I think the user should specify if he was to show only mood value or causes of the mood too, because sometime it can be something personal, like relationship fact, health, money troubles, lottery win, etc..

Maybe this could be a setting per time or a global privacy setting

when I can choose the dimension of the posted notification, Id could choose between one line, with the actual mood, a medium with actual and the modification (lower, highter, stable than) from previous, and full, with the graph image

maybe that the user choose this, I update the mood a pair of times at day, in my case Id like to notify all that reads my stuffs (like I my photoes upload, my new photo tags, etc..) of this, every time.
I have facebook in italian and dont know the english for different notifications kind, I know they can be placed in my profile home and in other homepage feed, 2 different settings could be done

maybe facebook networks, location network and facebook direct linked friends network (one graph per user with average data from any friend with the application)

The base would probably be a good implementation of link between tags, there are standard graph implementation, I dont know the backend architecture but a link betheen 2 tags will probably end in the DB in a pair of 2 tags, tag A is correlated with B if there is an entry (A,B). Other entries can define the kind and weight of the repationship, allowing future extensions.

surely herarchic grouping is a step over and need a proper design, the same for a clustering algorithm, even if given a graph of tag with links between tags there are various standard algorithms to guess clusters.

Hierarchic grouping can be useful, for instance the user can say his mood is low for rain, and after his mood is low for snow, and then put rain and snow as subtag of "bad weather", so a statistical analysis can take snow and rain together.
An automatic clustering algorithm could be implemented after, such that it propose groups of linked tags to be labeled as single cluster by the user.

Im using now italian words as tag, and sometimes composite phrases, in general I think that an user-driven grouping method (like manual, on proposal, graphical, by multiple selection, etc..) could be more effective for all people who use complex phrases or not english words.

I can suggest to make a pair of google accounts and use them from 2 different browsers

Examples for previous:
> Single tag value to sum to the average value 5 obtaining the final value:
message:
+2 eaten, +1 seen my sister, -1 raining
total become: 5+2+1-1=7
> Single total resume with factors weights
7 2 eaten, 1 seen my sister, -1 raining
total:7
in this case each tag is recorded with its value but doesnt affect the total mood wich is taken from the first number

Id like to put all the partial moods in one chat message, like now, maybe with single relative values, will post some example to explain

When I post my mood and the cause of it, sometime there are good and bad things toghether, so the mood is the sum of both positive and negative values, It would be better to have some manner to express the mood as sum of factors that can be positive and negative

a predefined per-user set of tag will probably induce the user to produce data more suitable for statistic analysis, forcing the reuse of a single tag for different similar cases