Graffiti used to be one of the barometer antisocial behaviour crimes.

Like dog-fouling and fly-tipping, it can be seen as a visual measure of the quality of a town or city, and often has a disproportionate effect on the happiness of those living there.

This is why it was a feature of the "broken windows" theory of urban decay, and preventing it a central element of high-profile policies such as the Zero Tolerance approach of New York City mayor Rudi Guiliani in the early 1990s.

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Now, as Police Scotland chief Sir Stephen House points out, it is in decline in Scotland. Vandalism of all kinds is a dying crime. However it is being replaced, Sir Stephen argues, by a new social menace: Twitter abuse.

The evidence for this is all around us. We have seen how the fringes of the independence referendum debate have been dogged on both sides by unpleasant character assassination. Children fear online cyber-bullying almost as urgently as its real-world parallel. Celebrities in particular are treated as fair game for sometimes pretty appalling public commentary.

It is clear that a number of factors lie behind the change. Young people are less likely to be on the streets than in previous generations. When they are at home, they have easy access to the communications technologies with which to air their views, unpleasant or otherwise. Meanwhile urban environments have become less vulnerable to vandalism - anti-graffiti paint, CCTV and "smart" design have had an impact.

Graffiti's "move" online raises several questions. The first is whether it is better to have such scrawlings carried out via the keyboard or the aerosol and the marker-pen?

While some would argue it is possible to turn the computer off and ignore online unpleasantness, it does not appear that way to young people, for whom life online is increasingly important.

Meanwhile the shocking case of Facebook "troll" Stewart McInroy, who was sentenced to 10 months yesterday for pretending to have killed missing Glenrothes man Allan Bryant Jr, and taunting his family, shows that online offensiveness can be just as damaging as writing on walls, or more damaging.

So, especially when aggravated by misogyny (as with the abuse of campaigners calling for more women on banknotes) racism, like that suffered by Glasgow councillor Feargal Dalton, or homophobia, such actions are not trivial.

The next question is whether it is appropriate for the police to intervene. It clearly is.

Not all online conflict is antisocial, still less criminal behaviour. Political debate and freedom of speech must be protected.

Policing this issue will plainly involve the force in some grey areas, where officers will have to determine what is name-calling and fair comment and what is beyond the pale.

But the same judgments already have to be made with hate crimes. As Sir Stephen argues, abusing someone online is no different from doing so offline. If it is antisocial behaviour in one medium, then the same law should apply in another.