This paper concerns fundamental identities in the study of age of information (AoI) in gossip networks. We recover known recursive identities for arbitrary kth moments of the age process based on the recent connection between AoI and first passage percolation. Apart from the connection to percolation, our proofs are more concise and can be followed using only elementary facts from probability. Our argument generalizes some techniques known in the statistical physics community, and we remark on connections to the Eden model.