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Does anyone have any thoughts on the SEO value of outbound links on a local search site?
Can I change that question to...
Does anyone have any thoughts on the value of outbound links on a local search site?
Yes, Outbound Links or OBLs are part of the overall process. I like to spread them out across many pages as opposed to focusing them all on one page. And, at times, I will include the actual URI (web address) below the Anchor Text (the text in the link). This is for printing purposes. I want to make sure those printing an article or something else where the visible URI is necessary, have a web address to reference on their printout.
Just think of it as one part of the overall equation. You have IBLs (Inbound Links) and you have OBLs (Outbound Links). One compliments the other. And, don't expect people to link back to you just because you've linked to them. If you feel the resource you are providing is of value to your visitor, then by all means, link directly to the page of the resource when applicable. If you are sending referrals to that resource and they are tracking activity on their site, they will see it and investigate. ;)
...value of outbound links on a local search site? Eg, all things being equal, does a business profile page with a url outperform the same page without the url?
I think by "url" here you may be talking about having active links (as opposed just to the urls on the page or the names of other sites without links).
Two considerations, I think... If your active links are to useful sites, then you will be increasing the utility of your page over time for the user, and that will be affecting other factors (like inbound links) that will cause your page to do better over time.
In the here-and-now-snapshot of this very moment, all other things being equal... outbound links to a group of on-topic, high quality sites will, I feel, in themselves be seen as quality factors, and will increase your "hub" score. There are a great many factors in the ranking equation, and I'm guessing that the hub score is one of them.
A hub has extraordinary value...hence the value of outbound links. But you don't get to that level immediately. It takes time and lots of inbounds initially, IMHO.
Regardless to be a valuable resource for your visitors posting urls is worthwhile.