43 research outputs found

    Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

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    Two graphs GG and HH are homomorphism indistinguishable over a class of graphs F\mathcal{F} if for all graphs F∈FF \in \mathcal{F} the number of homomorphisms from FF to GG is equal to the number of homomorphisms from FF to HH. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl

    Improving Expressivity of Graph Neural Networks using Localization

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    In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of k−k-WL for any kk. We analyze the power of Local k−k-WL and prove that it is more expressive than k−k-WL and at most as expressive as (k+1)−(k+1)-WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local k−k-WL equivalent. We also introduce two variants of k−k-WL: Layer k−k-WL and recursive k−k-WL. These methods are more time and space efficient than applying k−k-WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just 1−1-WL. The same idea can be extended further for larger patterns using k>1k>1. We also compare the expressive power of Local k−k-WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a]

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    When do homomorphism counts help in query algorithms?

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    A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B.Comment: 24 page

    The Weisfeiler-Leman Dimension of Existential Conjunctive Queries

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    The Weisfeiler-Leman (WL) dimension of a graph parameter ff is the minimum kk such that, if G1G_1 and G2G_2 are indistinguishable by the kk-dimensional WL-algorithm then f(G1)=f(G2)f(G_1)=f(G_2). The WL-dimension of ff is ∞\infty if no such kk exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ\varphi, we quantify the WL-dimension of the function that maps every graph GG to the number of answers of φ\varphi in GG. The works of Dvor\'ak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ\varphi, the WL-dimension is equal to the treewidth of the Gaifman graph of φ\varphi. In this work, we give a characterisation that applies to all conjunctive qureies. Given any conjunctive query φ\varphi, we prove that its WL-dimension is equal to the semantic extension width sew(φ)\mathsf{sew}(\varphi), a novel width measure that can be thought of as a combination of the treewidth of φ\varphi and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ\varphi are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ\varphi cannot be computed by GNNs of order smaller than sew(φ)\mathsf{sew}(\varphi).Comment: 36 pages, 4 figures, abstract shortened due to ArXiv requirement

    Limitations of Game Comonads via Homomorphism Indistinguishability

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    Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for k-variable counting logic and thereby initiated a line of work that imports category theoretic machinery to finite model theory. Such game comonads have been developed for various logics, yielding characterisations of logical equivalences in terms of isomorphisms in the associated co-Kleisli category. We show a first limitation of this approach by studying linear-algebraic logic, which is strictly more expressive than first-order counting logic and whose k-variable logical equivalence relations are known as invertible-map equivalences (IM). We show that there exists no finite-rank comonad on the category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence, answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this result by ruling out a characterisation of IM-equivalence in terms of homomorphism indistinguishability and employing the Lov\'asz-type theorems for game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are homomorphism indistinguishable over a graph class if they admit the same number of homomorphisms from every graph in the class. The IM-equivalences cannot be characterised in this way, neither when counting homomorphisms in the natural numbers, nor in any finite prime field.Comment: Minor corrections in Section

    Expectation-Complete Graph Representations with Homomorphisms

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    We investigate novel random graph embeddings that can be computed in expected polynomial time and that are able to distinguish all non-isomorphic graphs in expectation. Previous graph embeddings have limited expressiveness and either cannot distinguish all graphs or cannot be computed efficiently for every graph. To be able to approximate arbitrary functions on graphs, we are interested in efficient alternatives that become arbitrarily expressive with increasing resources. Our approach is based on Lov\'asz' characterisation of graph isomorphism through an infinite dimensional vector of homomorphism counts. Our empirical evaluation shows competitive results on several benchmark graph learning tasks.Comment: accepted for publication at ICML 202

    Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation

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    We introduce the 2-sorted counting logic GCkGC^k that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to express that a vertex v is contained in a hyperedge e. We show that two hypergraphs H, H' satisfy the same sentences of the logic GCkGC^k if, and only if, they are homomorphism indistinguishable over the class of hypergraphs of generalised hypertree width at most k. Here, H, H' are called homomorphism indistinguishable over a class C if for every hypergraph G in C the number of homomorphisms from G to H equals the number of homomorphisms from G to H'. This result can be viewed as a generalisation (from graphs to hypergraphs) of a result by Dvorak (2010) stating that any two (undirected, simple, finite) graphs H, H' are indistinguishable by the (k+1)-variable counting logic Ck+1C^{k+1} if, and only if, they are homomorphism indistinguishable on the class of graphs of tree width at most k

    Equivariant Polynomials for Graph Neural Networks

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    Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this hierarchy has propelled significant advances in GNN analysis and architecture developments, it suffers from several significant limitations. These include a complex definition that lacks direct guidance for model improvement and a WL hierarchy that is too coarse to study current GNNs. This paper introduces an alternative expressive power hierarchy based on the ability of GNNs to calculate equivariant polynomials of a certain degree. As a first step, we provide a full characterization of all equivariant graph polynomials by introducing a concrete basis, significantly generalizing previous results. Each basis element corresponds to a specific multi-graph, and its computation over some graph data input corresponds to a tensor contraction problem. Second, we propose algorithmic tools for evaluating the expressiveness of GNNs using tensor contraction sequences, and calculate the expressive power of popular GNNs. Finally, we enhance the expressivity of common GNN architectures by adding polynomial features or additional operations / aggregations inspired by our theory. These enhanced GNNs demonstrate state-of-the-art results in experiments across multiple graph learning benchmarks

    Fundamentals

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    Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters
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