  A vertex of a randomly growing graph is called a persistent hub if at all but
finitely many moments of time it has the maximal degree in the graph. We
establish the existence of a persistent hub in the Barab\'asi--Albert random
graph model with probability one. We also extend this result to the class of
convex preferential attachment graphs, where a vertex of degree $k$ gets a new
edge with probability proportional to some convex function of $k$.

Probability