Metrics: Attention, Focus, and Gravity
How the cybergraph measures what matters -- from individual attention to collective focus to structural gravity.
Attention vs Focus
Attention and focus are distinct but related.
Attention is individual. It is how much a neuron projects onto a target particle or axon. A neuron directs attention by creating cyberlinks and allocating conviction. Attention is the cause -- the input signal from each participant.
Focus is collective. It is the aggregated result of all neurons' attention, processed by the tri-kernel into a single probability distribution over all particles. Focus is the effect -- the output of the collective computation.
Individual neurons direct attention. The cybergraph aggregates all attention into focus. Every particle's focus score $\phi^*_i$ reflects the contributions of all neurons, weighted by their tokens and filtered by the epistemic layer.
Attention is produced by two mechanisms: will (broad auto-distribution across all cyberlinks) and fine-tuning (manual per-target weight adjustment). Both produce the same thing at the receiving end -- attention at the target particle.
The Transformer Attention Connection
The transformer attention mechanism computes, for each position in the context, a weighted average of all other positions:
$$\text{Attn}(Q, K, V) = \text{softmax}\!\left(\frac{QK^\top}{\sqrt{d}}\right)V$$
Three projections: queries $Q = XW_Q$ ask "what am I looking for?", keys $K = XW_K$ announce "what do I contain?", values $V = XW_V$ provide "what information do I carry?". The dot product $QK^\top$ scores compatibility. The softmax converts scores to a probability distribution -- the Boltzmann distribution with temperature $\sqrt{d}$.
The softmax is the same operation as the LMSR price function and the tri-kernel diffusion step. All three are exponentiated scores normalized to sum to 1.
Transformer attention is one step of the tri-kernel diffusion operator $D$ applied to the current context window. Probability mass flows from each query position toward compatible key positions -- exactly the random walk dynamics that the tri-kernel uses to compute focus over the cybergraph.
Deep Equilibrium Models showed that iterating a transformer layer to convergence reaches the same fixed point as the tri-kernel: $\phi^*$ restricted to the context window. $L$ layers of attention = $L$ steps of diffusion toward that fixed point.
Attention as Bayesian Query
Attention answers: given my current state (query), what posterior weight should I assign to each position (key)?
The softmax is the posterior $P(\text{position } j \mid \text{query } i)$ under a uniform prior and an exponential likelihood $\exp(q_i \cdot k_j / \sqrt{d})$. The query-key product is the log-likelihood under this model. The softmax is the Bayes-normalized posterior. Attention is Bayesian inference over the context.
Multi-Head and Dialects
Through multi-head attention, different heads learn different relation types. Head $h$ with projection $W_Q^{(h)}, W_K^{(h)}$ captures one dialect -- one pattern of connectivity in the cybergraph.
The graph-native-transformer derivation proves that the minimum number of heads equals the number of distinct dialect types in the graph. Each head specializes in one kind of relationship: "is-a", "contradicts", "extends", "cites". The graph's link topology determines how many attention heads the compiled transformer needs.
Gravity
Gravity is a node-level metric. Like physical gravity, it is a property of the node itself -- a massive body warps space around it and attracts everything, regardless of what is nearby.
$$G_i = \phi^*_i \cdot \sum_{j \neq i} \frac{\phi^*_j}{d(i,j)^2}$$
where $\phi^*_i$ is the node's own focus probability, $\phi^*_j$ are focus probabilities of all other nodes, and $d(i,j)$ is the shortest path length in the cyberlink graph.
A node's gravity is its focus mass multiplied by the total attention field it experiences from the rest of the graph. High-focus node surrounded by other high-focus nodes = enormous gravity. High-focus node on the periphery = less gravity despite its own mass.
Physical Analogy
A planet curves spacetime by its mass alone. The gravitational potential of a body in a field of other masses:
$$\Phi_i = m_i \cdot \sum_{j} \frac{m_j}{r_{ij}^2}$$
| physics | knowledge graph |
|---|---|
| mass $m$ | focus probability $\phi^*$ |
| distance $r$ | graph distance $d(i,j)$ |
| gravitational potential $\Phi$ | node gravity $G_i$ |
The node does not choose what to attract. It simply has mass (focus), and everything within graph distance falls toward it proportionally.
Gravity Spectrum
| gravity | profile | meaning |
|---|---|---|
| high | high $\phi^*$, surrounded by high-$\phi^*$ neighbors | core attractor -- holds the graph together |
| medium | moderate $\phi^*$, or high $\phi^*$ but few neighbors | regional hub -- local structure anchor |
| low | low $\phi^*$, or isolated from high-$\phi^*$ nodes | peripheral -- structurally weightless |
Applications
Skeleton extraction: nodes with the highest gravity form the structural skeleton of the knowledge graph. Remove them and the graph fragments.
Peripheral detection: nodes with high focus but low gravity are isolated attractors -- they have mass but sit far from other massive nodes. Connecting them to the core would dramatically restructure the graph.
Cohesion measurement: total graph gravity $G_{\text{total}} = \sum G_i$ measures how tightly the knowledge core is packed. A graph with high total gravity has its attention concentrated in a dense, interconnected core. Low total gravity means focus is scattered.
Pairwise Force
The force between any two specific nodes:
$$F_{ij} = \frac{\phi^*_i \cdot \phi^*_j}{d(i,j)^2}$$
The highest $F_{ij}$ pairs are the structural bonds of the graph. Pairs with high $\phi^*_i \cdot \phi^*_j$ but large $d(i,j)$ are the most valuable missing cyberlinks -- creating them collapses distance and unlocks attention flow.
How Ranking Works in Practice
Cyberank is focus materialized as a per-particle score. It is the probability of being observed by a randomly walking neuron -- the fixed point of the tri-kernel.
The tru computes cyberank every block. The score feeds into:
- Karma: accumulated track record of a neuron's contributions
- Syntropy: the organizational quality of the focus distribution ($J = D_{\text{KL}}(\phi^* \| u)$ -- how far attention deviates from uniform noise)
- Inference: the probability distribution that drives query responses and autoregressive generation
- Sorting in cyb: the user-facing ordering of search results
Luminosity = size x $\phi^*$ -- what a node radiates (knowledge output). Gravity = $\phi^*$ x $\sum(\phi^*_j/d^2)$ -- how strongly a node attracts (structural pull). A healthy graph needs both: high-luminosity nodes that radiate knowledge, with high-gravity nodes that hold the structure together.
See attention for allocation strategies. See focus flow computation for the global process. See tri-kernel for the diffusion connection. See gravity for the full metric specification.