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%%%%%%%%%%%%%% NOTES %%%%%%%%%%%%%%%%%%%  %\appendix \appendix  \section{Irrational individuals?} 
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By multiplying the two matrices, a new opinion matrix is generated, representing the opinion state of agents after one influence step (for instance after they exchange information with each other). DeGroot, using the associativity of matrix multiplication, shows that whether the opinions of a group of agents converge in the limit depends on the underlying influence graph. If multiplying the influence matrix by itself converge in the limit, then any distribution of opinions of the agents will converge too.  %as Jackson puts it: no matter what beliefs p(0) the agents start with, they end up with limiting beliefs corresponding to the entries of $p(\infinite)=lim_t T_tp(0)$.   More precisely, convergence depends on (each connected and closed component) of the influence graph being aperiodic (that is, the greatest common divisor of all directed cycles lenghts is $1$.): ``convergence in overall society hold if an only if each closed and strongly connected set of nodes converges, which happens if and only if each such set is aperiodic." (jackson, p.233) \cite[p.233]{jackson08social}  %Moreover, any strongly connected and closed group reaches a consensus for any initial opinion vector if and only if it is aperiodic (jackson, p.234). (corollary 8.1) An overall consensus is reached iff there is exactly one strongly connected and closed group of agents and it is aperiodic. (Corollary 8.2, Berger) A consensus is reached iff there exists n such that some column of the influence matrix multiplied $n$ times by itself has all positive entries. Once a column is all positive, it stays positive forever! (a column with all positive entries represent the fact that at some point one agent influences all other agents, showing that all agents must have a path to some agent and so there is exactly one strongly connected closed group.