NoteForMu.md

1. Class and Probability of Edges

1.1 Whole network

Let us ignore the vertices and their status for a moment, and focus on edges and their binary status, which means whether the edge has t transmitted the infection or not.

For simplicity later, I use the terminology for occupation status from percolation theory, at any time $t$:

• If the transmission has already passed through the edge (i.e. the edge has already transmitted the infection), the edge is occupied.
• If the transmission has not yet passed through the edge (i.e. the edge has not yet transmitted the infection), the edge is unoccupied. For our SIR model with no reinfection, an occupied edge can not switch back to unoccupied.

At any moment $t$, MSV framework classify all edges in the whole network (with proportion/probability $=1$) into two large classes using its core variable $\phi(t)$:

• $\phi(t)$ proportion of edges has not yet transmitted the infection (unoccupied)
• $1-\phi(t)$ proportion of edges has already transmitted the infection (occupied) As we are modeling random networks, the proportions for edges can also be treated as probabilities:
• $\phi(t)=\mathbb{P}(\text{A randomly selected edge is occupied at } t)$
• $1-\phi(t)=\mathbb{P}(\text{A randomly selected edge is unoccupied at } t)$

1.2 A Random Focal Vertex $a$

Lets uniformly randomly pick a focal vertex $a$, regardless of the its status, with a random variable $K$ as its degree. For this random vertex $a$ in random network, choose a random edge $E$ connected to vertex $a$ is the same as choose a random edge from the whole network.

So at any time $t$, this edge $E$ should also have probability $\phi(t)$ being unoccupied and probability $1-\phi(t)$ being occupied.

1.3 A Neighbour Vertex $b$ of $a$, with unoccupied edge $E$

For the unoccupied probability $\phi(t)$ of $E$, we can further parse it finer based on the status (S, I or R) of neighbour vertex $b$ connect to $E$ on the other side than $a$.

For this neighbour vertex $b$, as we know it is connected to an unoccupied edge $E$, so it is not the same as a random chosen vertex from whole network. Therefore, the probability that neighbour vertex $b$ is susceptible, infected and recovered respectfully should not be $S(t), I(t), R(t)$, i.e. the proportion of each compartment.

1.4 Two perspective

==I think here is the key issue and confusion point: == The vertex $b$ has two equivalent identities in random network, but looks different from the perspective of vertex or edge:

1. $b$ is a random vertex connected to an unoccupied edge $E$. ($a$ is not involved).
2. From randomly chosen focal vertex $a$, $E$ is a random unoccupied edge connected to $a$, then $b$ is the neighbour connect to $a$ through $E$.

NOTE: In probabilities, AND is denoted by $\wedge$ and condition is denoted by $|$.

MSV framework use $\phi_S(t), \phi_I(t), \phi_R(t)$ to denote the probability that a vertex connected to unoccupied edge $E$ AND being susceptible, infected or recovered. So based on the two perspective, we have two event with the same probability. $$\begin{aligned} \phi_S(t) = & \ \mathbb{P}(\text{A random vertex } b \text{ is susceptible at time } t \ & \wedge b \text{ attached to the unoccupied edge } E \text{ attached to } a) \ = & \ \mathbb{P}(\text{For random vertex } a \text{, one of its random edge } E \text{ is unoccupied } \ & \wedge E \text{ attached to a suseptible neighbour } b) \end{aligned}$$Similar for $\phi_I$ and $\phi_R$.

Use the same language for $\phi(t)$, we get the partition: $$\begin{aligned} \phi(t) &= \mathbb{P}(\text{A random edge } E \text{ is unoccupied at } t ) \ & = \mathbb{P}(\text{For random vertex } a \text{, one of its random edge } E \text{ is unoccupied}) \ & = \phi_S(t) + \phi_I(t) + \phi_R(t) \end{aligned}$$ And further we have the full partition of all edges in the network: $$\ 1 = (1-\phi(t)) + \phi(t) = (1-\phi(t)) + (\phi_S(t) + \phi_I(t) + \phi_R(t)) $$

1.5 Definition of $\phi_S$

The expression for $\phi_S$ is derived from the definition without $a$ involved: $$\begin{aligned} \phi_S(t) & = \mathbb{P}(\text{A random vertex } b \text{ is susceptible at time } t \wedge b \text{ attached to a unoccupied edge } E ) \ & = \frac{G'_p(\phi(t))}{\delta}=\frac{G'_p(\phi(t))}{G'_p(1)} \ & = \sum_k \frac{p_k k}{\delta} \phi(t)^{k-1}=\frac{1}{\delta}\sum_k k (p_k \phi^{k-1}) \ & =\frac{1}{\delta}\sum_k k (\frac{ \phi \times p_k \times\phi^{k-1}}{\phi}) \ & = \frac{1}{\delta}\sum_k k (\frac{\mathbb{P}(E \text{ is unoccupied} \wedge b \text{ has degree k } \wedge b \text{ has no other occupied edge}) }{\mathbb{P}(E \text{ is unoccupied})}) \end{aligned}$$ So indeed as Todd point out, this is conditioned on the focal edge $E$ is unoccupied, but only from $b$ and $E$'s perspective.