Connections has not been tested with WordPress Multisite, but is has been reported by several users that Connections does work without issue. Please follow this process to properly activate Connections on WordPress Multisite.
If you have activated Connections for the network, you have to disable the network activation. Then go into the multisite settings and check the box for plugin administration. Next go into individual sites you wish to use Connections, you can then choose which non-network activated plugins you want to activate. Connections should show up in this list, activate it.

Hi… I’ve recently had contact with you on the WP.org forum.
I’m developing a site for a university dept, which is hosted by the university IT org, so I don’t have direct control over most parts of the (multisite) hosting environment. I have asked them to look into installing Connections for me, but they were reluctant, because of issues related to multisite. Here’s what they said:
“I just took a quick look and in particular generated thumbnails are have
been showing up as broken images. Also the plugin uses static
directories to store things like images and templates so every site
using the plugin share the same directories when storing files. This
has the potential for users affecting or having access to files from
other people’s sites.”
I’m not sure what the part about broken images means, but maybe you can make sense out of it. Is there any hope that you might make changes in the near future that would make my IT org more comfortable about installing this for me? Thanks.
It functions but does not carry across sites. You can add and manage contacts from the main site in the multisite network but it will not share those connections across sites.
It used to work well for me in Multisite until the recent updates. My issue now is that the directory shows for public users but not for anyone logged in as an author. All of my sub ste authors receive a “No Results” page when looking at the directory. I have also changed the short code to [connections public_override='true'] Doesn’t work here.
@Kevin
Can you please check the settings on the Connections : Settings admin page under the General tab and the Connections : Roles admin page?