Graphs are a very general mathematical abstraction for real world networks such as social-, sensor-, biological- or traffic-networks. These types of networks often generate large quantities of observational data, and using machine learning techniques to build predictive models for them is an area of substantial current interest. Graphs also arise as abstract models for knowledge, e.g., in the form of semantic models for a logic knowledge base, or directly as a knowledge graph. In most cases, an appropriate representation as a graph will require that one goes beyond the fundamental graph model, and allows that nodes are annotated with attributes, and that there are several distinct edge relation.

Evidently, in machine learning the main interest is in learning from graph data, whereas in knowledge representation and reasoning the primary focus is on deductive reasoning. However, both learning and reasoning play a role in all disciplines: making class label predictions from a learned model is a highly specialized (and limited) form of reasoning, and learning logic rules from examples has a long history in symbolic AI. It is the goal of this review to survey the large and diverse area of approaches for learning and reasoning with graphs in different areas of AI and adjacent fields of mathematics. Given the scope of the subject, our discussion will be mostly at a high, conceptual level. For more technical details, and more comprehensive literature reviews, we will point to relevant specialized surveys. The main objective of this review is to establish a coherent formal framework that facilitates a unified analysis of a wide variety of learning and reasoning frameworks. Even though this review aims to cover a broad range of methods and disciplines, there will be a certain focus on graph neural networks (GNNs) and statistical relational learning (SRL), whereas the fields of graph kernels and purely logic-based approaches receive a little less attention than they deserve.

The following examples illustrate the range of modeling, learning, and reasoning approaches that we aim to cover in this review. Each example describes a general task, a concrete instance of that task, and a particular approach for solving the task. The examples should not be construed in the way that the described solution approach is the only or even most suited one to deal with the given task. The intention is to illustrate the diversity of tasks and solution techniques.

Example 1.1. (Node classification with graph neural networks) One of the most common tasks in machine learning with graphs is node classification, i.e., predicting an unobserved node label. A standard instance of such a classification task is subject prediction in a bibliographic data graph: nodes are scientific papers that are connected by citation links, and the node labels consist of a subject area classification of the paper. In the inductive version of this task, one is given one or several training graphs containing labeled training nodes. The task is to learn a model that allows one to predict class labels of unlabeled nodes that are not already contained in the training graphs. For example, the training graph may consist of the current version of a bibliographic database, whereas the unlabeled nodes are new publications when they are added to the database. In the transductive version of the task, both the labeled training nodes and the unlabeled test nodes reside in the same graph, which is already fully known at the time of learning. This is the case when an incomplete subject area labeling is to be completed for a given bibliographic database. Graph neural networks are a state-of-the-art approach to solve such classification tasks (e.g., Niepert et al., 2016; Hamilton et al., 2017; Welling and Kipf, 2017; Veličković et al., 2018).

Example 1.2. (Link prediction via node embeddings) Here the task is to predict whether two nodes are connected by an edge. This prediction problem is usually considered in a transductive setting, where all the nodes and some of the edges are given, and edges between certain test pairs of nodes have to be predicted. Bibliographic data graphs are again a popular testbed for link prediction approaches (Kipf and Welling, 2016; Pan et al., 2021). Recommender systems also can be seen as handling a link prediction problem: the underlying graph here contains user and product nodes, and edges connect users with products that the user has bought (or provided some other type of positive feedback for). The link prediction problem then amounts to predicting positive user/product relationships that have not yet been observed. Numerous different link prediction approaches exist (Kumar et al., 2020 gives a comprehensive survey). A variety of different approaches is based on constructing for each node in the graph a d-dimensional real-valued embedding vector, and to score the likelihood of the existence of an edge between two nodes by considering the proximity (according to a suitable metric) of their embedding vectors. This general paradigm encompasses approaches such as matrix factorization (Koren and Bell, 2015) and random walk based approaches (Perozzi et al., 2014; Grover and Leskovec, 2016).

A good and concise monograph that covers the modern machine learning methods described in the preceding two examples is (Hamilton, 2020).

Example 1.3. (Graph Classification with inductive logic programming) One may also want to predict a class label associated with a whole graph. A classic example is predicting properties of molecules, where molecules are represented as graphs consisting of nodes representing the atoms, and links representing bonds between the atoms. The famous Mutagenesis dataset (Srinivasan et al., 1996), for example, consists of 188 molecules with a Boolean mutagenic class label. A predictor for this label may be given in the form of a logic program, such as the following:

(this program here is purely given for expository purposes, and does not resemble realistic classification programs for this task). A particular molecule specified by a list of ground facts, such as carbon(at_1), carbon(at_2), nitrogen(at_3), …, bond(at_1, at_3), would then be classified as mutagenic, if mutagenic can be proven by the program from the ground facts. This will be the case if and only if the molecule contains a cycle consisting of carbon atoms. Parts of the program (e.g., the definitions of carbon path and cycle) may be provided by experts as background knowledge, whereas other parts (e.g., the dependence of mutagenic on the existence of a carbon cycle) would be learned from labeled examples.

Example 1.4. (Graph similarity and classification with graph kernels) The problem of graph classification (especially in bio-molecular application domains) has also extensively been approached with kernel techniques. Graph kernels (see Kriege et al., 2020 for an excellent survey) are functions k that map pairs of graphs G, H to a value k(G, H)∈ℝ that is usually interpreted as a similarity measure for G and H, and which must be of the form k(G, H) = ϕ(G)·ϕ(H) for some finite or infinite dimensional real-valued feature vectors ϕ(G), ϕ(H). The graph kernel can then be used for graph classification by using it as input for a support vector machine classifier. Based on its interpretation as a similarity measure, graph kernels can also support other types of similarity-based analyses, e.g., clustering. Most graph kernels are defined by an explicit definition of the mapping from graphs G to feature vectors ϕ(G). Important examples are the Weisfeiler-Lehman kernel (WLK) (Shervashidze et al., 2011), the graphlet kernel (Shervashidze et al., 2009), and the random walk kernel (Gärtner et al., 2003). The components of the feature vectors in the first two contain statistics on the occurrence of local neighborhood structures in G, whereas the features of the random walk kernel represent statistics on node label sequences that are generated by random walks on the graph.

The previous examples were concerned with specific prediction tasks. In the following examples we move toward more general forms of reasoning.

Example 1.5. (Probabilistic inference with SRL) The task is to learn a probabilistic graph model that supports a rich class of queries. An example is a probabilistic model for the genotypes of people in a pedigree graph. The model should support a spectrum of queries. A basic type are conditional probability queries of the form: given (partial) genotype information for some individuals, what are the probabilities for the genotypes of the other individuals? This is still very similar to the node classification task of Example 1.1. A query that goes beyond what has been described in previous examples is a most probable explanation (MPE) query: again, given partial genotype information, what is the most probable joint configuration of the genotypes for all individuals? Statistical relational learning (SRL) approaches such as Relational Bayesian Networks (RBNs) (Jaeger, 1997), Markov logic networks (MLNs) (Richardson and Domingos, 2006), or ProbLog (De Raedt et al., 2007) provide modeling and inference frameworks for solving such tasks.

Example 1.6. (Logical reasoning with first-order logic) Going beyond the flexible, but still rather structured type of queries considered in Example 3.6, we can consider more general logical reasoning tasks. There are no standard example instances of this, so we illustrate this task and its solution by deduction in first-order logic by the following example: given the following knowledge about a social network:

Does this knowledge imply that there must be a user who has at least two followers, or there must be at least four different users:

Considering first all finite graphs, one finds that when (1) and (2) are true there must be a node with at least two incoming edges, i.e., the first part of the disjunction in (3) is true. When the graph is infinite, then this implication no longer holds, but then the second disjunct of (3) will be true (where the number 4 may be replaced with any natural number). Thus, we find that (1) and (2) logically entail (3). This inference can be performed by automated theorem provers. For example the SPASS prover (Weidenbach et al., 2009) can answer our query.

Example 1.7. (Limit behavior of random graphs) Many probabilistic models have been developed for the temporal evolution of growing networks. The simplest possible model is to assume that every new node is connected to the already existing nodes with a fixed probability p, and these connections are formed independently of each other. One may then ask how the probabilities of certain graph properties develop, as the size of the graph grows to infinity. Classic results of the Erdős-Rényi random graph theory establish, for example, that the limiting probability for the evolving graph to be connected is 1 (Erdős and Rényi, 1960) [the actual results of the Erdős-Rényi random graph theory are actually much more sophisticated, as they pertain to models where the edge probability is a function p(n) of the number n of vertices]. Similarly, it is known that the probability of every property that can be expressed in first-order logic converges to either 0 or 1 (Fagin, 1976). Reasoning about such limiting probabilities goes beyond reasoning about all graphs as in Example 1.6, since we now consider probability distributions over infinite sequences of graphs. Some types of queries about the limit behavior of random graphs are formally decidable. This is the case, for example, for the limit probability of a first-order sentence (Grandjean, 1983). However, the computational complexity of these reasoning tasks puts them outside the range of practically feasible implementations.

Figure 1 arranges the landscape of reasoning scenarios we have considered in the preceding examples in two dimensions: one dimension characterizes the domain of graphs that we reason about: at the bottom of this dimension is the transductive setting of Examples 1.1 and 1.2 in which reasoning is limited to a single given graph. At the next level, labeled “one graph at a time,” any single reasoning task is about a specific graph, but this graph under consideration can vary without the need to change or retrain the model. This corresponds to the inductive setting in Example 1.1, as well as the tasks described in Examples 1.4 and 3.6. At the third level, the reasoning concerns several or all graphs at once (Example 1.6). Finally, we may go beyond reasoning about properties of individual graphs, and consider global properties of the space of all graphs. This is exemplified by Example 1.7, where the reasoning pertains to the relationship between different probability distributions on graphs. These informal distinctions about the domain of reasoning will be partly formalized by technical definitions in Section 5.

The second dimension in Figure 1 is correlated with the first, but describes the type of reasoning that is performed. The most restricted type here is to perform a narrowly defined prediction task such as node classification or link prediction. Based on the distinction between “model checking” and “theorem proving” promoted by Halpern and Vardi (1991) we label the next level as “model checking”: this refers to evaluating properties of a single given structure, where the class of properties that can be queried is defined by a rich and flexible query language. “Deduction” then refers to (logical) inference about a class of structures, and in “model theory” such a class of structure becomes the object of reasoning itself. Within this two-dimensional schema, Figure 1 indicates what areas of reasoning are covered by different types of frameworks. The reasoning landscape here delineated extends beyond the boundaries of what can currently be tackled with automated, algorithmic reasoning methods in AI, and extends to human mathematical inference. In the remainder of this review we will focus on algorithmic reasoning. However, it is an always open challenge to extend the scope of what can be accomplished by algorithmic means.

Figure 1 focuses on reasoning rather than learning. This is for two reasons: first, reasoning covers a somewhat wider ground that includes scenarios (e.g., logical deduction) not yet supported by learning methods. Second, a taxonomy of learning scenarios in terms of data requirements and learning objectives will naturally follow from the reasoning taxonomy (cf. Section 6). The link between reasoning and learning is a model that can be learned from data and that supports the reasoning tasks under consideration. In this review we use a general probabilistic concept of a model formally defined in in Section 3, that allows us to express a wide range of reasoning tasks as different forms of querying a model (Section 5). Similarly, a wide range of learning scenarios can be understood in a coherent manner as constructing models from data under a maximum likelihood objective (Section 6). This review is partly based on Jaeger (2022), which already contains an in-depth exploration of GNN and SRL methods for learning and reasoning with graphs. The current review takes a much broader high-level perspective. It contains fewer technical details on GNNs and SRL, and instead develops a general conceptual framework for describing a much wider range of learning and reasoning frameworks.

In this section we establish the basic definitions and terminology we use for graphs. Table 1 collects the main notations. Our graphs will actually be multi-relational, attributed hyper-graphs, but we just say shortly “graphs.”

Different attributes and (hyper-) edge relations are collected in a signature: a set R={r1,…,rm} of relation symbols. Each relation symbol has an arity(r_i)∈{0, 1, …}. Relations of arity 0 are global graph attributes, such as toxic for a graph representing a chemical molecule. Relations of arity 1 are node attributes. Relations of arities ≥3, can in principle be reduced to a set of binary relations by materializing tuples as nodes. For example, the 3-ary relation shortest_path(a, b, c) representing that node b lies on the shortest path from a to c can be encoded by creating a new shortest path node sp, and three binary relations start, on, end, so that start(sp, a), on(sp, b), end(sp, c) are true. However, this leads to very un-natural encodings, and therefore we allow relations of arities ≥3.

A R-graph is a structure (V, R), where V is a finite or countably infinite set of nodes (also referred to as a domain), and R = (R₁, …, R_m) are the interpretations of the relation symbols: Ri:Varity(ri)→{0,1}. In the case of arity(r_i) = 0 this is just a constant 0 or 1.

We write IntV(r) for the set of possible interpretations of the relation symbol r∈R over the domain V, and IntV(R)=×r∈RIntV(r) for the set of all interpretations of the whole signature R. In most cases it is sufficient to consider the domains V = [n]: = {1, …, n} for n∈ℕ. We use i, j, … as generic symbols for nodes, and bold face i, j, … for tuples of nodes. We here use somewhat logic-inspired terminology and notation. Taking the logic conventions even further, we can equivalently define an interpretation of r as an assignment of true/false values to ground atoms r(i) [i∈|V|^arity(r)]. Going in the opposite direction toward the terminological conventions of neural networks, such an interpretation also can be seen as a |V| × ⋯ × |V|-dimensional (arity(r) many factors) tensor.

In the following, we usually take the signature R as given by the context, and do not refer to it explicitly, thus saying “graph” rather than R-graph, and also abbreviating IntV(R) by IntV Also, when the intended meaning is clear from the context, we use the simple term “relation” to either refer to a relation symbol r, or an interpretation R in a specific graph.

Note that according to our definitions all relations are directed. Undirected edges/relations are obtained as the special case where the interpretation R_i for a tuple i only depends on the elements of i, not their order. Furthermore, only Boolean node attributes are permitted. Multi-valued attributes can be represented by multiple Boolean ones in a one-hot encoding.

For a given domain V and signature R we denote with G(V,R) the set of all R-graphs with domain V, with G(<∞,R) the set of all finite R-graphs, and with G(R) the set of all graphs, also allowing (countably) infinite domains V. As a basis for probabilistic graph models, we denote with ΔG(V,R) the set of probability distributions over G(V,R) (in the case of infinite V this must be based on suitable measure-theoretic definitions that we do not elaborate here).

We introduce a general concept of a model, so that all reasoning tasks become different forms of querying a model. Ours will be a high-level, purely semantic notion of a model that imposes no restrictions on how models are represented, implemented or constructed. Our model definition is probabilistic in nature. As we shall see, this still allows us to capture purely qualitative, logic-based frameworks, although at the cost of casting them in a slightly contrived way into the probabilistic mold via extreme 0, 1-valued probabilities. Roughly speaking, a model in our sense will be a mapping from partly specified graphs to probability distributions over their possible completions.

A partial graph is a structure (V,R˜) where V is a finite or countably infinite domain, and R~=(R~1,…,R~m) are partial interpretations of the relation symbols: R~i:Varity(ri)→{0,1,?}. We write R~i≼R~i′ (R˜i′R˜i extends R~i) nif

If R~i≼R~i′ for i = 1, …, m we write R~i≼R~i′.

G~(V,R) denotes the set of all partial R-graphs with domain V, and G~(<∞,R) the set of all finite partial R-graphs. Finally, Int(V,R~)(R) denotes the set of complete interpretations R∈IntVℛ with R˜i≼R.

Definition 3.1. A R-model M=(I,μ) consists of

defined for all (V,R~)∈ℐ, such that

In the case where the input just consists of a domain V (i.e., ℛ~ is completely unspecified), we simply write P_V for P(V,R˜).

Definition 3.1 has several important special cases: when a model is for classification of a specific label relation l, then I=G(<∞,ℛ∖ {l}).. The model then defines for every input graph with complete specifications of relations r∈R\{l} a distribution over interpretations L for l. We refer to such models as discriminative. A model that, in contrast, takes as input only a finite domain, i.e., I=G(<∞,∅), and thereby maps a domain V to a probability distribution over G(V,R) is called fully generative. In between these two extremes are models where R is partitioned into a set of input relations R_in and output relations R_out, and I=G(<∞,Rin). This is the typical case for SRL models. We refer to such models as conditionally generative (discriminative then is just a borderline case of conditionally generative). In all these special cases the input graph contains complete specifications of a selected set of relations. This covers many, but not all models used in practice: e.g., models for link prediction (cf. Example 3.3 below) operate on inputs with partial specifications of the edge relation. Transductive models are characterized as the special case |I| = 1.

The abstract definition 3.1 accommodates a multitude of concrete approaches for learning and reasoning about graphs. We call a modeling framework any approach that provides computational tools for representing, learning, and reasoning with models. Figure 2 gives an overview of several classes of modeling frameworks, and representatives for each class. The selection of representatives does not attempt a complete coverage of even the most important examples. In the following we continue the examples of Section 1 to illustrate how these frameworks indeed fit into our general Definition 3.1.

Example 3.2. (Node classification; GNNs) In the standard node classification scenario, the signature R consists of one binary edge relation e, observed node attributes a = a₁, …, a_l, and a node label l. A partial input graph has the form (V,R~), where in the partial interpretation R~=(E,A,L~) the edge relation and node attributes are fully observed, and the node label is unobserved for some (possibly all) nodes. Given a Graph neural networks then define label probabilities

for the unlabeled nodes. Since these predictions are independent for all nodes, they define a distribution over completions L of L~ via

In Figure 2, the class of GNN frameworks is divided into the sub-classes of message-passing and recurrent GNNs. MP-GNNs compute node feature vectors in a fixed number of iterations, whereas R-GNNs perform feature updates until a fixed point is reached. We return to this distinction in Section 4.2.

Example 3.3. (Link prediction; shallow embeddings) In its most basic form, a transductive link prediction problem is given by a single input graph I={(V,Ẽ)} with an partially observed edge relation. Usually, all edges that are not observed as existent are candidates for being predicted, so that Ẽ(i, j)∈{1, ?} for all i, j∈V (in practice, however, represented in the form Ẽ(i, j)∈{1, 0}, with the understanding that Ẽ(i, j) = 0 still allows the prediction E(i, j) = 1). Shallow embeddings construct for each node i∈V an embedding vector em(i), and define a scoring function score[em(i), em(j)] for the likelihood of E(i, j) = 1. As the objective usually is to just rank candidates edges, this score need not necessarily be probabilistic [concrete examples are the dot product or cosine similarity of em(i), em(j)]. However, turning the scores into probabilities by applying a sigmoid function does not affect the ranking and hence we may assume that the link prediction model defines edge probabilities P[e(i, j)|(V, Ẽ)] (i, j:Ẽ(i, j) = ?). Via the same implicit independence assumptions as in the previous example, then a distribution in the sense of Definition 3.1 is defined as

Example 3.4. (ILP) A logic program as shown in Example 1.3 can be interpreted as a model in our sense. For any input (V,R~), where R~ is specified by a list of true ground facts, the program defines the unique interpretation R^* in which a ground fact r(i) is true, iff it is provable from the program and R~. This can be expressed in the form of a degenerate probability distribution with

The set I of possible inputs for this model consists of all (V,R~) where R~ is defined by positive ground facts only, i.e., R~(i)∈{1,?} for all R~,i.

Example 3.5. (Graph Classification; graph kernels) A graph kernel k(G, H) alone is not a model in our sense. However, a support-vector machine classifier built on the kernel (denoted k-SVM) is such a model: a k-SVM maps graphs to label values in {0, 1}, which similar to Example 3.4 can be seen as a degenerate probabilistic model. Alternatively, since the k-SVM actually produces numeric scores for the labels (as the distance to the decision boundary), one can transform the score by a sigmoid function to non-degenerate probability values.

Example 3.6. (SRL) To solve the probabilistic inference tasks of Example an SRL model will be defined for input structures (V, E) with a fully observed edge relation e defining the pedigree structure. The model will then define a distribution P_{(V, E)}(A) over interpretations A of node attributes a. In a typical solution for this type of problem this will be done by defining a marginal distribution over genotypes for the individuals in V whose parents are not included in V, and a conditional distribution over child genotypes given parent genotypes. In order to solve the reasoning tasks described in Example, the framework must support queries about conditional probabilities of the form P(V,E)(a(i)|A~) (genotype probabilities for individual i given partial information A~ about genotypes in the pedigree) and arg maxAP(V,E)(A|A˜) (MPE inference).

The model outlined here falls into the sub-class of directed SRL frameworks, where the distributions defined by a model can be represented in the form of directed probabilistic graphical models. Other important sub-classes of SRL distinguished in Figure 2 are frameworks based on undirected probabilistic graphical models, and probabilistic generalizations of inductive logic programming. Because of their close relationship with the most popular undirected SRL framework, Markov logic networks, Figure 2 also lists exponential random graph models under the U-SRL class, even though in terms of historical background and applications, exponential random graphs rather fall into the random graph model category.

Example 3.7. (First-order logic) A logical knowledge base KB as exemplified by (1) and (2) in Example 1.6 can be seen as a discriminative model in our sense: for any graph G = (V, R)∈G(R) the semantics of the logic defines whether KB is true in G.¹ To formalize this as a discriminative model we augment the signature R with a binary graph label l_KB, set I = G(R), and P_G(l_KB = 1) = 1 if KB is true in G, and P_G(l_KB = 0) = 1, otherwise. Logical inference as described in Example 1.6 then amounts to determining whether for all graphs G in which (3) is false one has P_G(l_KB = 0) = 1. This is a probabilistic rendition of the task of automated theorem proving.

Example 3.8. (Random graph models) Classical random graph models are generative models in our sense for the signature R = {e} containing a single edge relation.

The preceding examples show that the quite simple Definition 3.1 is sufficient as a unifying semantic foundation for a large variety of frameworks for reasoning about graphs. Of course, most of these frameworks also (or primarily) are designed for learning models from graph data, but our initial focus here is on modeling and reasoning, before we turn to learning in Section 6.

Our Definition 3.1 of a model is completely abstract and does not entail any assumptions or prescriptions about the syntactic form and computational tools for model specification and reasoning. In the following, we consider several key modeling elements that are used across multiple concrete frameworks.

Given a model in the sense of Definition 3.1, we consider several classes of reasoning tasks. Narrowing down the range of possible reasoning tasks considered in Section 1, we are now only concerned with algorithmic reasoning.

So far our discussion has largely focused on modeling and reasoning. Our formal definitions in Sections 3 and 5 draw a close link between types of models and the reasoning tasks they support (discriminative, generative, transductive, inductive, …). Turning now to the question of how a model is learned from data, a similar close linkage arises between model types and learning scenarios. Based on the unifying probabilistic view of models according to Definition 3.1, different learning scenarios are essentially just distinguished by the structure of the training data, but unified by a common maximum likelihood learning principle.

The class of input structures I of a model I = (I, μ) characterizes its basic functionality and will be assumed to be fixed a-priori by the learning/reasoning task at hand. What is to be learned is the mapping μ. For this one needs training data of the following form.

Definition 6.1. Let ℐ⊆G˜(<∞,ℛ) be given. A dataset for learning the mapping μ consists of a set of examples

where (Vn,R˜n)∈ℐ, and R˜n+∈G˜(Vn,ℛ) complements R~n in the sense that R~n,i+(i)≠?⇒R~n,i(i)=? (i = 1, …, m).

The unifying learning principle is to find the mapping μ that maximizes the log-likelihood

In practice, the pure likelihood (20) will often be modified by regularization terms or prior probabilities, which, to simplify matters, we do not consider here. Furthermore, some learning objectives, such as the max-margin objective in learning a kernel-SVM, are not based on the log-likelihood as the central element at all. However, as the following examples show, Equation (20) still covers a fairly wide range of learning approaches.

Example 6.2. (Node classification with GNN) Consider R consisting of a single binary edge relation, node attributes a₁, …, a_k, and a node label l. For training a discriminative model with R_in = {edge, a₁, …, a_k}, and R_out = {l}, the training examples consist of complete input graphs (V_n, R_n)∈G(< ∞, R_in), and R~n+ is a partial interpretation L~n of l. In the transductive setting, N = 1, and (V₁, R₁) is equal to the single input graph for which the model is defined. Under the factorization (7) the log likelihood (20) then becomes

The usual training objective for GNNs of minimizing the log-loss is equivalent to maximizing this log-likelihood.

Example 6.3. (Generative models from incomplete data) The discriminative learning scenario of Example 6.2 requires training data in which input relations R_in are completely observed. When also some of the attributes a_i, and maybe the edge relation are only incompletely observed in the training data, then no valid input structure for a discriminative model is given. However, no matter which relations are fully or partially observed, a partial interpretation R~n can always be written as a valid training example

for a fully generative model. Without any assumptions on the factorization of the distributions P_n, the log-likelihood now takes the general form

While always well-defined, this likelihood may be intractable for optimization. An explicit summation over all R∈Int(Vn,Rn)(R) is almost always infeasible. When optimization of the complete data likelihood μ(V_n, ∅)(R) for R∈IntV(R) is tractable, then the expectation-maximization strategy can be applied, where one iteratively imputes expected completions R_n for the incomplete observations R_n under a current model μ, and then optimizes μ under the likelihood induced by this complete data. For U-SRL models, even the complete data likelihood usually is intractable due to its dependence on the partition function Z = Z(μ) in (8). In this case the true likelihood may be approximated by a pseudo-likelihood (Besag, 1975; Richardson and Domingos, 2006).

Example 6.4. (Learning logic theories) Consider now the case of a logical framework where a model is a knowledge base KB that we consider as a discriminative model in the sense of Example 3.7. Learning logical theories or concepts is usually framed in terms of learning from positive and negative examples, where the example data can consist of full interpretations, logical statements, or even proofs (De Raedt, 1997). The learning from interpretations setting fits most closely our general data and learning setup: examples then are fully observed graphs G_n = (V_n, R_n) together with a label R~n+=Ln∈{0,1}, and optimizing (20) amounts to finding a knowledge base KB such that KB is true in all G_n with L_n = 1, and false for G_n with L_n = 0, i.e., the standard objective in logic concept learning.

In view of the complementary benefits of symbolic and numeric models, it is natural to aim for combinations of both elements that optimally exploit the strengths of each. Emphasizing neural network frameworks as the prime representatives for numeric approaches, these efforts are currently mostly pursued under the name of neuro-symbolic integration (Sarker et al., 2021). An important example in the context of this review is the integration of the I-SRL ProbLog framework with deep neural networks (not specifically GNNs). The underlying philosophy for the proposed DeepProbLog framework (Manhaeve et al., 2018) is that neural frameworks excel at solving low-level “perceptual” reasoning tasks, whereas symbolic frameworks support higher-level reasoning. The integration therefore consists of two layers, where the lower (neural) layer provides inputs to the higher (symbolic) layer.

The unifying perspective on model semantics and model structure we have developed in Sections 3 and 4 gives rise to more homogeneous integration perspectives: a conditionally generative model that is constructed via the factorization (6), (7) consists of individual discriminative models (7) for each relation R_k as building blocks. In principle, different such building blocks can be constructed in different frameworks, and combined into a single model via (6). Moreover, this approach will be consistent in the sense that if all constructions of the component discriminative models use (20) as the objective, then the combination of these objectives is equivalent to the maximizing the overall log-likelihood μ(V_n, R_in)(R_out) directly for the resulting conditional generative model. Here we deliberately speak of “constructing” rather than “learning” component models in order to emphasize the possibility that some model components may be built by manual design, whereas others can be learned from data.

Piecing together a combined model from heterogeneous model components only is useful when the resulting model then can be used to perform inference tasks. This will limit the reasoning capabilities to tasks that can be broken down into a combination of tasks supported by the component models. A possible alternative is to compile all individual components into a representation in a common framework with high expressivity and flexible reasoning capabilities. As shown in Jaeger (2022), quite general GNN architectures for representing discriminative models can be compiled into an RBN representation, and integrated as components into a bigger generative model. While theoretically sound, it is still an open question whether this approach is practically feasible for GNN models of the size needed to obtain high accuracy on their specialized discriminative tasks.

We have given a broad overview over modeling, reasoning, and learning with graphs. The main objective of this review was to view the area from a broader perspective than more specialized existing surveys, while at the same time developing a coherent conceptual framework that emphasizes the commonalities between very diverse approaches to dealing with graph data. Our central definitions of models, reasoning types and learning tasks cover a wide range of different frameworks and approaches. While the uniformity we thereby obtain sometimes is a bit contrived (notably by casting logical concepts in probabilistic terms), it still may be useful to elucidate the common ground among quite disparate traditions and approaches, to provide a basis for further theoretical (comparative) analyses, and to facilitate the combination and integration of different frameworks.

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