A few types of graph cycles.

Transferics uses graph theory to analyze and organize flows of resources. Graph theory is a rich field of mathematics that allows structures to be described visually, algorithmically, and more abstractly, using linear algebra. This study is called the flow network model or graph transferics.


The basic element of graph transferics is the flow.  If a graph vertex represents an agent, trophic level, location, or other entity, the graph edge represents a flow, which is a transfer of a resource from one entity to another.  Flow is directional, which means flow graphs are directed graphs (digraphs).  Trade, by contrast, always involves a two-way exchange, and thus is undirected.

There are only two final outcomes possible for any flow in an transfery: recycling or dissipative loss.  Industrial ecologist Robert Ayres describes, based on this knowledge, a measure of ecological sustainability:

"A strong implication of the analysis sketched above is that a long-term (sustainable) steady-state industrial economy would necessarily be characterized by near-total recycling of plastics, paper, and other materials whose disposal constitutes an environmental problem.  Heavy metals are among the materials that would have to be almost totally recycled to satisfy the sustainability criterion.  The fraction of current metal supply needed to replace dissipative losses (i.e. production from virgin ores needed to maintain a stable level of consumption) is thus a useful surrogate measure of "distance" from a steady-state condition [...] of long-run sustainability."1

Waste streams must be kept as much as possible out of the biogeosphere, as anthropogenic input of heavy metals into the biogeosphere have long since exceeded that of all natural processes on Earth, and now approach multiple to hundreds of times greater.


A cycle, in the context of a VIAAC, would be any cost that is covered through a closed path of flow. This can involve unidirectional as well as bidirectional flow. As an example, consider a farmer, a landowner, and a rain harvester. All three have to eat, a cost which the farmer can cover. All three have to drink, which the rain harvester can cover. The farmer and the rain harvester need land, and the landowner is willing to let the other two work the land. This forms a closed cycle, because each of the three needs the other two to have a stable existence.


  • Cyclical flows that do not require nonrenewable resource inputs are considered sustainable cycles
  • A cycle can be either continuous or disjoint.  A continuous cycle is one where there is an uninterrupted flow of resources through the cycle. A disjoint cycle is one where the flow of resources is halted at some point.
    • A disjoint cycle that depends on a specific condition for the cycle to continue is a conditional disjunction.
    • One that requires a sequence of discrete, ordered steps is a temporal disjunction.

Relationship to VIAAC Goals:[]

Part of the goal of a VIAAC is attain sustainability.  Resource flow graphs give us an easy way to describe this goal; A VIAAC is fully sustainable if all its flows are sustainable cycles.


1. Allenby, B. R., Richards, D. J. & National Academy of Engineering. The greening of industrial ecosystems. (National Academy Press, 1994).