Great post. Learned a lot from this. Keep writing more.

WITH UserFriends AS (
    SELECT user_id, friend_id
    FROM google_friends_network
)
SELECT 
    uf.user_id,
    COUNT(DISTINCT gf.friend_id) AS mutual_friend_count
FROM 
 UserFriends uf
JOIN 
    google_friends_network gf ON uf.friend_id = gf.user_id
GROUP BY 
    uf.user_id;

How about

WITH UserFriends AS (
    SELECT user_id, friend_id
    FROM google_friends_network
), FriendsOfFriends AS (
    SELECT uf.user_id, gf.friend_id AS friend_of_friend
    FROM UserFriends uf
    JOIN google_friends_network gf ON uf.friend_id = gf.user_id
)
SELECT user_id, COUNT(DISTINCT friend_of_friend) AS mutual_friend_count
FROM FriendsOfFriends
GROUP BY user_id;

You can use below query in SQL.

WITH UserFriends AS (
    SELECT user_id, friend_id
    FROM google_friends_network
),
FriendsOfFriends AS (
    SELECT uf.user_id, gf.friend_id AS friend_of_friend
    FROM UserFriends uf
    JOIN google_friends_network gf ON uf.friend_id = gf.user_id
),
MutualFriends AS (
    SELECT fof.user_id, COUNT(1) AS mutual_friend_count
    FROM FriendsOfFriends fof
    JOIN google_friends_network uf ON fof.user_id = uf.user_id AND fof.friend_of_friend = uf.friend_id
    GROUP BY fof.user_id
)
SELECT user_id, mutual_friend_count
FROM MutualFriends;