Auditing bipartite motif interpretations: a worked example with conservation checks and open-path decomposition

Abstract

Motif profiles of bipartite agent-object networks, such as tourist-site visits and customer-item transactions, are read as evidence about structural roles and about differences between networks, often without asking what the two degree sequences already fix. In a simple bipartite graph the induced k-fan count on one node type is a sum of degree combinations, so it has zero variance under a null that preserves both degree sequences. We apply this known result to a reconstructed tourism rating network of 17 tourists, 80 sites and 637 edges, the sole inferential worked example, and, as a provenance-limited illustration, to published motif-instance aggregates over 36 monthly luxury customer-item networks. The four fan classes are exact functions of the degree sequences: in the tourism network the raw fan counts and the size-3 two-fan ratio (84.7% fan-out) restate those sequences. The published luxury counts require at least 89,502 customer-item edges against 26,451 reported transactions, so their 99.8% fan-in is reported as a descriptive value only. Against a hard bipartite configuration null, the four-cycle count is degree-consistent (z about +1.0) and the open path is deficient (z about -5.8) by 2,243 instances, 2.4% of the null mean; the deficit survives every leave-one-tourist-out re-run (z -4.5 to -8.1). An exact identity splits it at the point estimate into 64.4% mixing and 35.6% four-cycle, but that split is not an attribution: the observed mixing term lies below all 500 null samples, the two components are almost collinear under the null (r = 0.968), and the mixing share ranges from 36.6% to 116.5% under leave-one-tourist-out deletion. The deficit is extreme relative to the sampled null, while its class-level interpretation is undetermined and unstable. We give a four-step pre-interpretation check and a reference implementation.