Network Connectivity

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One of the defining features to a network is going to be 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 qualitative change as we will try to demonstrate in this short lecture.



One of the defining features to a network is going to be 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 qualitative change as we have previously tried to note, it marks a shift from a component based regime, where we need to firstly think about the components to 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 interconnect.
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, we might call this ease of two nodes interacting or coupling, the coupling parameter to the network.
So 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 become more dense 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. So when we increase our coupling parameter we are increasing the density of the network and the average degree of connectivity.