La solution consistait à ajouter un argument de port à la fonction mysqli_connect.
$db_host = 'MY.IP.ADD.RESS';
$db_user = 'user';
$db_pass = 'password';
$db_database = 'database';
$db_port = '3306';
$link = mysqli_connect($db_host,$db_user,$db_pass,$db_database,$db_port);