According to the second law of thermodynamics a system, left to its own devices, will eventually lose order: hot coffee poured into cold will dissipate its heat until all the coffee in the cup is of the same temperature; matter breaks down over time when exposed to the elements; and systems lose structure and differentiation. The same is not true for complex systems. They gain order and structure over time.

What constitutes a flow?

In general, we can conceptualize flows as some form of energy that helps drive or push the system. But what do we mean by energy? And what kinds of energy flows should we pay attention to in the context of complexity?

In some cases, the source of system energy aligns with a strictly technical definition of what we think of when we say 'energy'. Such is the case in the classic example of 'Benard rolls' (see Open / Dissipative for a video of this phenomena). Here, a coherent, emergent 'roll' pattern is generated by exciting water molecules by means of a heat source.  It becomes relatively straightforward to identify thermal energy as the flow driving the system: heat enters the water system from below, dissipates to the environment above, and drives emergent water roll activity in between.

But there are a host of different kinds of complex systems where we see all kinds of driving flows that do not necessarily have their dynamics directed in accordances with this strict conception of 'energy'.

Example:

In an academic citation network, citations could be perceived as the 'energy' or flow that drives the system towards self-organization. As more citations are gathered, a scholar's reputation is enhanced, and more citations flow towards that scholar.  A pattern of scholarly achievement emerges (that follows a {{power-law}} distribution), due to the way in which the 'energy flows' of scholarly recognition (citations), are distributed within the system. While we tend to think that citations are based on merit, a number of studies have been able to replicate patterns that echo citation distribution ratios using only the kinds of mechanisms we would expect to see within a complex system - with no inherent merit required (see also Preferential Attachment!).

Similarly, the stock market can be considered as a complex adaptive system, with stock prices forming the flow which helps to steer system behavior; the world wide web can be considered as a complex adaptive system, with the number of website clicks serving as a key flow; the ways in which Netflix organizes recommendations can be considered as a complex adaptive systems, with movies watched serving as the flow that directs the system towards new recommendations.

Clearly, it is helpful to understand the specific nature of the driving flows within any given complex system, as this is what helps push the system along a particular trajectory. For ants, (who form emergent trails), food is the energy driving the system. The ants adjust their behaviors in order to gain access to differential flows (or sources) of food in the most effective way possible given the knowledge of the colony. In this case, the global caloric value of food stocks found is a good way to track the effectiveness of ant behavior.

If we look at different systems, we should be able to somehow 'count' how flow is directed and processed: citation counts, stock prices, website clicks, movies watched.

Multiple Flows:

Often complex systems are subject to more than one kind of flow that steers dynamics. For example, we can look at the complex population dynamics of a species within an ecosystem with a limited carrying capacity. Here, two flows are of interest: the intensity of reproduction (or the flow of new entrants into the environmental context), and the flow of food supplies (that limits how much population can be sustained). Here one flow rate drives the system (reproductive rate), while another flow rate chokes the system (carrying capacity). This interactions between two input flows (one driving and the other constraining), produces very interesting emergent dynamics that lead the system to oscillate or move periodically from one 'state' (or Attractor States) to another. A more colloquial way of thinking about this periodic cycling is captured in the idea of 'boom' and 'bust' cycles, although there are other kinds of cycles that involve moving between not just two, but many additional cycling regimes (see Bifurcations for more!).

Go with the flow:

Flow is the source of energy that drives self-organizating processes. A complex system is a collection of agents that are operating within a kind of loose or Open / Dissipative boundary, and flow is what comes in from the outside and is then processed by these agents.  Food is not part of the ant colony system, but it is what drives colony dynamics. The magic of self-organization is that, rather than each agent needing to independently figure out how best to access and optimize this external flow, each agent can learn from what its neighbors are doing.

Accordingly, there are two kinds of flows in a complex system - the external flow that needs to be internalized and processed, and the internal flows amongst agents that help signal the best way to perform within a given environment (and thereby process these external flows). The act of generating these signals is what Pierre Paul Grasse describes as Stigmergy -  a process that in some way marks are alters the shared environment of all agents in ways that can thereby steer agent behavior. For example, ants depositing pheromones on a path leading to food is an example a stigmergic signal.

An environment characterized by stigmergic signals is no longer neutral - it has areas or zones of intensity that affect all agents in the system that are in proximity to these signals. Thus, although agents may function  in random ways, stigmergy shifts the probability that agents in a system will behave in one way versus another:  the more intensity a particular zone of stigmergy has, the more likely agents will be drawn into the behavior directed by that zone.

Using stigmergy signals to help direct the processing of flows,  agents gradually move into regimes that process these flows utilizing minimal energy requirements: through Positive Feedback they draw other agents along into similar regimes of behavior making the system, as a whole, an efficient energy processor.

Its all about difference:

Every complex system channels its own specific form of driving flow.

In every case, it is important to look beyond technical definitions of energy flows in complex systems, to instead understand these as the differences that matter to the agents in the system, or as Gregory Bateson states 'the difference that makes a difference'. All complex systems involve some sort of differential, and this differential is regulated by an imbalance of flows, that thereby steers subsequent agent actions.  As the system realigns itself through  attuning to these differentials, new behaviors or patterns emerge that, in some way, optimize behaviors.

Inherent Laziness: Its everywhere!

A nice way to think about this is to imagine that everything in the world is essentially trying to do the least possible work - particularly when being pushed around by some outside force. The Driving Flows are the outside force, which are basically come into the agent territory.

Responsive Agents, Differential Flows:

Sometimes, all the agents really care about is basically shaking off the disturbance: the liquid molecules being heated in the Benard Rolls were happily drifting about, only to have some annoying heat energy start to come along that they now need to contend with. At that point, the regime that allows the heat to pass through the system and rise to the top is for the molecules to get into neater alignments of rolls that allows these currents to go through with less overall disruption. The same is true in the action of sand grains forming ridges, in response to the driving flows of the winds. In both cases, the agents, left to themselves, are not driving flows in and of themselves.

Active Agents, Differential Flows:

At other times, the agents are themselves a kind of driving force, that need and external driving flow to achieve a goal (eat, reproduce, etc.), but they are trying to figure out how to claim the prize without wasted effort. Unlike the agitated fluid, or the disturbed sand, the ants will move to seek the driving flow, whether or not it is present, ultimately running out of steam. We can see here that the ants are different from the sand grains, because the sand grains are passive without the external input, whereas the ant behavior actively seeks out the external input. A tree growing does the same thing - its roots look for nutrients, its branches and leaves extend towards the sun - the environment and the agent work together to minimize the effort of the growing tree to get what it needs without expending unnecessary resources.

Flowing Agents, Differential Context

A final example inverts the situation entirely, where the driving flow is coming strictly from an agent in an environment. Imagine I want to walk up a hill. My drive is to get to the top, but I want to do so expending the least amount of energy in terms of the parameters of both time and effort. I can charge directly upwards - using the principle that the shortest distance between two points is a straight line. But while this might initially appear to be a good solution, I quickly discover that the effort of the direct vertical path takes a toll on my energy level. Instead, if I extend the distance of travel, but reduce the slope (thereby moving at a lateral incline), my energy of each step is reduced. That said, the more I reduce the energy of movement, the longer the lateral inclines - meaning that more time to get to the top is extending. Our bodies make a balanced calculation to find the zig-zagging path up the hill that is able to account both for the time parameter and the energy parameter. The path is an emergent outcome of this calculation. The best solution for reaching the top while expending the minimum amount of resources (as a function of both time and energy). It is still worth noting that this activity is still happening in an environment with a differential - the differential this time being the slope of the terrain - but this differential is not one that is being produced by a flow moving into the system (like the heat differential in Benard Rolls), it is instead that we have an agent trying to flow through a differential environment.