Research article · Open frontierOpen frontier
An elephant uses far more energy per second than a mouse. It does not use energy in direct proportion to its mass.
Take an animal that is one hundred times heavier. A proportional law would predict one hundred times the energy use at rest. The relationship made famous by Max Kleiber predicts roughly thirty-two times the energy use.
Energy used per unit time while an animal rests is called basal metabolic power. The relationship is called Kleiber’s law.
KLEIBER’S LAW
A larger animal uses more energy as a whole, but less energy for each unit of body mass.
B is the animal’s basal metabolic power: energy used per unit time while the animal is at rest. M is its body mass. The exponent 3/4 tells us how strongly power rises with mass.
Read it as: Double the body mass and total metabolic power rises by about 68 percent, not 100 percent.
Three-quarters is a famous biological benchmark, not an exact ruler obeyed by every organism under every condition. Measured exponents vary across animal groups. Its importance lies in the pattern. Life appears to process energy through size in a regular but less-than-proportional way.
One influential explanation traces this pattern to branching transport networks—the vessels that carry blood and nutrients through a body. A nearby exponent has now appeared in a growing digital network through a different mechanism.
The digital result
Here a Xypher is a graph together with rules for valuing and choosing changes to that graph.
In the Xypher growth simulations, nodes trade across a changing graph. Some actions open more future connection routes than others. The system assigns an internal price to that expansion. We call it pgrowth: the accounting price of opening one more unit of future possibility.
The simulations began from five graph sizes between 25 and 500 nodes, with three independently randomized runs and 500 turns at each starting size. The graphs continued to grow, so the fit used their average size during the second half of each run. Across three agent configurations, the price followed a stable power law: every proportional increase in network size produced a regular proportional increase in price.
Larger networks produced a predictably higher internal price for future-opening growth.
N is the number of nodes. The exponent says that doubling the network multiplied the growth price by roughly 1.74 to 1.80.
Read it as: The same increase in network size produced nearly the same increase in price throughout the tested range.
This is close to Kleiber’s three-quarter exponent. A digital graph has no lungs, blood, or cells, yet an internal quantity followed a scaling curve in the same neighborhood as biological metabolism.
The resemblance is exciting. The mechanism underneath it is even more useful, because we can see where the exponent comes from.
The first ingredient: a thicker market
A market with more participants can offer more useful trading partners. The simulation represents this advantage with a square-root multiplier. If the network contains N nodes, matched trade value receives a factor of √N.
Here surplus means the value received from completed simulated trades minus the cost paid for them.
The square-root multiplier is an assumption of the model, not a measured result.
Its consequence is simple. If each of N participants gains opportunities on the scale of √N, then total surplus grows on the scale of
N participants × √N opportunity per participant = N3/2 total surplus.
The measured runs preserved this relationship closely. Surplus per participant scaled approximately as N0.495, giving total surplus near N1.495—almost N3/2.
The thicker-market premise supplied the first part of the curve. The Xypher’s accounting feedback supplied the second.
The second ingredient: a price inside its own feedback
The model records each positive increase in future possibility as an expansion receipt. The growth price helps determine how much value the system issues in those receipts. That issued amount then helps determine the next growth price.
The price therefore appears on both sides of its own update.
At a stable balance, the relationship becomes:
THE GROWTH-PRICE FEEDBACK
The feedback makes the balanced growth price a square root.
The top of the fraction is the economic surplus created during one turn. The bottom adds every increase in future connection possibility produced during that turn. It does not subtract actions that close possibilities elsewhere.
Read it as: More surplus pulls the price up. More expansion receipts spread that value across more growth and pull the price down.
Now the three-quarter exponent has a clear origin. If total surplus grows as N3/2 and the amount of future opened per turn stays constant, the square-root feedback gives
√(N3/2) = N3/4.
Three-quarters is the clean limit of the model. It follows from a square-root market advantage passing through a square-root price feedback.
What the graph added
The amount of future opened per turn did not stay perfectly constant.
As the simulated graphs became larger, the total increase in future connection possibility per turn declined weakly. This rate fell approximately as N−0.13 to N−0.14. Because that rate sits at the bottom of the price equation, its decline pushed the final price exponent slightly above three-quarters, toward 0.82.
The experiment repeated the measurement across three agent configurations. The reader should look for two things in the table: the exponents remain in a narrow range, and the fitted power law remains extremely regular.
| Agent configuration | Measured price exponent | Regularity of fit |
|---|---|---|
| One direct measure of future connection possibility | 0.804 | R² ≥ 0.995 |
| Five measurements kept separate | 0.850 | R² ≥ 0.995 |
| Five measurements combined before acting | 0.838 | R² ≥ 0.995 |
R² measures how closely the simulated points follow one power-law curve. A value of one would be a perfect fit. Every run family above was at least 0.995.
The square-root market premise was put into the model. The weak decline in future-opening rate, the survival of the scaling through the configured graph-search dynamics, and the narrow range across the three agent configurations came out of the runs.
The elephant and the thermometer
Matching exponents do not automatically mean matching laws. We must compare the quantities on both sides.
| Kleiber’s biological relationship | The Xypher simulation | |
|---|---|---|
| Size | Body mass, M | Node count, N |
| Measured quantity | Metabolic power: energy per time | Growth-accounting price |
| Observed exponent | Near the three-quarter benchmark | 0.80–0.85 over the tested range |
An elephant processes far more energy per second than a mouse. It is not proportionally hotter.
That distinction matters here. In the Xypher thermodynamic account, temperature acts as an energy price per nat. Throughput is a flow. A nat is one unit of entropy measured with the natural logarithm.
A Xypher’s temperature tells us how much energy it takes to multiply the number of complete states the graph can occupy. Metabolic power tells us how much energy the organism processes per unit time. A growth-accounting price is closer in kind to temperature than to power, but similarity is not identity. It must be read by an independent thermometer before the two can be connected.
In a separate digital experiment, the complete graph-and-energy-store states were few enough to count exhaustively. One thermometer counted how their number changed with energy. A second compared each allowed state change with its exact reverse. They agreed.
The quantity that can be compared with metabolism
At each network size, measure two things independently:
- The Xypher’s temperature, read from the complete states it can occupy.
- The positive expansion rate: how much future connection possibility the graph opens per unit time.
Temperature prices one unit of state-opening change. Multiply that price by the amount opened per turn, and the result is an energy-valued flow. We write that flow as:
ENERGY-VALUED EXPANSION FLOW
Temperature gives expansion its energetic price. The expansion rate tells us how much is processed per unit time.
T(N) is the independently measured temperature at size N. R+(N) is the positive future possibility opened per turn, measured in the same entropy units. Preceipt(N) is their energy-valued flow per turn, which we call receipt throughput.
Read it as: How much energy-valued expansion does this Xypher process over time?
This is the digital quantity with the same basic shape as metabolic power: energy-valued change per unit time. We can identify it with metabolic power only if every energy transfer during growth—including supplied work and released heat—is accounted for, that account shows the receipt flow is the corresponding energy throughput, and one turn represents the same unit of time at every size.
The experiment that decides
The next experiment does not need another fitted resemblance. It needs separate instruments.
For every network size, we should:
- Read temperature from complete state counts.
- Read it again from reversible state traffic, without giving the second instrument the first answer.
- Measure positive future opened per turn.
- Calculate receipt throughput from temperature multiplied by that rate.
- Record the growth-accounting price as a separate quantity.
The experiment must also state what network size represents and keep its unit of time fixed. A node count is not automatically the same thing as biological mass.
One control is decisive: remove the model’s √N market multiplier and see whether the exponent survives.
Then repeat the experiment with several known size multipliers. If the final exponent simply follows the multiplier we supplied, we have mapped an accounting consequence of the premise. If a stable exponent remains with no size multiplier, it is not inherited solely from the square-root assumption; the remaining parts of the model must then be isolated.
The outcomes will be easy to distinguish:
- If only the growth price scales, we have a digital accounting law.
- If independently measured receipt throughput scales, we have a digital throughput law.
- If that throughput approaches three-quarters after the explicit size multiplier is removed, we have stronger evidence for a digital analogue of Kleiber scaling.
This is a stronger frontier than matching two exponents after the fact. It tests whether the law survives without the explicit size premise while independent instruments watch where the law lives.
What we have found
The current result already contains real structure.
A thicker-market premise produces aggregate surplus near N3/2. A self-referential receipt price takes its square root. The configured graph-search dynamics add a small, measured correction. Together they produce a stable growth-price exponent near 0.82 across the tested range.
Most of the three-quarter exponent comes from the model’s explicit market-size premise. The configured graph-search dynamics add the smaller correction.
The next experiment will tell us whether this mechanism belongs only to the price of growth, or whether a growing Xypher with a closed energy account also develops the less-than-proportional energy flow we recognize in living systems.
Research trail
- Max Kleiber, “Body Size and Metabolism”, Hilgardia 6(11), 315–353 (1932).
- Geoffrey B. West, James H. Brown, and Brian J. Enquist, “A General Model for the Origin of Allometric Scaling Laws in Biology”, Science 276, 122–126 (1997).
- Nick J. B. Isaac and Chris Carbone, “Why Are Metabolic Scaling Exponents So Controversial?”, Ecology Letters 13, 728–735 (2010).
- Xyphers research, the independent digital thermometer and the growth-scaling frontier.
- Xyphers research, scaling derivation and simulation record.