"expand their network" is correct and usable in written English.
You can use it when you want to refer to the growth of a person's or organization's connections, contacts, and/or relationships. For example, "The company has made it a priority to expand their network of international partners over the past two years."
Some big companies, meanwhile, are seeking to expand their network of small suppliers.
The government is offering a financial incentive for the grid to expand their network".
Since then, their channel and popularity skyrocketed, allowing them to expand their network to include several more personalities, each with their own YouTube channel.
Their immediate goal was to move out of the dorms and pay off the credit card debt they had amassed trying to expand their network.
Domestic code-share agreements became popular between major airlines and regional feeder carriers as a way for major carriers to expand their network of flights without having to serve low-volume commuter markets directly at high costs.
He explained that for carriers, buying spectrum is the easiest way for them to expand their network, but newer technologies, like improved antennas and techniques for offloading mobile traffic to Wi-Fi networks, could multiply the number of mobile devices that carriers can serve by at least tenfold.
Outsiders might expect insurance companies to expand their networks to treat additional patients next year.
The deal underscores the aggressive efforts by content and telecommunications companies to expand their networks, in preparation for potential advertising and e-commerce opportunities over broadband networks.
Over the years, code-share affiliations were formed among United States and foreign airlines, as carriers searched for ways to expand their networks internationally.
