Bigraph independence : a mixture of the five natural independences

We introduce a notion of non-commutative joint independence for multiple algebras in a non-commutative probability space. The pairwise relationships between these algebras are encoded by a graph with two edge sets -- a combinatorial structure we call a bigraph -- and naturally encompass the five fundamental types of independence: tensor, free, (anti)monotone, and Boolean. It subsumes the BMT independence of Arizmendi--Mendoza--Vazquez-Becerra (when all pairwise relationships are Boolean, (anti)monotone, or tensor) and the $ε$ or $Λ$-independence of Mlotkowski (when the pairwise relationships are tensor and free). We present explicit combinatorial moment formulas, a Hilbert space construction, and natural associativity relations within this setting. Furthermore, we demonstrate that bigraph independence emerges in the asymptotic behavior of tensor product random matrix models with respect to a vector state, encompassing the Charlesworth--Collins model for $\varepsilon$-independence as a special case and offering a random matrix perspective on BMT independence.