Problem $1$

Consider network $1.$ The dashed lines are weak ties and the solid lines are strong ties. Pink and white represent

female and male.

a$)$ How many connected components are there? $(1$ point$)$

b$)$ Perform the breadth$-$first search for node $7.(5$ points$)$

c$)$ What is the maximum distance of two nodes in the graph?

$(2$ points$)$

d$)$ Use your answer to part c$)$ to determine whether the small$-$

world phenomenon holds for this graph. $(2$ points$)$

e$)$ Does node $4$ satisfy the Strong Triadic Closure Property?

Explain why. $(5$ points$)$

f$)$ Is there evidence of homophily? Provide the calculations.

$(10$ points$)$

g$)$ Is it possible to label the unlabeled links with $+$ and $-\text{'}$s so

that the network becomes weakly balanced but not balanced? If

yes, label the links. $(5$ points$)$

h$)$ Find all Nash equilibria of the following game with strate$-$

gies $S,T,$ and V: $(10$ points$)$

i$)$ Assume that strategy $S$ is no longer available. Find all

evolutionarily stable strategies of the resulting game. $(10$ points$)$