Can you explain the network effects? I'm not sure I agree.
While a GDS is technically a "2-sided network," I wouldn't say that they have 'network effects' - each GDS offers to a travel agent the same access to the same air inventory. These effects used to be true when there were no alternatives for a travel agent booking airfare (pre-internet), but today, the Amadeus GDS offers little that's different from Sabre or Travelport's GDS. Separately, the GDS itself is in secular decline. It's an outdated system that rips off the airlines, who are rushing to adopt NDC to work around the GDS. The only reason the GDS still exists is that travel agents earn a kickback when they use it.
IMO their software business is their crown jewel - Amadeus is viewed as the tech leader in airline/hotel customer reservation system (CRS) software and they've gained share over time. It's a super sticky business with nice economics.