If I understand your question right, you want to visualize your own data as a network. In this case, you would clearly not want to use a web resource like STRING (of which I am one of the developers). Whereas STRING can show networks, it is first and foremost a database of functional interactions among proteins, and the purpose of the STRING network viewer is to allow the user to view the data from STRING as a network.
However, I think you are putting the cart before the horse by saying that you want to visualize your data in VisAnt, which says which tool you want to use rather than how you want your example data shown. What should be the nodes in your network? Is each protein a node? Or is each complex a node? What are the edges? If the proteins are nodes, do you want to represent each complex as a fully connected component (also known as matrix representation)? Or is this in fact pull-down data for which the first protein in each line is the bait (in which case the spoke representation would be more appropriate)? Or are the edges interactions among complexes rather than proteins?
Only once we know those things can we tell 1) which tool is best suited for the job (which might be Cytoscape instead of VisAnt) and 2) how to prepare the data to make the end result look the way you want.
Edit after answer to questions posted above
If you want to show each complex in a matrix representation, what you need to do is to turn your file of complexes into a file of binary interactions, i.e. a file in which each line represents an edge in your network. For your example, that would be:
Q13439 Q9NV70
Q13439 Q9UPN3
Q9NV70 Q9UPN3
P21980 P12277
O43318 O15111
O43318 Q13748
O15111 Q13748
P21980 P51178
To do this, I would write a simple AWK script, but it could be done easily in any scripting language.
Once you have the file, you can simply load the network into Cytoscape using the File->Import->Network->File...
menu point. You can probably load the same file into VisAnt or other network viewers, but I am not familiar with them. After that you will almost certainly want to load in some mapping between UniProt accession numbers to label, e.g., P21980 as TGM2 in the your visualization.
Given the discussion on my answer, I think the best would be that you post a new question in which you describe: 1) What is the actual data that you started from? I suspect it is a large-scale AP-MS study. 2) Which analysis steps you've already done to arrive at complexes? I assume you have run some clustering algorithm on some representation of the data. 3) What is the actual goal here? I guess it is to make a figure that gives an overview of the complete dataset as a network. The figures from other papers that you refer to cannot be made from this representation of the complexes alone. I thus think you got derailed somewhere along the way from the raw data to producing a figure.