IBM Ponder This #264 · April 2020

This month's challenge is inspired by the COVID-19 pandemic.
Suppose that the world has five cities and they are all connected to one another as depicted by the five-verticed graph in the following picture:

Let's assume that the infection passes daily along the edges. So, if on day t, "D" is infected and "C" is healthy, then "C" has a 10% chance of getting infected, by "D", on day t+1.
If "A" is infected at time 0, after ten days, there is about a 29.16521896% probability that all five will be infected.

Find a graph with no more than eight vertices that gives a probability of 70.00% (accurate to the second decimal digit after the decimal point) of all cities/vertices being infected after ten days.
Give your answer as an adjacency matrix.
For example, the adjacency matrix of the graph depicted above is

01110
10001
10010
10101
01010

Bonus '*' for getting the closest to 70% in 19 days; and ’**' for getting even closer. Current best is less than 1.9E-7 away from 0.7.

Solution

Best opened after a real attempt

To be added.