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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 19 "GRAPHICAL EVOLUTION" }}{PARA 19 
"" 0 "" {TEXT -1 53 "By Esben Sloth Andersen\nAalborg University, 10 J
ul 97" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 332 
"The mathematical theory of random graphs concerns the changes in the \+
characteristics of a graph with a given number of nodes (also called v
ertices). A stochastic process is randomly adding branches (also calle
d edges) to the graph. The process starts with the first branch betwee
n two randomly chosen nodes, and it stops when there " }}{PARA 0 "" 0 
"" {TEXT -1 53 "is a branch between all pairs of nodes in the graph. \+
" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "A supplementary procedure" }}
{PARA 0 "" 0 "" {TEXT -1 100 "To study random graphs, you need to load
 Maple's networks package. Press <enter> to load Maple code:" }
{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(net
works):" }}}{PARA 0 "" 0 "" {TEXT -1 135 "Maple's procedures do not fo
llow the sequential approach to random graphs. Therefore, a modified p
rocedure has been used. Press <enter>" }{MPLTEXT 1 0 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Random := proc(N,K,seed)" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "global _seed;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "local die, edge, G, i, q, success;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "  if K/N > 2 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "     ERROR(`this procedure is unfit for high K/N ratios`); " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "  _seed := seed;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "  G := \+
new();" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  addvertex(\{seq(i,i=1..
N)\},G);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  die := rand(1..N);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  for i from 1 to K do\011success \+
:= false;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "     while success <> \+
true do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "        edge := convert(
[die(),die()],set);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "        if n
ot member(edge, ends(G)) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  \+
         if nops(edge) = 2 then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
              addedge(edge,G);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " \+
             success := true;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  \+
         fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "        fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "     od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  RETURN(G
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}}{SECT 0 {PARA 256 "
" 0 "" {TEXT -1 18 "An unfolding graph" }}{PARA 0 "" 0 "" {TEXT -1 79 
"Here we try to hand simulate a process of graphical evolution. To get
 the same " }}{PARA 0 "" 0 "" {TEXT -1 85 "sequence of edges each time
, we define the seed value of the random generator (to 5)." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Gr1 := Random(20, 3, 5):  component
s(Gr1); draw(Gr1); # K/N = 0.15" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "Gr2 := Random(20, 7, 5):  components(Gr2); draw(Gr2); # K/N = \+
0.35" }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 66 "Gr3 := Random(20,
 11, 5): components(Gr3); draw(Gr3); # K/N = 0.55" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 66 "Gr4 := Random(20, 15, 5): components(Gr4); d
raw(Gr4); # K/N = 0.65" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 65 
"Gr5 := random(20, prob=1):components(Gr5); draw(Gr5); # K/N = 9.5" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 
"References" }}{PARA 0 "" 0 "" {TEXT -1 16 "Further reading:" }}{PARA 
0 "" 0 "" {TEXT -1 71 "Edgar M. Palmer (1985), \"Graphical Evolution\"
, Wiley & Sons, New York. " }}{PARA 0 "" 0 "" {TEXT -1 71 "Stuart A. K
auffman (1993), \"The Origins of Order: Self-Organization and" }}
{PARA 0 "" 0 "" {TEXT -1 82 "Selection in Evolution\", New York and Ox
ford, Oxford University Press, pp. 307 ff." }}}}{MARK "7 4 0" 5 }
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