All models and theories are like windows onto the world. None of them are perfect. They all enable us to see some things but also inhibit us from seeing others, and more importantly, they all rest upon some set of assumptions. This set of methods and assumptions that support a particular scientific domain is called a paradigm, and network theory is based upon a paradigm that has a number of features to it that we can identify.
Firstly, networks are all about connectivity. Within systems whose components are relatively isolated, we can focus our interest on the individual components of the system. By analyzing their properties, we can gain an understanding of how the whole system works. For example, say I have put together a financial portfolio of different assets. Well, if the risk on these different assets is not related in any way, that is to say, they are all from very different sectors of the economy, well I can calculate the overall risk of my portfolio by simply analyzing the properties of each asset and then summing them up to formulate the total value of the overall portfolio.
But what if many of the assets in my portfolio are correlated? If I have acquired many investments within both say the food processing industry and agriculture, well the risk on both is correlated, or assets in both logistics and retail are again interconnected. Because of these correlations, the value of the assets will move together. Thus, the real risk-return ratio of my portfolio is no longer defined by that of each asset in isolation, but now by these correlations, that is to say, what is connected to what and in what way are they connected comes to now define the whole system. We often spend a lot of time analyzing individual components and then assume that the whole system is simply an additive function of these parts. But when we turn up the connectivity within a system it is increasingly the relations between components that come to define the overall system, and this is where networks theory finds its relevance as it is all about connectivity.
Networks have a very different kind of space to the one we are used to. We have spent our lives walking around in a 3-dimensional space, what we called a Euclidean geometry. That is deeply intuitive to us but we need to forget about this, because it is the geometry of things or object, not of connectivity. The geometry of networks is what we call their network topology, and this topology stretches and bends our three-dimensional space around it. For example, say one lives in Istanbul Turkey. If we pull out a map, we will see that Budapest, the capital of Hungary, is much closer than London to Istanbul. But because London is a major hub in the global air transportation system while Budapest in only a minor one, it is much easier to make a connection to London instead of Budapest. So if we put something in this network at Istanbul because of the network’s topology, London is essentially closer to it than Budapest. This should give some insight into how a network operates in a different type of geometry than the one we are used to, and thus they cut across our traditional domains, not just in space as in this example, but in all areas making the study of networks a truly interdisciplinary one.
Thirdly, networks represent a very organic type of structure that often emerges from the bottom up, but also has some environmental constraints imposed upon it. Examples of this might be the trading routes that have emerged at different periods in history. During the Middle Ages, traders from Asia would exchange goods with merchants in India and the Middle East, who would, in turn, bring goods to Europe and vice versa; thus emerging an almost global network of trade routes out of the local interactions between merchants. The same can be seen within an ant colony where individual ants leave a trail of pheromones to food sources. Here again, a network emerges from the different ant trails. But none of these connections and the networks that emerge out of them are for free. They cost something to maintain, and thus many of these networks are the product of an interaction between the local elements that are creating the connections and the environment that is placing some resource constraint on its development.
Lastly, complexity and nonlinearity are inherent features of networks. As the number of elements in a network may just grow in a linear fashion as in 1, 2, 3 and so on, the number of connections between them may grow exponentially. So if you take just a small group of people, say 10 or 20, there can be literally billions of different types of networks between them. Nonlinearity is a reoccurring theme with networks, and we can only approach this type of exponential complexity with the use of computers. When we combine this new way of describing the world that network theory gives us with the powerful tools of computation and a flood of new data sources that we now have available to us, we get a mini-revolution and a new approach called network science that is starting to having a major impact on many areas of science. Networks give us a very intuitive way of visually representing complexity. The model of a network is distinctly visual and this can provide a quick and intuitive overview to a complex system. By just looking at it, we can get a quick sense of how connected it is, what are the key nodes and other critical information to understanding the whole system.
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