How something spreads across a network is a key question of interest when analyzing many different networks, the classical example of this being the diffusion of a disease through some population. But we might be talking about how the loss of one species in an ecosystem has an effect on others, the spread of financial contagion from one institution to another, 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. Firstly, there is some infectiousness metric, where we are talking about how infectious the phenomenon that is spreading on the network is.
A corollary to this is asking how resistant are the nodes to this contagion; which gives us a resistance parameter. Next, one needs 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, one needs to consider if this diffusion is taking place strategically or at random; this ties back to topology because some network topologies are more susceptible to strategic influence than others.
Infectiousness refers to something that is likely to spread or influence others in a rapid manner, irrespective of the type of network 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 a disease is or a number of other metrics, but we also need to ask how many nodes a given node can infect in 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 instance, thus enabling a much more rapid contagion rate. Inversely, we need to consider how resistant the nodes in the network are to the spreading of this phenomenon. Imagine trying to promote gay marriage in some conservative rural community. No matter how infectious your campaign is, 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 phenomenon. We could also add time to our model here, capturing how nodes may be affected for only a brief period of time before recovery; as would be the case with the spread of many diseases or some trend in fashion.
The network’s topology is a key consideration in understanding 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 to get an idea of how many edges any phenomenon would have to traverse in order to affect the whole system. For example, modern broadcast media has arisen hand-in-hand with the modern nation state, as it is only through these centralized hubs that uniform information can be rapidly disseminated to a large population, and thus a key component in creating a sense of nation culture and cohesion. Without these centralized hubs, diffusion can be a lot slower and become heterogeneous.
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 modeled 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 of financial contagion. Toxic assets do not themselves choose where to end up in the network and which nodes to effect most. 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 many dependent nodes, and thus have a greater effect than simply choosing to attack any node at random.
Complex contagion is the process in which multiple sources of exposure to a phenomenon are required before an individual adopts the change in its 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 other 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 could be trying to model how an individual will vote for two different candidates in an election based upon the social network they are a part of. We would then be defining some variable as to how many of the node’s neighbors need to vote for a particular party before they would cast their vote for the same 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. There is feedback, as when the node changes its state it will affect the choices of others around it also. All of this means that this more complex form of contagion is nonlinear, with the possibility of exponential cascades forming. Real-world diffusion across something like a social network is a complex process that may require multiple network models, that is, allowing the network model to have multiple different connections between nodes; in order to capture how different types of connections and networks interact to enable or resist the diffusion of some phenomena. Within the voting example, we might have to take into account economic factors and other social relations in order to capture the true dynamics at play.
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3. Hodas, N.O. and Lerman, K. (2014). The Simple Rules of Social Contagion. Scientific Reports, [online] 4(1). Available at: https://www.nature.com/articles/srep04343 [Accessed 4 Sep. 2020].