34 research outputs found
On the power of graph neural networks and the role of the activation function
In this article we present new results about the expressivity of Graph Neural
Networks (GNNs). We prove that for any GNN with piecewise polynomial
activations, whose architecture size does not grow with the graph input sizes,
there exists a pair of non-isomorphic rooted trees of depth two such that the
GNN cannot distinguish their root vertex up to an arbitrary number of
iterations. The proof relies on tools from the algebra of symmetric
polynomials. In contrast, it was already known that unbounded GNNs (those whose
size is allowed to change with the graph sizes) with piecewise polynomial
activations can distinguish these vertices in only two iterations. Our results
imply a strict separation between bounded and unbounded size GNNs, answering an
open question formulated by [Grohe, 2021]. We next prove that if one allows
activations that are not piecewise polynomial, then in two iterations a single
neuron perceptron can distinguish the root vertices of any pair of
nonisomorphic trees of depth two (our results hold for activations like the
sigmoid, hyperbolic tan and others). This shows how the power of graph neural
networks can change drastically if one changes the activation function of the
neural networks. The proof of this result utilizes the Lindemann-Weierstrauss
theorem from transcendental number theory
Graded modal logic and counting message passing automata
We examine the relationship of graded (multi)modal logic to counting
(multichannel) message passing automata with applications to the
Weisfeiler-Leman algorithm. We introduce the notion of graded multimodal types,
which are formulae of graded multimodal logic that encode the local information
of a pointed Kripke-model. We also introduce message passing automata that
carry out a generalization of the Weisfeiler-Leman algorithm for distinguishing
non-isomorphic graph nodes. We show that the classes of pointed Kripke-models
recognizable by these automata are definable by a countable (possibly infinite)
disjunction of graded multimodal formulae and vice versa. In particular, this
equivalence also holds between recursively enumerable disjunctions and
recursively enumerable automata. We also show a way of carrying out the
Weisfeiler-Leman algorithm with a formula of first order logic that has been
augmented with H\"artig's quantifier and greatest fixed points
Single-Node Attack for Fooling Graph Neural Networks
Graph neural networks (GNNs) have shown broad applicability in a variety of
domains. Some of these domains, such as social networks and product
recommendations, are fertile ground for malicious users and behavior. In this
paper, we show that GNNs are vulnerable to the extremely limited scenario of a
single-node adversarial example, where the node cannot be picked by the
attacker. That is, an attacker can force the GNN to classify any target node to
a chosen label by only slightly perturbing another single arbitrary node in the
graph, even when not being able to pick that specific attacker node. When the
adversary is allowed to pick a specific attacker node, the attack is even more
effective. We show that this attack is effective across various GNN types, such
as GraphSAGE, GCN, GAT, and GIN, across a variety of real-world datasets, and
as a targeted and a non-targeted attack. Our code is available at
https://github.com/benfinkelshtein/SINGLE
Adaptive Multi-grained Graph Neural Networks
Graph Neural Networks (GNNs) have been increasingly deployed in a multitude
of different applications that involve node-wise and graph-level tasks. The
existing literature usually studies these questions independently while they
are inherently correlated. We propose in this work a unified model, Adaptive
Multi-grained GNN (AdamGNN), to learn node and graph level representation
interactively. Compared with the existing GNN models and pooling methods,
AdamGNN enhances node representation with multi-grained semantics and avoids
node feature and graph structure information loss during pooling. More
specifically, a differentiable pooling operator in AdamGNN is used to obtain a
multi-grained structure that involves node-wise and meso/macro level semantic
information. The unpooling and flyback aggregators in AdamGNN is to leverage
the multi-grained semantics to enhance node representation. The updated node
representation can further enrich the generated graph representation in the
next iteration. Experimental results on twelve real-world graphs demonstrate
the effectiveness of AdamGNN on multiple tasks, compared with several competing
methods. In addition, the ablation and empirical studies confirm the
effectiveness of different components in AdamGNN
Decidability of graph neural networks via logical characterizations
We present results concerning the expressiveness and decidability of a popular graph learning formalism, graph neural networks (GNNs), exploiting connections with logic. We use a family of recently-discovered decidable logics involving ``Presburger quantifiers''. We show how to use these logics to
measure the expressiveness of classes of GNNs, in some cases getting exact correspondences between the expressiveness of logics and GNNs. We also employ the logics, and the techniques used to analyze them, to obtain decision procedures
for verification problems over GNNs. We complement this with undecidability results for static analysis problems involving the logics, as well as for GNN verification problems
Descriptive Complexity for Distributed Computing with Circuits
We consider distributed algorithms in the realistic scenario where distributed message passing is operated by circuits. We show that within this setting, modal substitution calculus MSC precisely captures the expressive power of circuits. The result is established via constructing translations that are highly efficient in relation to size. We also observe that the coloring algorithm based on Cole-Vishkin can be specified by logarithmic size programs (and thus also logarithmic size circuits) in the bounded-degree scenario