Notation

TL;DR

MENoBiS models directed non-binary networks: to every ordered node pair \((i,j)\) we assign an integer occupation number \(t_{ij}\) counting events. Everything else — families, ensembles, constraints, filtering — is defined on top of this single object.

Occupations and support

Let \(t_{ij}\in\{0,1,2,\ldots\}\) be the occupation number of the ordered pair \((i,j)\). The network is directed: \((i,j)\) and \((j,i)\) are different coordinates with their own occupation numbers.

The binary support of the network is the occupation indicator

\[ a_{ij}=\Theta(t_{ij})=\mathbf 1[t_{ij}>0]. \]

A pair with \(a_{ij}=1\) is an occupied pair.

The canonical MENoBiS sparse representation lists only occupied pairs as three columns source target occ_num; unoccupied pairs are implicit.

Symbols

Symbol	Meaning
\(N\)	number of nodes
\(t_{ij}\)	occupation number of ordered pair \((i,j)\)
\(a_{ij}\)	binary support indicator, \(\mathbf 1[t_{ij}>0]\)
\(T\)	total occupation \(\sum_{ij} t_{ij}\) (total number of events)
\(E\)	occupied-pair count \(\sum_{ij} a_{ij}\)
\(s^{\mathrm{out}}_i,\ s^{\mathrm{in}}_i\)	out/in strength sequences
\(k^{\mathrm{out}}_i,\ k^{\mathrm{in}}_i\)	out/in binary degree sequences
\(d_{ij}\)	pair cost (e.g. Euclidean distance)
\(C(t)\)	total cost \(\sum_{ij} t_{ij}d_{ij}\)
\(M\)	B/W layer/degeneracy parameter
\(q_{ij}\)	pair fugacity/intensity
\(\ell_{ij}\)	support fugacity (zero-inflated models)

Derived quantities

Out and in strengths:

\[ s_i^{\mathrm{out}}=\sum_j t_{ij}, \qquad s_j^{\mathrm{in}}=\sum_i t_{ij}. \]

Out and in binary degrees:

\[ k_i^{\mathrm{out}}=\sum_j a_{ij}, \qquad k_j^{\mathrm{in}}=\sum_i a_{ij}. \]

Total occupation and occupied-pair count:

\[ T=\sum_{ij} t_{ij}, \qquad E=\sum_{ij} a_{ij}. \]

Admissible pairs and self loops

The basic coordinates are ordered pairs. The self-loop policy changes which pairs are admissible:

\[ \text{no self loops:}\quad L=N(N-1), \qquad \text{self loops allowed:}\quad L=N^2, \]

where \(L\) is the number of candidate pairs.

Terminology

Preferred MENoBiS vocabulary:

• non-binary network — a network with integer occupation numbers;
• occupation number — the integer event count \(t_{ij}\);
• occupied pair — a pair with \(a_{ij}=1\);
• event — one unit of occupation;
• binary support — the occupation indicator \(a_{ij}\);
• strength — out/in event sums;
• degree — out/in support sums.

The phrase weighted network appears in the historical literature (thesis and related papers) as a synonym for non-binary networks. Within MENoBiS documentation it is treated as historical literature terminology; the primary terms are the ones above.