Let
- a = assortative
- d = disassortative
- c = core
- p = periphery
- n = none (if within- and between-block densities between this block and the other block are all equal)
For example, if two blocks are assortative communities (higher within- than between block edge densities), than
Blocks can take on different roles in different block-pair relationships. For example, block
Note that in the case of (dis-)assortative relationships
Blocks are categorise into roles with respect to all other blocks based on the elements of the edge density matrix
$\min(\omega_{rr}, \omega_{ss}) > \omega_{rs} \iff R_{rs} = \text{a}$ $\omega_{rr} \geq \omega_{rs} > \omega_{ss} \text{ or } \omega_{rr} > \omega_{rs} \geq \omega_{ss} \iff R_{rs} = \text{c}, R_{sr} = \text{p}$ $\omega_{ss} \geq \omega_{rs} > \omega_{rr} \text{ or } \omega_{ss} > \omega_{rs} \geq \omega_{rr} \iff R_{rs} = \text{p}, R_{sr} = \text{c}$ $\omega_{rs} > \max(\omega_{rr}, \omega_{ss}) \iff R_{rs} = \text{d}$