Information Technology Reference
In-Depth Information
5
1
4
3
2
1
0.01
0
0
0.5
1
W
0.0001
0
1
2
3
4
5
W
Fig. 6.2. Histogram of energy barrier heights for the three analyzed sequences. Empty circles
correspond to the homopolymer, lled squares and crosses respectively to the slow{folding and
fast{folding heteropolymers. In the inset the rightmost part of the histogram is reported in lin{lin
scale.
The dierence in the eects of renormalization on connectivities deserves to be
investigated in deeper detail. Indeed, while the number of nodes clearly decreases
with renormalization, it is much less obvious how should behave an intensive quan-
tity as the local connectivity. Analyzing the details of the renormalization process
one might notice that, at each renormalization step, the average number of con-
nections might both increase or keep unchanged according to the amount of shared
connections between the two coalescing nodes. If the coalescing nodes have k and
k
0
connections of which they share m, the resulting renormalized node will have
k + k
0
m2, where the2 term takes into account the disappearing connection
between the two nodes. The two opposite cases that might take place are m = 0
(no shared connections) and m = k1 = k
0
1 (the two nodes share all connec-
tions) which lead to a nal node with k + k
0
2 and k1 connections respectively.
During renormalization on a realistic graph m will in principle take all possible
values. Therefore, if m is small in average the connectivity of the graph will tend
to increase, while it will decrease for large m values.
In order to conrm that the observed drop in the connectivity of heteropolymers
actually depends on an high percentage of shared connections between coalescing
nodes we analyze the relation between node connectivity and their size, that is the
number of zero{temperature nodes that coalesce on a given node. In Fig. 6.3 we
plot the degree of the nodes of the renormalized graphs of the three sequences under
study at T = 0:08 versus their size. Data have been averaged at every node size for
display purposes. A least{square t with a power law (dashed lines in the gure)
highlights a clear dierence between homo and heteropolymers. While in the rst
case the best tting exponent is 1, in the other two cases it is signicantly smaller,
indicating that renormalization in the heteropolymers causes the deletion of many
connections.
