Problem $10(1\mathrm{.}5$ points$)$

For your favorite voting system, list:

$($a$)$ One weakness. $(0\mathrm{.}5$ point$)$

$($b$)$ One strength. $(0\mathrm{.}5$ point$)$

$($c$)$ Identify an alternative system that addresses the weakness but doesn't have the same strength. Explain briefly why this system has these properties. $(0\mathrm{.}5$ point$)$

Problem $11(1\mathrm{.}5$ points$)$

Below we depict a social network in Figure $4$ showing interactions at different times. All edges depict the timespan when two nodes interacted. For two of these edges, the b$-$d edge and the d$-$e edge, we don't know the ending time of the interaction. We will study how this impacts the spread of an Figure $4$: Network depicting social interactions at different times. If node $1$ is infected at time $\backslash (\backslash$mathbf$\{$t$\}\backslash ),$ and it shares an edge $\backslash ([\backslash$mathbf$\{$a$\},\backslash$mathbf$\{$b$\}]\backslash )$ with node $2,$ such that $\backslash (\backslash$mathbf$\{$a$\}\backslash$leq $\backslash$mathbf$\{$t$\}\backslash$leq $\backslash$mathbf$\{$b$\}\backslash ),$ then node $2$ will with a probability $\backslash (\backslash$mathrm$\{$p$\}>0\backslash )$ be infected by node $1.$

infectious disease, that spread with probability $\backslash (\backslash$mathrm$\{$p$\}>0\backslash ).$ We assume that node $\backslash ($ a $\backslash )$ is infected at time $\backslash ($ t$=0\backslash ).$

$($a$)$ Assume first that $\backslash ($ x$=1\backslash )$ and $\backslash ($ y$=1\backslash ).$ Could the entire network become infected with these values? $(0\mathrm{.}5$ point$)$

$($b$)$ Find $($some other$)$ values of $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ such that the entire network could become infected. Additionally list for each node the earliest point it could be infected $(0\mathrm{.}5$ point$)$

$($c$)$ We learn that in fact $\backslash ($ x$=1\backslash )$ an $\backslash ($ y$=1\backslash ),$ however we also learn that we are missing an edge with start time and stop time $[13,20].$ Can you use this edge to connect two nodes, such that the entire network could become infected? $(0\mathrm{.}5$ point$)$