Information Technology Reference
In-Depth Information
information obtained from well-connected individuals than from less well-connected
individuals.
The Widrow-Hoff “delta” learning rule was modified by removing the learning rate
and replacing it with the function
f
(
s, r
),where
s
and
r
are the sender and recipient of
a piece of information (a shout).
⎧
⎨
0
.
3+
0
.
2log
E
(
s
)
E
(
r
)
log(
R
max
)
:
E
(
s
)
≥
E
(
r
)
f
(
s, r
)=
⎩
0
.
2log
E
(
r
)
E
(
s
)
log(
R
max
)
0
.
3
−
:
E
(
s
)
<E
(
r
)
The function,
E
, gives the number of neighbours (degree) of an agent, and
R
max
is
the largest ratio of edges between two adjacent agents within the market. This adaptive
learning rate weights information according to relative connectivity within the market,
i.e., the ratio of the sender's connectivity to the recipient's connectivity determines the
learning rate. When the sender is more highly connected than the receiver the informa-
tion received is more likely to be accurate and so more adaptation occurs. When the
receiver is more connected, the receiver's current picture of the market state is likely
to be more accurate than the senders and so less adaptation occurs. The value is nor-
malised by the maximum ratio present in the market in order to ensure that the learning
rate remains within the same bounds as standard ZIP traders. Connectivity ratios are
log-scaled to ensure that learning rate adaptation is sensitive to the small differences
in connectivity that characterise most sender-recipient pairs in a network generated by
a preferential attachment process (where there will be only a few very well-connected
individuals).
4R su s
Figure 1(right) shows the results obtained with the modified learning rule. All other
parameters are the same as the previous scenario. As before the deviation of the val-
uations decreases over time. Again the most connected agents converge more quickly
than the least connected agents. Figure 2(left) affords an easier comparison between
the two studies. The least well-connected agents converge more quickly when using the
modified learning rule than when using a fixed learning rate. At all times they have a
lower deviation from the optimum price than agents using a fixed learning rate. By the
end of the market they are significantly closer to the optimum price than those using
fixed learning rates (t-test,
p<
0
.
0001). Over longer experiments they may, however,
eventually converge to the same value. The convergence of the better-connected agents
is very similar, with or without the presence of the modified learning rule, although
convergence is slightly retarded in the former case. By the end of the market, however,
both groups have attained very similar values. It must be remembered that, as a con-
sequence of the preferential attachment scheme that generates the market network, the
distribution of agent connectivities exhibits a power law. Poorly-connected individuals
vastly outnumber well-connected agents. As a result, even if the adaptive learning rate
does significantly retard the convergence of well-connected agents, only a very small
