Is there a way to see a graphical representation of a website's backlink history?
I have tried Magestic SEO, but the data looks inaccurate:
Does Google Webmaster Tools come with such a data visualisation tool?
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Unfortunately, Google Webmaster Tools doesn't show this kind of history chart.
However, many services online show backlinks history like Ahrefs.com. This is my favourite because there is many useful charts visible for free. Moreover, free version can be enough to analyze backlinks of a website. Then, you can pay to see more data.
Keep in mind that the Majestic graph you are looking at says that it is non-cumulative. According to them you were getting a couple links per day for a long while and then suddenly got a bunch all at once. Check to see if they have a cumulative view that would slope up more gradually.
Google Webmaster Tools does not show you history of your inbound links over time.
Seomoz's Open Site Explorer (Free 30 day trial and then $99/month) claims as a feature: Historical Link Analysis -- A historical view of your competitive domain metrics over time