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, structure, and information.
This is because such systems, while operating as bounded 'wholes', are not entirely bounded. They remain open to the environment, and the environment, in some fashion, 'feeds' or 'drives' the system: providing energy that can be used by the system to build and retain structure. Thus complex systems violate the second law of thermodynamics and, rather then tending towards disorder (entropy), they are pushed towards order (negentropy).
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). 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'.
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 operate within a complex system - with no 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 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 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.
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 environment 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) 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 more than two regimes (see Bifurcations for more!).
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 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 get the job done. As agents move into regimes that process flows in ways that minimize energy requirements, they draw other agents along into similar regimes of behavior making the system, as a whole, an efficient energy processor.
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. All complex systems involve some sort of differential, and this differential is regulated by an imbalance of flows, that thereby steer subsequent agent actions. As the system realigns itself through attuning to these differentials, new behaviors or patterns emerge that, in some way, optimize behaviors.