Is there a graph showing the growth and growth rate of electronic
connexions on the planet, ideally since the invention of the telegraph
to today (including phones, cell phones, faxes and all computers and
networks of).

Brunews -- good to see you back at Google Answers!  Market research
and/or census numbers will reveal more commonplace statistics, such as
the number of telephones or the number of fax machines.  In emerging
markets, penetration analyses are are routinely done -- doing such
things as comparing how quickly radio/TV/VCRs/cellphones/PCs penetrate
homes. To the great surprise of many, saturation of consumer
electronics markets takes 10-15 years -- but then homes end up with 2,
5 or even 10 of each item.  (How many CD players are in your house,
counting the cars?)

Reading into where you might be going -- penetration analyses and
copious statistics exist for Internet usage.  I might recommend
looking at Nielsen's Net Ratings page for a brief example:
http://www.nielsen-netratings.com/

Best regards,

Omnivorous-GA

Omnivorous: thank you for the link.

I know that research about the number of connections and their growth
rate is being done by telcos. I will find it.

Thank you for your help.