Examples A straightforward example of a network system with variable topology is that of a family of graphs generated over a fixed set of vertices by changing the graph edges or connections between its vertices.
The idea of a varying topology has been introduced to describe possible topological distinctions in bio-molecular organisms through stages of
development, evolution, neo-plasticity, etc. This is indicated
schematically in the diagram
below where we have an
-stage
dynamic evolution (through complexity) of categories
where the vertical arrows denote the assignment of topologies
to the class of objects of the
along
with functors
, for
:
In this way a variable topology
can be realized through such
-levels of complexity of the development of an organism.
Another example is that of cell/network topologies in a categorical approach involving concepts such as the free groupoid over a graph (Brown, 2006). Thus a varying graph system clearly induces an accompanying system of variable groupoids. As suggested by Golubitsky and Stewart (2006), symmetry groupoids of various cell networks would appear relevant to the physiology of animal locomotion as one example.
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