Executable Semantic Graphs for Enterprise Digital Twins

Enterprise architecture (EA) modeling frameworks traditionally enforce a strict division between static domain documentation and dynamic runtime execution. We present a formal framework for compiling heterogeneous architectural views into a single, strongly-typed Executable Semantic Graph (ESG).

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1. Formal Mathematical Definition

Let an Executable Semantic Graph $G$ be defined as a 5-tuple:

$$G = (V, E, \mathbf{S}, \mathbf{W}, \delta)$$

Where:

• $V = \{v_1, v_2, \dots, v_n\}$ represents the set of operational semantic nodes.
• $E \subseteq V \times V$ represents directed execution edges.
• $\mathbf{S}: V \rightarrow S_{state}$ maps each node to its internal queue state matrix.
• $\mathbf{W}: E \rightarrow \mathbb{R}^+$ defines transition weights (routing probabilities).
• $\delta: (V, S_{state}, \Delta t) \rightarrow S_{state}'$ is the state transition transfer function over time quantum $\Delta t$.

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2. Structural Layer Compaction

To eliminate data redundancy across separate enterprise modeling standards, ESG compiles visual projections into unified node primitives[^1]:

Source Model View	Primary Primitive	ESG Mapping Target	Runtime Behavioral Trait
BPMN 2.0	Service Task / Sub-Process	Node.Compute	Latency distribution & Queue bounds
ArchiMate 3.1	Application Interface	Node.Gateway	Multiplexing & Rate limiting
UML 2.5	State Transition	Edge.Conditional	Boolean route evaluation
VSM (Value Stream)	Processing Time ($T_{p}$)	Node.Metric	Accumulator & Wastage tracking

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3. Haskell Type System Specification

The core graph evaluation kernel operates on immutable algebraic data types to guarantee deterministic state transitions across parallel worker threads:

module EdgeNode.Core.GraphEngine where

import Data.Map.Strict (Map)
import Data.Text (Text)

type NodeId = Text
type Weight = Double

data NodeState 
  = Active { queueLength :: !Int, avgLatencyMs :: !Double }
  | Degrading { errorRate :: !Double }
  | Saturated
  deriving (Eq, Show)

data SemanticNode = SemanticNode
  { nodeId    :: !NodeId
  , nodeLabel :: !Text
  , state     :: !NodeState
  , capacity  :: !Int
  } deriving (Eq, Show)

-- | Evaluates next state transition vector over time step Delta T
stepNode :: SemanticNode -> Double -> SemanticNode
stepNode node deltaT
  | capacity node <= 0 = node { state = Saturated }
  | otherwise          = node { state = Active (capacity node - 1) (deltaT * 0.85) }