There were lots of ways to raise your profile if you had ambitions in the Roman world, especially if you came from an upper-class family. Organising (and paying for!) public games was a good tactic for example. Some individuals clearly had very high public profiles and probably had a great deal of influence, both in and out of office.
It also helped to be well connected. You might be able to influence those who held power, even if you didn’t win elections yourself. Wealthy and powerful member of society would have ‘clients’ (poorer, less powerful members of society) who benefitted from their support or influence. Clients could ask their ‘patron’ for assistance, and patrons could insist that their clients supported them in return.
The dinner parties which were a central part of Pompeian social life were also opportunities to meet important people and discuss key issues. There were no strict rules about who got invited, so the guest list could be varied and different people could meet together.