Table 1 Sparse balance theory conditions: definitions of sparse structural balance theory that generalize existing definitions of balance.

∀ i, j, k ∈ V, for every combination:

Classical⁹

A1, A2, A3, A4

if 7D2 e_i k ≠ 0 and e_k j ≠ 0

then e_i j = e_i ke_k j should be valid

∀ i, j, k ∈ V, for every combination:

Clustering¹⁸

A1, A2, A3

if e_i k ≠ 0 and e_k j ≠ 0

and (e_i k > 0 or e_k j > 0)

then e_i j = e_i ke_k j should be valid

∀ i, j, k ∈ V, for every combination:

Transitivity¹⁹

A1

and e_i k > 0 and e_k j > 0

then e_i j = e_i ke_k j should be valid