Indeed, Walker believes Aberdeen have a strong squad and 'more intensity' about them this term.
That said, he feels Celtic won't be repeating their silly mistakes of last campaign, when they dropped points to the likes of Hamilton Academical and Inverness Caledonian Thistle early on.
Walker told Sky Sports News: "If you remember at the beginning of last season Celtic were losing to Hamilton, losing to Inverness and were dropping other silly points and I just can't see that happening too often this season."
And, on Celtic's main rivals for the title, he added: "I think Aberdeen have got a wee bit more intensity about them this season.
"I think Derek McInnes has put together a really good squad... and I think they will again pose the biggest threat to Celtic."
Walker certainly has more faith in the Hoops' competition for the title than Hull City midfielder David Meyler, who recently joked on Twitter that Celtic should just be handed the Scottish Premiership trophy already.
Interestingly, it's Hearts that are top of the table after four games, with the league leaders the only side to win all four of their opening fixtures. Aberdeen have so far won three out of three.
Walker also spoke highly of Hearts but feels, long term, the Dons are Celtic's biggest challengers.
In terms of 'silly points' dropped, the Bhoys drew 2-2 at Kilmarnock recently but that was far from the frustration of last season's loss to Hamilton.