Every node in the network has some output depending on what inputs it receives. That output is forwarded to some number of other nodes, and is multiplied by the weight. I.e. if a single node is outputting 1 for example, but the weight for the connection from that node to another node is 0.3, then the input that other node receives from that connection is 0.3. So essentially it changes how important the output of one node is for the effect it has on a specific other node.
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