In a recent work we have developed an analysis system based on valued abstract argumentation to model and reason about the accepted and rejected tweets of a Twitter discussion. Given a Twitter discussion, the system builds an argument graph where each node denotes a tweet and each edge denotes a criticism relationship between a pair of tweets of the discussion. In the social network Twitter, a tweet always refers to previous tweets in the discussion, so the obtained underlying argument graph is acyclic. In this work we introduce and investigate a natural extension of the system, referred as author-centered analysis system, in which tweets within a discussion are grouped by authors, such that tweets of a same author represent his/her opinion in the discussion with a single node in the graph, and criticism relationships denote controversies between the opinions of Twitter users in the discussion. With this new approach, the interactions between authors can give rise to circular criticism relationships and thus the underlying argument graph can contain cycles. The output of the author-centered analysis system is the set of authors such that their opinions are consistent and are globally accepted within a Twitter discussion. The system can be of special relevance for assessing Twitter discussions in fields where identifying groups of authors whose opinions are globally compatible or consistent, but at the same time are widely accepted, is of particular interest.