If you need to look up an index, there are a couple of ways of doing it:
// use a variable
$prop = 'my-crazy-property';
$obj->$prop;
// use {}
$obj->{'my-crazy-property'};
// get_object_vars (better with a lot of crazy properties)
$vars = get_object_vars($obj);
$vars['my-crazy-property'];
// you can cast to an array directly
$arr = (array)$obj;
$arr['my-crazy-property'];
If you need to work within a string (which is not your best idea, you should be using manual concatenation where possible as it is faster and parsed strings is unnecessary), then you should use {} to basically escape the entire sequence:
$foo = new stdClass();
$foo->{"my-crazy-property"} = 1;
var_dump("my crazy property is {$foo->{"my-crazy-property"}}";
Since you mentioned that this is LinkedIn's API which, I believe, has the option of returning XML, it might be faster (and possibly cleaner/clearer) to use the XML method calls and not use the objects themselves. Food for thought.