Social Automata• Agent-based models– Contrast to global descriptive models– Local interactions by agents• Assumptions– Agents are autonomous: bottom-up control of system– Agents are interdependent– Agents follow simple rules– Agents adapt, but are not optimalSchelling’s Segregation Model• Mild preference to be close to others similar to oneselfleads to dramatic segregation– Conflict between local preferences and global solution– Nobody may want a segregated community, but it occurs anyway• Model– 2-D lattice with Moore neighborhoods– Two types of individuals– If < 37% of neighbors are of an agent’s type, then the agent moves to alocation where at least 37% of its neighbors are of its typeSchelling’s Segregation ModelA perfectly integrated, butimprobable, communityA random starting communitywith some discontentSchelling’s Segregation ModelA community after several iterations ofdistcontented people movingLarge events are rare, small events commonThe population of a city is inversely proportional to its rankLog (rank of city) = Log (population)-P P=1Log (rank of word) = Log (probability)-P P=1Situations where log-log linear plots are found• Gutenberg-Richter law– X: the size of an earthquake, Y: the frequency of the earthquake• Words (Zipf’s law)– X: rank of a word, Y: frequency of word• Citations– X: rank of paper in terms of citations, Y: number of citations• Animal sizes– X: Size of a species (Kg), Y: number of species with this sizeThree related lawsZipf’s Law: the size of the Rth largest occurrence of an event isinversely proportional to its rankPareto’s law: probability of having an income greater than x† size µ rank-B,B ª 1† P(X > x) µ x-kPower law distribution† P(X = x) µ x-(k +1)= x-aThis PDF is the derivative of Pareto’s law’s CDFPower lawsA mathematical relation that forms linear plots when datais transformed into log-log coordinatesVery few sites have many usersPower lawsA power function with slope =-2.07 fits the AOL data wellP(site has X visitors) = CX-2.07Zipf’s lawNumber of unique visitors is inversely proportional tothe rank of the siteThe Rth largest site has N visitors is equivalent to:R sites have N or more visitors (as in Pareto’s law)From Zipf’s Law to Power Function† size µ rank-B,B ª 1† P(X = x) = x-a† a = 1+1B† P X ≥ rank-B( )= rankThere are rank variables withvalues of at least rank-B† P X ≥ x( )= x-1BSwitch variables† P X = x( )= x- 1+1bÊ Ë Á ˆ ¯ ˜ Take derivative to move fromCDF to PDFPower Law in BibliometricsLn (citations to Goldstone) = Ln (rank of citation)-1.59Ln (Rank)3 .53.02 .52 .01.51 .0.50 .0-.576543210-1ObservedY=X^-1.59Ln(citations)Power Law in Baby namesSocial Security data, 1990sMichael 21243 Ashley 14108Christopher 16421 Jessica 14090Matthew 15851 Emily 10345Joshua 14973 Sarah 10109Jacob 13086 Samantha 10096Andrew 12281 Brittany 9016Daniel 12178 Amanda 8982Nicholas 12072 Elizabeth 7745Tyler 11739 Taylor 7329Joseph 11646 Megan 7266Power Law in Baby namesSocial Security data, 1990sLNGIRLLNR ANK76543210-11 41 21 0864O bser vedLinea rLNBOYLNR ANK76543210-11 41 21 08642O bser vedLinea rApparent departure from power law, but only for a verysmall number of most common names. Ln(10) = 2.3, soonly 10 data points out of 100 make up deviationPower Law in Baby namesRANK1 2001 0008 006004 002 000-2003 0002 0001 0000Obser vedLinea rLogari thmicPowerExponentia lLogisticRANK1 2001 0008 006004 002 000-2003 0002 0001 0000Obser vedLinea rLogari thmicPowerExponentia lLogisticFrequency of NameBoy names Girl namesUntransformed data (Cutting out the top four names)Power law fits bestBoy exponent = -1.34, Girl exponent = -1.11Naming for boys is more “elitist” than girls (faster drop-off of frequency with rank)What causes a power law relation?• Objects you are studying grow over time• Growth rate of any individual object is random– Typically normally distributed• Expected rate of growth is independent of scale– Same X% growth for small and large objects• Herb Simon’s model of city development– With probability P, form a new city with added group G– With probability (1-P) attach G to existing city– Probability that any city attracts G is proportional to its population• No added advantage to large city• Each person in every city acts as an independent attractor† P Choosei( )=CiCjjÂCi=# people in City iDiffusion Limited Aggregation for Population GrowthSugarscape (Epstein & Axtell)• Explain social and economic behaviors at large scale throughindividual behaviors (bottom-up economics)• Agents– Vision: high is good– Metabolism: low is good• Movement: move to cell within vision with greatest sugar• GR: grow sugar back with rate R• Replacement: Replace dead agent with random new agentWealth Distribution• Uniform random assignments of vision and metabolism still resultsin unequal, pyramidal distribution of wealth• Start simulation with number of agents at the carrying capacity• Random life spans within a range, and death from starvation• Replace dead agent with new agent with random new agentWealth DistributionNumber of AgentsWealth Bin<----- Time ------Wealth Distribution<----- Time ------% of Population% of WealthLorenz CurvesGini ratio: pink area/(area below 45 degree line)† G =1- Yi+1+ Yi( )i= 0k-1ÂXi+1- Xi( )Y = cumulated proportion of wealthX = cumulated proportion of populationG = 0: everybody has same wealthG = 1: All is owned by one individualWhy an Unequal Distribution of Wealth?• Epstein & Axtell: “Agents having wealth above the mean frequentlyhave both high vision and low metabolism. In order to become oneof the very wealthiest agents one must also be born high on thesugarscape and live a long life.”• This is part of the story, but not completely satisfying if vision andmetabolism variables are uniformly or normally distributed• multiplicative effect of variables?• Binomial and Poisson distributions– Binomial function describes the probability of obtaining xoccurrences of event A when each of N events is independentof the others, and the probability of event A on any trial is P:† F x( )=N!x! N - x( )!Px1- P( )N -x- Poisson distribution approximates Binomial if P is small and N is large (e.g.accidents, prairie dogs, customers). The probability of obtaining x occurrencesof A when the average number of occurrences is l