A linear network (LN), also called a chain network, consists of \(n+1\) nodes (parties) labeled by \(A_1,\ldots ,A_{n+1}\) together with n sources labeled by \(S_1,\ldots ,S_n\) . To explore the correlation of the LN, each party \(A_i\) performs \(m_i\) measurements labeled by \(x_i\in [m_i]\) on its system and gets \(o_i\) outcomes denoted by \(a_i\in [o_i]\) . The probabilities \(P(\mathbf{{a}}|\mathbf{{x}})=P(a_1,\ldots ,a_{n+1}|x_1,\ldots ,x_{n+1})\) form a tensor \(\mathbf{{P}}=\llbracket P(\mathbf{{a}}|\mathbf{{x}})\rrbracket \) , called an \(n+1\) -partite correlation tensor (CT) over the index set \(\Delta _{n+1}.\) Particularly, when each party performs just one fixed measurement (also called without input setting), the resulted CT is said to be an \(n+1\) -probability tensor (PT). In this work, we aim to characterize n-chain-locality of \(n+1\) -CTs based on an LN. By introducing D-n-chain-locality (resp. C-n-chain-locality) of an \(n+1\) -CT in light of the existence of a discrete (resp. continuous) n-chain-local hidden variable model, we prove that an \(n+1\) -CT \(\mathbf{{P}}\) is D-n-chain-local if and only if it can be realized physically by n shared separable states and a set of local POVMs. Importantly, we also prove that C-n-chain-locality and D-n-chain-locality of an \(n+1\) -CT are the same, so that we can call them n-chain-locality. The corresponding conclusions are obtained for \(n+1\) -PTs, and the relationships between D-n-chain-local (resp. C-n-chain-local) CTs and D-n-chain-local (resp. C-n-chain-local) PTs are established. Lastly, we prove that the set consisting of all n-chain-local CTs over \(\Delta _{n+1}\) forms a nonconvex compact subset in the Hilbert space of all real tensors over \(\Delta _{n+1}.\)
