Protocol Graph Model¶

Overview¶

The protocol graph models a multi-protocol agent mesh as a weighted, labeled graph. Each agent is a node with capabilities, each connection is an edge annotated with supported protocols, and routing algorithms navigate the graph while respecting protocol constraints and minimizing translation costs. The formal model is:

\[G = (V, E, P, \Phi, \Gamma)\]

This abstraction enables protocol-aware routing, automatic protocol translation, and workflow optimization across heterogeneous agent networks where different agents speak different protocols.

Key Definitions¶

$V$: Agent nodes. Each node $v \in V$ has a set of capabilities $\text{caps}(v)$ and a set of supported protocols $\text{protos}(v) \subseteq P$.
$E$: Edges. Each edge $(u, v) \in E$ represents a communication link between agents $u$ and $v$, with an associated weight $w(u, v) \in \mathbb{R}^+$ representing latency or cost.
$P$: Protocol universe. The set of all protocols in the mesh, e.g., $P = \{\text{MCP}, \text{A2A}, \text{ANP}, \ldots\}$.
$\Phi$: Protocol assignment. A function $\Phi : E \to 2^P$ mapping each edge to the set of protocols available on that link.
$\Gamma$: Protocol affinity. A function $\Gamma : V \times P \to [0, 1]$ scoring how well each agent supports each protocol. An affinity of $1.0$ means native support; lower values indicate degraded or translated support.

Node Types¶

Type	Description	Typical Protocols
`LLM_AGENT`	Language model agent with reasoning capabilities	MCP, A2A
`TOOL_SERVER`	Stateless tool execution endpoint	MCP
`COORDINATOR`	Orchestration node managing multi-agent workflows	A2A, ANP
`GATEWAY`	Protocol translation bridge between network segments	All protocols

Routing Algorithms¶

Four routing algorithms are provided, each optimizing for a different objective.

Algorithm	Class	Strategy	Complexity
Dijkstra	`ProtocolAwareDijkstra`	Minimum-cost path with automatic protocol translation at boundaries	$O((V + E) \log V)$
BFS	`ProtocolConstrainedBFS`	Minimum-hop path restricted to edges supporting a required protocol	$O(V + E)$
Resilient	`ResilientRouter`	Failover routing with degraded topology -- reroutes around failed nodes	$O((V + E) \log V)$
Semantic	`SemanticRouter`	Capability-similarity matching weighted by protocol affinity $\Gamma$	$O(V \cdot C)$ where $C$ is capability dimension

All routers return a Route object containing the path, total cost, protocol transitions, and any translations required.

Path Cost Function¶

The cost of a route is defined by the path cost function (Definition 12):

\[\text{cost}(\pi, \sigma) = \sum_{i} w_{e_i}(p_i) + \sum_{i} \tau(p_i, p_{i+1}, v_i)\]

where:

$\pi = (v_0, v_1, \ldots, v_k)$ is the path through agents
$\sigma = (p_0, p_1, \ldots, p_{k-1})$ is the protocol used on each edge
$w_{e_i}(p_i)$ is the weight of edge $e_i = (v_i, v_{i+1})$ using protocol $p_i$
$\tau(p_i, p_{i+1}, v_i)$ is the translation cost incurred at node $v_i$ when switching from protocol $p_i$ to $p_{i+1}$

The translation cost decomposes as $\tau = \tau_{\text{base}} + \tau_{\text{semantic}}$, where $\tau_{\text{base}}$ is the fixed overhead and $\tau_{\text{semantic}}$ penalizes information loss. When $p_i = p_{i+1}$ (no protocol change), $\tau = 0$.

Protocol Transformers¶

When a message must cross a protocol boundary (e.g., from an MCP tool server to an A2A coordinator), a protocol transformer converts the message format.

A transformer is a partial function:

\[T : M_p \to M_{p'} \cup \{\bot\}\]

where $M_p$ is the message space of protocol $p$, $M_{p'}$ is the message space of protocol $p'$, and $\bot$ indicates a failed transformation (the message cannot be represented in the target protocol).

Transformer Classifications¶

Classification	Description	Information Loss
Lossless	Bijective mapping -- no information lost or gained	None
Lossy	Surjective mapping -- some metadata or structure is dropped	Bounded
Destructive	Significant structural changes -- may alter semantics	Unbounded

Transformer Composition¶

Transformers compose sequentially. A path requiring two protocol hops uses:

\[T_{p \to p''} = T_{p' \to p''} \circ T_{p \to p'}\]

The composition is valid only if neither component returns $\bot$. The total information loss is the sum of individual losses, making shorter protocol paths preferable.

Workflow Model¶

A workflow over the protocol graph is defined as:

\[W = (T, \prec, \rho)\]