Equations

TL;DR

Grand-canonical MENoBiS models are independent over node pairs. Families share multipliers but differ in the expected occupation equation.

Pair parameter

For every allowed ordered pair:

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

where x and y are strength multipliers. Without cost, \(f_{ij}=1\). With cost, \(f_{ij}=\exp(-\gamma d_{ij})\).

Non-zero-inflated families

Family	MENoBiS	Expected occupation	Domain
ME	Poisson	\(\mathbb{E}[t_{ij}]=q_{ij}\)	\(q>0\)
B	Binomial(M)	\(\mathbb{E}[t_{ij}]=Mq_{ij}/(1+q_{ij})\)	\(q>0\)
W	NegBin(M)	\(\mathbb{E}[t_{ij}]=Mq_{ij}/(1-q_{ij})\)	\(0<q<1\)

For W, M=1 is the geometric case.

Event nature matters

The same strength or degree constraints generate different statistics when events are distinguishable, aggregated binary layers, or indistinguishable.

Zero-inflated constraints

Strength-edges and strength-degree constraints also control binary occupation. They use a raw binary multiplier \(\ell_{ij}\) and a positive-support factor \(G_F(q)\):

Family	\(G_F(q)\)
ME	\(e^q-1\)
B	\((1+q)^M-1\)
W	\((1-q)^{-M}-1\)

The occupation probability is:

\[ \mathbb{E}[\Theta(t_{ij}>0)] = \frac{\ell_{ij}G_F(q_{ij})}{1+\ell_{ij}G_F(q_{ij})}. \]

The expected occupation is:

\[ \mathbb{E}[t_{ij}] = \frac{\ell_{ij}q_{ij}G'_F(q_{ij})}{1+\ell_{ij}G_F(q_{ij})}. \]

Constraint map

MENoBiS constraint	Matched expectation
STRENGTH	outgoing and incoming strengths
STRENGTH_COST	strengths plus total cost \(\sum_{ij}\mathbb{E}[t_{ij}]d_{ij}\)
STRENGTH_EDGES	strengths plus total binary edges
STRENGTH_DEGREE	strengths plus in/out degrees
DEGREE_EVENTS	in/out degrees plus total events
partial variants	parent constraints after subtracting frozen pairs

Important rule

ME, B, and W are not interchangeable. B and W must use their own equations and solvers; they must not call the ME solution and relabel the result.