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18.5: Mean-Field Approximation on Random Networks

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If we can assume that the network topology is random with connection probability pe, then infection occurs with a joint probability of three events: that a node is connected to another neighbor node (pe), that the neighbor node is infected by the disease (q), and that the disease is actually transmitted to the node (pi). Therefore, 1peqpi is the probability that the node is not infected by another node. In order for a susceptible node to remain susceptible to the next time step, it must avoid infection in this way n1 times, i.e., for all other nodes in the network. The probability for this to occur is thus (\(1 − p_{e}qp_{i})^{n−1}. Using this result, all possible scenarios of state transitions can be summarized as shown in Table 18.5.1.

Table 18.5.1: Possible scenarios of state transitions in the network SIS model.

Current state Next state Probabiltiy of this transition
0 (susceptible) 0 (suceptible) (1q)(1peqpi)n1
0 (susceptible) 1 (infected) (1q)(1(1peqpi)n1)
1 (infected) 0 (susceptible) qpr
1 (infected) 1 (infected) q(1pr)

We can combine the probabilities of the transitions that turn the next state into 1, to write the following difference equation for qt (whose subscript is omitted on the right hand side for simplicity):

qt+1=(1q)(1(1peqpi)n1)q(1pr)=1q(1q)(1peqpi)n1+qqpr1(1q)(1(n1)peqpi)qpr(because peqpi is small)=q((1q)(n1)pepi+1pr)=q((1q)s+1pr)=f(q)(with s=(n1)pepi)

Now this is a simple iterative map about qt, which we already know how to study. By solving f(qeq)=qeq, we can easily find that there are the following two equilibrium points:

qeq=0,1prs

And the stability of each of these points can be studied by calculating the derivative of f(q):

df(q)dq=1pr+(12q)s

df(q)dq|q=0=1pr+s


df(q)dq|q=1pr/s=1+prs

So, it looks like prs is playing an important role here. Note that 0pr1 because it is a probability, and also 0s1 because (1s) is an approximation of (1peqpi)n1, which is also a probability. Therefore, the valid range of prs is between -1 and 1. By comparing the absolute values of Eqs. ??? and ??? to 1 within this range, we find the stabilities of those two equilibrium points as summarized in Table 18.5.2.

Table 18.5.2: Stabilities of the two equilibrium points in the network SIS model.

Equilibrium point 1prs0
0<prs1
q=0
unstable stable
q=1Prs
stable unstable

Now we know that there is a critical epidemic threshold between the two regimes. If pr>s=(n1)pepi, the equilibrium point qeq=0 becomes stable, so the disease should go away quickly. But otherwise, the other equilibrium point becomes stable instead, which means that the disease will never go away from the network. This epidemic threshold is often written in terms of the infection probability, as
pi>pr(n1)pe=prk

as a condition for the disease to persist, where k is the average degree. It is an important characteristic of epidemic models on random networks that there is a positive lower bound for the disease’s infection probability. In other words, a disease needs to be “contagious enough” in order to survive in a randomly connected social network.

Exercise 18.5.1

Modify Code 16.6 to implement a synchronous, simultaneous updating version of the network SIS model. Then simulate its dynamics on an Erd˝osR´enyi random network for the following parameter settings:

• n = 100, pe=0.1, pi=0.5, pr=0.5(pr<(n1)pepi)

• n = 100, pe=0.1, pi=0.04, pr=0.5(pr>(n1)pepi)

• n = 200, pe=0.1, pi=0.04, pr=0.5(pr<(n1)pepi)

• n = 200, pe=0.05, pi=0.04, pr=0.5(pr>(n1)pepi)

Discuss how the results compare to the predictions made by the mean-field approximation.

As you can see in the exercise above, the mean-field approximation works much better on random networks than on CA. This is because the topologies of random networks are not locally clustered. Edges connect nodes that are randomly chosen from the entire network, so each edge serves as a global bridge to mix the states of the system, effectively bringing the system closer to the “mean-field” state. This, of course, will break down if the network topology is not random but locally clustered, such as that of the Watts-Strogatz small-world networks. You should keep this limitation in mind when you apply mean-field approximation.

Exercise 18.5.2

If you run the simulation using the original Code 16.6 with asynchronous updating, the result may be different from the one obtained with synchronous updating. Conduct simulations using the original code for the same parameter settings as those used in the previous exercise. Compare the results obtained using the two versions of the model, and discuss why they are so different.


This page titled 18.5: Mean-Field Approximation on Random Networks is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Hiroki Sayama (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.

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