Exercise $3.$ A food chain is a directed graph with $n$ nodes. The directed edges go from $1$ to $2,2$

to $3,4$ to $5$ and so on$.$

$($a$)$ What are the degree centralities of nodes $1,2,$dotsn $?($Remember that for a directed graph,

degree centrality is usually identified with $k_i^{in}.$

$($b$)$ What are the eigenvalue centralities of nodes $1,2,$dotsn $?$ Explain your answer.

$($c$)$ Now calculate the Katz centralities of each node. $($Hint: it is easier to start with the

definition of Katz centrality, than the matrix inversion formula that we derived in class. Your

answer will depend on the parameter $)..$

$($d$)(*$ Optional question$)$ In class we determined that the Katz centrality is defined for $<mfrac\; id="EditorElement\_519460"><mrow\; id="EditorElement\_519453"><mn\; id="EditorElement\_519450">1</mn></mrow><mrow\; id="EditorElement\_519459"><mi\; id="EditorElement\_519456"></mi></mrow></mfrac>$

where $$ is the largest real eigenvalue of $A.$ What is the condition for $$ in the Katz centrality

calculation you did in part $($c$)?$ Explain why $$ can be arbitrarily large for this network.