What is a relationship with total hash rate(hash power) and average block generation time? This chart - hash rate means total hash rate of All minor?

I think the most significant factor for difficulty and block generation time is the hash rate of the biggest mining pool not total hash rate of all of minors.

I want to assure it that if there is a mining pool have 30% of total hash rate and this mining pool is the biggest pool.

Then this mining pool will succeed to find target hash value faster than others(stochastically).

Therefore, I think difficulty for block generation time will be set by this mining pool.

As a result, if other mining pools don't have a enough hash rate for example,  10% of total hash rate, then stochastically, it will takes 3 time more to find a traget value.

So i think the most significient factor is hash rate of the biggest mining pool

Am i thinking well?

And it seems like out of title, but i have one more question.

If fork is happened, Mining pool can recognize fork is happened?
This chart - hash rate means total hash rate of All minor?


No, it represents the hash rate of the network. It is determined by using the time it took to find the block and the current difficulty of the Bitcoin blockchain. To cite an example, the current difficulty bits are 0x1729fb45, which means we need to find a block header hash that is less than '0x00000000000000000029fb450000000000000000000000000000000000000000. This represents a one in 2.88e+22 chance of finding the header hash. Assuming block time of 10 min, this is 2.88e+22/600 = 47e+18 hashes/sec. If the block are found faster than 10 minutes the hash rate is higher, if it is found in more than 10 minutes, the hash rate is lower. That is what you see in the hash rate graph. 


  I want to assure it that if there is a mining pool have 30% of total hash rate and this mining pool is the biggest pool. Then this mining pool will succeed to find target hash value faster than others(stochastically).


No. Your big mining pool will only have a 30% influence over the difficulty adjustment. The rest of the network will have a 70% influence, which is more than what a solo miner has. The block time adjustement is based on how fast the blocks are mined (the algo tends to keep the average time at 10 min) not how much percent hash power each miner has.


  Can a miner find if fork happened?


If miner receives two different blocks at the same height then he will know that the network has forked as half of the network will be seeing a different tip versus others. When the next block will be mined, on top of either of them, the network will converge again.