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Let's say peer A creates a smart contract and deploys it. If peer B wants to sign that contract how he will be able to do it? Also if peer A wants to send a transaction how he will be able to sign the transaction using his private key? If peer B does not sign the contract created and deployed by peer A,does it mean peer B won't be part of the network.

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Ethereum handles abstracts all the signing logic away from you. Ethereum allows you to define a smart contract which is as restricted set of mutations you can make to the global verified state. All you need to do is create a contract with the correct restrictions. To be a part of the network, all nodes need to do is collect information from other peers and send information to them. Participating in contracts is voluntary.

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The contract is "signed" when it is deployed (the corresponding trnsaction is created and "mined" in block). All the other nodes just can verify it (the block), but not sign it. So the question is not "Would be some node a part of the network", but "would be this block a part of the network". Usually you should wait some time (or some number of blocks generated, after "your" block was generated) to be sure, that blockchain became stable. For Ethereum that stabilization time is considered 12 blocks (or ~3 minutes)

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