Relation

Prerequisites: Appendix B (compact Lie groups, bi-invariant metric).

2.1 Axiomatic Definition#

Motivation#

In this theory, a relation is the most fundamental unit of the world. Nodes (individuals) are secondary; they acquire meaning only as endpoints of relations.

Every relation has two aspects:

• Scalar intensity: how strong the relation is (a non-negative real number).
• Group-valued transit: what kind of transformation the relation carries (an element of a compact Lie group $G$).

Definition#

Definition 2.1 (Relation). Over a finite set $V$, a relation at time $t\in \mathbb{T}$ is a tuple $(i,j,w_t(i,j),g_t(i,j))$ where:

• $i,j\in V$: source and target nodes,
• $w_t(i,j)\in {\mathbb{R}}_{\ge 0}$: relation weight (scalar, non-negative),
• $g_t(i,j)\in G$: relation transit (parallel transport from $i$ to $j$).

Physical interpretation.

• $w_t(i,j)$: confidence, capacity, or influence of the relation from $i$ to $j$.
• $g_t(i,j)$: a "translation rule" for comparing the internal states of $i$ and $j$.

Geometric interpretation. $g_t(i,j)$ is a discrete analogue of parallel transport in differential geometry. Discretising a connection on a vector bundle assigns a group element to each edge---exactly the structure captured here.

2.2 Conventions#

Convention 2.2 (No self-relation). $w_t(i,i)=0,g_t(i,i)=e.$ Self-reference is excluded from the scope of relations.

Rationale. Allowing self-loops creates unnecessary complications in the conductance definition and is conceptually unnatural: a relation is inherently "between two".

Convention 2.3 (Asymmetry). In general: $w_t(i,j)\ne w_t(j,i),g_t(i,j)\ne g_t(j,i{)}^{-1}.$ Relations are directed; the $i\to j$ relation is independent of the $j\to i$ relation.

Remark. When $g_t(i,j)=g_t(j,i{)}^{-1}$, we say the relation is compatible (traversing $i\to j\to i$ returns to the identity). This is not assumed in general.

2.3 Symmetrised Weight#

From the asymmetric weights we extract a direction-free scalar intensity.

Definition 2.4 (Symmetric weight). $W_t(i,j):=\frac{1}{2}(w_t(i,j)+w_t(j,i)).$ Properties: $W_t(i,j)=W_t(j,i)\ge 0$ and $W_t(i,i)=0$.

Motivation. Graph-theoretic quantities (conductance, volume, cut) are defined via $W_t$. While $w_t$ preserves directional information, the topological structure---which nodes cluster together---is determined by $W_t$ alone.

Definition 2.5 (Degree). $d_t(i):={\sum }_{j\in V}{W}_t(i,j).$ The total relational capacity of node $i$ with respect to the entire set $V$.

Proposition 2.6 (Basic properties of degree).

(i) $d_t(i)\ge 0$; equality iff $i$ is completely isolated.

(ii) ${\sum }_{i\in V}{d}_t(i)=2{\sum }_{i<j}{W}_t(i,j)$ (double counting).

(iii) $d_t(i)$ is a global quantity: it requires summation over all of $V\setminus \{i\}$.

Proof. (i) Sum of non-negative reals. (ii) Symmetry of $W_t$. (iii) By definition. $\square$

Key observation (foreshadowing Chapter 6): $d_t(i)$ cannot be computed from a subset $F\ni i$ alone. Whether to include $d_t(i)$ in the "intrinsic data" of a fruit is the central modelling choice of Chapter 6.

2.4 Categorical Perspective (Remark)#

The data $(V,w_t,g_t)$ can be viewed categorically:

• Nodes of $V$ are objects.
• Positive-weight relations $(i,j)$ are morphisms.
• Transits $g_t(i,j)$ are $G$-valued labels on morphisms.

This makes the relational field a finite version of a $G$-enriched weighted directed graph. Since no composition rule is postulated, it is not a category in the strict sense.