Evidence contributions in component network meta-analysis from the shortest-path approach

Component network meta-analysis (CNMA) decomposes the overall effect of a multicomponent intervention into the effects of its constituent components. It is important to quantify the contribution of each single studies (or comparisons) to the individual component effect obtained from the CNMA model. However, evidence for a single component is often distributed across comparisons of multicomponent interventions, making it difficult to trace graph‑theoretic based paths of evidence in a standard network plot. We propose a two-stage algorithm to quantify evidence contributions in CNMA. First, as component-level evidence is not encoded as connected topological paths in the network of standard NMA, we introduce the concept of pseudo-paths. A pseudo‑path for a target component is defined as a set of directed edges whose linear combination—with non‑negative coefficients—yields a vector that isolates the effect of that component (i.e., equals 1 for the target component and 0 for all others). All pseudo-paths are identified by solving a non‑negative linear feasibility problem based on the CNMA design matrix $${{\varvec{X}}}_{{\varvec{a}}}$$. Second, we adapt the iterative logic of the shortest‑path approach to allocate evidence flow to these pseudo‑paths. Starting from the pseudo‑path with the fewest edges, we assign a flow on each edge is given by the corresponding absolute entry of the component-level hat matrix $${{\varvec{H}}}_{{\varvec{a}}}$$. After each allocation, the residual flows on the involved edges are updated, and the process repeats until all flow is exhausted. The algorithm generalizes the shortest‑path approach to an algebraic setting where paths are defined by linear combinations of edges with potentially fractional coefficients, and the flow is distributed proportionally to these coefficients, rather than equally as in standard NMA. We illustrated this approach using both a hypothetical example and real-world datasets. In both real-world data networks, the two-stage algorithm systematically identified and quantified the contributions of the pseudo-paths. The flow-weighted sum of pseudo-path–derived estimates matched exactly (within numerical tolerance) the overall component effect estimated by the CNMA model. This confirms that the proposed algorithm correctly decomposes and then recomposes the evidence structure that gives rise to the component effect estimate. This study adapts the shortest‑path approach for use in CNMA, providing a quantitative method to trace evidence contributions to component‑level estimates. By introducing pseudo‑paths and a corresponding flow‑allocation algorithm, the method extends path‑based contribution analysis from standard NMA to the CNMA setting, enabling transparent decomposition of how evidence from multicomponent interventions synthesizes into component effects.