There seem to be two proposals, neither of which seem like particularly good ideas:

1) Proposal for hybrid POS/PON: Non-linear proof of stake rewards. Rewards are effectively per account up to a threshold. Seems ripe for exploitation by botnets.
2) Proposal for POI: Seems to be the Google Page Rank algorithm (i.e. Principal Components) with transactions replacing links between webpages, and the sum of coin ages replacing the content quality measure (i.e. the prior distribution over nodes). Seems equally vulnerable to exploitation by botnets. (Suppose there are N legitimate nodes, all trading, N botnet nodes, not trading, and 1 master node for the botnet. The botnet nodes have arbitrarily old coins so receive substantial weight, despite their lack of any trading except with the master. All of this weight then flows to the 1 master node for the botnet, who must then get the highest "PageRank" in the network.) In any case, rewarding old coins is the opposite of what you want to do. The existence of old coins implies that the currency isn't actually being used! By rewarding old coins you're effectively taxing transactions, which imposes a large efficiency cost.