A Path Algebra for Multi-Relational Graphs

The adjacency of vertex $i$ and vertex $j$ is defined by the edge $(i,j)$. A structure of this form is called a graph and is usually defined as $\ddot{G}=(\ddot{V},\ddot{E})$, where $i,j\in\ddot{V}$ are vertices and $(i,j)\in\ddot{E}$ is the edge adjoining those vertices.¹¹1The “high dot” notation denotes that $\ddot{G}\neq\dot{G}\neq G$, where $G$ is the main definition used throughout the article. When the only distinguishing characteristic between two edges is the vertices they join, the graph is called single-relational. The reason for this is that there is only a single type of relation in the graph—namely, the binary relation $\ddot{E}\subseteq(\ddot{V}\times\ddot{V})$. Single-relational graphs have been used widely to model various systems of homogenous elements related by a single type of relation and as such, have numerous algorithms associated with their analysis [1].

When the domain of discourse is variegated by a heterogeneous set of relations, then the multi-relational graph becomes the more applicable construct. A multi-relational graph can be defined as $\dot{G}=(\dot{V},\dot{\mathbb{E}})$, where $\dot{\mathbb{E}}$ is a family of edge sets and $\dot{\mathbb{E}}=\{\dot{E_{1}},\dot{E_{2}},\ldots,\dot{E_{m}}\subseteq(\dot{V}\times\dot{V})\}$. When $m>1$, then there are multiple relations between the vertices of $\dot{V}$. Multi-relational graphs not only specify which vertices are adjacent to one another, they also specify the way in which they are adjacent. With respect to the formalisms of this article and without loss of generality, a multi-relational graph can also be represented as $G=(V,E)$, where $E$ is ternary relation, $E\subseteq(V\times\Omega\times V)$, and $\Omega$ is a set of edge labels (i.e. relation types). Thus, in reference to the structure $\dot{G}=(\dot{V},\dot{\mathbb{E}})$, $|\dot{\mathbb{E}}|=|\Omega|$ and $\sum_{n=1}^{n\leq|\dot{\mathbb{E}}|}|\dot{E}_{n}|=|E|:\dot{E}_{n}\in\dot{\mathbb{E}}$. The ternary relation model is the multi-relational graph structure used throughput this article. The reason for the use of this particular $G$ definition will be explained in §II.

Given the growing use of multi-relational graphs in computing [2] and the lack of graph techniques for such structures (relative to single-relational graphs), an algebraic model for traversing multi-relational graphs is presented. This article can be interpreted as a convergence of the $n$-ary relational algebra of [3], the concatenative single-relational path algebra in [4], and the multi-relational tensor algebra presented in [5]. However, unlike [3], the presented algebra is tied specifically to path construction by means of graph traversals as in [5] and [4]. Next, unlike the algebra in [4], which is oriented primarily towards single-relational graphs, the presented algebra conveniently supports multiple relations as in [3] and [5]. Finally, unlike [5], the presented algebra is a concatenative, order-preserving variation of the relational algebra in [3] and, as such, more aligned with [4].

The operations presented are summarized in the itemization below and are provided here as a consolidated summary for ease of reference.

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  $\|a\|$: the path length of path $a$.

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  $\circ:E^{*}\times E^{*}\rightarrow E^{*}$ : the concatenation of two paths.²²2The unary Kleene star operation ^∗ forms the free monoid $E^{*}=\bigcup_{n=0}^{\infty}E^{i}$, where $E^{0}=\{\epsilon\}$ and $\epsilon$ is the empty/identity element.

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  $\sigma:E^{*}\times\mathbb{N}^{+}\rightarrow E$: the projection of the $n^{\text{th}}$ edge of a path.

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  $\gamma^{-}:E^{*}\rightarrow V$: the projection of the tail (first element) of a path.

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  $\gamma^{+}:E^{*}\rightarrow V$: the projection of the head (last element) of a path.

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  $\omega:E\rightarrow\Omega$: the projection of the label of an edge.

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  $\cup:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$: the union of two path sets.

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  $\bowtie_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$: the concatenative join of two path sets.

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  $\times_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$: the concatenative product of two path sets.

Definitions of these operations are provided in §II. The use of these operations to represent basic traversal idioms is presented in §III. In §IV, regular paths can be recognized and generated as demonstrated in §IV-A and §IV-B, respectively. Making use of the algebra to evaluate single-relational graph algorithms is presented in §IV-C. The algebra provides a set of core operations for constructing a multi-relational graph traversal engine that is founded on monoid, automata, and formal language theory.

Traversing a graph is the process of moving over the edges specified in $E$. During a traversal, paths are derived and properties of those paths can be extracted.

The binary operation $\circ:E^{*}\times E^{*}\rightarrow E^{*}$ is the concatenation of two paths into a new path such that if $(i,\alpha,j)$ and $(j,\beta,k)$ are two edges in $E$, then their concatenation is the path $(i,\alpha,j,j,\beta,k)$, where $i,j,k\in V$ and $\alpha,\beta\in\Omega$. Concatenation is associative (i.e. $(a\circ b)\circ c=a\circ(b\circ c)$), not commutative (i.e. it is generally true that $a\circ b\neq b\circ a$), and $\epsilon$ serves as an identity (i.e. $\epsilon\circ a=a=a\circ\epsilon$).

Operations exist to extract information out of a path. The operation $\sigma:E^{*}\times\mathbb{N}^{+}\rightarrow E$ is a projection that maps a path to the $n^{\text{th}}$ edge in that path. For example, if $a=(i,\alpha,j,j,\beta,k)$, then $\sigma(a,1)=(i,\alpha,j)$ and $\sigma(a,2)=(j,\beta,k)$. Next, for any path, $\gamma^{-}:E^{*}\rightarrow V$ projects the tail (first vertex) of the path such that $\gamma^{-}((i,\alpha,j))=i$. Likewise, $\gamma^{+}:E^{*}\rightarrow V$, where $\gamma^{+}((i,\alpha,j))=j$. Similarly, for edge labels, $\omega:E\rightarrow\Omega$, where $\omega((i,\alpha,j))=\alpha$.³³3All projection operations can be reduced to a single string indexing operation, but for the sake of clarity in the following discussion, they are presented as being atomic.

The binary operation $\cup:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$ is standard set union. The binary operation $\bowtie_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$ is the concatenative join of two sets of paths such that if $A,B\in\mathcal{P}(E^{*})$, then

	$\displaystyle A\bowtie_{\circ}B=$	$\displaystyle\;\{a\circ b\;|\;a\in A\;\wedge\;b\in B$
		$\displaystyle\;\;\wedge\;\left(a=\epsilon\;\vee\;b=\epsilon\;\vee\;\gamma^{+}(a)=\gamma^{-}(b)\right)\},$

where $\gamma^{+}(a)=\gamma^{-}(b)$ ensures that only joint (i.e. adjacent) paths are concatenated.⁴⁴4The defined concatenative join is analogous to the $\theta$-join in [3], where $\begin{array}[]{c}A\bowtie B\\ \gamma^{+}(a)=\gamma^{-}(b)\end{array}$. In this form, its known as an equijoin. A discussion relating concatenative join and the relational algebra is found in [6]. For example, if

$$A=\left\{(i,\alpha,j),(j,\beta,k,k,\alpha,j)\right\}$$

and

$$B=\left\{(j,\beta,j),(j,\beta,i,i,\alpha,k),(i,\beta,k)\right\},$$

then

	$\displaystyle A\bowtie_{\circ}B=$	$\displaystyle\;\{(i,\alpha,j,j,\beta,j),(i,\alpha,j,j,\beta,i,i,\alpha,k),$
		$\displaystyle\;\;(j,\beta,k,k,\alpha,j,j,\beta,j),$
		$\displaystyle\;\;(j,\beta,k,k,\alpha,j,j,\beta,i,i,\alpha,k)\},$

where $i,j,k\in V$, $\alpha,\beta\in\Omega$, and $(i,\alpha,j),(j,\beta,k),(k,\alpha,j),\\ (j,\beta,j),(j,\beta,i),(i,\alpha,k),(i,\beta,k)\in E$. Given that $\bowtie_{\circ}$ is based on $\circ$, $\bowtie_{\circ}$ is associative, but not commutative.

The binary operation $\bowtie_{\circ}$ constructs joint paths. It may be the case that traversing disjoint paths is desirable.⁵⁵5For example, priors-based algorithms require the concept of “teleportation” in order to make a disjoint jump in the graph. The Cartesian product supports the concatenation of potentially disjoint paths. As such, $\times_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$, where $A\times_{\circ}B=\{a\circ b\;|\;a\in A\;\wedge\;b\in B\}$.

Finally, to conclude this section, the reason why the $\dot{G}=(\dot{V},\dot{\mathbb{E}}=\{\dot{E_{1}},\dot{E_{2}},\ldots,\dot{E_{m}}\subseteq(\dot{V}\times\dot{V})\})$ definition of a multi-relational graph is not used is because when evaluating concatenative joins over binary relations, the edge label information is lost and thus, the path label can not be determined. In other words, if $e$ and $f$ are edges from two different binary relations, then $e\circ f$ would only provide a sequence of vertices and as such would not specify from which relations the join was constructed. This is a deficiency of the algebra in [4], where binary relations are used and $\circ:V^{*}\times V^{*}\rightarrow V^{*}$ as opposed to $\circ:E^{*}\times E^{*}\rightarrow E^{*}$, where $E=(V\times\Omega\times V)$. While the algebra in [4] is applicable to multi-relational graphs (as any two relations can be joined), it was specifically intended for single-relational graphs, where problems involving path labels are not considered. In contrast, the specification defined in this article preserves path labels.

From the explicit adjacencies (edges) defined in the edge set $E$, there exists implicit adjacencies (paths) defined by $e\circ f$, where $e,f\in E$ and $e\circ f\in E^{*}$. Given the previously defined operations, different types of common traversal idioms can be affected.

The basic traversals defined in §III can be mixed and matched to yield different types of joint paths in $E^{*}$. This section will introduce some typical applications of the presented multi-relational path algebra to problems that are specific to multi-relational graphs—focusing primarily on problems involving regular paths.⁶⁶6For the sake of simplicity, only regular paths are discussed. However, with more machinery (e.g. memory structures), more complex traversals can be expressed using the core operations presented in §II.

This article defined a path algebra for multi-relational graphs represented as $G=(V,E\subseteq(V\times\Omega\times V)$. The core traversal types (complete, source, destination, and labeled) allow for the expression of more expressive traversals through the restriction of the join set $E$. Applications to regular path recognizers (§IV-A), generators (§IV-B), and “semantically-rich” single-relational graph construction (§IV-C) were presented. Generally, the algebra has applicability to the construction of a multi-relational graph traversal engine.