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how is the relation betweeen network hash rate to healthyness of bitcoin, such as bitcoin value compared to USD, or the growth rate of this currency?

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The network hash-rate is mostly reflective of Bitcoin's exchange rate. If the exchange rate goes up, then mining's hash rate will follow. It will lag, but it will mostly follow the direction and to a lesser degree, the magnitude of the change in the exchange rate.

While there are some who will mine regardless of profitability, each additional bit of hashing has the effect of lessening profitability -- displacing those who are mining for economic gain.

So as a whole, the network hash rate itself doesn't say much about the currency's health. Now for each individual that goes and spends $20K worth on mining hardware, you can assume that one individual believes in bitcoin's future enough to risk that kind of an investment, if profit was the reason for pursing such a venture.

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You can find graphs charting this relationship at http://bitcoinx.com/charts/. (scroll down)

Network hash rate and exchange rate are certainly correlated, but the relationship is very complex, I think it would be inaccurate to say that one depends on the other. In general, if price increases due to an outside factor (eg. new adopters buying Bitcoins) hash rate will go up, as you can see in the chart at the above link around July 2011, when price spiked and hash rate followed.

Impact of hash rate changes due to outside factors on price is harder to predict. Hash rate increases can lead to increased investor confidence in the currency (which would increase price), but they can also drive miners on the edge of profitability out of mining and potentially even Bitcoin, decreasing the price. (less participants in a market generally leads to price decreases, though not always)

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    Well put. I would add that the marginal cost of acquiring a Bitcoin, has a floor set by the current hash rate using the most efficient source of computation (energy costs). A rational actor will choose to mine for Bitcoins rather than purchase them, if the cost of mining is lower than the cost of exchange. But getting involved in mining has rather large fixed costs that make it difficult for most individuals to choose mining over purchasing; so one can expect the cost of Bitcoin to be at a premium over energy costs of mining for the near future.
    – mckoss
    Commented Mar 17, 2012 at 20:01
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Bitcoin hash rate is:

25 BTC / (physical cost to perform one hash in BTC) / 10 minutes.

There are two factors in this formula which are subject to change:

  • The physical cost to perform one hash in BTC is obviously linked (linearly) to the healthiness (trade value) of the currency. It's also subject to change with hardware improvements, but this has no effect on the overall security of the system. (Attackers would also benefit from it.)

  • Last, 25 BTC is in fact not fixed: it's an approximation of today's block reward + transaction fees miners get mining one block. In particular, every 4 years this reward is scheduled to be divided by two. Therefore, if the trade value of bitcoins is not multiplied by two every four years, the total effort spent on mining (to secure the chain) could decrease.

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