Biology Reference
In-Depth Information
for nectar. Graphs consist of
N
elements forming nodes with a connect-
edness
K,
the average number of arrowheads terminating at each ele-
mental node. Connections can be regular, random, or however you want
to make them. Figure 2.3 has
N
= 5 and
K
= 5 (with symmetry of every
element connecting to every other element, including itself). Each ele-
ment, in turn, must have a “decision” function to determine whether it is
on or of on the basis of the inputs it gets from the other elements to
which it is connected. In set notation from high-school algebra, the set
of all functions assigned to all elements is
{F}.
One kind of function is
a threshold, for example, “If three or more of the elements that are
connected to me are on, then I will be of .” h erefore, I have a thresh-
old of 3. h
reshold functions are a special subset of Boolean functions
(Table 2.1).
Our stone-soup ensemble model was developed for computer simu-
lation. We randomly assigned a threshold decision function to each of
the elements. h resholds were drawn from a predetermined distribu-
tion of the subset of Boolean threshold functions. For example, each
element was randomly assigned a number between 0 and
N.
If the
number of “on” inputs to the node (rel ecting the states of the nodes to
which the inputs are connected) is equal to or greater than the assigned
number, then the element turns of . If the number is less than the as-
Table 2.1
Boolean functions for
K
= 2
Boolean functions
X
1
X
2
1 23456789 0 1 2 3 4 5 6
00
0
1
0
0
0
1
1
1
0
0
0
1
1
1
0
1
01
0
0
1
0
0
1
0
0
1
1
0
1
0
1
1
1
10
0
0
0
1
0
0
1
0
1
0
1
1
1
0
1
1
11
0
0
0
0
1
0
0
1
0
1
1
0
1
1
1
1
Note:
Boolean logic is based on sets of functions that can exist in only two states, 0 or 1. In this
example, “of ” is assigned to the case of 0 and “on” to the case of 1. h ere are only two inputs to each
node,
X
1
and
X
2
. h ere are four possible input states for nodes with two Boolean inputs, as shown in the
column on the let : 00, 01, 10, and 11. h ere are 16 possible functions. For example, function 1 is “of ”
regardless of inputs, while function 3 returns a 1, “on,” only when
X
1
= 0 and
X
2
= 1. Function 2 is a
threshold function that returns a 0 if either of the two inputs is 1.
