O que substituir mysql_result para MySQLi

if($query_run = mysqli_query($link, $query)) {
    // This gets you one row at a time, use a while if there are multiple rows
    // while($row = mysqli_fetch_assoc($query_run)){}
    $row = mysqli_fetch_assoc($query_run);
    $count = $row['count'];
    // Do whatever with $count
}