It is
claimed that the Glasgow accent is spreading to Edinburgh:
In his new book, The English Language In Scotland: An Introduction To Scots, Professor Jones notes "the introduction of Glaswegian pronunciations and other linguistic features across the central belt" and "a levelling between Glasgow and Edinburgh pronunciations"
According to the Lord Provost of Edinburgh:
In a "working class west Edinburgh accent", Eric Milligan, lord provost of Edinburgh, said people from the city would be very surprised to hear they sounded Glaswegian because "it's very different through here"
I don't normally agree with him but I think the LP is correct although he and I may not meet enough of the younger generation that is supposedly being colonised by the western linguistic aggressors.