Problem $11($Financial Network Risk Management$).$ In risk management, Bellman$-$Ford can model financial transactions between institutions, where $[10$ marks$]$:

$-$ Each institution is a vertex, and each transaction or loan is an edge with a weight representing risk exposure.

$-$ Negative weights may indicate secure transactions with low or no risk, while positive weights represent risky or high$-$exposure transactions.

$($a$)$ Describe how Bellman$-$Ford would identify the least risky transaction paths across institutions.

$($b$)$ Explain the significance of a negative cycle in this network and how Bellman$-$Ford can help manage it$.$

$($c$)$ What insights does Bellman$-$Ford provide to risk managers in a financial network?