Hello Fellow Magicians!

Over the past few weeks, we’ve been exploring graphs as a data structure and how they serve as a powerful medium for machine learning and AI in the AEC industries. Today, I’d like to take a step back and provide a broader introduction to what graphs are and why they matter.

Graphs

When most people hear the word "graph," they likely think of bar charts, pie charts, or line graphs—the kind you might create in Excel to visualize data. These are great tools for making data easier to interpret, but that’s not the kind of graph we’re talking about.

Instead, we’re diving into a completely different concept rooted in graph theory, a branch of mathematics that focuses on relationships and connections. These graphs aren’t about plotting data points—they’re about representing structures of nodes (things) and edges (relationships). This foundational concept is what allows us to model complex systems, from social networks to building designs, in elegant and meaningful ways.

Let’s take a look at where this all started and how it shapes the way we work today.

The Story of Leonhard Euler and the Seven Bridges of Königsberg: A Gateway to Graph Theory

The story of graphs begins with a problem that sounds almost whimsical: a walk through the city of Königsberg (now Kaliningrad, Russia), with its seven bridges crisscrossing the Pregel River. The city’s residents often wondered:

This seemingly simple riddle stumped everyone—until Swiss mathematician Leonhard Euler tackled it in 1736. In doing so, he laid the foundation for a new branch of mathematics: graph theory.

The Königsberg Puzzle

The city of Königsberg was divided into four distinct landmasses: two islands and two mainland areas, connected by seven bridges. The challenge was to find a continuous path that started and ended at the same place while crossing each bridge once.

To make sense of the problem, Euler stripped away everything unnecessary—the riverbanks, the city streets, even the bridges themselves—and reduced it to an abstract network. He replaced each landmass with a node (or vertex) and each bridge with a line (or edge) connecting the nodes. The result was a simple graph.

Euler’s Revolutionary Insight

Euler realized that the problem wasn’t about the specific layout of the city but about the connections between nodes and edges. He discovered a crucial property:

• For a path to traverse every edge exactly once, the graph must either:

  1. Have exactly two nodes with an odd number of edges (in which case the path can start and end at these two nodes), or

  2. Have no nodes with an odd number of edges (in which case the path forms a circuit and starts/ends at the same node).

When Euler examined the Königsberg graph, he found that all four nodes had an odd number of edges—a dead giveaway that the task was impossible. No such path existed!

Why This Matters

Euler’s solution wasn’t just about solving the Königsberg bridge problem; it was about creating a new way of thinking. He introduced the concept of a graph—a mathematical abstraction of relationships—and proved that such structures could be analyzed using logic and rules - and that such an analysis would successfully describe people’s experience of the world around them. His work opened the door to studying connections and networks, which have since become essential in countless disciplines.

What Makes a Graph?

At its core, a graph is a mathematical structure made up of two main elements: nodes and edges. These elements form a network that represents relationships between entities. Let’s break it down:

Nodes and Edges

• Nodes (or Vertices):
  Nodes are the "things" in your graph. They can represent anything—people, buildings, cities, molecules, or even abstract concepts. In the Königsberg example, the nodes represented landmasses.

• Edges:
  Edges are the "connections" between nodes. These can represent relationships like friendships (in a social network), roads (in a transportation map), or dependencies (in a project plan). In Königsberg, the edges were the bridges connecting the landmasses.

Attaching Information to Nodes and Edges

What makes graphs so powerful is their ability to hold information. Both nodes and edges can store data, turning a simple structure into a rich, dynamic model of the real world.

• Attributes on Nodes:
  Nodes can carry information about the entities they represent. For example:

  • A node in a social network might store a person’s name, age, and interests.

  • A node in a building model might store a room’s name, size, and function.

• Attributes on Edges:
  Edges can store information about the relationship between nodes. For instance:

  • In a transportation network, an edge might include the distance between two cities or the speed limit of a road.

  • In a design model, an edge might store the type of relationship between components, like "is above" or "is adjacent to."

By attaching this metadata, graphs transform into more than just a web of connections—they become a detailed, queryable database.

Modern Applications of Graphs

Euler’s abstract nodes and edges are now the basis for solving some of today’s most complex problems:

• Maps and Routes: GPS navigation systems calculate the shortest path between locations using graphs.

• Networks: Social media platforms, internet routing, and communication systems are modeled as graphs.

• Project Planning: Dependency graphs help manage tasks and deadlines in complex projects.

• Biology: Molecular interactions and brain networks are studied using graph-based approaches.

Timeless Legacy

Euler’s genius transformed a playful puzzle into a mathematical framework that continues to shape how we understand and navigate relationships in our world. From the bridges of Königsberg to the algorithms that guide your daily commute, the legacy of this brilliant insight is everywhere. Maybe in how we might develop power full AI for AEC..

So next time you cross a bridge—navigate a building—or even scroll through your social media feed—remember: the story of graphs began with a question as simple as a stroll through a city.