Mature heads of Royal Institution promote educational networks (5)
I believe the answer is:
ripen
'mature' is the definition.
(to ripen is to mature)
'heads of royal institution promote educational networks' is the wordplay.
'heads of' suggests taking the first letters.
The initial letters of 'royal institution promote educational networks' is 'RIPEN'.
(Other definitions for ripen that I've seen before include "Become ready for picking" , "come near to falling" , "Become ready to harvest" , "Mature, develop" , "Grow to maturity like fruit" .)