Mean pairwise federation similarity

Read the equation

There are n(n−1)/2 unordered pairs among n agents, so multiplying their sum by 2/[n(n−1)] yields the mean pairwise cosine similarity. For valid cosine inputs the result lies in [−1,1]. It summarizes representational alignment, including anticorrelation, rather than directly measuring collective task performance.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

• At least two agents and nonzero vectors for the constituent cosine similarities are required.

Original context

## 11.3 Federation Coherence

```
Coherence_fed = (2/(n(n-1))) · Σ_{i<j} C_ij
```

**Explanation**: Average pairwise coherence across all agent pairs in the federation.

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# 12. Activation Functions

## 12.1 Softsign

```
softsign(x) = x / (1 + |x|)
```

**Range**: (-1, 1)

**Explanation**: Bounded activation that preserves sign and approaches ±1 asymptotically. Smoother than tanh at origin.

## 12.2 Sigmoid

Coherence_fed = (2/(n(n-1))) · Σ_{i<j} C_ij

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