43 research outputs found
Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors
Two graphs and are homomorphism indistinguishable over a class of
graphs if for all graphs the number of
homomorphisms from to is equal to the number of homomorphisms from
to . 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
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 WL for any . We analyze the
power of Local WL and prove that it is more expressive than WL and at
most as expressive as WL. We give a characterization of patterns whose
count as a subgraph and induced subgraph are invariant if two graphs are Local
WL equivalent. We also introduce two variants of WL: Layer WL and
recursive WL. These methods are more time and space efficient than applying
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 WL. The same idea can be extended further for larger patterns using
. We also compare the expressive power of Local 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
LIPIcs, Volume 261, ICALP 2023, Complete Volum
When do homomorphism counts help in query algorithms?
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
The Weisfeiler-Leman (WL) dimension of a graph parameter is the minimum
such that, if and are indistinguishable by the -dimensional
WL-algorithm then . The WL-dimension of is if no
such 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 , we quantify the WL-dimension of the function that
maps every graph to the number of answers of in .
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 , the WL-dimension is equal to the
treewidth of the Gaifman graph of .
In this work, we give a characterisation that applies to all conjunctive
qureies. Given any conjunctive query , we prove that its WL-dimension
is equal to the semantic extension width , a novel width
measure that can be thought of as a combination of the treewidth of
and its quantified star size, an invariant introduced by Durand and Mengel
(ICDT 2013) describing how the existentially quantified variables of
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 cannot be computed
by GNNs of order smaller than .Comment: 36 pages, 4 figures, abstract shortened due to ArXiv requirement
Limitations of Game Comonads via Homomorphism Indistinguishability
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
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
We introduce the 2-sorted counting logic 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 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
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
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
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