I think I maybe rather late, but here is my code for getting shares,likes and comments for URL, please observe that it doesn't require an access token, hope this helps in some way!.
string url = "http://www.youtube.com";
string QUrl = "https://graph.facebook.com/?fields=id,share,og_object{engagement{count},likes.summary(true).limit(0),comments.limit(0).summary(true)}&id=" + url;
System.Net.HttpWebRequest Request = (System.Net.HttpWebRequest)System.Net.WebRequest.Create(QUrl);
Request.ContentType = "text/json";
Request.Timeout = 10000;
Request.Method = "GET";
string content;
using (WebResponse myResponse = Request.GetResponse())
{
using (System.IO.StreamReader sr = new System.IO.StreamReader(myResponse.GetResponseStream(), System.Text.Encoding.UTF8))
{
content = sr.ReadToEnd();
}
};
var json = JObject.Parse(content);
var like_count = json["og_object"]["likes"]["summary"]["total_count"];
Console.WriteLine("Like Count :" + like_count);
var share_count = json["share"]["share_count"];
Console.WriteLine("Share Count :" + share_count);
var comment_count = json["og_object"]["comments"]["summary"]["total_count"];
Console.WriteLine("Comment Count :" + comment_count);