Meaning of the Idiom: The phrase "Cock of the walk" refers to someone who dominates a group or thinks they are the most important person in a particular circle. It originates from the behavior of a dominant rooster (cock) in a farmyard (walk).
Why the others are incorrect:
A bone of contention (Incorrect): Refers to a subject or issue over which there is a continuing disagreement.
A man of his words (Incorrect): Refers to a person who keeps their promises and is reliable.
A white elephant (Incorrect): Refers to a possession that is useless or troublesome, especially one that is expensive to maintain.
Contextual Fit: The sentence describes people who find themselves "dominating" others in a group, which perfectly matches the definition of being the "Cock of the walk."
