A Similarity network is one that expressed how similar entities (expressed as the nodes) are. The degree of similarity being the weight of the node.

The weight matrix $A_{ij}$ represents level of similarity between entities $i$ and $j$ in the network. A similarity network is almost always complete (the only deviation from completeness is from nodes that can't be compared for some reason).

For example, if we have a matrix of votes, we can define $A$ as:

$A=\frac{\text{times i and j voted same ways}}{\text{total number of times both i and j voted on same measure}}$