It doesn't look like you can get this information using the v3 API at this time, but I believe what socialblade is calling networks is the content owner or 'Partner Name'.
In an API response, a feed entry contains a partner video if the entry contains a <media:credit>
tag for which the value of the yt:type attribute is partner.
<media:credit role='uploader' scheme='urn:youtube' yt:display='partner_name'
yt:type='partner'>
partner_name</media:credit>
Described here: https://developers.google.com/youtube/2.0/reference#Video_Feeds
It's probably just a matter of assigning channels which only contain videos by a distinct partner to that partner's 'network'. (Or just as likely any videos.)
Edit: Since that demonstrably appears not to be accurate, I'll point out that it appears the analytics API does have that information, but from the description it sounds like it only allows reports on channels your own "network". That being said, SocialBlade has to be getting their information about other networks from somewhere:
Content owners who participate in the YouTube Partner Program can also use the API to retrieve reports that aggregate Analytics data for all YouTube channels that are linked to their content owner account.