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In multiparty session types, interconnection networks identify which roles in a session engage in communication (i.e., two roles are connected if they exchange a message). In session-based interpretations of linear logic the analogue notion corresponds to determining which processes are composed, or cut, using compatible channels typed by linear propositions. In this work, we show that well-formed interactions represented in a session-based interpretation of classical linear logic (CLL) form strictly less-expressive interconnection networks than those of a multiparty session calculus. To achieve this result, we introduce a new compositional synthesis property dubbed partial multiparty compatibility (PMC), enabling us to build a global type denoting the interactions obtained by iterated composition of well-typed CLL threads. We then show that CLL composition induces PMC global types without circular interconnections between three (or more) participants. PMC is then used to define a new CLL composition rule that can form circular interconnections but preserves the deadlock-freedom of CLL.