what is the relationship between bandwidth and hash rate? What is the relationship between bandwidth and hash rate?
I have an AntMiner S4 mining at 2.1GHashes.
I understand that the miner has no hard drive, and does its operations purely in memory, and that the device sends and receives data when calculations are complete. 

My question is what does running a Bitcoin Miner do to network load (I am assuming network load is a function of Hash rate).
Hash rate measures how fast your miner can attempt to find a nonce that satisfies the proof of work.  The speed at which it performs this task is not related to network bandwidth.  It does not perform a network operation per hash.

Bitcoin mining still requires network usage, however.  Your miner will request a unit of work from the mining pool, which is really just a block header.  This is only 608 bits (not bytes) of data.  Your miner's job is to fill in the remaining 32 bits of the block header (total size is 640 bits), called the "nonce", and hash it to determine if the proof of work is satisfied.  If you have exhausted all 2^32 options, then you will request a new unit of work.

So your miner will request units of work more often the faster it can hash, but this amount of network traffic is small.  Since each unit of work is only 608 bits, this is a very negligible bandwidth usage.  You won't see a significant increase in profitability by increasing bandwidth.

You may want to check out this answer to How do mining pools distribute work effectively?, specifically the comments.