(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 256127, 6496]*) (*NotebookOutlinePosition[ 280199, 7401]*) (* CellTagsIndexPosition[ 280155, 7397]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " in Education and Research " }], "Subtitle", TextAlignment->Center], Cell[TextData[{ "\"Social Simulations\" Column\n", StyleBox["by", FontSize->14], "\nLouis J. D'Andria" }], "Subtitle", TextAlignment->Center] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Transmitting Culture Part II\ \>", "Title", TextAlignment->Center], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ In our previous column [Gaylord and D'Andria, 1998] we presented a \ mobile version of the Axelrod model of social learning through the binary \ transmission of cultural values [Axelrod, 1997]. In this column, we will look \ at a simpler variant of that model and then an even simpler version of the \ model in which social status determines the direction of meme transmission. \ \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Closer We Agree, the Closer Our Agreement Becomes", "Section"], Cell["\<\ In this model of cultural transmission, the number of meme values \ that are shared between two facing individuals is ignored and instead, a meme \ value is randomly chosen and changed based on how similar the chosen meme \ value is for the two individuals.\ \>", "Text"], Cell[CellGroupData[{ Cell["The System", "Subsection"], Cell[TextData[{ "Our model [Gaylord and D'Andria, 1998] uses an ", StyleBox["n", "IT"], " by ", StyleBox["n", "IT"], " square lattice with wraparound boundary conditions. There is a population \ density ", StyleBox["p", "IT"], " of individuals occupying lattice sites and the remaining sites are empty. \ Each person is characterized by the direction he is facing and a ", StyleBox["meme", FontSlant->"Italic"], " list with ", StyleBox["s", "IT"], " elements. The system evolves over a given number of time steps or until \ all of the people have identical meme lists." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Populating Society", "Subsection"], Cell["We first need to specify the values of the lattice sites.", "Text"], Cell["An empty site has value 0. ", "BulletListItem"], Cell[TextData[{ "A site that is occupied by an individual has a value of an ordered pair in \ which the first component is a randomly chosen integer between 1 and 4 \ (indicating that the individual is facing north, east, south, or west, \ respectively), and the second component is a meme list of ", StyleBox["s", "IT"], " elements, each of which has an integer value between 1 and ", StyleBox["m", "IT"], " (giving each of the ", StyleBox["s", "IT"], " memes of an individual a measure)." }], "BulletListItem"], Cell["\<\ The code for creating this system is given by [Gaylord and \ D'Andria, 1998]:\ \>", "Text"], Cell[TextData[ "RND := Random[Integer, {1, 4}]\nsociety = Table[Floor[p + Random[]], {n}, \ {n}] /. \n 1 \[RuleDelayed] {RND, Table[Random[Integer, {1, m}], \ {s}]}"], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Executing a Time Step", "Subsection"], Cell[TextData[{ "The time step will be executed in three consecutive ", StyleBox["partial-steps", FontSlant->"Italic"], ", the first two involving the cultural exchange between individuals facing \ one another on adjacent sites and the last one involving the movement of \ individuals." }], "Text"], Cell[CellGroupData[{ Cell["Deciding and Exchanging ", "Subsubsection"], Cell[TextData[{ "Any individual who is facing another individual on an adjacent site (e.g., \ a south (east, north, west) facing individual who is face-to-face with a \ north (west, south, east) facing individual on the adjacent site) interacts \ culturally with that individual. The cultural exchange that occurs between \ facing individuals is a two-way interaction between the two individuals, \ allowing both to be changed by the interaction. A comparison is made between \ the values of corresponding memes of the two individuals. If all of the \ values of the corresponding memes are the same for each individual, nothing \ happens. If any of the values of the corresponding memes of the two \ individuals differ, then cultural transmission or ", StyleBox["social learning", FontSlant->"Italic"], " occurs by randomly choosing one of the memes whose value differs for the \ two individuals and changing that value for ", StyleBox["both", FontSlant->"Italic"], " individuals. This process will be carried out in two stages, the first \ being to decide what meme to change and what the new meme value will be, and \ the second being to carry out the change." }], "Text"], Cell["\<\ The lattice sites are first updated using the following set of \ rules.\ \>", "Text"], Cell["\<\ decide[{3, y_}, _, _, {1, y_}, _] := {3, y} decide[{2, y_}, _, {4, y_}, _, _] := {2, y} decide[{2, y_}, _, {4, z_}, _, _] := Module[{r = Random[Integer, {1, s}]}, {2, y, Random[Integer, {y[[r]], z[[r]]}], r} /; Random[Integer,{1, m}] > Abs[y[[r]] - z[[r]]]] decide[{3, y_}, _, _, {1, z_}, _] := Module[{r = Random[Integer, {1, s}]}, {3, y, Random[Integer, {y[[r]], z[[r]]}], r} /; Random[Integer,{1, m}] > Abs[y[[r]] - z[[r]]]] decide[a_, _, _, _, _] := a\ \>", "Input"], Cell["\<\ The first two decide rules apply to a south (east) facing \ individual having a north (west) facing individual on the adjacent southern \ (eastern) site, when the meme lists of both are identical. In this case the \ decision is to make no change in any of the meme values.\ \>", "Text"], Cell[TextData[{ "The third and fourth decide rules apply to a south (east) facing \ individual having a north (west) facing individual on the adjacent southern \ (eastern) site, when the meme lists of the two are not identical. Since, as \ in a fair coin toss where it does not matter who flips the coin and who makes \ the call of heads or tails, it is irrelevant which of the two interacting \ individuals determines the cultural exchange for both individuals. We let \ each south (east) facing individual who is face-to-face with another \ individual on an adjacent site randomly select a meme in his meme list to \ change and, with a probability equal to the ", StyleBox["closeness", FontSlant->"Italic"], " of the meme's value for the two individuals, determine what the new meme \ value will be by randomly choosing a new value between the meme values for \ the two individuals." }], "Text"], Cell["\<\ The fifth decide rule applies to all other sites (either empty or \ occupied by an individual) and leaves them unchanged.\ \>", "Text"], Cell["The decide rules are applied to the lattice sites using", "Text"], Cell["vonNeumann[decide, #]&", "Input"], Cell["where", "Text"], Cell["\<\ vonNeumann[func_, lat_] := MapThread[func, Map[RotateRight[lat, #]&, {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1}}], 2]\ \>", "Input"], Cell["\<\ Having decided what cultural exchange to make, we next carry out \ the exchange by having each individual who is face-to-face with another \ individual on an adjacent site make the change in one of his meme values that \ was determined by the south (east) facing neighbor. This is done by updating \ the lattice sites using the rules:\ \>", "Text"], Cell["\<\ exchange[{3, y_, c_, d_}, _, _, _, _] := {3, ReplacePart[y, c, d]} exchange[{2, y_, c_, d_}, _, _, _, _] := {2, ReplacePart[y, c, d]} exchange[{1, z_}, {3, y_, c_, d_}, _, _, _] := {1, ReplacePart[z, c, d]} exchange[{4, z_}, _, _, _, {2, y_, c_, d_}] := {4, ReplacePart[z, c, d]} exchange[a_, _, _, _, _] := a\ \>", "Input"], Cell["\<\ In the preceding exchange rules, the individuals with four-element \ lists are the ones who have made the decision on what meme to change and what \ the new meme value should be. The first and second exchange rules use these \ values, which are given by the third and fourth list elements, to make the \ change in these persons. The third and fourth exchange rules use these same \ values to make the changes in the meme lists of the people facing the \ decision makers. The fifth rule applies to people who do not exchange values \ and to empty sites.\ \>", "Text"], Cell["\<\ The exchange rules are applied to the lattice sites using the \ anonymous function\ \>", "Text"], Cell["vonNeumann[exchange, vonNeumann[decide, #]]&", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Moving ", "Subsubsection"], Cell["\<\ The 28 rules for the movement of individuals are given by the set \ of walk rules derived in a previous column [Gaylord and D'Andria, 1998]\ \>", "Text"], Cell["\<\ walk[{1,a___},0,_,_,_,{4,___},_,_,_,_,_,_,_] := {RND,a} walk[{1,a___},0,_,_,_,_,_,_,{2,___},_,_,_,_] := {RND,a} walk[{1,a___},0,_,_,_,_,_,_,_,{3,___},_,_,_] := {RND,a} walk[{1,a___},0,_,_,_,_,_,_,_,_,_,_,_] := 0 walk[{2,a___},_,0,_,_,{3,___},_,_,_,_,_,_,_] := {RND,a} walk[{2,a___},_,0,_,_,_,{1,___},_,_,_,_,_,_] := {RND,a} walk[{2,a___},_,0,_,_,_,_,_,_,_,{4,___},_,_] := {RND,a} walk[{2,a___},_,0,_,_,_,_,_,_,_,_,_,_] := 0 walk[{3,a___},_,_,0,_,_,{4,___},_,_,_,_,_,_] := {RND,a} walk[{3,a___},_,_,0,_,_,_,{2,___},_,_,_,_,_] := {RND,a} walk[{3,a___},_,_,0,_,_,_,_,_,_,_,{1,___},_] := {RND,a} walk[{3,a___},_,_,0,_,_,_,_,_,_,_,_,_] := 0 walk[{4,a___},_,_,_,0,_,_,{1,___},_,_,_,_,_] := {RND,a} walk[{4,a___},_,_,_,0,_,_,_,{3,___},_,_,_,_] := {RND,a} walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,{2,___}] := {RND,a} walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,_] := 0 walk[{_,a___},_,_,_,_,_,_,_,_,_,_,_,_] := {RND,a} walk[0,{3,___},{4,___},_,_,_,_,_,_,_,_,_,_] := 0 walk[0,{3,___},_,{1,___},_,_,_,_,_,_,_,_,_] := 0 walk[0,{3,___},_,_,{2,___},_,_,_,_,_,_,_,_] := 0 walk[0,_,{4,___},{1,___},_,_,_,_,_,_,_,_,_] := 0 walk[0,_,{4,___},_,{2,___},_,_,_,_,_,_,_,_] := 0 walk[0,_,_,{1,___},{2,___},_,_,_,_,_,_,_,_] := 0 walk[0,{3,a___},_,_,_,_,_,_,_,_,_,_,_] := {RND,a} walk[0,_,{4,a___},_,_,_,_,_,_,_,_,_,_] := {RND,a} walk[0,_,_,{1,a___},_,_,_,_,_,_,_,_,_] := {RND,a} walk[0,_,_,_,{2,a___},_,_,_,_,_,_,_,_] := {RND,a} walk[0,_,_,_,_,_,_,_,_,_,_,_,_] := 0\ \>", "Input"], Cell["\<\ The walk rules are applied to the lattice sites using the anonymous \ function\ \>", "Text"], Cell["GN[walk, vonNeumann[exchange, vonNeumann[decide, #]]]&", "Input"], Cell["where", "Text"], Cell["\<\ GN[func_, lat_] := MapThread[func, Map[RotateRight[lat, #]&, {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1}, {1, -1}, {-1, -1}, {-1, 1}, {1, 1}, {2, 0}, {0, -2}, {-2, 0}, {0, 2}}], 2]\ \>", "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Evolving the System", "Subsection"], Cell[TextData[{ "The system evolves over ", StyleBox["t", "IT"], " time steps, starting with the initial lattice configuration, society, \ using the following nesting operation." }], "Text"], Cell["\<\ NestList[GN[walk, vonNeumann[exchange, vonNeumann[decide, #]]]&, society, t]\ \>", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The Program", "Subsection"], Cell[TextData[{ "cultureSpreadingCloseness[n_, p_, s_, m_, t_] :=\nModule[{society, RND, \ decide, exchange,\n walk, vonNeumann, GN},\ \n RND:= Random[Integer, {1, 4}];\n society = Table[Floor[p + Random[]], \ {n}, {n}] /. \n 1 \[RuleDelayed] {RND, Table[Random[Integer, {1, \ m}], {s}]};\n \n decide[{3, y_}, _, _, {1, y_}, _] := {3, y};\n decide[{2, \ y_}, _, {4, y_}, _, _] := {2, y};\n decide[{2, y_}, _, {4, z_}, _, _] := \n \ Module[{r = Random[Integer, {1, s}]},\n {2, y, Random[Integer, \ {y[[r]], z[[r]]}], r} /;\n Random[Integer,{1, m}] > Abs[y[[r]] - \ z[[r]]]];\n decide[{3, y_}, _, _, {1, z_}, _] := \n Module[{r = \ Random[Integer, {1, s}]},\n {3, y, Random[Integer, {y[[r]], \ z[[r]]}], r} /;\n Random[Integer,{1, m}] > Abs[y[[r]] - z[[r]]]];\n \ decide[a_, _, _, _, _] := a;\n\n exchange[{3, y_, c_, d_}, _, _, _, _] :=\n \ {3, ReplacePart[y, c, d]};\n exchange[{2, y_, \ c_, d_}, _, _, _, _] :=\n {2, ReplacePart[y, c, \ d]};\n exchange[{1, z_}, {3, y_, c_, d_}, _, _, _] :=\n \ {1, ReplacePart[z, c, d]};\n exchange[{4, z_}, _, _, _, {2, y_, c_, \ d_}] :=\n {4, ReplacePart[z, c, d]};\n \ exchange[a_, _, _, _, _] := a;\n \n ", "walk[{1,a___},0,_,_,_,{4,___},_,_,_,_,_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,{2,___},_,_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,_,{3,___},_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{2,a___},_,0,_,_,{3,___},_,_,_,_,_,_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,{1,___},_,_,_,_,_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,_,_,_,_,{4,___},_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{3,a___},_,_,0,_,_,{4,___},_,_,_,_,_,_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,{2,___},_,_,_,_,_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,_,_,_,_,{1,___},_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{4,a___},_,_,_,0,_,_,{1,___},_,_,_,_,_] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,{3,___},_,_,_,_] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,{2,___}] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,_] := 0;\n \ walk[{_,a___},_,_,_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,{3,___},{4,___},_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,___},_,{1,___},_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,___},_,_,{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,{4,___},{1,___},_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,{4,___},_,{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,_,{1,___},{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,a___},_,_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,{4,a___},_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,{1,a___},_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,_,{2,a___},_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,_,_,_,_,_,_,_,_,_,_] := 0;", "\n \n vonNeumann[func_, lat_] := \n MapThread[func, \ Map[RotateRight[lat, #]&,\n {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, \ 1}}], 2];", "\n", " GN[func_, lat_] :=\n MapThread[func, Map[RotateRight[lat, #]&,\n \ {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1},\n {1, -1}, {-1, \ -1}, {-1, 1}, {1, 1}, {2, 0},\n {0, -2}, {-2, 0}, {0, 2}}], 2];\n \ \n NestList[GN[walk, vonNeumann[exchange,\n \ vonNeumann[decide, #]]]&, society, t]]" }], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Running the Simulation", "Subsection"], Cell["\<\ We run the cultureSpreadingCloseness program on a 25 by 25 lattice \ having a 70% population of individuals with two memes, each of which can have \ two possible values, over 500 time steps. \ \>", "Text"], Cell["\<\ SeedRandom[5]; results = cultureSpreadingCloseness[25, 0.7, 2, 2, 500];\ \>", "Input"], Cell["\<\ We can look look at a percentile bar chart of the number of \ individuals with common meme lists.\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ <", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.0238095 0.047619 0.0147151 0.588604 [ [.02381 .00222 -3 -9 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ogoo0?mood]oo`001Woo00<007ooOol00Woo00D007ooOomoo`0000=oo`05001oogooOol00004Ool2 0009Ool00`00Oomoo`3oOom;Ool000Ioo`03001oogoo009oo`05001oogooOol00003Ool01@00Oomo ogoo00000Woo0`0000=oo`00Ool0ogooEGoo0006Ool00`00Oomoo`02Ool01@00Oomoogoo00000goo 00D007ooOomoo`00009oo`03001oo`000?mooeQoo`0017oo0`001Goo0`001Goo0`000goo0`00ogoo F7oo003oOomeOol00?moogEoo`00ogooMGoo003oOomeOol00001\ \>"], ImageRangeCache->{{{0, 371.875}, {229.312, 0}} -> {-1.97172, -0.109149, 0.078897, 0.00638289}}] }, Open ]], Cell["\<\ Comparing the figures obtained from the cultureSpreadingCloseness \ model to the figures obtained from the cultureSpreadingShared model [Gaylord \ and D'Andria, 1998], we see that the two models give similar results, \ indicating that there is no tendency for the population to adopt a common set \ of shared values. One way that convergence towards a shared value set might \ be achieved is to have a bias in the cultural exchange process, so that only \ one of the individuals changes one of his meme's values.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Social Status and Role Models", "Section"], Cell["\<\ When two people of unequal social status interact culturally, it is \ more likely that the individual with the lower status will adopt a meme value \ of the individual with a higher social status while the meme value of the \ higher social status person will remain unchanged. This is most evident \ amongst young people who often adopt the attitudes of a so-called role model \ (hence, the famous advertising phrase \"be like Mike\"). We can model this \ effect using a simpler version of the previous model because of the one-way \ nature of the cultural interaction. The necessary code changes follow.\ \>", "Text"], Cell[CellGroupData[{ Cell["Creating People with Varying Social Status", "Subsubsection"], Cell[TextData[{ "The value of a site that is occupied by an individual is a triplet of the \ form {direction, status, memelist}. The first component is a randomly chosen \ integer between 1 and 4, indicating that the individual is facing north, \ east, south, or west, respectively. The second component is a random real \ number between 0 and 1, representing the social status of the person. The \ third component is a meme list of ", StyleBox["s", "IT"], " elements, each of which has an integer value between 1 and ", StyleBox["m", "IT"], "." }], "Text"], Cell[TextData[{ StyleBox["Note:", FontWeight->"Bold"], " Although we use randomly chosen fixed social status values for each \ individual in this simple model, it is straightforward to use one or more \ changeable attributes of an individual, such as wealth, to determine social \ status. We can then change an individual's status over time as that attribute \ value changes endogeneously. It is also possible to consider a model in which \ nonchangeable attributes, such as race or gender, play a role in determining \ status. Finally, we can combine these two by considering some combination of \ changeable and unchangeable attributes to determine social status. For \ example, a changeable attribute that is important in determining the social \ status of people who are in a dominant position in a society, such as WASPS, \ may become less important in determining the social status of people who are \ not in the that position, such as blacks and women." }], "Note"], Cell[TextData[{ "The code for creating an ", StyleBox["n", "IT"], " by ", StyleBox["n", "IT"], " lattice with a population density ", StyleBox["p", "IT"], " of individuals with varying social status is given by:" }], "Text"], Cell[TextData[ "RND := Random[Integer, {1, 4}]\nsociety = Table[Floor[p + Random[]], {n}, \ {n}] /. 1 \[RuleDelayed] \n {RND, Random[], Table[Random[Integer, {1, \ m}], {s}]}"], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Deciding and Exchanging", "Subsubsection"], Cell["\<\ No decide rules are needed in this model because the decision to \ change the value of a (randomly chosen) meme can be incorporated into the \ exchange rules by simply carrying out the exchange only if the person has a \ lower status than the individual he is facing. The exchange rules are given \ by:\ \>", "Text"], Cell["\<\ exchange[{1, a_, u_}, {3, b_, v_}, _, _, _] := {1, a, ReplacePart[u, v[[#]], #]&[ Random[Integer, {1, s}]]} /; a < b exchange[{2, a_, u_}, _, {4, b_, v_}, _, _] := {2, a, ReplacePart[u, v[[#]], #]&[ Random[Integer, {1, s}]]} /; a < b exchange[{3, a_, u_}, _, _, {1, b_, v_}, _] := {3, a, ReplacePart[u, v[[#]], #]&[ Random[Integer, {1, s}]]} /; a < b exchange[{4, a_, u_}, _, _, _, {2, b_, v_}] := {4, a, ReplacePart[u, v[[#]], #]&[ Random[Integer, {1, s}]]} /; a < b\ \>", "Input"], Cell["All other individuals and empty sites remain unchanged.", "Text"], Cell["exchange[z_, _, _, _, _] := z", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["The Program", "Subsection"], Cell[TextData[{ "charlesBarkley[n_, p_, s_, m_, t_] :=\nModule[{society, RND, exchange, \ walk, vonNeumann, GN},\n\n RND := Random[Integer, {1, 4}];\n society = \ Table[Floor[p + Random[]], {n}, {n}] /. 1 \[RuleDelayed] \n {RND, \ Random[], Table[Random[Integer, {1, m}], {s}]};\n \n exchange[{1, a_, u_}, \ {3, b_, v_}, _, _, _] := \n {1, a, ReplacePart[u, v[[#]], #]&[\n \ Random[Integer, {1, s}]]} /; a < b;\n exchange[{2, a_, u_}, _, \ {4, b_, v_}, _, _] :=\n {2, a, ReplacePart[u, v[[#]], #]&[\n \ Random[Integer, {1, s}]]} /; a < b;\n exchange[{3, a_, u_}, _, _, {1, \ b_, v_}, _] :=\n {3, a, ReplacePart[u, v[[#]], #]&[\n \ Random[Integer, {1, s}]]} /; a < b;\n exchange[{4, a_, u_}, _, _, _, {2, b_, \ v_}] :=\n {4, a, ReplacePart[u, v[[#]], #]&[\n \ Random[Integer, {1, s}]]} /; a < b;\n exchange[z_, _, _, _, _] := z;\n \n \ ", "walk[{1,a___},0,_,_,_,{4,___},_,_,_,_,_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,{2,___},_,_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,_,{3,___},_,_,_] := {RND,a};\n \ walk[{1,a___},0,_,_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{2,a___},_,0,_,_,{3,___},_,_,_,_,_,_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,{1,___},_,_,_,_,_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,_,_,_,_,{4,___},_,_] := {RND,a};\n \ walk[{2,a___},_,0,_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{3,a___},_,_,0,_,_,{4,___},_,_,_,_,_,_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,{2,___},_,_,_,_,_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,_,_,_,_,{1,___},_] := {RND,a};\n \ walk[{3,a___},_,_,0,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[{4,a___},_,_,_,0,_,_,{1,___},_,_,_,_,_] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,{3,___},_,_,_,_] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,{2,___}] := {RND,a};\n \ walk[{4,a___},_,_,_,0,_,_,_,_,_,_,_,_] := 0;\n \ walk[{_,a___},_,_,_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,{3,___},{4,___},_,_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,___},_,{1,___},_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,___},_,_,{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,{4,___},{1,___},_,_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,{4,___},_,{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,_,_,{1,___},{2,___},_,_,_,_,_,_,_,_] := 0;\n \ walk[0,{3,a___},_,_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,{4,a___},_,_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,{1,a___},_,_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,_,{2,a___},_,_,_,_,_,_,_,_] := {RND,a};\n \ walk[0,_,_,_,_,_,_,_,_,_,_,_,_] := 0;", "\n ", "\n", " vonNeumann[func_, lat_] := \n MapThread[func, Map[RotateRight[lat, \ #]&,\n {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1}}], 2];\n\n \ GN[func_, lat_] :=\n MapThread[func, Map[RotateRight[lat, #]&,\n \ {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1},\n {1, -1}, {-1, -1}, \ {-1, 1}, {1, 1}, {2, 0},\n {0, -2}, {-2, 0}, {0, 2}}], 2];\n \n \ NestList[GN[walk, vonNeumann[exchange, #]]&, society, t]]" }], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Running the Simulation", "Subsection"], Cell["\<\ We run the charlesBarkley program on a 25 by 25 lattice having a \ 70% population of individuals with two memes, each of which can have two \ possible values, over 500 time steps, and look at the resulting graphics of \ the initial and final states of the system.\ \>", "Text"], Cell["\<\ SeedRandom[7] results = charlesBarkley[25, 0.7, 2, 2, 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charlesBarkley program.\ \>", "Text"], Cell[TextData[{ "Thus, the presence of individuals who can serve as role models for others \ seems to dramatically increase the convergence of the society towards a \ common meme list and shared values. 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This can be done with the following rewrite \ rules.\ \>", "Text"], Cell[TextData[ "renumber[0, __] := 0\nrenumber[{a_, memes_List, c_}, res__] := \n {a, \ memes, Max[c, Cases[{res}, {_, memes, d_} \[RuleDelayed] d]]}"], "Input"], Cell["The renumber rules are applied to the lattice using", "Text"], Cell["vonNeumann[renumber, latCluster]", "Input"], Cell["where", "Text"], Cell["\<\ vonNeumann[func_, lat_] := MapThread[func, Map[RotateRight[lat, #]&, {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, 1}}], 2]\ \>", "Input"], Cell["\<\ If we perform this relabeling of cluster numbers repeatedly until \ the lattice configuration no longer changes, each individual in a cluster of \ individuals with the same meme list will then have the same cluster number as \ the highest numbered individual in that cluster. This can be done by \ employing the built-in function FixedPoint.\ \>", "Text"], Cell["FixedPoint[vonNeumann[renumber, #]&, latCluster]", "Input"], Cell[TextData[{ StyleBox["Note:", FontWeight->"Bold"], " The clusters are not numbered consecutively (1, 2, 3, \[Ellipsis]) by \ this procedure, although it is very simple to create a set of transformation \ rules to produce consecutive cluster numbers if that is desirable." }], "Note"], Cell["\<\ Combining these code fragments, we can create the folowing cluster \ identification function.\ \>", "Text"], Cell[TextData[ "clusterNumbering[lattice_] :=\nModule[{k, latCluster, renumber, vonNeumann},\ \n k = 0;\n latCluster = lattice /. {a_, memes_List} \[RuleDelayed] {a, \ memes, ++k};\n \n renumber[0, __] := 0;\n renumber[{a_, memes_List, c_}, \ res__] :=\n {a, memes, Max[c, Cases[{res}, {_, memes, d_} \[RuleDelayed] \ d]]};\n\n vonNeumann[func_, lat_] := \n MapThread[func, \ Map[RotateRight[lat, #]&,\n {{0, 0}, {1, 0}, {0, -1}, {-1, 0}, {0, \ 1}}], 2];\n \n FixedPoint[vonNeumann[renumber, #]&, latCluster]]"], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "Axelrod, Robert. 1997. ", StyleBox[ "The Complexity of Cooperation: Agent-Based Models of Competition and \ Cooperation", FontSlant->"Italic"], ". Princeton, NJ: Princeton University Press.\nGaylord, Richard J. and \ Louis J. D'Andria. 1998. ", StyleBox[ "SIMULATING SOCIETY: A Mathematica Toolkit for Modeling Socioeconomic \ Behavior", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". New York: TELOS/Springer-Verlag. \n", FontWeight->"Plain"], "Gaylord, Richard J. and Louis J. D'Andria. 1998. Modeling a Society of \ Mobile Heterogeneous Individuals, ", StyleBox["Mathematica in Education and Research", FontSlant->"Italic"], ", 7(3), xx - yy. \nGaylord, Richard J. and Louis J. D'Andria. 1998. \ Transmitting Culture Part I, ", StyleBox["Mathematica in Education and Research", FontSlant->"Italic"], ", 7(4), xx - yy. 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