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i.e., that they form a closed triangle:
h is measure can be intuitively understood with respect to recurrences in
the phase space. We identify a recurrence of states by a network link: close
points on the phase space trajectory are connected by a link. h reeconnected
points form a triple, but only if all three points recur closely to each other,
thus forming a triangle. Such a triangular coni guration will remain along
the phase space trajectory if the dynamic is very regular (recurring states
remain recurring over a long period of time). However, if the dynamic is
chaotic, then parts of the phase space trajectory that were initially close
will subsequently diverge and the triangular coni guration will break down,
although the corresponding triple nodes might remain interconnected for
some time. h e probability of i nding triangles is therefore higher for regular
dynamics but lower for chaotic dynamics. h is explanation is, of course,
rather simplii ed but a theoretically substantiated explanation can be found
in Donner et al. (2011).
In order to calculate the probability that triples also form triangles we need
to compute the number of connected triples and the number of triangles,
which can be achieved directly from the recurrence plot but excluding the
main diagonal.
A = R - eye(size(R));
h e number of triangles and triples is then
numTripl = sum(sum(A * A));
numTria = trace(A * A * A);
and i nally, the transitivity coei cient is the fraction
Trans = numTria/numTripl
which yields
Trans =
0.5930
h is number means that the system does not have regular dynamics (which
would yield a transitivity coei cient close to one).
Changes in the dynamics, such as transition points and regime changes,
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