Ensembles

TL;DR

An ensemble is how the constraints enter the probability law.

• Grand canonical (GC): constraints are matched in expectation.
• Canonical: total occupation \(T\) is fixed exactly; the remaining fitted structure stays probabilistic.
• Microcanonical (MC): the specified hard constraints are fixed exactly, realization by realization.

flowchart LR
    OBS[Observed constraints] --> GC[Grand canonical]
    OBS --> CAN[Canonical]
    OBS --> MC[Microcanonical]

    GC --> GCT[Constraints matched in expectation]
    CAN --> CANT[Total occupation T exact; remaining fitted structure soft]
    MC --> MCT[Hard constraints define a fiber]

    MCT --> INIT[Construct one feasible state]
    INIT --> SAMPLE[Sample target measure on / traced onto the fiber]
    SAMPLE --> DIAG[Burn-in, mixing and diagnostics where MCMC is used]

The strength+cost microcanonical route is hybrid: strengths are exact while cost is controlled in expectation through the cost multiplier.

Grand canonical

Use the generic sufficient-statistics form

\[ P_{\mathrm{GC}}(t)=\frac{1}{Z(\theta)} \left[\prod_{ij}d_F(t_{ij})\right] e^{-\theta\cdot C(t)}, \]

where \(d_F\) is the pair degeneracy of the chosen family and \(C(t)\) collects the constrained statistics. Implemented GC routes have independent pair statistics, so sampling factorizes over admissible pairs.

GC constraints fluctuate around their fitted expectations across sampled networks. That fluctuation is part of the null hypothesis: a GC null asks whether the observed network is unusual given that the constrained quantities average to the observed values.

Canonical

MENoBiS canonical sampling fixes total occupation \(T\) exactly. The remaining fitted structure is encoded probabilistically rather than fixed realization by realization:

\[ P_{\mathrm{CAN}}(t)=P_{\mathrm{GC}}(t\mid T(t)=T^\star). \]

Canonical is not exact-strength sampling. Strengths remain soft, fitted quantities; only \(T\) is exact. Canonical is currently implemented for family ME with the STRENGTH constraint (the fitted strengths supply the multinomial weights and \(T^\star\) supplies the fixed total).

Microcanonical

For hard constraints \(C(t)=C^\star\),

\[ P_{\mathrm{MC}}(t\mid C^\star) = \frac{ \left[\prod_{ij}d_F(t_{ij})\right] \mathbf 1[C(t)=C^\star] }{ \Omega_F(C^\star) }, \]

with

\[ \Omega_F(C^\star) = \sum_t \left[\prod_{ij}d_F(t_{ij})\right] \mathbf 1[C(t)=C^\star]. \]

The implemented microcanonical routes share a two-stage philosophy:

1. construct one feasible state on the constraint fiber;
2. sample the target measure on (or traced onto) that fiber.

Different constraints use different constructors and different exact sampling kernels. The high-level philosophy is shared; the concrete algorithms are route-specific and are documented in the contributor algorithm index.

Hybrid cost route

The microcanonical STRENGTH_COST route is hybrid:

• strengths are exact;
• cost is controlled in expectation through the cost multiplier \(\gamma\) (f_{ij}=e^{-\gamma d_{ij}}).

Its exactness label is accordingly "exact stationary MCMC with cost matched in expectation", never "all constraints exact".

Which combinations exist?

Only a subset of family × ensemble × constraint combinations is implemented. The authoritative list is the generated capability matrix, including its per-route exactness semantics.