What characteristic behaviour does our community exhibit? Is it possible to capture this through some statistic or visually through graphs?
Certainly! There is a host of information available through the SEDE. This post is meant to collate some of the discussions contained within How can we explore the content of TeX.SX? as a showcase of the community's behaviour on an annual basis. For those interested in contributing statistics, considering adding an answer to the linked post.
Please provide an answer in the form of a SEDE query and if not, explain in as much detail as possible the type of statistic you'd like to see. Consult the Database schema documentation for the public data dump and SEDE for information on the variables and content. Well-formed constructions will make its way into this post's answer section shortly after New Year.
The idea is to highlight activity of the community and not to isolate individual behaviour (although this may be the case naturally when it comes to "outlier users" - like those with highest reputation, for example). This is not an absolute rule; exclusions may apply.
Examples of good inquiries/statistics:
What is the total number of badges awarded by class over time?
This statistic attempts to see whether merit (through posts, say) and participation (through moderation, say) act similarly. Does this change over time?
Is there a better time to post a question based on community activity?
Facebook is a good example of this, where posts aired at the time when most of your friends "wake up" (in the morning, hopefully) receives far more attention than those posted when everyone's asleep.
Examples of bad inquiries/statistics:
When will I receive my next
<type>badge? Or, what badges will I receive next?
This statistic is specific to an individual (again, exclusions may apply...) or doesn't attempt to address community behaviour.
What time of day do most community members vote?
Voting is (and always will be) anonymous. One way of protecting this anonymity is by removing time-stamps from votes in the SEDE (keeping only date-stamps). As such, it's not possible to ascertain this effectively.