On Edge-MetaGraph

1. Preliminaries

This section establishes notation and summarizes the set-theoretic and combinatorial notions needed in the sequel. Throughout, all sets that serve as vertex domains are assumed to be finite, and we write ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$.

1.1. MetaGraph (Graph of Graphs)

Graph theory investigates mathematical structures consisting of vertices and edges to model relationships and connectivity [1,2]. A MetaGraph is a graph whose vertices are themselves graphs, with edges representing specified relations between those graphs (cf.[3,4,5,6]).

Definition 1.1.

(Metagraph (graph of graphs)).  [7] Fix a nonempty universe $\mathfrak{G}$ of finite graphs (undirected, loopless by default) and a nonempty family of binary relations

$$\mathcal{R}\subseteq \mathcal{P}\left(\mathfrak{G}\times \mathfrak{G}\right).$$

A metagraph over $(\mathfrak{G},\mathcal{R})$ is a directed, labelled multigraph

$$M=(V,E,s,t,\lambda )$$

with

$$V\subseteq \mathfrak{G},s,t:E\to V,\lambda :E\to \mathcal{R},$$

satisfying the incidence constraint

$$\forall e\in E:\left(s\left(e\right),t\left(e\right)\right)\in \lambda \left(e\right).$$

Elements of V are meta-vertices (each is a graph $G\in \mathfrak{G}$). For $e\in E$ with $\lambda \left(e\right)=R$, we write $s\left(e\right)\stackrel{R}{\to }t\left(e\right)$ and call e a meta-edge. If $\mathcal{R}=\left\{R\right\}$ is a singleton, labels may be omitted. If every $R\in \mathcal{R}$ is symmetric, M can be viewed as an undirected labelled multigraph.

Remark 1.1.

We fix a common base universe $\mathfrak{G}$ of finite, loopless graphs that encode biochemical modules. Each $G\in \mathfrak{G}$ comes with a vertex-labeling ${\ell }_V:V\left(G\right)\to {\Sigma }_V$ (e.g. metabolite/protein/gene names) and, when needed, an edge-labeling ${\ell }_E:E\left(G\right)\to {\Sigma }_E$ (e.g. reaction/interaction types). We use the following chemically meaningful binary relations on $\mathfrak{G}$:

$$\begin{array}{cc}\hfill R_{\mathrm{shareM}}(G,H)& :\iff \left\{\mathrm{metabolite}\mathrm{names}\mathrm{in}G\right\}\cap \left\{\mathrm{metabolite}\mathrm{names}\mathrm{in}H\right\}\ne \varnothing ,\hfill \\ \hfill R_{\mathrm{shareP}}(G,H)& :\iff \left\{\mathrm{protein}\mathrm{names}\mathrm{in}G\right\}\cap \left\{\mathrm{protein}\mathrm{names}\mathrm{in}H\right\}\ne \varnothing ,\hfill \\ \hfill R_{\mathrm{GRN}\Delta }(G,H)& :\iff E\left(G\right)▵E\left(H\right)\ne \varnothing \left(\mathrm{directed}\mathrm{GRNs};\mathrm{activation}/\mathrm{inhibition}\mathrm{labels}\mathrm{allowed}\right).\hfill \end{array}$$

Set $\mathcal{R}:=\{R_{\mathrm{shareM}},R_{\mathrm{shareP}},R_{\mathrm{GRN}\Delta }\}$.

1.2. Iterated MetaGraph (Graph of Graphs of ... of Graphs)

An Iterated MetaGraph is a graph whose vertices are metagraphs, recursively extending graph-of-graphs structure to multiple hierarchical levels [7,8,9]. Related concepts, such as MetaStructures [10,11], are also known.

Definition 1.2

(Unit metagraph embedding). [7] For $X\in \mathfrak{G}$ define the unit metagraph

$$\mathsf{U}\left(X\right):=\left(\left\{X\right\},\varnothing ,\_,\_,\_\right).$$

This gives an injective map $\mathsf{U}:\mathfrak{G}\hookrightarrow \mathrm{Obj}\left(\mathrm{Meta}(\mathfrak{G},\mathcal{R})\right)$.

Definition 1.3

(Relation lifting). Given $\mathcal{R}$ on $\mathfrak{G}$, define its lift${\mathcal{R}}^{\uparrow }$ on finite metagraphs over $(\mathfrak{G},\mathcal{R})$ by

$$\forall R\in \mathcal{R},(M_1,M_2)\in R^{\uparrow }\iff \exists x\in V\left(M_1\right),y\in V\left(M_2\right):(x,y)\in R.$$

Set ${\mathcal{R}}^{\uparrow }:=\{R^{\uparrow }:R\in \mathcal{R}\}$.

Definition 1.4

(Iterated object and relation universes). Define recursively for $t\in N_0$:

$${\mathfrak{G}}^{\left(0\right)}:=\mathfrak{G},{\mathcal{R}}^{\left(0\right)}:=\mathcal{R},$$

$${\mathfrak{G}}^{(t+1)}:=\left\{\mathit{finite}\mathit{metagraphs}\mathit{over}\left({\mathfrak{G}}^{\left(t\right)},{\mathcal{R}}^{\left(t\right)}\right)\right\},{\mathcal{R}}^{(t+1)}:={\left({\mathcal{R}}^{\left(t\right)}\right)}^{\uparrow }.$$

Definition 1.5

(Iterated MetaGraph of depth t). For $t\in N_0$, an iterated metagraph of depth t is a metagraph

$$M^{\left(t\right)}=(V^{\left(t\right)},E^{\left(t\right)},s^{\left(t\right)},t^{\left(t\right)},{\lambda }^{\left(t\right)})$$

over $\left({\mathfrak{G}}^{\left(t\right)},{\mathcal{R}}^{\left(t\right)}\right)$, i.e., $V^{\left(t\right)}\subseteq {\mathfrak{G}}^{\left(t\right)}$, ${\lambda }^{\left(t\right)}:E^{\left(t\right)}\to {\mathcal{R}}^{\left(t\right)}$ and

$$\forall e\in E^{\left(t\right)}:\left(s^{\left(t\right)}\left(e\right),t^{\left(t\right)}\left(e\right)\right)\in {\lambda }^{\left(t\right)}\left(e\right).$$

Remark 1.2

We use the relation lifting of the preliminaries: for $R\in \mathcal{R}$ and metagraphs ${\mathsf{M}}_1,{\mathsf{M}}_2$ over $(\mathfrak{G},\mathcal{R})$,

$$({\mathsf{M}}_1,{\mathsf{M}}_2)\in R^{\uparrow }\iff \exists X\in V\left({\mathsf{M}}_1\right),\exists Y\in V\left({\mathsf{M}}_2\right)\mathit{such}\mathit{that}(X,Y)\in R.$$

Example 1.1

(A concrete iterated metagraph (depth $t=1$) with a biomedical interpretation). We give an explicit example of an iterated metagraph of depth $t=1$ (i.e., a graph whose vertices are metagraphs of base graphs).

Base level (${\mathfrak{G}}^{\left(0\right)}=\mathfrak{G}$).Let $\mathfrak{G}$ contain three pathway graphs

$$G_1=\mathit{Glycolysis},G_2=\mathit{Pyruvate\; Oxidation},G_3=\mathit{TCA\; Cycle}.$$

Let ${\mathcal{R}}^{\left(0\right)}=\left\{R_{\mathrm{shareM}}\right\}$, where

$$(G,H)\in R_{\mathit{shareM}}\iff \mathit{the}\mathit{pathways}G\mathit{and}H\mathit{share}\mathrm{at}\mathit{least}\mathit{one}\mathit{metabolite}.$$

Assume $(G_1,G_2)\in R_{\mathrm{shareM}}$ (shared metabolite: pyruvate) and $(G_2,G_3)\in R_{\mathrm{shareM}}$ (shared metabolite: acetyl-CoA).

Metagraphs of graphs (objects in ${\mathfrak{G}}^{\left(1\right)}$).Define two metagraphs over $({\mathfrak{G}}^{\left(0\right)},{\mathcal{R}}^{\left(0\right)})$:

$$M_A^{\left(0\right)}:V\left(M_A^{\left(0\right)}\right)=\{G_1,G_2\},\mathrm{one}\mathrm{meta}-\mathrm{edge}{G}_1\stackrel{R_{\mathrm{shareM}}}{\to }{G}_2,$$

$$M_B^{\left(0\right)}:V\left(M_B^{\left(0\right)}\right)=\{G_2,G_3\},\mathrm{one}\mathrm{meta}-\mathrm{edge}{G}_2\stackrel{R_{\mathrm{shareM}}}{\to }{G}_3.$$

By construction, $M_A^{\left(0\right)},M_B^{\left(0\right)}\in {\mathfrak{G}}^{\left(1\right)}$.

Lifted relation and depth-1 iterated metagraph.Since $(G_2,G_3)\in R_{\mathrm{shareM}}$ with $G_2\in V\left(M_A^{\left(0\right)}\right)$ and $G_3\in V\left(M_B^{\left(0\right)}\right)$, the lifted relation satisfies

$$\left(M_A^{\left(0\right)},M_B^{\left(0\right)}\right)\in R_{\mathrm{shareM}}^{\uparrow }\in {\mathcal{R}}^{\left(1\right)}.$$

Hence the depth-1 iterated metagraph

$$M^{\left(1\right)}=(V^{\left(1\right)},E^{\left(1\right)},s^{\left(1\right)},t^{\left(1\right)},{\lambda }^{\left(1\right)})$$

can be defined by

$$V^{\left(1\right)}=\{M_A^{\left(0\right)},M_B^{\left(0\right)}\},E^{\left(1\right)}=\left\{e\right\},s^{\left(1\right)}\left(e\right)=M_A^{\left(0\right)},t^{\left(1\right)}\left(e\right)=M_B^{\left(0\right)},{\lambda }^{\left(1\right)}\left(e\right)=R_{\mathrm{shareM}}^{\uparrow }.$$

Figure 1.1 depicts this construction.

2. Results

The results of this paper are presented in this section.

2.1. Edge-MetaGraph (Graph-Labeled Edges)

Edge-MetaGraph is a graph whose edges carry two-ported internal graphs, enabling edge-substitution expansion by gluing ports to endpoints.

Definition 2.1

(Two-ported graphs). Fix a nonempty universe $\mathfrak{G}$ of finite (undirected, loopless by default) graphs. Atwo-ported graphis a triple

$$H^{\partial }=(H,p^{-},p^{+}),$$

where $H\in \mathfrak{G}$ and $p^{-},p^{+}\in V\left(H\right)$ are distinguished vertices called thetail-portandhead-port, respectively. We write $Tail\left(H^{\partial }\right):=p^{-}$ and $Head\left(H^{\partial }\right):=p^{+}$, and denote by

$${\mathfrak{G}}^{\partial 2}:=\{(H,p^{-},p^{+}):H\in \mathfrak{G},p^{-},p^{+}\in V\left(H\right)\}$$

the class of all two-ported graphs over $\mathfrak{G}$.

Definition 2.2

(Edge-MetaGraph). Let ${\mathfrak{G}}^{\partial 2}$ be as in Definition 2.1. AnEdge-MetaGraph over $\mathfrak{G}$is a directed multigraph equipped with a two-ported-graph label on each edge, i.e., a tuple

$$\mathbb{E}=(V,E,s,t,\ell ),$$

where $(V,E,s,t)$ is a directed multigraph (loops and parallel edges allowed) and

$$\ell :E\to {\mathfrak{G}}^{\partial 2}$$

is a labeling map. For $e\in E$, writing $\ell \left(e\right)=(H_e,p_e^{-},p_e^{+})$, the intended meaning is: the edge e from $s\left(e\right)$ to $t\left(e\right)$ carries the internal graph $H_e$, whose ports $p_e^{-},p_e^{+}$ will be attached to the endpoints $s\left(e\right),t\left(e\right)$, respectively.

Definition 2.3

(Expansion / edge-substitution). Let $\mathbb{E}=(V,E,s,t,\ell )$ be an Edge-MetaGraph and write $\ell \left(e\right)=(H_e,p_e^{-},p_e^{+})$. Define a graph $Expand\left(\mathbb{E}\right)$ (the expansion of $\mathbb{E}$) by gluing each labeled graph $H_e$ along its ports to the endpoints of e as follows.

Take the disjoint union of vertex sets

$$W:=V\bigsqcup \underset{e\in E}{⨆}V\left(H_e\right),$$

and let ∼ be the smallest equivalence relation on W such that for every $e\in E$,

$$p_e^{-}\sim s\left(e\right),p_e^{+}\sim t\left(e\right).$$

Set $V(Expand(\mathbb{E}\left)\right):=W/\sim$. For edges, take the disjoint union

$$F:=\underset{e\in E}{⨆}E\left(H_e\right),$$

and for $f=\{x,y\}\in E\left(H_e\right)$ define its endpoints in $V(Expand(\mathbb{E}\left)\right)$ to be

$$\left({\left[x\right]}_{\sim },{\left[y\right]}_{\sim }\right).$$

Then $Expand\left(\mathbb{E}\right)$ is the (multi)graph with vertex set $W/\sim$ and edge multiset induced from F.

(If $\mathfrak{G}$ is a universe of directed graphs, replace $\{x,y\}$ by ordered pairs $(x,y)$ throughout.)

Example 2.1

(Real-world example: software deployment pipeline with edge-level internal dependencies). Consider a company operating a microservice platform. At a coarse operational level, we describe the deployment pipeline as a directed multigraph

$$(V,E,s,t),$$

whose vertices are macro-stages:

$$V=\{\mathit{Code},\mathit{Build},\mathit{Test},\mathit{Deploy},\mathit{Monitor}\}.$$

A directed edge $e:\mathsf{Build}\to \mathsf{Test}$ represents the handoff “send build artifacts to the test stage”. However, this handoff is not atomic: it involves multiple tools and checks (artifact registry, signature verification, test orchestration, reporting).

To model such hidden internal structure, define an Edge–MetaGraph

$$\mathbb{E}=(V,E,s,t,\ell ),\ell :E\to {\mathfrak{G}}^{\partial 2},$$

where each edge $e\in E$ is labeled by a two-ported internal graph

$$\ell \left(e\right)=(H_e,p_e^{-},p_e^{+}).$$

For example, for the edge $e:\mathsf{Build}\to \mathsf{Test}$, take $H_e$ to be the graph whose vertices are

$$V\left(H_e\right)=\{\mathit{ArtifactRepo},\mathit{SignatureCheck},\mathit{TestRunner},\mathit{ReportStore}\},$$

and whose edges represent required interactions (e.g. artifact fetch, signature verification, executing tests, storing reports). Choose ports

$$p_e^{-}=\mathit{ArtifactRepo}\in V\left(H_e\right),p_e^{+}=\mathit{ReportStore}\in V\left(H_e\right),$$

so that $p_e^{-}$ is the entry interface to the internal process (receiving artifacts) and $p_e^{+}$ is the exit interface (publishing test reports).

Similarly, for the edge $\mathit{Deploy}\to \mathit{Monitor}$, the internal graph $H_{e^{\prime }}$ may encode

$$\mathit{DeployController}\to \mathit{MetricsAgent}\to \mathit{AlertManager},$$

with ports identifying the start/end of the deployment-to-observability integration.

Figure 2.1 visualizes the Edge–MetaGraph viewpoint: the outer pipeline is shown together with two illustrative internal graphs attached to edges. By Definition 2.3, the expanded graph $Expand\left(\mathbb{E}\right)$ is obtained by gluing each port $p_e^{-}$ to $s\left(e\right)$ and each port $p_e^{+}$ to $t\left(e\right)$.

2.2. Iterated Edge-MetaGraphs

Iterated Edge-MetaGraphs recursively label each edge by a two-ported Edge-MetaGraph, creating multi-level edge-substitution and expansion hierarchy.

Definition 2.4

(Two-ported iterated Edge-MetaGraphs). Let $t\in {\mathbb{N}}_0$ and let ${\mathfrak{E}}^{\left(t\right)}$ be a class of finite objects each equipped with a finite vertex set $V\left(X\right)$. A two-ported iterated Edge-MetaGraph of depth t is a triple

$$X^{\partial }=(X,p^{-},p^{+}),X\in {\mathfrak{E}}^{\left(t\right)},p^{-},p^{+}\in V\left(X\right),$$

where $p^{-}$ and $p^{+}$ are distinguished vertices called the tail-port and head-port. We write $Tail\left(X^{\partial }\right):=p^{-}$ and $Head\left(X^{\partial }\right):=p^{+}$, and set

$${\left({\mathfrak{E}}^{\left(t\right)}\right)}^{\partial 2}:=\{(X,p^{-},p^{+}):X\in {\mathfrak{E}}^{\left(t\right)},p^{-},p^{+}\in V\left(X\right)\}.$$

Definition 2.5

(Iterated Edge-MetaGraph universes). Fix a nonempty universe $\mathfrak{G}$ of finite graphs (as in the preliminaries). Define recursively a hierarchy ${\left({\mathfrak{E}}^{\left(t\right)}\right)}_{t\in {\mathbb{N}}_0}$ by

$${\mathfrak{E}}^{\left(0\right)}:=\mathfrak{G},$$

and for each $t\ge 0$,

$${\mathfrak{E}}^{(t+1)}:=\left\{(V,E,s,t,\ell )|(V,E,s,t)\mathrm{is}\mathrm{a}\mathit{finite}\mathit{directed}\mathit{multigraph}\mathit{and}\ell :E\to {\left({\mathfrak{E}}^{\left(t\right)}\right)}^{\partial 2}\right\}.$$

Thus, an object of ${\mathfrak{E}}^{(t+1)}$ is an Edge-MetaGraph whose edge labels are two-ported objects from the previous level ${\mathfrak{E}}^{\left(t\right)}$.

Definition 2.6

(Iterated Edge-MetaGraph of depth t). Let $t\in {\mathbb{N}}_0$. An iterated Edge-MetaGraph of depth t is an element of ${\mathfrak{E}}^{\left(t\right)}$ from Definition 2.5.

Equivalently, for $t\ge 1$ it is a tuple

$${\mathbb{E}}^{\left(t\right)}=(V^{\left(t\right)},E^{\left(t\right)},s^{\left(t\right)},t^{\left(t\right)},{\ell }^{\left(t\right)})$$

such that $(V^{\left(t\right)},E^{\left(t\right)},s^{\left(t\right)},t^{\left(t\right)})$ is a finite directed multigraph and

$${\ell }^{\left(t\right)}:E^{\left(t\right)}⟶{\left({\mathfrak{E}}^{(t-1)}\right)}^{\partial 2}.$$

In particular, depth 1 coincides with the (non-iterated) Edge-MetaGraph concept where edges are labeled by two-ported base graphs.

Remark 2.1

(Recursive expansion (optional notation)).  Using the expansion operator Expand already defined for Edge-MetaGraphs, one may define a depth-t flattening ${Expand}^{\left(t\right)}$ recursively by

$${Expand}^{\left(0\right)}\left(G\right):=G,{Expand}^{(t+1)}\left(\mathbb{E}\right):=Expand\left(V,E,s,t,e\mapsto \left({Expand}^{\left(t\right)}\left(X_e\right),p_e^{-},p_e^{+}\right)\right),$$

where $\ell \left(e\right)=(X_e,p_e^{-},p_e^{+})$. This yields an ordinary graph obtained by expanding all nested edge labels down to depth 0.

Example 2.2

(Real-world example: multi-layer logistics & service dependencies). Consider an international e-commerce company that ships products from a factory to a customer. At the top level (depth 2), we model the end-to-end process as a directed multigraph

$$(V^{\left(2\right)},E^{\left(2\right)},s^{\left(2\right)},t^{\left(2\right)}),$$

whose vertices are major stages:

$$V^{\left(2\right)}=\{\mathit{Factory},\mathit{ExportHub},\mathit{ImportHub},\mathit{LastMile},\mathit{Customer}\}.$$

An edge $e\in E^{\left(2\right)}$ represents a contractual handoff such as $\mathit{ExportHub}\to \mathit{ImportHub}$ or $\mathit{LastMile}\to \mathit{Customer}$.

Each such handoff is not atomic in practice: it is implemented by a nested workflow involving multiple carriers, checkpoints, and IT services. Hence we label every top-level edge e by a two-ported object

$${\ell }^{\left(2\right)}\left(e\right)=(X_e,p_e^{-},p_e^{+})\in {\left({\mathfrak{E}}^{\left(1\right)}\right)}^{\partial 2},$$

where $X_e\in {\mathfrak{E}}^{\left(1\right)}$ is a depth-1 Edge–MetaGraph describing the internal sub-process realizing the handoff, and the ports $p_e^{-},p_e^{+}$ indicate the entry/exit interface of that sub-process.

For instance, the edge $e:\mathsf{ExportHub}\to \mathsf{ImportHub}$ may be labeled by a depth-1 Edge–MetaGraph $X_e$ whose vertices represent operational checkpoints

$$V\left(X_e\right)=\{\mathit{PickUp},\mathit{OriginCustoms},\mathit{Airline},\mathit{DestinationCustoms},\mathit{Unload}\},$$

and whose edges correspond to concrete tasks (document verification, container scanning, flight booking, etc.). Crucially, each task edge $f\in E\left(X_e\right)$ is again labeled by a two-ported base graph

$${\ell }_{X_e}\left(f\right)=(H_f,q_f^{-},q_f^{+})\in {\left({\mathfrak{E}}^{\left(0\right)}\right)}^{\partial 2}={\mathfrak{G}}^{\partial 2},$$

where $H_f$ is a small graph encoding the micro-structure of the task, e.g. a dependency graph between IT services:

$$\mathit{OrderDB}\to \mathit{RiskScoring}\to \mathit{CustomsAPI}\to \mathit{LabelPrinter}.$$

Thus, the overall system forms an iterated Edge–MetaGraph of depth 2: top-level shipment handoffs are edges labeled by depth-1 Edge–MetaGraphs, and inside those, task edges are labeled by two-ported base graphs capturing service-level dependencies. Expanding the structure (recursively) yields a single flattened dependency graph for the entire factory-to-customer delivery pipeline.