if you want to update this in table. you need something like
UPDATE t
SET t.edge_weight = tmp.edge_weight
FROM [socialDB].[dbo].[final] t
INNER JOIN
(
SELECT [socialDB].[dbo].[final].[author_key_from] ,
[socialDB].[dbo].[final].[author_key_to],
COUNT(*) AS "edge_weight"
FROM [socialDB].[dbo].[final]
GROUP BY [socialDB].[dbo].[final].[author_key_from] ,
[socialDB].[dbo].[final].[author_key_to]
)tmp
ON tmp.[author_key_from] = t.[author_key_from]
AND tmp.[author_key_to] = t.[author_key_to]
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SQL Server MVP
http://visakhm.blogspot.com/