Being so serious-minded, it's also quite exclusive -- almost snobby. It's not like Twitter where you can immediately start following people, confident in the knowledge that some will follow you back. If you want to connect with someone you already have to know each other to some degree (well, usually). So building a network of followers is a bit more involved than it is on other social media. This is where Linkedin groups are really useful.
They are without a doubt the best way of getting on the radar of lots of other Linkedin users in your niche. Really, all you have to do is participate in them. Just liking stuff that's shared will keep your photo popping up on the right hand side. Commenting -- and thoughtfully -- is even better, of course. (Just on this point: I think that using a photo is a better option than a logo. Sure, Linkedin is selective and professionally oriented, but it's still a social network. So people want to get to know you as a person first and foremost.)
Do this on several of them and you're sure to get some connection requests. I know this from experience. After every burst of activity I have on the groups, the invitations follow.
You can also be proactive, and initiate connections with people you see in the groups. But I would make a point of establishing some contact in the discussions first either by liking or commenting on their posts (or both). Make sure that they respond to you in some way (preferably positive) in these threads. Then you'll know for sure that they're aware of who you are. So they'll be much more likely to accept your offer.
Just on this aspect of familiarity: Twitter, Facebook, and Google Plus are also good places to get to know people who are on Linkedin. I've sent a couple of Linkedin connection requests to people I've been communicating with on these sites, and they've quickly accepted. So, that's another reason to be active on several social networking sites.
As well as being excellent for growing your network, Linkedin groups can also be a great traffic source. People are very interested in the content you submit to them. Every time I've shared blog posts on the groups I've received several clicks. Of course you don't want to be spamming these groups with your own blog posts all the time. You should share the work of others, too. But if you promote your own content occasionally you'll be getting some extra, quality traffic to it and building your authority at the same time.