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\begin{document}
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\centerline{Math 36 --- Weekly Homework}
\centerline{Assigned: 10/8}
\centerline{Due: 10/15}
\begin{questions}
\question
Show that a degree sequence does not uniquely determine a network by giving two networks with the same degree sequence that are not isomorphic.
\vspace{2 in}
\question
Give the degree, closeness, betweenness, and eigenvector centralities for the following network.
\vspace{3 in}
\question
Add one new vertex (with as many neighbors as you want) to the network from 2 to make vertices 4, 5, and 6 as unimportant as possible. Explain your decision.
\vspace{3 in}
\question
Here we will examine the Prisoner's Dilemma on a grid and determine what payoff values favor cooperation or defection or both.
Use a population on a square lattice. Each round, all pairs of neighbors play the game with their neighbors. Then an individual will be chose uniformly to die. The neighbors of the deceased individual compete to fill the spot. Select a neighbor proportional to fitness (which is a function of payoff). Repeat this until all individuals are playing one strategy.
The details:
\begin{itemize}
\item Use a population of $N=400$ individuals on a 20 $\times$ 20 square lattice.
\item Use weak selection. The fitness of an individual will be $f = 1 - w + w(\text{payoff})$, where $w = 0.01$.
\item The payoff matrix takes the form
\begin{table}[h]
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\begin{tabular}{cc|c|c|}
& \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$}\\
& \multicolumn{1}{c}{} & \multicolumn{1}{c}{$A$} & \multicolumn{1}{c}{$B$} \\\cline{3-4}
\multirow{2}*{Player $1$} & $A$ & $b-c$ & $-c$ \\\cline{3-4}
& $B$ & $b$ & $0$ \\\cline{3-4}
\end{tabular}
\end{table}
\end{itemize}
For each of the following sets of values for $b$ and $c$, determine if the probability of invasion by a single mutant (cooperator or defector) is above or below $1/N$.
\begin{enumerate}[label = (\alph*)]
\item $b = 8$, $c = 1$
\item $b = 4$, $c = 1$
\item $b = 4$, $c = 2$
\end{enumerate}
\end{questions}
\end{document}
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