 How something spreads across a network is a key question we'll be interested in asking when analyzing many different types of networks. The classical example of this being the diffusion of some disease through a population, but we might be talking about how the loss of one species in an ecosystem has an effect on others or the spread of some information within a group of people. More formally, we call this spreading on a network propagation or diffusion. How diffusion happens and how long it takes is defined by a number of different parameters. We'll just list the primary factors involved here before looking at them individually. Firstly, we have some infectiousness metric where we're talking about how infectious is the phenomena that is spreading on the network. A corollary to this is asking how resistant are the nodes to this contagion, giving us a resistance parameter. Next, we need to consider the topology to the network. Obviously, this diffusion is taking place along the connections within the network, meaning different structures to the connections and different degree distributions will be another defining factor when considering diffusion. Lastly, we need to consider if this diffusion is taking place strategically or randomly. And this ties back to topology because as we've already discussed, some network topologies are more susceptible to strategic influence than others. So firstly, to discuss infectiousness. By infectiousness, we mean something that is likely to spread or influence others in a rapid manner, irrespective of the type of network that it is spreading on. If you hear about some important piece of news, you feel driven to tell others and that is infectiousness. It is like an outward force that is pushing the phenomena across local connections and out over the network. We may be able to quantify this in terms of money or how contagious the disease is or a number of other metrics, but we also need to ask how many nodes a given node can infect within any given time interval. A mosquito can only bite one other creature at a time, but a person can broadcast a message to possibly millions of other people at any given instant, thus enabling a much more rapid contagion rate. Inversely, we need to think about how resistant the nodes in the network are to the spreading of this phenomena. Imagine trying to promote gay marriage in some conservative rural community. No matter how infectious your campaign might be, it is unlikely to take off, and this is due not to your failures, but to the resistance of the other nodes in the network to this particular phenomena. We might also add time to our model here, capturing how nodes may be infected for only a brief period of time before recovery, as would be the case with the spreading of many diseases or some trend in fashion. Next, we need to look at the topology of the network to understand how something is likely to spread across it. The primary factor here being simply the overall degree of connectivity to the network. Obviously, the more connected it is, the faster something should spread across it, but also we would need to look at the average shortest path length to get an idea of how many edges a phenomena would have to traverse in order to affect the whole system. We also need to analyse the degree distribution to understand how centralised the network is, as centralised networks with major hubs can enable rapid local and global diffusion. For example, modern broadcast media has arisen hand in hand with the modern nation state, as it is only through these centralised hubs that uniform information can rapidly disseminate to a large population, and thus a key component in creating a sense of national culture and cohesion. Without these centralised hubs, diffusion can be a lot slower and become heterogeneous as it spreads out. We also need to ask whether this dissemination is random or strategic. That is, whether there is some logic behind the promotion and dissemination aimed at strategically affecting nodes that have a high degree of connectivity and thus enabling a more rapid diffusion. Many forms of diffusion can be modelled as random. A virus has no logic telling it to attack creatures that have lots of physical contact with others. We might say the same at financial contagion. Toxic assets don't themselves choose where to end up in the network and which nodes to affect. These are factors that are defined by other dynamics. But some diffusion processes are strategic. For example, military strategy is often specifically designed to attack a critical node in an opposition's military or infrastructure network. In the hope that this shock to a critical node will then propagate to its dependent nodes and thus have a greater effect than simply choosing to attack any node at random. Lastly, we'll just touch upon the topic of complex contagion. The simple contagion model we've been describing so far is essentially binary, meaning either a node is affected or not. And within this model, all that matters is whether one other node affects it or not. Complex contagion, in contrary, is the process in which multiple sources of exposure to a phenomena are required before an individual adopts that change in state. An example of this might be the adoption of some new technology or innovation, which is costly, especially for early adopters, but less so for those who wait. We can then model this as a form of complex contagion, asking how many nodes need to adopt the innovation before a given node will do likewise. There might also be two competing events propagating across the network. An example of this might be trying to model how an individual will vote for two different candidates in an election based upon the social network they're part of. We could then define some variable as to how many of the nodes' neighbors need to vote for a particular party before they would also cast their vote for that party. These complex models have many interacting parts, thus there will be tipping points as a node will not do anything until a threshold value is met. And there is feedback as when the node changes its state, it will affect the choice of others around it. All of this means that this more complex form of contagion is non-linear, with the possibility of exponential cascades forming. We've only just begun to touch upon the most salient metrics affecting network diffusion. Real world diffusion across something like a social network is a complex process that may require multiplex models, that is, allowing our model to have multiple different connections between the nodes. In order to capture how different types of connections and networks interact to enable or resist the diffusion of some phenomena. So within our voting example, we might have to take into account economic factors and relations in order to capture the true dynamics at play. But this is beyond the scope of this course. So we'll wrap up here and move on to talk about the closely related subject of network robustness in the next module.