Graph Neural Networks struggle on random graphs without node identifiers.
problem Graph Neural Networks' limitations on random graphs without node identifiers.
method Study of Graph Neural Networks and Structural Graph Neural Networks convergence on large random graphs.
result Structural Graph Neural Networks are more powerful and universal than Graph Neural Networks on random graphs.
GraphMoE generates random graphs using neural networks and graphlets.
problem Learning generative models for random graphs.
method GraphMoE uses a neural network trained with graphlets and subgraph counts to match the distribution of random graphs.
result GraphMoE can generate graphs that mimic various real-world datasets and fool graph classifiers.
New method for faster graph parameter inference from large random Kronecker graphs.
problem Efficiently infer graph parameters from large random Kronecker graphs.
method Decompose adjacency matrix into signal and noise components, then use denoising and solving approach.
result Proposed method achieves comparable or better performance than existing methods at lower computational cost.
Topology helps estimate chromatic numbers of random graphs on spheres.
problem Estimating chromatic numbers of random graphs on spheres.
method Topology, specifically connectivity of Lóvasz's neighborhood complex.
result Connectivity bound is useful in dimensions 1 and 2, but generally poor.
Two types of nonidentifiability in latent position graphs identified and characterized.
problem Identifying and characterizing nonidentifiability in latent position random graph models.
method Defined and examined subspace nonidentifiability and model-based nonidentifiability, providing examples and characterizing limits.
result Characterized the limits of model-based nonidentifiability and obtained additional limiting results for specific graph models.
This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs. VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the n…
Novel graph neural network combines random walks with local message passing.
problem Graph neural networks struggle with long-range dependencies.
method Combines random walks with local message passing in a novel architecture.
result Significant performance improvements on graph benchmarks.
GCNs converge and remain stable on large random graphs, revealing geometric insights.
problem Understanding the behavior of GCNs on large, sparse random graphs.
method Analysis of GCNs on random graph models with latent variables and geometric edge probabilities.
result GCNs converge to their continuous counterparts as graph size increases, and are stable to small graph deformations.
IDPGs extend RDPGs with a Poisson process for random latent positions.
problem Modeling randomness in latent positions for graph structure.
method Introduce IDPGs using Poisson point processes on latent Euclidean space.
result Continuous analogues of adjacency matrices link latent structure to observed graphs.
While a wide range of interpretable generative procedures for graphs exist, matching observed graph topologies with such procedures and choices for its parameters remains an open problem. Devising generative models that closely reproduce real-world graphs requires domain knowledge and time-consuming simulation. While e…
A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability …
Study on signed graphs with random signs, focusing on community detection.
problem Community detection in signed stochastic block models.
method Strong concentration inequalities for adjacency and Laplacian matrices, applied to signed Laplacian matrix.
result The sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for balanced community detection.
New model learns from random graph samples to estimate graph parameters.
problem Scalability issues in graph learning methods for large graphs.
method Develops a graph classification model working on randomly sampled subgraphs.
result Validates mini-batch learning on graphs and provides generalization bounds.
Graph neural networks (GNNs) are powerful machine learning models for various graph learning tasks. Recently, the limitations of the expressive power of various GNN models have been revealed. For example, GNNs cannot distinguish some non-isomorphic graphs and they cannot learn efficient graph algorithms. In this paper,…
Extends random dot product graph model to handle multiple graphs.
problem Modeling and analyzing multiple graphs with shared nodes.
method Jointly embed adjacency matrices into a latent space.
result Node representations converge to latent positions with Gaussian error.
A new test for IRG models using KSD for small networks.
problem Testing goodness-of-fit for inhomogeneous random graph models.
method Kernelised Stein Discrepancy (KSD) test for IRG models.
result The test is effective for small networks and provides theoretical guarantees.
GNNs with random node initialization are shown to be universally expressive.
problem Limitations of standard GNNs in distinguishing graphs.
method Random node initialization (RNI) to enhance GNNs' expressive power.
result GNNs with RNI are proven to be universally expressive.
New model calculates logarithmic surface diameter.
problem Calculating diameter of random hyperbolic surfaces.
method Exploration process inspired by graph breadth-first search.
result Diameter is logarithmic in surface genus.
Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about t…
Study evaluates RKHS choices for assessing graph models using KSD tests.
problem Effect of RKHS choice on KSD tests for graph model assessment.
method Investigated power performance and computational runtime of KSD tests for ERGMs and synthetic graph generators.
result Different RKHS choices affect KSD test performance and computational runtime.
New method for identifying graph shift operators using vertex-time autoregressive models.
problem Identifying graph shift operators from graph signals.
method Online optimization using vertex-time autoregressive model and stochastic gradient projection.
result Successful recovery of graph shift operators from graph signals.
In this work, we consider an extension of graphical models to random graphs, trees, and other objects. To do this, many fundamental concepts for multivariate random variables (e.g., marginal variables, Gibbs distribution, Markov properties) must be extended to other mathematical objects; it turns out that this extensio…
In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erdös-Renyi random graphs, geometric random graphs, expanders and random k-regular graphs. Moreover we show that …
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
We introduce a new framework for comparing parametric network families.
problem Comparing and analyzing data modeled as parameterized families of networks.
method A Gromov-Wasserstein variant of optimal transport for defining distances.
result Established foundational properties and theoretical approximation guarantees for the new distances.
AgraSSt assesses graph generators using Stein operators and kernel discrepancies.
problem Assessing the quality of graph generators that are implicit or not in explicit form.
method AgraSSt uses Stein operators and kernel discrepancies to assess graph generators, providing interpretable criticisms.
result Theoretical guarantees and empirical validation for various graph models.
Paper explores embedding methods for detecting pseudo-cliques in random graphs, showing limitations and potential.
problem Detecting planted pseudo-cliques in random dot product graphs.
method Adjacency Spectral Embedding (ASE) and Graph Encoder Embedding (GEE).
result These methods can localize pseudo-cliques with additional clean network data, but not without it.
The random dot product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random graphs, from relatively simple stochastic block models to complex latent position graphs. In this survey paper, we describ…
TTERGM models improve social network predictions by incorporating triadic relationships.
problem Lack of models capturing triadic relationships and social learning theories in temporal network data.
method Introduced TTERGM, a generative model that includes triadic relationships and social learning theory as additional probability distributions. Parameters are estimated via Monte Carlo maximum likelihood.
result TTERGM achieves improved accuracy and fidelity compared to existing models on social network data.
It has been shown recently that graph signals with small total variation can be accurately recovered from only few samples if the sampling set satisfies a certain condition, referred to as the network nullspace property. Based on this recovery condition, we propose a sampling strategy for smooth graph signals based on …
This paper explores GNN functions on random graphs, highlighting the importance of node Positional Encodings.
problem Understanding the expressive power of GNNs on large random graphs.
method General convergence notions, input node features, and Positional Encodings (PEs).
result GNNs can converge to certain functions on large random graphs, emphasizing the role of PEs.
Graphs can be fooled by small edge changes, but this work protects them.
problem Adversaries can manipulate graph data to mislead graph classification models.
method We introduce a smoothed graph classification model with a robustness guarantee.
result The smoothed model maintains consistent predictions under small adversarial perturbations.
New algorithms improve community detection and parameter estimation for PABM.
problem Improving community detection and parameter estimation for PABM.
method Connecting PABM to GRDPG, constructing new algorithms, and deriving asymptotic properties.
result Absolute number of community detection errors tends to zero as graph vertices increase.
Brooks and Makover introduced an approach to random Riemann surfaces based on associating a dense set of them - Belyi surfaces - with random cubic graphs. In this paper, using Bollobas model for random regular graphs, we examine the topological structure of these surfaces, obtaining in particular an estimate for the ex…
The paper shows that relaxing assumptions about causal graphs can lead to exponentially large equivalence classes.
problem The size of Markov equivalence classes under relaxed assumptions.
method Analytical proofs for three settings: sparse random directed acyclic graphs, uniformly random acyclic directed mixed graphs, and uniformly random directed cyclic graphs.
result Exponentially large lower bounds for the expected size of Markov equivalence classes.
Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of the embeddings of their sample graphs. In particular, the graph models that we c…
New framework models epistemic uncertainty in GNNs using random sets.
problem Uncertainty in graph neural network predictions.
method Introduces a belief function (random set) approach to model epistemic uncertainty in GNNs.
result Demonstrates superior uncertainty quantification on various graph datasets.
Sharp threshold found for Frechet mean of inhomogeneous graphs.
problem Finding the Frechet mean of inhomogeneous Erdos-Renyi random graphs.
method Thresholding the expected adjacency matrix of the ensemble.
result The Frechet mean graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Graph embedding methods represent nodes in a continuous vector space, preserving information from the graph (e.g. by sampling random walks). There are many hyper-parameters to these methods (such as random walk length) which have to be manually tuned for every graph. In this paper, we replace random walk hyper-paramete…
New methods learn from single graphs, improving transductive node classification.
problem Statistical foundations of transductive learning for single graphs.
method Developed new concentration-of-measure tools for large graphs.
result Achieved optimal nonparametric rate of N−1/2 for single graph learning. SteinGen generates diverse graph samples from a single example.
problem Generating graphs with characteristic structures and diversity from a single example.
method Combines Stein's method and MCMC with Glauber dynamics and re-estimation of the Stein operator.
result High distributional similarity to the original data, combined with high sample diversity.
Proposes BGCN-NRWS for semi-supervised node classification with reduced overfitting.
problem Uncertainty in graph structure for semi-supervised node classification.
method Bayesian Graph Convolutional Network using Neighborhood Random Walk Sampling (BGCN-NRWS) with MCMC graph sampling and variational inference.
result Consistently competitive classification results compared to state-of-the-art.
Given an edge-independent random graph G(n,p), we determine various facts about the cohomology of graph products of groups for the graph G(n,p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n goes to infinity. This includes random ri…
Study on convergence of graph neural networks on random graphs.
problem Convergence of message passing graph neural networks on large random graphs.
method Extended convergence results to a broad class of aggregation functions using McDiarmid inequality.
result Non-asymptotic bounds for convergence quantified with high probability.
A scalable deep GMRF model for general graphs improves predictions and uncertainty estimates.
problem Handling generally structured data on graphs efficiently.
method A new multi-layer structure of Deep GMRFs designed for general graphs, enabling efficient training and close-to-exact Bayesian inference.
result Close-to-exact Bayesian inference for latent field predictions with uncertainty estimates.
Estimates latent norms and Gram matrices for graphs on Euclidean balls.
problem Estimating latent points and their relationships in graphs on Euclidean balls.
method Estimates latent norms and Gram matrices using observed graph data.
result Graphs on Euclidean balls can have power-law degree distributions.
Random projections help in representing sparse graphs efficiently.
problem Efficiently representing sparse graphs of varying sizes and vertex sets.
method Random projection of adjacency matrices to retain graph functionality and properties.
result Random projections can accurately represent graphs of different sizes and vertex sets in the same space.