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This links panel was originally suggested and added by another person within the company, to improve SEO and drive traffic across company sites.
Probably not a good idea then.
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From a user prospective, common sense tells me that an aerospace executive is unlikely to be looking for news on agriculture for example
More confirmation that it's probably not a good idea.
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so removing, or streamlining the links panel, makes more sense to me.
I think streamlining might very well be the way to go. Figure out ways where it makes sense to the human visitors. Certainly there's nothing wrong with a company promoting their other business areas. And just by saying something like "we also have websites in these industries" and then linking to them, should not be a problem.
It's doubtful that those types of links would need to be sitewide, however. It would make more sense to link to them from a Corporate page, or an about us, or something like that.
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I guess my main concern is that linking to non-relevant content may damage, rather than enhance the ranking of our pages. Our sites are related because we are part of the same company, but they are not related by actual content.
By being part of the company, that IS a good relationship. There's nothing wrong with links to stuff that is in a different industry. Relationships between sites is not something that is easily detected. For instance, almost everyone travels, right? So is it unrelated to have a link to travel information on a site selling diapers? It seems like it would be, but if you have the same target market then it's more related then you might think.
Are TV shows wrong to show commercials for products that have nothing to do with what the show is all about? Why is an ad for a Hummer related to the show "Friends"?
In the same vein, what if I had a "Friends" website. Would it be wrong of me to link to the Hummer site? Or vice-versa.
You can't be worried too much about relevance, because there's no real way of judging it, nor should there be. We don't know what's in people's heads. What they might be interested in may not be what someone else thinks is relevant. But that won't ever be a problem to search engines. There's no way they would go around banning sites because they link to something that's a completely different industry.
It just would defy common sense completely. Search engineers are way smarter than that.
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I was under the impression (maybe wrongly) that having all our sites on the same IP C block would suggest an un-natural network, and links would be penalised or discounted by SE's.
Well see in my opinion, by putting them on separate C's you've now suggested an unnatural network where there was a natural one before. Natural that one company would link to all their own sites. It's much more unnatural for separate companies to link to completely unrelated stuff.
Hope this provides you and others with some food for thought.