Chapter 1: The Stench of the Past

In July of the mid-1800s, Londoners were met with an unbearable odor that left them feeling nauseous. Those with wealth would often perfume their handkerchiefs to mask the smell, while many less fortunate chose to escape to the countryside in search of relief. This event, now known as the Great Stink of 1858, marked a low point in the city's history.

The River Thames had become a cesspool, overflowing with centuries of untreated human waste from the old wooden sewer systems. The intense summer heat exacerbated the situation, causing the polluted waters to emit a foul miasma that lingered for miles.

Recognizing the urgent need for change, the City of London Corporation called for proposals to design a new sewer system. Joseph Bazalgette’s plan was ultimately selected, and he is now celebrated as a key figure in improving public health in London. His engineering expertise led to the construction of a sewer network that not only enhanced hygiene but also contributed to the decline of cholera in the city.

Bazalgette’s sewer system continues to function effectively, transporting waste from millions of residents toward treatment plants near the Thames estuary. This monumental engineering project included 22,000 kilometers of sewers, utilized 318 million bricks, and excavated 2.7 million cubic meters of soil.

Bazalgette was meticulous, ensuring that every aspect of the system was future-proofed. He focused on the slope and gravity of the pipes and the sizes of the tunnels. Two critical questions guided his planning: how to minimize the sewer route between points and identify the most crucial junctions.

Bazalgette’s work exemplified early applications of graph theory, emphasizing key concepts we still analyze in networks today: the distance between nodes and the significance of those nodes.

Chapter 2: Understanding Distance in Graphs

Distance is a fundamental yet vital concept in graph theory. A graph comprises vertices and edges that connect pairs of these vertices. The distance between any two vertices is determined by the number of edges in the shortest path linking them. This is often referred to as 'geodesic distance,' which is considered 'infinite' if no path exists.

For instance, in a basic graph, the distance from vertex 2 to vertex 6 is 3, meaning there are two paths of this length available.

Minimizing distance is crucial in various engineering contexts. Additionally, in social networks, understanding minimum distance is essential; the concept of six degrees of separation reflects this. Studies on platforms like Facebook show an average minimum distance of 4.57 between users.

Conversely, maximum distance can reveal unfamiliarity and diversity. For example, in a corporate setting, analyzing past collaborations can help create diverse discussion groups by ensuring participants have not previously worked together.

This video, "The Great Stink of 1858," delves into the historical context of this event, exploring its implications for public health.

Chapter 3: Evaluating Vertex Importance

In any graph, some vertices hold more significance than others. In Bazalgette’s sewer system, certain junctions required closer monitoring due to their potential impact on the overall network. Similarly, in social networks, some individuals exert more influence based on their connectivity.

One way to measure this importance is through valence, or the number of edges connected to a vertex. However, a more comprehensive measure is betweenness centrality, which counts how often a vertex lies on the shortest path between two other vertices. Vertices with high betweenness centrality are crucial for information dissemination and maintaining network connectivity.

Understanding betweenness centrality is essential in social networks, allowing for strategic investments in individuals who can amplify messages or foster connections. It also helps assess the potential impact of losing key individuals from the network.

Calculating betweenness centrality can be computationally intensive, especially in large networks, but there are efficient data science tools available, like the igraph package in R and the networkx library in Python.

If you're interested in a deeper exploration of these concepts, including practical examples in R and Python, refer to Chapters 5 and 6 of my textbook on Network Analysis.

The second video, "Smelly Facts About London's The Great Stink of 1858," provides intriguing insights and lesser-known details about this pivotal moment in history.