Begg family Gills Caithness

[Max39]

There are a sizeable number of individuals with the surname "Begg" in the Gills, Upper Gills, Warse and Canisbay area. Are they related?
John Begg was a miller in Gills from 1841 census. He married Margaret Moodie from Dunnet in 1780. Their son Alexander was a miller in Auckengill, Freswick.
Canisbay Parish seems to be a sparsely populated area and I think that would be the case in early 1800's?
Any comments and information would be most welcome as I search for my maternal ancestors.