July 2008

Contagious disease are spread (generally) when one person comes in contact with another. Thus, the number of links in a network (the number of connections one has) will go a long way to determining how fast diseases are spread.

One question which needs to be answered is whether a hub-and-spoke network or a more diffused network will spread diseases faster. On the one hand, if the hub gets infected, it is very likely that everyone else gets infected in the hub and spoke diagram. On the other hand, if the hub does not get infected, then a more diffused network likely will spread diseases more quickly.

A paper by Jackson and Rogers (2007) uses the concept of stochastic dominance to demonstrate which types of networks spread diseases the quickest. Today, I will summarize their model.

Model

All nodes (think of a node as a person) in a network can either be infected (have the disease) or susceptible (do not have the disease and are not immune). We will ignore immunity in this model. The probability a node is infected is: ν(d_iθ_i + x) where ν ∈ (0,1) describes the infection rate. The variable d_i represents the degree (number of connections) that node i has, θ_i ∈ [0,1] is the fraction of i‘s neighbors who are infected and x is a non-negative scalar representing the rate at which infection sprouts up independent of social connections.

An individual recovers from a disease with probability δ.

Now we want to characterize how diseases spread through different social networks. Let P(d) be the probability a randomly chosen node has d connections (degree d). If ρ(d) equal the average infection rate among nodes with degree d, then the average infection rate can be calculated as:

• θ=(Σ_dρ(d)P(d)d)/(Σ_dP(d)d)

The variable θ is the average neighbor infection rate. We can estimate the change in the infection rate over time for nodes of degree d with the following equation:

• ∂ρ(d)/∂t=[1-ρ(d)] ν(θd+x) – ρ(d)δ

The first part of the fraction shows how quickly susceptible nodes (i.e.; 1-ρ(d)) are infected and the second part show how quickly infected nodes (i.e., ρ(d)) are cured. In steady state [i.e., ∂ρ(d)/∂t=0], we have that the average infection rate is:

• θ=m^-1Σ_d[(ν(θd²+xd)P(d)/δ]/[1+ν(θd +x)/δ]]
• m = Σ_dP(d)d

Network Comparisons

Let us now define networks according to the concept of stochastic dominance. Network P’ has first order stochastic dominance over P if Σ_{(d=0 to Y)} P’(d) ≤ Σ_{(d=0 to Y}) P(d) ∀ Y, with Σ_{(d=0 to Y})P’(d) < Σ_{(d=0 to Y)} P(d) for some Y. This means that network P’ has a higher fraction of nodes with lots of connections compared to network P. Jackson and Rogers prove the following:

1. If P’ strictly first order stochastically dominates P, then the steady state θ’ > θ and the steady state ρ’ > ρ.
2. If P’ is a strict mean-preserving spread of P, then θ’ > θ.

Theory (1) implies that if a network has more connections, it will have a higher steady state average neighbor infection rate (θ), and a higher overall average infection rate (ρ). This makes perfect sense.

One the other hand, theory (2) shows what happens as we move towards a hub and spoke system (i.e., a mean-preserving spread in P). A mean preserving spread means the average number of connections between nodes stays the same, but there are more likely to be nodes with very few connection or very many connection. Thus, a hub and spoke system will have a higher neighborly infection rate, but this does not mean that the average infection rate will be higher.

The authors expound on theory (2) in more detail below:

The change in infection rate due to a change in the degree distribution comes from countervailing sources, as more extreme distributions have relatively more very high degree nodes and very low degree nodes. Very high degree nodes have high infection rates and serve as conduits for infection, thus putting upward pressure on average infection. Very low degree nodes have fewer neighbors to become infected by and thus have relatively low infection rates. Which of these two forces is the more important one depends on the ratio λ=ν/δ, i.e., the effective spreading rate. For low λ, the first effect is the more important one, as nodes recover relatively rapidly, and so there must be nodes with many neighbors in order keep the infection from dying out. In contrast, when λ is high, then nodes become infected more quickly than they recover. Here the more important effect is the second one, as most nodes tend to have high infection rates, and so how many neighbors a given node has is more important than how well those neighbors are connected.

Conclusion

For fast spreading disease where people recover slowly, a diffuse network increases the average infection rate. For slow spread diseases, or diseases where people recover relatively quickly, a hub-and-spoke system increases the average infection rate.

• Jackson, Matthew O. and Rogers, Brian W. (2007) “Relating Network Structure to Diffusion Properties through Stochastic Dominance,” The B.E. Journal of Theoretical Economics: Vol. 7 : Iss. 1 (Advances), Article 6.