To do so effectively you need to bear the following in mind:
ONE: Provide the help they most want...
It's very easy to think you're being helpful while actually making life harder for your network or creating distractions. Be clear that the solution you are offering is specific to their challenge, not something that's loosely related. If you're introducing them to someone else, be sure that it's relevant before making the connection.
If necessary ask them to clarify exactly what they are looking for before taking action.
TWO: ..when they most need it
I once spoke to a printer who said that he could 'go the extra mile' for his customers by delivering ahead of their deadlines on a regular basis. However, he said, he would both lose the impact of the extra service by doing so regularly, plus he may inconvenience them if they didn't have the storage space.
Maximise the impact of the help you offer by timing it for when the people in your network most need your support and when it will make the biggest difference.
THREE: No surprises
If you're making an introduction between people in your network, make sure that both parties expect to and are interested in being connected.
Failure to get buy in to the purpose of the connection can lead to negative responses, confusion and wasted time. That's no help to anyone and will reflect poorly on you.
FOUR: Stay in the loop
If it's possible, keep yourself in the loop. Find out if your help, advice or introductions were helpful. Get a progress report and ascertain whether your contact has progressed and if there is any more help you can offer.
Show genuine interest and concern rather than simply ticking boxes.
FIVE: Learn from your experiences
Remember what help you've offered has had the biggest impact and make a mental note of the type of support each person in your network most appreciates.
Obviously, people will be looking for different things at different times; so you need to be aware of their circumstances and needs at any particular juncture, but you can certainly learn the type of offers to which each individual is likely to respond positively.