A Primer on Network Sciences
Citation Networks of two academic journals, Biochimica et Biophysica Acta and Neurobiology of Disease, where the Y axis is publication date. This was made using the icite database of academic metadata
One of the more interesting areas I’ve come across is the field of Network Science and the ways it begins to clarify and transform the otherwise impenetrable organizational problems around us. Social networks, slime-molds, legal or academic networks, recommendation algorithms for YouTube and Amazon, logistics and GPS systems, metabolic systems, neural networks, economic systems, Patent Citations, subway systems, and many other organizational systems can be characterized and analyzed using the common language of network analysis. These are all examples of complex systems that evolve over time and have many interacting parts — and as such, they’re all instances where traditional, reductive problem-solving doesn’t work.
Networks have been discovered and rediscovered many times in many domains, and as such the terms go by many names. In mathematics, graphs are composed of “nodes” and “links.” In sociology, social networks are composed of “Actors” and “Ties” — in Neural Networks, we talk of “Neurons” and “Synapses” — Ecosystems tend to be described by “Species” and “Predation” — bibliographic networks use the terms “Article” and “Citation,” and so on. Fundamental organizational principles of nature are a bit like a root-language informing the jargon of these many specializations. As these specialized fields develop, we also develop our comprehension of the underlying organizational principles.
A Primer on Network Sciences
Here the red nodes represent destinations or houses on our island
There are two ways we can talk about networks - one is generalized, talking about the abstract organizations and patterns of information, while the other is to talk about specific cases around us in the world. Learning from abstractions can be a bit like hearing the rules of a board-game without actually playing, so let’s use specific examples to build an intuitive understanding which then generalizes. I think the example of roads or paths between houses is really intuitive, so let’s start there.
Let us imagine that we are homesteading on an island with some of our best friends. Each friend explores the beautiful forests and shores and chooses a spot to build a house. There are many landmarks like swimming holes or campsites of friends that stand out to us from the landscape. Initially these landmarks and sites are unconnected, there are no roads or paths - they do not form a network. When we want to visit the campsite of our friend, we have to trudge through the underbrush and find our way through the forest. There are a bunch of unconnected nodes, interesting destinations on the island.
The simplest solution to this navigational problem would be to create straight paths from every destination to every other destination, covering the island in paths.
Many direct paths between nearby nodes.
Intuitively we know this is not an optimal solution: it ignores how hard it is to create, maintain, and traverse our paths in the real world. Additionally, paths like this are rarely created all at once — they are usually created slowly over time as houses are built and destinations are navigated to. Early paths form between early settlers and important destinations like the shore, the spring, and other friends. These paths are then reused as much as possible, and new paths are only made when the cost of the exploration, creation and maintenance is justified.
To clear a new, long path, it usually needs to be useful to multiple people and for multiple purposes. How do we strike a balance between maintenance cost and network efficiency? Nature has a solution for this which we find everywhere (from botanical systems to river formation to our lungs and circulatory systems:) we reduce the costs of a system by bundling and reusing connections. When we bundle the edges in a system, we can reduce maintenance and initial costs and create efficiencies or economies of scale. This is often called “Graph Bundling.”
We tend to see more pressure for “graph bundling” when it is more costly to create and maintain edges. For example, people easily take many unique paths through a flat and grassy park, but tend to reuse long paths through mountainous or difficult terrain.
We can think of graph bundling as being like a force that pulls every path towards every other path. Changing the pull of this force will determine how much paths are “reused.”
Over time, the most used paths become more trodden and formalized — they become easier to follow, but they also become harder to change. The way paths become entrenched into roads is much like a river formation: the more water tends to flow down into a valley, the deeper it carves the valley, making it more likely for water to flow into the valley. For a road, the more people follow a path, the more the ground is trodden, the more obvious the path becomes, the more it is known and maintained. The path on our island that goes from important destinations like the dock to the well turn into a gravel road — links tend to be reinforced the more they are used.
A side effect is that if people want to gather, they will tend to gather somewhere along this main road, because that tends to be central and accessible. If a market space or community center is to be created, it will tend to be in these central locations, benefiting from high traffic and easy accessibility. This is an example of Preferential Attachment: highly connected things are more likely to form new connections, and things that benefit more from connectedness will experience more of this pressure to be centralized.
Preferential attachment means that the postwoman who says hello to each person in the morning will be a great source of news and information, as each day she visits a little with every other person in town. She becomes quite popular, and as more people get to know her, she becomes a better and better source of information. This is known as a network effect, when the more connected something is, the more valuable that source becomes. A strong community will have strong network effects, where the connections inside the community increases the value of the community, thereby increasing the connections further.
Preferential attachment, the tendency to connect to highly connected nodes, often leads to network effects — disproportionately connected connections; these concepts are closely related. A couple of people on the island know almost every other person, such as our postwoman. This distribution is common enough that we call it a “scale free” network - when most people on the island know only a couple others, but a couple people on the island know everybody.
In a scale free network, a couple locations are used by almost everybody, like the post-office, school, or store, while most destinations like houses are used by only a few individuals. This kind of network gives rise to the “six degrees of separation” dynamic. On our island of friends, each person might be connected through a single postwoman, so each person would have just one degree of separation - they could ask the postwoman how any other person is doing and likely get some meaningful information.
Network science goes a lot deeper than these concepts however, and this article is just the first part of a larger series intended to explore complex systems, network science, and organizational principles in nature.
