📄 models.ps
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h(for)e(\(pure\))g(random)h(graphs)f(to)g(this)h(mo)q(del.)28b(An)o(y)18 b(\014nite)h(n)o(um)o(b)q(er)f(of)f(categories)301175 y(are)e(allo)o(w)o(ed,)g(giving)h(a)f(reasonable)h(amoun)o(t)e(of)h(con)o(trol)g(o)o(v)o(er)f(the)h(top)q(ologies)h(generated.)1011232 y(T)l(o)k(summarize,)i(T)l(able)g(1)e(indicates)j(the)e(probabilit)o(y)h(of)e(an)h(edge)g(b)q(et)o(w)o(een)g(t)o(w)o(o)e(v)o(ertices)j(at)30 1288 y(Euclidean)17 b(distance)f Fi(d)fFj(for)g(eac)o(h)g(mo)q(del)h(describ)q(ed)h(ab)q(o)o(v)o(e.)301410 y Fk(Hierarc)n(hical)g(Mo)r(dels)30 1496 y Fj(The)f(random)f(and)g(regular)h(mo)q(dels)g(represen)o(t)g(extremes)f(in)h(the)g(sense)g(that)e(random)h(mo)q(dels)i(o\013er)30 1552 y(v)o(ery)h(little)h(con)o(trol)f(o)o(v)o(er)f(the)h(structure)g(of)g(the)g(resulting)h(top)q(ologies,)g(while)g(regular)f(mo)q(dels)h(are)30 1609y(extremely)h(rigid)h(in)g(their)f(structure.)34 b(Neither)20b(t)o(yp)q(e)g(of)g(mo)q(del)g(captures)g(the)g(hierarc)o(h)o(y)g(that)f(is)30 1665 y(presen)o(t)h(in)h(real)f(in)o(ternet)o(w)o(orks,)g(though)f(b)q(oth)h(ma)o(y)g(re\015ect)g(some)f(notion)i(of)e(lo)q(calit)o(y)i(if)g(certain)30 1722 y(no)q(des)i(are)f(more)h(lik)o(ely)h(to)e(b)q(e)h(connected)h(than)e(others.)41 b(W)l(e)23b(no)o(w)f(describ)q(e)i(t)o(w)o(o)e(metho)q(ds)g(of)301778 y(creating)16 b(hierarc)o(hical)i(top)q(ologies)e(b)o(y)g(connecting)h(smaller)g(comp)q(onen)o(ts)f(together)f(according)i(to)e(a)30 1835 y(particular)h(structure.)30 1955 y Fc(N-Lev)o(el)302040 y Fj(The)k(N-lev)o(el)g(hierarc)o(hical)h(mo)q(del)g(constructs)d(a)h(top)q(ology)g(recursiv)o(ely)l(.)34 b(Beginning)21b(with)f(a)f(con-)30 2097 y(nected)e(graph,)g(at)f(eac)o(h)h(step)f(in)i(the)f(recursion,)g(eac)o(h)g(no)q(de)g(in)h(the)f(curren)o(t)f(top)q(ology)h(is)g(replaced)30 2153 y(b)o(y)d(a)f(connected)i(graph.)k(The)14 b(edges)g(are)g(\\resolv)o(ed")g(in)g(v)m(arious)h(w)o(a)o(ys)e(\(e.g.,)f(b)o(y)i(selecting)h(no)q(des)g(at)30 2210y(random)g(in)h(the)f(replacemen)o(t)h(graphs\).)1012266 y(More)g(precisely)l(,)j(in)f(constructing)f(the)g(\015at)f(random)h(graphs,)f(w)o(e)h(divide)i(the)e(Euclidean)i(plane)302323 y(in)o(to)14 b(equal-sized)j(square)d(sectors,)g(the)g(n)o(um)o(b)q(er)h(of)f(whic)o(h)h(is)g(determined)h(b)o(y)e(a)hFb(sc)n(ale)h Fj(parameter)e Fi(S)s Fj(;)30 2379 y(eac)o(h)f(no)q(de)h(in)g(the)f(graph)g(is)h(then)g(assigned)g(to)e(one)h(of)g(the)hFi(S)1108 2363 y Ff(2)1140 2379 y Fj(squares.)19 b(In)14b(constructing)f(an)g(N-lev)o(el)30 2436 y(hierarc)o(hical)k(graph,)e(the)h(top-lev)o(el)h(graph)e(is)h(constructed)g(in)g(this)g(fashion)g(using)h(scale)f(parameter)30 2492 y Fi(S)58 2499 y Ff(1)782492 y Fj(.)k(Then)c(eac)o(h)g(square)f(con)o(taining)h(a)f(no)q(de)h(is)g(sub)q(divided)j(again,)c(according)h(to)e(the)i(second-lev)o(el)30 2549 y(scale)k(parameter)e(\()p Fi(S)410 2556 y Ff(2)4302549 y Fj(\),)h(and)g(a)g(graph)g(is)h(constructed)f(using)h(that)f(top-lev)o(el)h(square)f(as)g(the)g(unit)30 2605 y(plane.)i(Euclidean)16 b(co)q(ordinates)f(of)e(all)j(no)q(des)f(are)f(adjusted)g(to)g(the)g(new)h(scale,)f(in)h(whic)o(h)h Fi(n)e Fj(squares)302661 y(of)i(the)h(original)h(top-lev)o(el)g(plane)f(are)g(no)o(w)fFi(n)11 b Fe(\002)h Fi(S)924 2668 y Ff(2)960 2661 y Fj(squares.)24b(This)18 b(pro)q(cess)f(con)o(tin)o(ues)g(recursiv)o(ely)l(.)302718 y(The)f(result)g(is)g(that)f(the)h(scale)g(of)g(the)f(\014nal)i(graph)e(is)i(the)e(pro)q(duct)h(of)g(the)f(scales)i(of)e(the)h(individual)30 2774 y(lev)o(els,)f(and)f(edge)g(lengths)h(are)e(roughly)h(determined)i(b)o(y)d(edge)i(lev)o(el.)20 b(F)l(or)14b(example,)g(in)h(a)e(three-lev)o(el)919 2940 y(2)p eop%%Page: 3 33 2 bop 440 266 a 15458714 15392932 7762247 24141905 23220961 39534837 startTexFig 440 266 a%%BeginDocument: hierlayfig.ps/arrowhead {0 begintransform originalCTM itransform/taily exch def/tailx exch deftransform originalCTM itransform/tipy exch def/tipx exch def/dy tipy taily sub def/dx tipx tailx sub def/angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse defgsaveoriginalCTM setmatrixtipx tipy translateangle rotatenewpatharrowHeight neg arrowWidth 2 div moveto0 0 linetoarrowHeight neg arrowWidth 2 div neg linetopatternNone not {originalCTM setmatrix/padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mularrowWidth div def/padtail brushWidth 2 div deftipx tipy translateangle rotatepadtip 0 translatearrowHeight padtip add padtail add arrowHeight div dup scalearrowheadpathifill} ifbrushNone not {originalCTM setmatrixtipx tipy translateangle rotatearrowheadpathistroke} ifgrestoreend} dup 0 9 dict put def/arrowheadpath {newpatharrowHeight neg arrowWidth 2 div moveto0 0 linetoarrowHeight neg arrowWidth 2 div neg lineto} def/leftarrow {0 beginy exch get /taily exch defx exch get /tailx exch defy exch get /tipy exch defx exch get /tipx exch defbrushLeftArrow { tipx tipy tailx taily arrowhead } ifend} dup 0 4 dict put def/rightarrow {0 beginy exch get /tipy exch defx exch get /tipx exch defy exch get /taily exch defx exch get /tailx exch defbrushRightArrow { tipx tipy tailx taily arrowhead } ifend} dup 0 4 dict put def/arrowHeight 10 def/arrowWidth 5 def/IdrawDict 50 dict defIdrawDict begin/none null def/numGraphicParameters 17 def/stringLimit 65535 def/Begin {savenumGraphicParameters dict begin} def/End {endrestore} def/SetB {dup type /nulltype eq {popfalse /brushRightArrow ideffalse /brushLeftArrow ideftrue /brushNone idef} {/brushDashOffset idef/brushDashArray idef0 ne /brushRightArrow idef0 ne /brushLeftArrow idef/brushWidth ideffalse /brushNone idef} ifelse} def/SetCFg {/fgblue idef/fggreen idef/fgred idef} def/SetCBg {/bgblue idef/bggreen idef/bgred idef} def/SetF {/printSize idef/printFont idef} def/SetP {dup type /nulltype eq {pop true /patternNone idef} {dup -1 eq {/patternGrayLevel idef/patternString idef} {/patternGrayLevel idef} ifelsefalse /patternNone idef} ifelse} def/BSpl {0 beginstorexynnewpathn 1 gt {0 0 0 0 0 0 1 1 true subsplinen 2 gt {0 0 0 0 1 1 2 2 false subspline1 1 n 3 sub {/i exch defi 1 sub dup
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