Complexity science purports to shift the focus from describing states to describing processes, yet this conceptual shift has been much slower and more difficult than many would let on. The amount of self-anointed complexity research that still relies on equilibrium analysis reveals the limited degree to which the field has actually broken free from its methodological roots. Complex adaptive systems are exactly those that self organize in a way that allows them to maintain functionality, cohesion, or other such properties while being continuously in flux. If a system really reaches equilibrium then it’s a bad candidate for being a complex system. But we don’t need to throw out all our old concepts in the pursuit of new, dynamic ones; we can use them as a springboard for developing complexity science. This post is an attempt to analogize the equilibrium concept in a way that directly shifts the focus from states to processes.

An equilibrium state, which we might also call a fully stable state, is a state in which the system remains once it is reached. That's not the only going notion – other equilibrium concepts exist already, such as an equilibrium distribution. An example of an equilibrium distribution used commonly in Markov modeling is when the percentages of agents in each state remains constant although individual agents continue to transition from one state to another. This is not what I mean by process equilibrium, though it's a closer idea. An equilibrium distribution of this sort implies a cycling of the states, and the process by which this is achieved is static. This is certainly important, even for complexity analysis, but it doesn’t describe the process itself equilibrating.

A standing wave at first seems to be a similar concept to the equilibrium distribution. These are stereotypical emergent phenomena (and while this may not guarantee that it's a complex system, let go ahead and say that it is) in which the parts are continuously churning though the aggregate pattern is constant. Holding the water flow rate and rock locations constant, the wave phenomenon seems to be in equilibrium…so isn't that a complex system in equilibrium? No. It's the water particles that compose the complex system from which the wave emerges, and the water particles are not in equilibrium. Sloppy term usage can get one in trouble with these sorts of things, so let's be extra careful. When aggregate properties are constant through changes in the micro components that is not equilibrium; that's simply identifying a robust higher-level (aka emergent) phenomenon.

The concept of process equilibrium relates the concept of equilibrium analysis (and its partner concepts) to the processes themselves. Two different interpretations naturally come to mind: 1) the actual rules used by agents stop changing even though a mechanism exists through which they could change, or 2) the process (e.g. the set of agent actions) always stays the same although individual agents may change their actions (e.g. because they switch roles). The first interpretation is analogous to a stable state, but it's stability in the enacted process. The second is analogous to a stable distribution, but of actions instead of states.

These rule-related interpretations are perhaps novel, but they are not a far stretch and have probably already been explored to some degree. My thinking is that in some models the sets of behavior rules accessible to agents are constant and in some models the agents have the capability to change their behavior rules (through other rules, or indirectly by changing the context of future behaviors). If the behavior rules can’t change then the system is always in process equilibrium and that’s trivial and uninteresting. But if the agents can learn or adapt then the behavior rules can change; and if at some point the rules being used stop changing, then that’s interesting. If agents can learn but stop learning to do new things, or they can adapt but no longer do so, then that's a kind of system stability that we'd like to capture but aren't generally. The second type demonstrates a system-wide co-adaptation of behaviors…everything is still acting, perhaps even constantly changing their actual actions each time step, but the rule set becomes frozen through internal dynamics.

But we can capture behavior rules in many ways. In the Game of Life the agents can’t change their rules, but we can track which rules are actually used by each cell and mark when that set of actually used rules stop changing…or changes in cycles. Such tracking could distinguish configurations that create perpetual novelty from ones that repeat, and how long that cycle is. However, this is treating the used behavior rule as a property of each agent at each time-slice of the model…again falling into static thinking. That's not anything new.

So now we’re going to step it up one level. We’re going to take the set of deployable behavior rules as the unit of analysis here. This means that as long as the set of rules that agents can use continues to change then the system is not in process equilibrium. This concept of process equilibrium then becomes a way to analyze contingent behavior capabilities. It draws a line between doing something different only in different environments and being able to do something different with the same input at a later time. What I'd like to do is run the analogy one more step higher to separate learning and adaptation along another line. Perhaps define "learning" as being able to change the deployable behavior rules via a fixed set of update rules and define "adapting" as being able to change the updating rules. Then process equilibrium of the process equilibrium would also allow us to measure how adaptive a system is and identify systems that start out adaptive but sometimes converge into simpler learning dynamics…and when and why this happens.

There are some problems with this proposal, I think. One is that it isn't clear that adapting should be altering learning rules rather than being a different way to alter behavior rules directly. Perhaps the difference between learning and adapting is how the behavior rules change rather than which rules change. In my workshops I tell the story that a learning traffic light accepts the meaning of the red, yellow, and green and determines when and how long to shine each light. An adapting traffic light might add a different color or shine two at once or other such alterations to actual behaviors. No matter the difference between learning and adapting, they are both rule-driven activities separate from the behavior rules and thus the process equilibrium still does its first-tier work.