DTM.3 - Couple Change Within Carriers to Spread Between Them

Type: Method Status: Stable

DTM.3:1 - Problem frame

Use this when a practice changes its participants, and those changes affect whether, what or how they pass it on—or when wider adoption changes the conditions under which participants can use it.

A learner becomes able to demonstrate a method only after practice. Growing adoption can overload shared teaching support. An installed standard changes which exchanges need conversion. A model that holds these conditions fixed can therefore predict the wrong continuation even if its separate local descriptions are reasonable.

The first result is a coupled account: which participant state changes which transmission or uptake event, and which population change feeds back to participants. It may be two connected ordinary sentences or a small mathematical model. Use MMP.18 for the general work of reconciling mathematical descriptions once these subject relations have been identified.

If participant change has no material effect on the receiving question over its horizon, retain the simpler transmission model.

DTM.3:2 - Problem

Two common reductions erase the relation that drives the result. One treats every carrier as an unchanged source of identical transmission. Another computes a detailed participant trajectory but assumes its population consequences are just that trajectory multiplied by the number of participants.

“Carrier” here names the person, group, device or other participant in which the relevant variant is realized or retained for this model. It does not imply disease. A carrier’s state, its resources, the method it uses and the contact network remain different objects.

DTM.3:3 - Forces

DTM.3:4 - Solution

Find the relation crossing each modeling boundary, construct its effect on an event, then test what the chosen aggregation loses.

DTM.3:4.1 - Identify the participant state that can change transmission

Start from a receiving event defined by DTM.2. Ask what changes its occurrence or resulting variant: the source’s capability, time available to teach, quality of a demonstrated result, retention, equipment compatibility, or the receiver’s current preparation, for example.

Keep a state only if it can change the receiving outcome. Recover how that state changes through use, learning, wear, recovery, support or another mechanism of the actual practice. Describe the process before choosing its equation.

A person’s ability to coordinate a turn and a robot controller’s learned parameters can each affect a demonstration. That does not give them the same learning law. Obtain the internal process from the relevant human-development, movement or engineering methods.

DTM.3:4.2 - Convert the state into a transmission contribution

Determine the quantity delivered to the next model. Examples include demonstrations per period that meet a performance criterion, an uptake probability conditional on readiness, the composition of reconstructed variants, or the cost of exchange with an adopter.

Preserve the event selected in DTM.2. If each encounter is eligible for a repeatable success, the expected success count is the sum of its event probabilities. With n comparable encounters and common probability p, that count is np; this expectation does not require independence.

First uptake by a recipient is different: after it occurs, later encounters cannot create that recipient’s first uptake again. For n independent comparable opportunities before such uptake, its probability is 1−(1−p)^n. More generally, let p_k be the chance of first uptake at opportunity k conditional on no earlier uptake and the stated history. Compose those conditional chances along that history; if histories vary, also account for their probabilities. Alternatively, update the eligible recipient set as events occur.

Aggregate first-uptake probabilities over distinct eligible recipients, not encounters. For a continuous-time model, sum their hazards to obtain the population flow as in DTM.2. This preserves the quantity passed to DTM.4.

Use DTM.2’s conditional factors and resulting-variant rule where needed. Do not multiply a capability score by a contact count without explaining how that score changes the selected event probability or rate.

Then name the observation that could establish or challenge the proposed state-to-event relation. If two mechanisms predict different outcomes for the same state, retain both until the receiving decision warrants discrimination by MMP.16.

DTM.3:4.3 - Recover the feedback from wider spread

Ask what changes for a participant when more, fewer or different others use the variant. Common possibilities include access to compatible partners, demand on shared support, availability of teachers, recognition rules and a protective response.

Build only the feedback supported by the case. For example, increasing adoption may create more teachers and also consume their time. The net effect is a question about those quantities, not an automatic positive feedback.

Where the influence is one-way over the chosen horizon, state that reduction and its reason. A diagram with arrows in both directions is not a requirement to invent the second mechanism. Where both directions matter, trace the loop far enough to see which state or event the returning effect changes.

C.32.MWA helps keep the structures separate: the vertical of methods enacted in one performance, resource provision, control, contacts and units of selection need not share the same boundaries.

DTM.3:4.4 - Choose an aggregation that preserves the needed effect

Choose among individual states, a few relevant classes, a distribution over state or age since adoption, and a justified common state. Use the least detail that preserves the comparison.

When event contribution is q(x), the relevant population quantity is the appropriately weighted average of q(x), not automatically q of the average x. Weights come from the modeled events: an individual making many demonstrations can contribute more exposure than an individual making none. Population shares alone need not be the weights.

A time-since-adoption description is useful when newcomers and experienced users produce materially different events. An average state is more defensible when the effect is approximately linear over the relevant range or when variation is small enough for the decision. Test that approximation, rather than choosing a fixed number of classes.

Return to MMP.18 if shared quantities, units, conservation, state boundaries or incompatible assumptions prevent composition.

DTM.3:4.5 - Decide whether a fast process may be reduced

If local adjustment is much faster than changes in adoption, a local equilibrium may provide the needed output. First identify the relaxation time and the population-change time over the conditions being compared. Examine startup and interventions, not only a final stationary point.

Retain the local dynamic state when learning delay, fatigue, memory, periodic loading or loss of support changes the prediction. A fast measured response does not establish the absence of slower memory.

The result states what is passed across the boundary, the conditions under which it can be summarized, and the change that would require restoring detail. DTM.4 then investigates persistence and return using that coupled model.

DTM.3:5 - Archetypal Grounding

DTM.3:5.1 - Equal average readiness can imply different spread

Consider an illustrative readiness x between 0 and 1. Suppose a relevant demonstration occurs once per period and produces the receiving event with probability q(x)=x². This law is a declared hypothesis for the example, not a general law of learning.

In population A every source has readiness 0.5, giving q=0.25. In population B half have readiness 0.1 and half 0.9. Both have mean readiness 0.5, but population B has mean contribution (0.01+0.81)/2=0.41.

A model using only mean readiness would treat both populations identically. If the comparison is about transmitted successful practice, that reduction loses the effect. Preserve the readiness distribution or sufficient classes. If the actual event law were linear and contact opportunities equal, the same particular loss would disappear.

This case also separates evidence obligations. Measurements of readiness do not establish q(x)=x²; that link requires observations of the chosen receiving event.

DTM.3:5.2 - Growth can reduce the support available to each learner

Suppose local readiness follows the conditional model:

x' = α(f)(1−x) − δx
α(f) = α₀/(1+kf).

Here f is the fraction adopting, α is the effective support rate per learner, δ a loss rate, and k specifies how adoption loads the support arrangement. The signs and functional form express the assumed sharing mechanism; they would have to change if adoption added support faster than demand.

For α₀=1, δ=0.25 and k=3 in the chosen time unit, the stationary readiness is α/(α+δ). It is about 0.714 at f=0.2 and 0.541 at f=0.8. If q=x² remains the receiving-event law, the corresponding contributions are about 0.510 and 0.292.

The loop is now visible: adoption changes support per participant, readiness changes the receiving-event rate, and that rate changes adoption. These stationary substitutions are useful only when readiness adjusts fast enough. Immediately after a large influx of beginners, replacing their states with these equilibria can be wrong.

A trial to increase adoption might therefore need additional teaching support. The equation does not prove that such support exists or how to teach the missing operation; those are provider questions.

DTM.3:5.3 - Spread changes compatibility without changing a learner

Consider a group comparing two exchange standards. Let f be the share of its relevant partners on the new standard. Over the same horizon, the group compares switching with continuing. Suppose it pays conversion costs while its old-standard partners keep their process unchanged. Its gain from switching is b−c−l(1−f): improvement b, switching cost c and conversion cost l for the remaining old-standard partner share.

Here the feedback is through compatible exchanges. No readiness state or training equation is needed. DTM.2 constructs the actual adoption event; the resulting adoption changes f and thus the gain available to later groups. A broadcast announcement is not the same as an authorized, affordable transition.

Use ECO.8 to retain the coordination and cost-bearing relations. Use the engineering compatibility method to establish that an adapter actually works. The spread model couples those supplied results rather than replacing them.

DTM.3:5.4 - Repeated opportunities do not create repeated first uptake

For one initially eligible recipient, suppose two independent comparable opportunities each succeed with probability 0.8. The probability of first uptake during those opportunities is 1−0.2²=0.96. It can also be obtained as 0.8+0.2×0.8: the second term includes only the recipients still eligible after the first opportunity.

The product 2×0.8=1.6 instead gives the expected number of successes when both attempts are actually performed and success is repeatable. It is not a first-uptake probability. Either quantity can be useful, but they feed different receiving models. The participant-state coupling must preserve that distinction even when the same internal readiness supplies both probabilities.

DTM.3:6 - Bias-Annotation

A biological source can make “within” suggest a single organism and “between” a population of organisms. Recover the actual participant boundaries instead. A shared provider or a network may cross organizational boundaries.

A detailed internal model can also attract attention away from the only output the receiving question needs. Detail is justified by a changed prediction or decision, not by anatomical or organizational completeness.

DTM.3:7 - Conformance Checklist

• The account names the event whose law changes and the participant state that changes it.
• Each boundary-crossing quantity has a subject interpretation, units where relevant, and an identified receiving use.
• Any feedback from spread is a stated mechanism; an omitted feedback has a bounded reason.
• Event-weighted contributions are distinguished from unweighted participant averages.
• A stationary reduction has a timescale and startup boundary; material delay or memory remains represented.
• A change in aggregation or participant boundary returns to the relevant modeling provider.

DTM.3:8 - Common Anti-Patterns and How to Avoid Them

Everyone transmits like the average participant. Average the event contributions under the relevant contact weights and compare with the proposed reduction.

A vertical of methods is a contact hierarchy. Separate the simultaneously enacted operations from the network through which another participant encounters the practice.

Fast equilibrium from the first moment. Check newcomers and changed conditions before discarding internal time.

Always draw two arrows. Keep one-way coupling where justified; construct reverse influence only from an actual mechanism.

DTM.3:9 - Consequences

The model can explain why a locally successful method fails to spread, or why its spread changes the conditions that made it successful. The cost is additional state or data when simple aggregation fails. Some cases become simpler after the correct interface quantity is identified.

DTM.3:10 - Rationale

Cross-level modeling becomes useful when one process supplies a quantity that another process changes in return. The subject method must identify those quantities; general mathematical composition cannot choose them from a shared vocabulary alone. Explicit aggregation and time assumptions make the connection revisable.

DTM.3:11 - SoTA-Echoing

Zilio et al. (2026) motivate separating within-carrier interaction from co-circulation and the feedback through transmission conditions. Their biological mechanisms are not used as universal cultural laws. The readiness and compatibility examples above are separate conditional constructions.

The historical Katz–Shapiro model of compatibility and network effects (1985) supplies a contrasting mechanism in which others’ adoption changes a user’s outcome. Its equilibrium approach does not by itself provide the adoption or learning dynamics used here.

MMP.18 provides general coupling and reduction methods. C.32.MWA preserves distinctions between structures. DTM adds the construction connecting participant state, transmission contribution and conditions changed by spread.

DTM.3:12 - Relations

• DTM.2 identifies receiving events and variant changes; DTM.4 examines the resulting regimes.
• DTM.6 supplies interaction mechanisms that can change local or population conditions.
• DTM.8/.9 add protective interventions and response dynamics where they matter.
• MMP.18, MMP.16 and C.32.MWA supply mathematical composition, distinguishing observations and structural correspondence.
• HCD, relevant movement methods and engineering methods supply their own acquisition or physical mechanisms; DTM does not replace them.

DTM.3:End