So, what does cluster-01 exactly represent here? an local cluster IC ip, any data lif, or...
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To directly address the above question:
'cluster-01' (or 'cluster1-01' as in the subject) specifies a node that you want the ping request to be sent from. You can get a list of node names available in your cluster by using a "node show" or "cluster show" command.
For the network ping command you are saying that you want the outbound ping to be send from the "node" that has been identified in the command. But it can be confusing to understand exactly which lif it's being sent from.
If you want to be more granular you canl specify the "-lif-owner" and the "-lif" in the network ping command. Then you will know 100% what lif is being used. Try it out, I think you will like it.