Figure X | Ring structure is a distinctive but uncommon feature of Reactome
pathway networks, and is not captured by standard network descriptors.

Each human Reactome pathway was converted into a network as described in the
Methods (molecules as nodes; species annotated as simple chemicals, SBO:0000247,
removed; products linked to reactants within each reaction; the largest
connected component retained), and the persistent-homology ring score S was
computed for every pathway whose largest component contains at least 100 nodes.
Reactome's nested and aliased pathway definitions can produce byte-for-byte
identical networks; these were collapsed by largest-component identity, yielding
80 unique networks from 90 qualifying pathway definitions.

(a) Distribution of S across the 80 networks. A pronounced global ring (S > 0.6)
is the exception rather than the rule: most pathway networks have low-to-
intermediate S, and only 16 of the 80 networks exceed the 0.6 threshold (dark
bars; dashed line).

(b) The 16 networks with S > 0.6, ranked by S and named. Scores range from S =
0.61 (Interferon Signaling, the pathway shown in Fig. 1C) to S = 1.00
(Transcriptional Regulation by MECP2). One entry, labelled "x4", is a single
network shared by four overlapping Reactome cell-junction pathway definitions —
Cell-Cell communication, Cell junction organization, Cell-cell junction
organization and Adherens junctions interactions (S = 0.92, 168 nodes); a fifth
cell-junction-named pathway, Regulation of Homotypic Cell-Cell Adhesion (164
nodes), is a structurally distinct network with a near-identical score and is
listed separately. This redundancy is intrinsic to Reactome's hierarchical
annotation and explains why several pathway names can carry the same ring score.

(c) Ring score versus network size (number of nodes in the largest connected
component, logarithmic scale). S is essentially uncorrelated with size (Spearman
rho = -0.09, p = 0.43): networks with S > 0.6 (red rings). In (c) and (d), point
colour encodes network density on a shared logarithmic scale; the absence of a
colour gradient along S reflects the lack of association between density and
ring score.

(d) Ring score versus clustering coefficient (mean of local clustering).
Clustering does not track S (Spearman rho = -0.18, p = 0.12): it is uniformly
low across all pathway networks (median ~0.017, maximum 0.15 — these reaction
graphs are nearly tree- or chain-like), and the most-clustered networks (for
example Translation, Signaling by GPCR, and DNA-repair pathways) are among the
least ring-like. The ring score is likewise independent of network density (rho
= +0.03, p = 0.82). Within this homogeneous pathway collection the ring signal
is therefore not explained by size, density or clustering, consistent with
global ring structure being a distinct topological property rather than a by-
product of these standard descriptors.
