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I want to know how to calculate expected rank of a binary matrix with distribution. I.e., the generated matrix are not randomly selected but with distribution.

A more intuitive example is the LT code, it generates the matrix with the given degree distribution. And the expected rank of a random matrix is similar with the random linear code. And I have found some useful results for how to calculating the the expected rank of a random matrix. For example:

Expected rank of a random binary matrix?

and the paper: RANDOM BLOCK-ANGULAR MATRICES FOR DISTRIBUTED DATA STORAGE

I think the paper is a really useful one that not only give the result of the probability of getting the full rank in a random matrix, but how to derive the results.

For now, I want to know how to infer the probability of getting the full rank in a matrix with certain distribution. And then infer the expected rank of the matrix. More specifically, suppose that the distribution of the node's degree is given, which is the same as the LT code.

I have tried my best to derive the result from the above given paper, but I didn't work it out finally.

Any hints would be appreciate!

desword
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  • Do you need a closed form for this, or would a complicated expression that would have to be evaluated by computer be good enough? – joriki Jul 16 '18 at 09:49
  • To others who are interested in this question but don't want to delve into the details of LT codes: As I understand the question, it asks for the expected rank of a binary matrix whose rows are independently generated by (in the simplest case: uniformly) randomly selecting a column count and then uniformly randomly selecting that many columns to be $1$ (with the others $0$). (Please correct me if I got that wrong.) – joriki Jul 16 '18 at 09:56
  • The case where the column count is chosen uniformly seems slightly more tractable -- would that be good enough for a start? – joriki Jul 16 '18 at 09:59
  • thanks very much for your interests. a closed form expression would be good. However, any hints about how to translate the degree distribution of LT code into the estimation of the rank of the matrix with distribution would also be appreciate. – desword Jul 16 '18 at 12:25
  • is the case of "column count is chosen uniformly" estimating the expected number of rank in a uniformly chosen random matrix? If yes, then I think the reference in RANDOM BLOCK-ANGULAR MATRICES FOR DISTRIBUTED DATA STORAGE has given a good answer. – desword Jul 16 '18 at 12:28
  • And yes, starting from the case of "the column count is chosen uniformly " is good. – desword Jul 16 '18 at 12:29

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