Another Spread of Behavior

Consider the network shown below. Everyone starts with a behavior B$.$ Suppose we are interested in

changing the behavior of this entire population to a new behavior A $,$ and we know that actors will

switch to A if at least a fraction q$=1/2$ of its neighbors have adopted A$.$ How can we do it with a

minimal investment in$,$ i$.$e$.,$ a minimal number of early adopters?

$($a$)$ Find a set of three nodes in the network with the property that if they act as the three initial

adopters of A$,$ then it will cause a complete cascade $($spread A to all nodes$).$ List their node IDs.

Initial Adopters causing complete cascade:

$($b$)$ Find a different set of three entirely different initial adopters $($not listed above$)$ who could also

cause a cascade of A$.$ List their node IDs.

Different Initial Adopters causing complete cascade:

$($c$)$ Find three clusters in the network, each of density greater than $1//2,$ with the property that no node

belongs to more than one of these clusters. List the members of the three clusters.

Cluster:

Cluster:

Cluster:

$($d$)$ How does your answer to $($c$)$ help explain why there is no set consisting of only two nodes in the

network that would be capable of causing a complete cascade of adoptions of A $?$

Explanation: