One of the defining features of a network is its overall degree of connectivity, which might qualify as the defining feature. Going from a system with a low degree of connectivity to one with a high degree of connectivity is not just a quantitative change in the number of edges within the network. It is also a qualitative change. It marks a shift from a component based regime, where we need to firstly think about the components of the system, their properties in isolation, and their linear interactions; to a relational based regime where we need to first model how the system is interconnected. One way of contextualizing the degree of connectivity to a network is by talking about how easy or difficult it is for a node in the network to make a connection with another because the overall connectivity emerges out of the local actions of the nodes in the network. If we make it difficult for them to interact then there will be a lower overall connectivity.
If we take any network, say a logistics network, we can ask under what circumstances are the nodes more likely to interact. In this case, the nodes are producers, distributors, and consumers, and they will be more likely to interact as the cost of transportation and trade restrictions are reduced. The development of the global economy over the past few decades could be cited as an example here. Through the reduction in trade tariffs and advancements in transportation and communications, the ease of interaction between producers and distributors on a global level has increased, resulting in the increased density of logistics and trade networks as regional and national economies have become integrated into the global economy. Just to put some real figures to this, in America the logistic cost of transporting some freight is thought to be around 5% of the cost of the goods, whereas in China it is thought to be around 12% to 13%. You can then imagine how a business will factor this into their choices as to whether they supply to far off distributors or not, and thus whether the network becomes denser or more sparse.
One way of quantifying this concept of the overall connectivity to a network is with reference to its density. The density of a network is defined as a ratio of the number of edges to the number of possible edges, and this will also correlate to the average degree of connectivity to the nodes in the network.1 So when we increase our coupling parameter, we are increasing the density of the network and the average degree of connectivity. This coupling parameter to a system can, of course, be defined by many different things. Depending on the network we are dealing with, it may be economic as in our example above which was measured in terms of the financial cost of transportation. Or it may be measuring the climatic condition to an ecosystem, where we could quantify it in terms of the average environmental temperature. As we reduce the temperature, the number and density of interactions between creatures reduce as they hibernate in isolation. We could also think about the formality of a social setting as a parameter. As we reduce the formality of the setting, say by having an office party, people’s social inhibitions are reduced and they are more likely to interact.
As we turn this coupling parameter up or down, thus requiring the nodes to exert more or fewer resources in order to create a connection, we would expect the level of integration within the network to increase or decrease. The easier it is for elements to create a connection the more connections and the longer these connections can be, thus working to integrate the entire system. And inversely, the more resistance there is for nodes to create connections the less there will be and the network will disintegrate, with the most costly ones, that is those that are maintained over a greater distance, being the first to go. But this does not always change in a linear fashion.
The amount of connectivity within the network will be primarily defined by how much resources a node has to expend in order to make that connection. The amount of resources that a node will have to expend in order to create a connection will grow in a somewhat proportional fashion to the length of the relations. So if I am walking to the local shop, I will have to exert a certain amount of energy to do this. And if the shop is twice as far away, I will have to expend twice as much resources in order to create that connection. This is a linear progression. But this simple linear scaling is not always the case. Say I am taking an intercontinental flight, a journey of some 2000 kilometers. This does not require 10 times more effort on my behalf than a local flight of 200 kilometers. Because of this, the degree of connectivity in the system may not always grow in a simple linear fashion, but because of this nonlinearity the level of connectivity can grow or decay in an exponential and rapid fashion resulting in there being tipping points and phase transitions.