Let$$s consider the limiting values that result from the Basic PageRank Update Rule $($i$.$e$.$ the version where we don$$t introduce a scaling factor s$).$ In Chapter $14,$ these limiting values are described as capturing $$a kind of equilibrium based on direct endorsement: they are values that remain unchanged when everyone divides up their PageRank and passes it forward across their out$-$going links.$14\mathrm{.}7.$ EXERCISES $433$ This description gives a way to check whether an assignment of numbers to a set of Web pages forms an equilibrium set of PageRank values: the numbers should add up to $1,$ and they should remain unchanged when we apply the Basic PageRank Update Rule. For example, this is illustrated in Chapter $14$ via Figure $14\mathrm{.}6$: you can check that if we assign a PageRank of $4/13$ to page A$,2/13$ to each of B and C$,$ and $1/13$ to the five other pages, then these numbers add up to $1$ and they remain unchanged when we apply the Basic PageRank Update Rule. Hence they form an equilibrium set of PageRank values. For each of the following two networks, use this approach to check whether the numbers indicated in the figure form an equilibrium set of PageRank values. $($In cases where the numbers do not form an equilibrium set of PageRank values, you do not need to give numbers that do; you simply need to explain why the given numbers do not.$)$ A $1/103/102/10$ B C E $3/101/10$ D Figure $14\mathrm{.}19$: A network of Web pages. $($a$)$ Does the assignment of numbers to the nodes in Figure $14\mathrm{.}19$ form an equilibrium set of PageRank values for this network of Web pages? Give an explanation for your answer. $($b$)$ Does the assignment of numbers to the nodes in Figure $14\mathrm{.}20$ form an equilibrium set of PageRank values for this network of Web pages? Give an explanation for your answer.