Isomorphism Testing Research Articles

Efficient Isomorphism for $$S_d$$-Graphs and T-Graphs

An H-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró et al. (Discrete Mathematics 100:267–279, 1992). An H-graph is proper if the representing subgraphs of H can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for \(S_d\)-graphs and T-graphs, where \(S_d\) is the star with d rays and T is an arbitrary fixed tree. Answering an open problem of Chaplick et al. (2016, personal communication), we provide an FPT-time algorithm for testing isomorphism and computing the automorphism group of \(S_d\)-graphs when parameterized by d, which involves the classical group-computing machinery by Furst et al. (in Proceedings of 11th southeastern conference on combinatorics, graph theory, and computing, congressum numerantium 3, 1980). We also show that the isomorphism problem of \(S_d\)-graphs is at least as hard as the isomorphism problem of posets of bounded width, for which no efficient combinatorial-only algorithm is known to date. Then we extend our approach to an XP-time algorithm for isomorphism of T-graphs when parameterized by the size of T. Lastly, we contribute an FPT-time combinatorial algorithm for isomorphism testing in the special case of proper \(S_d\)- and T-graphs.

Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

Hypergraph Isomorphism for Groups with Restricted Composition Factors

We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V , and asking whether there is a permutation γ ε Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time ( n + m ) O ((log d ) c ) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time n O ( d ) m O (1) . As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K 3, h ( h ≥ 3) as a minor in time n O ((log h ) c ) . In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n O ((log g ) c ) .

Identification of Young High-Functioning Autism Individuals Based on Functional Connectome Using Graph Isomorphism Network: A Pilot Study.

Accumulated studies have determined the changes in functional connectivity in autism spectrum disorder (ASD) and spurred the application of machine learning for classifying ASD. Graph Neural Network provides a new method for network analysis in brain disorders to identify the underlying network features associated with functional deficits. Here, we proposed an improved model of Graph Isomorphism Network (GIN) that implements the Weisfeiler-Lehman (WL) graph isomorphism test to learn the graph features while taking into account the importance of each node in the classification to improve the interpretability of the algorithm. We applied the proposed method on multisite datasets of resting-state functional connectome from Autism Brain Imaging Data Exchange (ABIDE) after stringent quality control. The proposed method outperformed other commonly used classification methods on five different evaluation metrics. We also identified salient ROIs in visual and frontoparietal control networks, which could provide potential neuroimaging biomarkers for ASD identification.

KerGNNs: Interpretable Graph Neural Networks with Graph Kernels

Graph kernels are historically the most widely-used technique for graph classification tasks. However, these methods suffer from limited performance because of the hand-crafted combinatorial features of graphs. In recent years, graph neural networks (GNNs) have become the state-of-the-art method in downstream graph-related tasks due to their superior performance. Most GNNs are based on Message Passing Neural Network (MPNN) frameworks. However, recent studies show that MPNNs can not exceed the power of the Weisfeiler-Lehman (WL) algorithm in graph isomorphism test. To address the limitations of existing graph kernel and GNN methods, in this paper, we propose a novel GNN framework, termed Kernel Graph Neural Networks (KerGNNs), which integrates graph kernels into the message passing process of GNNs. Inspired by convolution filters in convolutional neural networks (CNNs), KerGNNs adopt trainable hidden graphs as graph filters which are combined with subgraphs to update node embeddings using graph kernels. In addition, we show that MPNNs can be viewed as special cases of KerGNNs. We apply KerGNNs to multiple graph-related tasks and use cross-validation to make fair comparisons with benchmarks. We show that our method achieves competitive performance compared with existing state-of-the-art methods, demonstrating the potential to increase the representation ability of GNNs. We also show that the trained graph filters in KerGNNs can reveal the local graph structures of the dataset, which significantly improves the model interpretability compared with conventional GNN models.

An Improved Order-Preserving Pattern Matching Algorithm Using Fingerprints

Two strings of the same length are order isomorphic if their relative orders are the same. The order-preserving pattern matching problem is to find all substrings of text T that are order isomorphic to pattern P when T(|T|=n) and P(|P|=m) are given. An O(mn+nqlogq+q!)-time algorithm using the O(m+q!) space for the order-preserving pattern matching problem has been proposed utilizing fingerprints of q-grams based on the factorial number system and the bad character heuristic. In this paper, we propose an O(mn+2q)-time algorithm using the O(m+2q) space for the order-preserving pattern matching problem, but utilizing fingerprints of q-grams converted to binary numbers. A comparative experiment using three types of time series data demonstrates that the proposed algorithm is faster than existing algorithms because it reduces the number of order isomorphism tests.

Isomorphism testing of read-once functions and polynomials

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of read-once formulas is complete for log-space. In contrast, we observe that the problem becomes polynomial-time equivalent to the graph isomorphism problem when the input formulas can be represented as OR of two or more read-once formulas.We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective [Patarin EUROCRYPT '96, and Kayal SODA '11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is at least as hard as the graph isomorphism problem.

Tensor isomorphism by conjugacy of Lie algebras

We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable we give a polynomial-time algorithm to solve a generalization of module isomorphism for a common class of Lie modules. As a consequence, we show that isomorphism testing is in polynomial time for tensors whose derivation algebras are classical Lie algebras and whose densor spaces are 1-dimensional. The method has been implemented in the Magma computer algebra system.

Formal Identification of Argumentation Patterns in Scientific Texts

In this paper we present the methods for the automatic identification of argumentation patterns and the results of their application. These methods have been employed to analyze argumentation annotations of 25 scientific texts from two thematic areas. Under study were the expert annotations constructed manually with the help of web tools for visualizing argumentative statements and argumentation schemes, as well as for modelling the argumentation structure of a text as an oriented graph. Such graphs contain two node types: information nodes denoting statements and their connecting reasoning models (schemas) from Walton's compendium. The regularly employed reasoning models and their structural combinations (argumentation subgraphs) form argumentation patterns. Patterns containing more than one scheme are unmarked in initial graphs. As a result of processing argumentation annotations from the collection, we have constructed a joint spectrum of argumentation patterns with their absolute and text frequencies. The methods of frequent subgraph mining have been used to identify argumentation patterns containing three or more schemes. The subgraph matching has been performed through the use of the NetworkX package which implements the VF2 algorithm for subgraph isomorphism testing. We have analyzed the calculated frequencies to identify both the general principles behind the use of argumentation patterns typical of scientific texts, as well as the specific tendencies of their functioning within distinct thematic areas (linguistics and computer science). These general principles regulate the use of both separate schemes and their structural combinations. The latter appear in two configuration types: either as sequencies of argumentation schemes (up to 5 elements) or as tree structures (which contain up to 8 nodes). Specifically, we demonstrate that branching within a tree-form pattern typically presupposes the parallel use of identical argumentation schemes. Additionally, branching of argumentation patterns occurs more often in proximity to the main thesis of a text, than near the initial premises. Finally, the thematic area can condition not only the use of separate schemes, but also their structural combination within complex patterns. The results obtained are applicable to the evaluation of text similarity based on argumentation patterns employed in them, which, in its turn, can improve the clustering and classification of texts, evaluation of their persuasiveness, as well as the formal synthesis of argumentation.

Graph kernels based on optimal node assignment

The success of kernel algorithms depends on the kernel it uses and hence the development of kernels for structured data like graphs is an active research field. We designed two graph kernels by making use of the optimal assignment kernel framework, which is a less explored graph framework compared to its counterpart, namely, R-convolution kernels. In our work, the bijection associated with the optimal assignment framework is defined between sets that consist of the nodes of the graph kernel arguments and hence we named the proposed kernels as optimal node assignment kernels (ONA). In ONA, the nodes of the given data are divided into groups named as neighbourhood sets on the basis of the labels generated by the Weisfeiler–Lehman (WL) test for graph isomorphism and a matrix representation is defined for them. A kernel is then defined over the domain that consists of the neighbourhood sets in terms of such matrices and an aggregate measure of those kernel values is used for defining ONA kernel. The validity of the proposed kernels is mathematically proved. The kernel calculation using the hierarchy structure associated with the optimal assignment framework is also discussed. The performance of the proposed ONA kernels was analysed using graph classification datasets and was found to be superior in comparison with the other state-of-the-art graph kernels.

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used technique for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. This implies that the dimension of the algorithm needed to distinguish two given non-isomorphic graphs is at most the dimension required to distinguish non-isomorphic 3-connected components of the graphs (assuming dimension at least 2). To obtain our decomposition result, we show that, for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of the class of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

Permute Me Softly: Learning Soft Permutations for Graph Representations.

Graph neural networks (GNNs) have recently emerged as a dominant paradigm for machine learning with graphs. Research on GNNs has mainly focused on the family of message passing neural networks (MPNNs). Similar to the Weisfeiler-Leman (WL) test of isomorphism, these models follow an iterative neighborhood aggregation procedure to update vertex representations, and they next compute graph representations by aggregating the representations of the vertices. Although very successful, MPNNs have been studied intensively in the past few years. Thus, there is a need for novel architectures which will allow research in the field to break away from MPNNs. In this paper, we propose a new graph neural network model, so-called π-GNN which learns a "soft" permutation (i. e., doubly stochastic) matrix for each graph, and thus projects all graphs into a common vector space. The learned matrices impose a "soft" ordering on the vertices of the input graphs, and based on this ordering, the adjacency matrices are mapped into vectors. These vectors can be fed into fully-connected or convolutional layers to deal with supervised learning tasks. In case of large graphs, to make the model more efficient in terms of running time and memory, we further relax the doubly stochastic matrices to row stochastic matrices. We empirically evaluate the model on graph classification and graph regression datasets and show that it achieves performance competitive with state-of-the-art models.

Računska metodologija za sintezu konfiguracija sistema epicikličkog zupčastog prenosa sa više planetarnih zupčanika

Epicyclic Gear Transmission Systems (EGTs) find a wide variety of applications in automatic transmission systems due to their flexibility to deliver a variety of speed ratios. The design of EGT involves the enumeration of all possible layouts, typically called structural synthesis. Thus, the structural synthesis of an EGT consists of defining a topology for EGT with coupled planetary gear trains (PGTs) along with constraining elements such as connections, brakes, and clutches while delivering a definite transmission ratio between the input and output shafts. Given the number of PGTs, permanent connection, clutches, and brakes, the methodology systematically introduces these elements of PGTs to deliver the given number of distinct speed ratios. Determining the number of distinct speed ratios for a given combination of brakes and clutches has been expressed as a transformation of a graph representation of the EGT layout. A computer program has been developed for the structural synthesis of EGTs and validated against the 3 epicyclic, 6-speed EGT systems. The viability aspects of the synthesized structure are carried out. Also, an isomorphism test is addressed in the synthesis methodology to detect unique EGT structures at every synthesis stage.

Breaking the Expression Bottleneck of Graph Neural Networks

Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to measure the expressiveness of graph neural networks (GNNs), showing that the neighborhood aggregation GNNs were at most as powerful as 1-WL test in distinguishing graph structures. There were also improvements proposed in analogy to k-WL test ( <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$k&gt;1$</tex></formula> ). However, the aggregations in these GNNs are far from injective as required by the WL test, and suffer from weak distinguishing strength, making it become the expression bottleneck. In this paper, we improve the expressiveness by exploring powerful aggregations. We reformulate an aggregation with the corresponding aggregation coefficient matrix, and then systematically analyze the requirements on this matrix for building more powerful and even injective aggregations. We also show the necessity of applying nonlinear units ahead of aggregations, which is different from most existing GNNs. Based on our theoretical analysis, we develop ExpandingConv. Experimental results show that our model significantly boosts performance, especially for large and densely connected graphs.

Graph Neural Networks With Parallel Neighborhood Aggregations for Graph Classification

We focus on graph classification using a graph neural network (GNN) model that precomputes the node features using a bank of neighborhood aggregation graph operators arranged in parallel. These GNN models have a natural advantage of reduced training and inference time due to the precomputations but are also fundamentally different from popular GNN variants that update node features through a sequential neighborhood aggregation procedure during training. We provide theoretical conditions under which a generic GNN model with parallel neighborhood aggregations (PA-GNNs, in short) are provably as powerful as the well-known Weisfeiler-Lehman (WL) graph isomorphism test in discriminating non-isomorphic graphs. Although PA-GNN models do not have an apparent relationship with the WL test, we show that the graph embeddings obtained from these two methods are injectively related. We then propose a specialized PA-GNN model, called SPIN, which obeys the developed conditions. We demonstrate via numerical experiments that the developed model achieves state-of-the-art performance on many diverse real-world datasets while maintaining the discriminative power of the WL test and the computational advantage of preprocessing graphs before the training process.

Local WL invariance and hidden shades of regularity

The k-dimensional Weisfeiler–Leman algorithm is a powerful tool in graph isomorphism testing. For an input graph G, the algorithm determines a canonical coloring of s-tuples of vertices of G for each s between 1 and k. We say that a numerical parameter of s-tuples is k-WL-invariant if it is determined by the tuple color. As an application of Dvořák’s result on k-WL-invariance of homomorphism counts in the case of k=2, we spot some non-obvious regularity properties of strongly regular graphs and related graph families. For example, if G is a strongly regular graph, then the number of 7-paths between vertices x and y in G depends only on whether or not x and y are adjacent (this is true also for shorter paths but no longer true for 8-paths). Likewise, the number of cycles of length 7 passing through a vertex x in G is the same for every x (where the length 7 is also optimal).

О некоторых алгоритмах квазиполиномиального времени

Since ancient times, algorithms are inextricably linked with mathematics. However, the theory of algorithms received independent development only in the first half of the 20th century. The initial period was associated with discovery of unsolvable as well as very difficult problems. So, in 1927 Gabriel Sudan published the first example of a computable function that is provably not primitive recursive. The practical significance of asymptotic estimates of computational complexity became apparent only in the second half of the 20th century. We consider some well-known decision problems solvable in quasipolynomial time, that is, problems presumably belonging to the intermediate class between polynomial time and exponential time. The collection comprises isomorphism testing, solution to the parity game, and (validated under the generalized Riemann hypothesis) factorization of univariate polynomials over finite fields. We also present some technical results, which can be used to create new quasipolynomial-time algorithms.

Identity-aware Graph Neural Networks

Message passing Graph Neural Networks (GNNs) provide a powerful modeling framework for relational data. However, the expressive power of existing GNNs is upper-bounded by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test, which means GNNs that are not able to predict node clustering coefficients and shortest path distances, and cannot differentiate between different d-regular graphs. Here we develop a class of message passing GNNs, named Identity-aware Graph Neural Networks (ID-GNNs), with greater expressive power than the 1-WL test. ID-GNN offers a minimal but powerful solution to limitations of existing GNNs. ID-GNN extends existing GNN architectures by inductively considering nodes’ identities during message passing. To embed a given node, ID-GNN first extracts the ego network centered at the node, then conducts rounds of heterogeneous message passing, where different sets of parameters are applied to the center node than to other surrounding nodes in the ego network. We further propose a simplified but faster version of ID-GNN that injects node identity information as augmented node features. Alto- gether, both versions of ID-GNN represent general extensions of message passing GNNs, where experiments show that transforming existing GNNs to ID-GNNs yields on average 40% accuracy improvement on challenging node, edge, and graph property prediction tasks; 3% accuracy improvement on node and graph classification benchmarks; and 15% ROC AUC improvement on real-world link prediction tasks. Additionally, ID-GNNs demonstrate improved or comparable performance over other task-specific graph networks.

Isomorphism, canonization, and definability for graphs of bounded rank width

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

Eigen-GNN: a Graph Structure Preserving Plug-in for GNNs

Graph Neural Networks (GNNs) are emerging machine learning models on graphs. Although sufficiently deep GNNs are shown theoretically capable of fully preserving graph structures, most existing GNN models in practice are shallow and essentially feature-centric. We show empirically and analytically that the existing shallow GNNs cannot preserve graph structures well. To overcome this fundamental challenge, we propose Eigen-GNN, a simple yet effective and general plug-in module to boost GNNs ability in preserving graph structures. Specifically, we integrate the eigenspace of graph structures with GNNs by treating GNNs as a type of dimensionality reduction and expanding the initial dimensionality reduction bases. Without needing to increase depths, Eigen-GNN possesses more flexibilities in handling both feature-driven and structure-driven tasks since the initial bases contain both node features and graph structures. We present extensive experimental results to demonstrate the effectiveness of Eigen-GNN for tasks including node classification, link prediction, and graph isomorphism tests.