data scientists in network analysis

Abstract: Persistence diagrams (PDs) represent multiscale topological information as multisets of birth-death pairs $(b,d)$. How best to use them with Transformers, however, remains unclear. Connecting PDs to Transformers requires choosing how topological information is tokenized and exposed to self-attention. Prior methods use specialized designs for tokenization, aggregation, Transformer blocks, or attention, but are hard to compare since reported results rarely share PD inputs, splits, or model-selection rules. We instead confine the diagram-specific design to point representation, mapping each birth-death pair through fixed random Fourier features while leaving the Transformer backbone unchanged, and call this recipe Persistence Positional Encoding (PPE). In matched controls on eight TU graph benchmarks (TU8), changing only point representation shifts macro accuracy by 5.49 to 6.47 percentage points (pp), against 0.47 to 1.45 pp from changing the downstream module, identifying point representation as the dominant design variable tested. We further reevaluate vectorizations, kernels, learned representations, and Transformer-based models under one protocol with shared PD inputs, outer folds, and validation-only selection. Among all evaluated PD methods and downstream configurations, PPE achieves the highest mean on most benchmarks, including TU8, ORBIT5K, and MCB-C. Finally, we prove that grid averaging leaves a reconstruction error at any resolution, whereas a Transformer with fixed random Fourier frequencies is almost surely universal. PPE thus offers a minimal, Transformer-native baseline for persistence diagrams, and a common protocol for future comparison.

Abstract: Standard diffusion processes with isotropic noise corruption are not spectrally neutral. They inherit a coarse-to-fine generation hierarchy from the nonuniform data spectrum, because each frequency band loses signal-to-noise ratio (SNR) at its own rate. For signals on a graph, the relevant spectrum is that of the graph shift operator (GSO). A natural construction, therefore, builds both the diffusion dynamics and the denoiser from graph filters of the GSO, and any such design also takes a position on this hierarchy. We expose this hierarchy as a tunable axis through a family of colored score-based diffusions. Fixing the terminal prior to a Gaussian colored by the GSO couples the drift and diffusion through a positive spectral rate, which acts as a clock for each graph-frequency band. The clock sets when a band loses and recovers information, and the forward processes of standard and certain graph-aware diffusions appear as members of this family. A GSO-valued EDM preconditioning splits the denoiser into an analytic Wiener head that carries the Gaussian second-order structure and a graph neural network (GNN) that learns the remaining non-Gaussian dependencies. Numerical experiments show that colored diffusions reach a given generation quality in far fewer sampling steps than standard diffusion.

Abstract: Consensus can make pseudo-label learning more reliable, but only when its predictors contribute genuinely different evidence. Multiple models that repeat the same boundary provide additional votes without additional information. We address this problem by con structing decision sources through controlled changes to within-class structure. Starting from a shared graph representation, we vary center granularity and neighborhood mixing, reproduce each resulting source to test its stability, and select a complementary subset using node pair coassignment. Unanimous predictions from the selected sources are then ranked for student training. On the public fixed splits of Cora, CiteSeer, and PubMed, evaluated with five random seeds, the constructed sources improve fixed-budget training pseudo-label precision by 1.19 to 4.39 percentage points over three conventionally initialized GCN sources. Under matched structural filters, three-source consensus is more precise than each constituent source in all 45 dataset slot eed comparisons. The gains are strongest in pseudo-label quality: downstream accuracy remains competitive but does not lead on every dataset. These results identify source construction rather than model count alone as an important design problem for consensus-based pseudo-label learning.

Abstract: Graph foundation models aim to transfer across graphs, feature spaces, relational schemas, and prediction tasks, yet existing approaches typically generalize only within particular graph modalities or tasks. We propose Wander, a graph foundation model designed to operate across these settings within a single pretrained checkpoint. Following the prior-predictive perspective, we formulate graph learning as completion of a partially observed graph. We realize this task-general view through a common interface based on random walks, allowing the same model to operate across homogeneous and multi-relational graphs with varying features, labels, and relational schemas. Wander can increase its structural context at inference time without changing its learned parameters and, under suitable assumptions, universally approximates the corresponding Bayes-optimal predictor on bounded connected graphs. Empirically, a single pretrained checkpoint achieves state-of-the-art or highly competitive results across node classification, homogeneous link prediction, and knowledge-graph link prediction. Moreover, joint pretraining across graph modalities and tasks preserves performance in specialized settings while enabling positive transfer and the composition of separately learned capabilities at inference time.

Abstract: Many node-classification GNNs apply a linear classifier to a nonnegative mixture of local messages. An error can reflect either poor mixture weights or a reachable logit set poorly positioned for the classifier. We separate these causes with an exact-mass linear program and two learned post-hoc repairs. Every reweighted prediction has an equivalent centered logit translation, but only translations in a message-induced displacement set are realizable by reweighting. Across eight datasets, eight GNN backbones, and ten splits, mean accuracy rises from 62.6% for the frozen models to 63.8% with reweighting and 65.3% with set-conditioned translation. A parameter-matched node-only translator reaches 64.6%, showing that translation explains most of the gain while the message set supplies a smaller additional benefit. Although oracle reweighting can correct many errors, label-free reweighting captures little of this potential: local evidence is often present but hard to select, and relaxing the evidence constraint is more effective than learning within it.