In recent decades, it technological development and the possibility of collecting huge amounts of data they have changed the way we look at the world. Today we have networks that can contain millions of interacting elements: networks of people, digital networks, biological networks, infrastructural networks.
These systems show non-intuitive behaviorsdifficult to predict by studying only their individual parts. To understand them, a different approach is needed, namely network theory (or network theory), based on the mathematical tools of graph theory. Let’s see what it is and why it is important to understand complex systems.
From graphs to complex networks: what are these structures for?
In mathematics, a graph is a structure composed of points or nodes (the elements) e lines (the relationships between elements). It is a very simple model, but incredibly effective for representing complex systems.
Despite their simplicity, graphs allow for extraction in a synthetic and objective way information from huge networksimpossible to analyze in detail element by element.
And this is precisely one of the crucial points of network science: it is not enough to know all the parts of a system to understand itbecause the overall behavior depends above all on how those parts interact.
This represents an overcoming of the traditional reductionist approach, according to which knowing every single piece should be enough to understand the whole. In real systems this is not the case: a complex system it is not the simple sum of its partsbut it is the result of their connections.
Because looking at connections changes everything
Network theory allows us to reconstruct the interaction map within a system: first we study its architecture – nodes and connections – then we analyze the dynamics that propagate through those connections.
This is exactly why very different phenomena can be studied with the same mathematical tools: the diffusion of electrical signals in neuronsthe propagation of information about social networksthe domino effect caused by the delay of a major airport, the transmission of a virus respiratory within a population.
These systems belong to different fields — biology, technology, transportation, sociology — but share common structural properties. It is one of the most important results of network science: many complex systems work according to the same mathematical rules.
What makes a network “real”
When observing real networks – biological, technological, ecological or social – four recurring characteristics emerge:
- High clustering. Knots tend to form highly connected local groups. It is the phenomenon whereby “friends of my friends” are often connected to each other. The same happens in neurons of the same cortical area or between companies in the same sector.
- Small network distances. Between any two nodes exist on average a few steps. For example, in the cerebral cortex this results in rapid communication pathways between distant regions.
- Unequal distribution of connections. In many networks, a few nodes accumulate a very large number of links — the hub — while most nodes have few. This applies to global airports (e.g. Atlanta, Dubai, Heathrow), to web pages, to specific areas of the brain.
- Dynamism. Nodes can arise, grow and disappear, but the overall structure of the network remains stable. These properties place complex networks in an intermediate zone between completely ordered systems and completely random systems.
The fundamental models of network science
In the 1990s two mathematical models revolutionized the understanding of real networks: the Small-World model (Watts & Strogatz, 1998), which shows that it is possible to have simultaneously: strong local connections (high clustering), short global distances between distant nodes. In short, just a few “shortcuts” are enough to make the world surprisingly small.
The second is the Scale-Free model (Barabási & Albert, 1999), which reveals how the distribution of connections in many networks a power law follows: a few nodes have many connections and most have few connections. This happens because real networks grow over time and favor nodes that are already highly connected (preferential attachment). This is the mechanism behind the growth of the Web, social networks and many biological networks.
Because network science has become central
Today the network science is applied in increasingly diversified fields:
- epidemiology: to model the spread of diseases;
- transportation engineering: to design more robust networks;
- informatics: to understand the functioning of the Internet, the Web, for artificial intelligence;
- neuroscience: to study brain connectivity and changes after trauma;
- economy: to analyze financial systems and supply chains;
- ecology: to map relationships between species and energy flows.
The strength of this approach lies in the fact that it allows extremely complex systems to be described through common properties, regardless of the nature of their components. This is why Barabási defined the science of networks as “the next scientific revolution”: a new point of view that allows us to understand the deep organization of real systems.
