Event families: ME, B, W

TL;DR

MENoBiS offers three occupation families. They share the same constraint machinery but differ in what a unit of occupation means and in the resulting pair statistics. Choose a family by the nature of the events, not by speed (see Choose a model).

What are ME, B and W?

Each family is defined by a pair degeneracy \(d_F(t)\): the number of microscopic arrangements of \(t\) events on pair \((i,j)\). The degeneracy drives the conditional occupation law given a pair fugacity \(q_{ij}\).

Family	Name	Degeneracy \(d_F(t)\)	Occupation range	Interpretation
ME	Multi-edge	\(1/t!\)	\(t\ge 0\)	distinguishable events
B	Aggregated binary layers	\(\binom Mt\)	\(0\le t\le M\)	aggregate of \(M\) binary layers/trials
W	Weighted	\(\binom{M+t-1}{t}\)	\(t\ge 0\)	indistinguishable events

ME: distinguishable events

\[ d_{\mathrm{ME}}(t)=\frac{1}{t!},\qquad t\ge 0. \]

\[ P_{\mathrm{ME}}(t_{ij}=t)=e^{-q_{ij}}\frac{q_{ij}^{t}}{t!}. \]

\[ \mathbb E[t_{ij}]=q_{ij}. \]

Interpretation: occupation counts distinguishable events (for instance, individual trips in an origin–destination table). The Poisson pair law is the maximum-entropy law given an expected occupation \(q_{ij}\).

B: aggregated binary layers

\[ d_{\mathrm B}(t)=\binom Mt,\qquad 0\le t\le M. \]

\[ P_{\mathrm B}(t_{ij}=t)=\binom Mt\frac{q_{ij}^{t}}{(1+q_{ij})^M}. \]

\[ \mathbb E[t_{ij}]=M\frac{q_{ij}}{1+q_{ij}}. \]

Interpretation: the occupation is the aggregate of \(M\) binary layers/trials, each open with odds \(q_{ij}\). Occupations are bounded by \(M\): \(t_{ij}\le M\). The special case \(M=1\) is a Bernoulli (binary) pair.

W: indistinguishable events

\[ d_{\mathrm W}(t)=\binom{M+t-1}{t},\qquad t\ge 0. \]

\[ P_{\mathrm W}(t_{ij}=t)=\binom{M+t-1}{t}(1-q_{ij})^M q_{ij}^{t}, \qquad 0<q_{ij}<1. \]

\[ \mathbb E[t_{ij}]=M\frac{q_{ij}}{1-q_{ij}}. \]

Interpretation: occupation counts indistinguishable events (e.g. multi-occupancy of a shared resource). The pair law is negative binomial; the special case \(M=1\) is the geometric distribution.

Parameter domain

The W family requires \(q_{ij}\in(0,1)\). Solvers keep every fitted pair parameter inside this domain; infeasible or degenerate inputs surface as solver status messages rather than silently out-of-domain parameters.

The B \(M=1\) invariant

For B with \(M=1\), \(t_{ij}\in\{0,1\}\), so support and occupation coincide:

\[ s_i^{\mathrm{out}}=k_i^{\mathrm{out}}, \qquad s_i^{\mathrm{in}}=k_i^{\mathrm{in}}. \]

This is a mathematical consequence of the Bernoulli pair law, not an implementation quirk. It matters for feasibility: a B \(M=1\) fixed-degree problem is only feasible when strengths equal degrees.

Pair parameterization

The pair fugacity is written generically as

\[ q_{ij}=x_i y_j f_{ij}. \]

For exponential spatial cost \(f_{ij}=e^{-\gamma d_{ij}}\) (see Spatial costs).

This factorization is a convenient parameterization, not a claim that every route needs distinct node-specific multipliers. Global-constraint models (notably EDGES_EVENTS) are special cases in which the multipliers collapse to constants or global parameters.

Comparison

Question	ME	B	W
Occupation bound	unbounded	\(\le M\)	unbounded
Fitted pair param range	\(q>0\)	\(q>0\)	\(0<q<1\)
\(M\) layer parameter	not used	required	used
\(M=1\) special case	—	Bernoulli	geometric
Interpretation	distinguishable events	M binary layers	indistinguishable events

Identical constraints do not make families interchangeable

The same strength sequence fitted under ME, B and W produces different pair statistics and different sampled networks. A comparison across families is a comparison of event interpretations, not a numerically interchangeable choice.

Family availability across ensembles and constraints is listed in the generated capability matrix, included into the supported-models guide and authoritative for what is supported today.