I can't quite distinguish them, because "communication" is always social? I guess 交际 is like my social skills and 沟通 is used with "dispute" but is that it and what about 交流？
I am sure all three terms can be found in dictionaries, but some of the definitions might overlap and cause new learners confusion, I think it is better to just list their main difference in my answer:
交流 = interact / exchange
交際 = socialize
溝通 = communicate
文化交流 (cultural exchange) ; 互相交流 ( interact with each other)
交際應酬 ( socialize and partying) ；與上流社會交際 (socialize with the upper class )
無法溝通 (unable to communicate); 以眼神溝通 ( communicate with looks in the eyes)
Informal speaking, “交流” is like exchanging idea and information.
I like using linedict.com for this, you can find example sentences which help us 老外 see how words are used.
Click the radio button Examples and enter a word. You get examples with English translations.
交流:talk in forum,exchange each other.
交际:make friends or something like that. or other mean talk in part-->noun,like 交际能力：ability of socialize
沟通:still means talk,but usual talk for some purpose ,like change other person mind
交流， communication / exchanging ideas
沟通, talking about something frankly
交流= vt.communicate, n. Communication 交际= n. Social networks 沟通= vt. Deeply communicate