A brief description of this measure is presented here, closely following the work of
Amaral et al. (2007).
Consider a system represented by a square matrix A, of order N and with all values non
negative. A part of the system of order m (m = 1, ..., N-1), is a square block A* of order m,
which has its main diagonal formed by m elements of the main diagonal of A.
Let A* be a part of the system. For example:
A*
a11 a12
a21 a22
A* can be considered a sub-system of system A. This sub-system is the more
autonomous (or, equivalently, the less dependent) the greater the values of its elements (a11 ,
a12 , a21 , a22) relative to the elements (a1j , a2j , aj1 aj2), for all j>2.
In order to measure the greater or lesser autonomy of the sub-system A*, the degree of
autonomy of sub-system A* can be defined as:
Ga (A*)
IA1
IA *1+1A **ll+11 a ***ll
where ∣∣M∣∣ represents the “sum of the elements of matrix M”, A** is the block of all the
elements of the columns belonging to A* with the exception of the elements of A*, and A***
represents the same thing for the rows. For example, if A* is the block defined above:
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