Graph Theory Transferics

A few types of graph cycles.

Natural economics 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.

Flow

The basic element of graph theory economics 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 economy: 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.

Cycle

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.

Definitions:

• 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.

Closeability

A VIAAC's closeability is a description of its ability to survive without external inputs.

Definitions

A VIAAC IS:

• Input closeable if it can survive with no inputs except its own (but with outputs to the outside). This can describe any subtype of closeability, e.g. fully input closeable.
• Output closeable if it can survive with no outputs except its own (but with inputs from the outside). This can describe any subtype of closeability, e.g. market output closeable.
• Pseudo-closeable (of degree n) if it can become closeable in n steps (or clock cycles). This can describe any subtype of closeability, e.g. pseudo-market closeable of degree n.
• Quasi-closeable if it can survive with only inputs or outputs from other VIAACs or non-market economies. A quasi-closeable VIAAC is pseudo-quasi-closeable of degree 0.
• Semi-closeable if it can survive with only inputs or outputs from other VIAACs. All semi-closeable VIAACs are also quasi-closeable.
• Fully-closeable if it can survive with no inputs or outputs except its own. All fully-closeable VIAACs are also semi-closeable.
• The edge from a VIAAC or VIAAC network to an outside economy (another VIAAC, VIAAC network, market, and so on) is critical.  Thus, closeability describes the presence or absence of critical edges.  Any flow of resources that contains a critical edge is called a critical path.  A critical path in which there is a path from a vertex to itself is a critical cycle.

Similarity to Autarky

A closeable VIAAC is similar to an autarky in that it is able to survive without relying on external inputs/outputs. However, an autarky is a description of both the ability and the intent of being closed off from external inputs/outputs, hence "closeability" rather than "closed". Having the ability to operate as a closed community provides disaster protection, but may not provide the same quality of life or performance as an open community. Closeability strikes a balance between the two, as well as providing more detailed qualification than just closed or open.

Which Type a VIAAC Should Target

The type of closeability that should be aimed for depends on the community's current status and environment. One in a harsh or disaster-prone environment should be fully-closeable. One that has economic activity with other ICs may wish to remain only quasi-closeable. Each VIAAC will have to analyze its individual situation, disaster risk, and the risk of disaster in adjacent areas to determine which goal will fit the best. It is likely that adjacent VIAACs will have similar closeability goals.

Network Closeability vs. Individual Closeability

Nearby VIAACs, assuming they themselves are stable and connected, provide an additional trophic level whose closeability on the aggregate can be different (usually stricter) than that of the components. This will be affected by logistic conditions, such as which VIAACs in a group are connected. If a VIAAC network contains no nodes of degree greater than 2, then the network and individual closeability conditions should remain the same. If an individual VIAAC has degree greater than two (it is connected to three or more others), then it may be able to relax its closeability conditions, such that a group of four that is fully closeable on the aggregate may be semi-closeable individually. For further development, a group of 9 VIAACs that form a web or antiweb (each VIAAC is connected to 4 others) can be degree 1 pseudo-closeable; If they form an antihole (each VIAAC is a chord/each VIAAC is connected to 6 others), this can be relaxed to degree 3 pseudo-closeable. This is an oversimplification, as each VIAAC may have different conditions that influence its goal, but it's an easy way to think about the network effects.

References

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