If you link out to irrelevant sites, Google may ignore the page, too – but again, it depends on the site in question. Who you link to, or HOW you link to, REALLY DOES MATTER – I expect Google to use your linking practices as a potential means by which to classify your site. Affiliate sites, for example, don’t do well in Google these days without some good quality backlinks and higher quality pages.
The above information does not need to feature on every page, more on a clearly accessible page. However – with Google Quality Raters rating web pages on quality based on Expertise, Authority and Trust (see my recent making high-quality websites post) – ANY signal you can send to an algorithm or human reviewer’s eyes that you are a legitimate business is probably a sensible move at this time (if you have nothing to hide, of course).
Ever heard of Maslow's hierarchy of needs? It's a theory of psychology that prioritizes the most fundamental human needs (like air, water, and physical safety) over more advanced needs (like esteem and social belonging). The theory is that you can't achieve the needs at the top without ensuring the more fundamental needs are met first. Love doesn't matter if you don't have food.