A Proposed Mathematical Model for Wardley Maps — Part 1

Below is one way to formalise the concepts behind a Wardley Map using mathematical notation. Real‐world Wardley Maps are more art than strict science — so this isn’t a “canonical” formula, but rather a demonstration of how you might model the two key axes (value chain and evolution) and the dependencies among components.

1. Define the Set of Components

Let

be the set of all components in your system (for example, “User,” “API,” “Cloud,” “Database,” etc.).

We also have a special component

that represents the end user (the top of the value chain).

2. Dependencies as a Directed Graph

We define a directed graph

where

is the set of dependencies. A directed edge

means that

depends on

In a Wardley Map diagram, you typically draw a line from a node to the node it consumes or depends on.

For instance, if the “User” depends on the “Application,” there would be an edge from User to Application.

3. Value Chain Axis (Vertical)

In a Wardley Map, the vertical axis goes from Invisible (bottom) to Visible (top), reflecting how obvious a component is to the user. One way to quantify this is via graph distance from the “User” node:

1. Reverse the edges or otherwise define a way to measure “distance to the user.”
2. Let

be the shortest path length from

to

(If no path exists, define

1. .)

“Normalise” these distances to get a value in

For example:

means the component is at distance 0 (it is the user).

means it’s as far away from the user as any component in the system (very “invisible”).

Hence, the vertical coordinate of each component

can be:

4. Evolution Axis (Horizontal)

Wardley Maps place components along a left-to-right axis that represents evolution: from new/novel/custom (left) to standardised commodity or utility (right). We can model it with a function

that returns a value in

Example Definition of

:

Suppose

is how long

has existed or how widely it’s adopted:

A simple log scale might be:

If

is very small,

If

is as large as any component in the system,

Hence, the horizontal coordinate of each component

can be:

5. Plotting the Wardley Map

Each component

is then mapped to a point

on a 2D plane:

(horizontal) = how evolved

is (from novel to commodity).

(vertical) = how visible

is to the user (from invisible to very visible).

The edges

remain the same: if

draw a line or arrow from

to

In symbolic form:

6. Interpretation and Caveats

Subjectivity

Deciding how “visible” a component is or how “evolved” it is involves subjective judgement. The functions

and

above are simply illustrative ways to encode those judgements.

Graph Topology

The “distance” calculation for

depends on how you structure dependencies. Some people cluster items by how close they are to the user’s needs or do a topological sort.

Evolving Over Time

A key principle of Wardley Mapping is that components move rightward over time as they evolve from “Genesis” to “Commodity.” In a dynamic model,

might change over time, shifting components to the right as they mature.

No Single ‘Correct’ Formula

Different teams use different heuristics or scoring systems. The equations above provide a mathematical scaffold to show how you might turn Wardley Map thinking into coordinates and a dependency graph.

Final Takeaway

A Wardley Map can be seen as:

A directed graph of components and their dependencies.

A pair of mapping functions —

for “visibility” and

for “evolution” — that place each component on a 2D plane:

A visualisation in which edges are drawn from each component to the components it depends on.

By combining value chain (vertical) and evolution (horizontal), you get a holistic view of where to innovate, where to commoditise, and how everything ties back to user needs.

What’s Next

Check out part 2, a better model