Crossword Genius

With a world wide network, we get big-headed (3)

Ross

I believe the answer is:

web

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'with a world wide network' is the definition.
I can't judge whether this definition defines the answer.

'we get big-headed' is the wordplay.
'get' says to put letters next to each other.
'big-headed' becomes 'b' (this could be a standard abbreviation which I don't know about).
'we'+'b'='WEB'

Can you help me to learn more?

(Other definitions for web that I've seen before include "Network of fine threads" , "The world wide one of the Internet" , "Interwoven structure" , "Network, may be World Wide" , "Lattice" .)

I've seen this clue in the Evening Standard.
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