Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): two graphs it cannot tell apart receive identical representations, however deep or wide the network. A common remedy augments node or edge features with precomputed structural descriptors, most often counts of a fixed small subgraph such as triangles or longer cycles, but such counts require committing in advance to the size of the substructure counted, a choice usually made blind to the data. We study a descriptor that avoids this choice. The edge-girth of an edge is the length of a shortest cycle through it, and its multiplicity is the number of such shortest cycles; together they form a per-edge invariant that reports cycles of arbitrary length, computable exactly by a single breadth-first search per edge. Injected into a gated message-passing architecture, EGAGNN, it reaches a test MAE a factor three below the closest gated comparator on the ZINC-12k regression benchmark at 104k parameters; against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary counting cycles up to length eight, using twice as many channels, while a dictionary capped at length four performs no better than no structural information at all. On graph discrimination we prove a matching limitation: on graphs where every edge sees the same number of shortest cycles of the same length, the descriptor becomes constant and any model built on it collapses back to the 1-WL bound. This holds without exception across all 400 pairs of the BREC benchmark: not one of the 90 such pairs is distinguished.
State-of-the-art temporal-link-prediction (TLP) models are, in essence, multi-channel information aggregators: they combine an interaction-history channel, a time-encoding channel, and a structure channel. The first two have been refined relentlessly; the structure channel remains a crude afterthought -- DyGFormer encodes it as a 1--2-bit neighbour-cooccurrence count. We begin with a measurement: on sparse temporal graphs the classical 1-hop common-neighbour signal is near-random (discriminative AUC $\approx 0.50$), because two nodes almost never share a direct neighbour; the genuinely discriminative signal lies one hop deeper -- the 2-hop cohesive bridge, whose discAUC reaches 0.73--0.98, on both bipartite and non-bipartite graphs. Motivated by this, we propose HAWKEYE, a cohesion-aware structural channel that incrementally maintains the classical k-family of cohesiveness indicators (degree $\to$ k-core $\to$ k-truss) and forms 2-hop cohesive-bridge features. HAWKEYE is a drop-in replacement for a temporal-graph model's native structure channel, with no change to the backbone. Swapping HAWKEYE into DyGFormer improves test AP/MRR over the cooccurrence channel by +0.6 to +10.8 points across six multi-seed-validated datasets (uci, enron, USLegis, CanParl, reddit, mooc). On the bipartite recommendation benchmark tgbl-subreddit, a 3-seed single-pass struct-only ablation shows HAWKEYE nearly doubling the baseline test MRR (0.103$\pm$0.003 $\to$ 0.204$\pm$0.005, +10.1 points across all three seeds); the streaming pipeline scales to the 67M-edge tgbl-flight in five minutes per pass. We further characterise when it helps: the gain tracks a graph's training-free 2-hop discAUC and vanishes on degenerate or saturated graphs -- a predictable boundary. All code, data, and figure-generation scripts are released.
Maryam Gholami Shiri, Ivana Krminac, Marko Djukanović +3cs.SI cs.LG cs.NE
Benchmarking shortest-path algorithms is commonly based on aggregate performance over heterogeneous graph sets, which limits insight into how different search paradigms react to instance structure. We adopt an instance-landscape view of graph benchmarking by embedding graphs into a low-cost structural feature space and clustering them into regions of similar structure. Three benchmark suites are studied: weighted Erdős--Rényi graphs, random geometric (wireless) graphs, and real-world road networks. We evaluate four representative shortest-path solvers spanning uninformed exact search (Dijkstra), bidirectional exact search (bidirectional Dijkstra), heuristic-guided exact search (A$^{*}$), and deque-based strategies (DEQ). Clustering robustness is analyzed under multiple feature-selection schemes, and runtime distributions are compared across landscape regions using non-parametric tests. While generator parameters induce stable structural regions, we find that feature-space similarity does not necessarily imply performance similarity: significant runtime shifts are frequently observed even within the same landscape region. A merged-suite analysis further shows that different benchmark families occupy largely disjoint regions. These results highlight both the potential and the limits of structural landscapes for the structure-aware benchmarking of shortest-path algorithms.