interact with it? I suppose I could end this now by saying a RNG could be writing numbers to the memory component, thus no dependence? – cardycakes Jun 12 at 19:21
  • Your edit made the distinction very clear. From your comment, you are saying most properties do influence the future of the system (and are thus what I am referring to as dependent). Correct me if I am wrong. – cardycakes Jun 12 at 19:24
  • surely the wider system that interacted with any memory component could introduce some form of dependence? – Sure, but not on A, just on the rest of the system (also see my new edit). — The memory component need no awareness of it's previous state(s) to have its state dependent […] on the actors or nodes that will interact with it? – Yes. — you are saying most properties do influence the future of the system – Yes. Otherwise they would typically not be considered part of the system (as they are not relevant to its analysis). – Wrzlprmft Jun 12 at 19:43

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