Unfortunately you can't query by the fields of referenced documents directly, it's better handled by relational databases. In MongoDB, this is of course possible but requires multiple queries: first you find the networks user belong to, then you should find the connections of the networks.
Of course, it's enough to get the list of network identifiers to make the connection query. This is easily done with the help of a query builder:
$networkQb = $this->dm->getRepository(Network::class)->createQueryBuilder();
$networkQb->field('owner.$id')->equals($userId);
$networkQb->select('_id'); // Limit results to the ID of the network only
$networkQb->hydrate(false); // Don't return Network objects but only plain array
$networkResults = $networkQb->getQuery()->toArray();
$networkIdList = array_map(function($result) { return $result['_id']; }, $networkResults); // This converts array(array("_id" => "1234"), array("_id" => "5678")) to array("1234","5678")
// Then we'll make the actual query for the connections, based on the id list of the networks
$connectionQb = $this->dm->getRepository(Connection::class)->createQueryBuilder();
$connectionQb->field('network.$id')->in($networkIdList);
$connections = $connectionQb->getQuery()->toArray();
This kind of query is still relatively fast as far as there are only a few networks per user.