Mobility scholars are increasingly turning to computational methods to analyze mobility tables. Most of these approaches start with the detection of mobility clusters, namely, sets of occupations within which the flow of workers is dense and across which the flow is sparse. Yet clustering is not the only way worker flows can be structured. This article shows how a degree-corrected stochastic blockmodel can detect patterns of mobility that are more general than clustering and consistent with the homogeneity criterion laid out by Goodman as well as the internal homogeneity thesis proposed by Breiger. Because of the intractable marginal likelihood of the model, parameters are estimated using a variational expectation maximization algorithm. Simulation results suggest the estimation algorithm successfully recovers (conditionally) stochastically equivalent mobility classes. Analysis of two real-world examples shows the model is able to detect meaningful mobility patterns, even in situations in which commonly used community detection algorithms fail.
BickelPeter J.ChenAiyou. 2009. “A Nonparametric View of Network Models and Newman–Girvan and Other Modularities.”Proceedings of the National Academy of Sciences106(50):21068–73.
5.
BickelPeter J.ChoiDavidChangXiangyuZhangHai. 2013. “Asymptotic Normality of Maximum Likelihood and Its Variational Approximation for Stochastic Blockmodels.”Annals of Statistics41(4):1922–43.
6.
BiernackiChristopheCeleuxGillesGovaertGérard. 2000. “Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood.”IEEE Transactions on Pattern Analysis and Machine Intelligence22:719–25.
7.
BishopChristopher M.2006. Pattern Recognition and Machine Learning. New York: Springer.
8.
BlauPeterDuncanOtis Dudley. 1967. The American Occupational Structure. New York: John Wiley.
9.
BleiDavid M.KucukelbirAlpMcAuliffeJon D.2017. “Variational Inference: A Review for Statisticians.”Journal of the American Statistical Association112(518):859–77.
10.
BlockPerStadtfeldChristophRobinsGarry. 2022. “A Statistical Model for the Analysis of Mobility Tables as Weighted Networks with an Application to Faculty Hiring Networks.”Social Networks68(2):264–78.
11.
BlondelVincent D.GuillaumeJean-LoupLambiotteRenaudLefebvreEtienne. 2008. “Fast Unfolding of Communities in Large Networks.”Journal of Statistical Mechanics: Theory and Experiment2008:P10008.
12.
BreigerRonald L.1981. “The Social Class Structure of Occupational Mobility.”American Journal of Sociology87(3):578–611.
13.
BreigerRonald L.1990. Social Mobility and Social Structure. Cambridge, UK: Cambridge University Press.
14.
BurtRonald S.1987. “Social Contagion and Innovation: Cohesion versus Structural Equivalence.”American Journal of Sociology92(6):1287–1335.
15.
CarlenJanede Dios PontJaumeMentusCassidyChangShyr-SheaWangStephaniePorterMason A.2022. “Role Detection in Bicycle-Sharing Networks Using Multilayer Stochastic Block Models.”Network Science10(1):46–81.
CelisseAlainDaudinJean-JacquesPierreLaurent. 2012. “Consistency of Maximum-Likelihood and Variational Estimators in the Stochastic Block Model.”Electronic Journal of Statistics6:1847–99.
18.
ChengSiweiParkBarum. 2020. “Flows and Boundaries: A Network Approach to Studying Occupational Mobility in the Labor Market.”American Journal of Sociology126(3):577–631.
19.
CloggClifford C.1981. “Latent Structure Models of Mobility.”American Journal of Sociology86(4):836–68.
20.
DanonLeonDiaz-GuileraAlbertDuchJordiArenasAlex. 2005. “Comparing Community Structure Identification.”Journal of Statistical Mechanics: Theory and Experiment2005:P09008.
21.
DaudinJ.-J.PicardFranckRobinStéphane. 2008. “A Mixture Model for Random Graphs.”Statistics and Computing18(1):173–83.
22.
DempsterArthur P.LairdNan M.RubinDonald B.1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.”Journal of the Royal Statistical Society: Series B (Methodological)39(1):1–22.
23.
DuncanOtis Dudley. 1979. “How Destination Depends on Origin in the Occupational Mobility Table.”American Journal of Sociology84(4):793–803.
24.
EricksonBonnie H.1988. “The Relational Basis of Attitudes.” Pp. 99–121 in Social Structures: A Network Approach, edited by WellmanB.BerkowitzS. D.Cambridge, UK: Cambridge University Press.
25.
FeathermanDavid L.HauserRobert Mason. 1978. Opportunity and Change. New York: Academic Press.
26.
FogelJamieModenesiBernardo. 2023. “What Is a Labor Market? Classifying Workers and Jobs Using Network Theory.” arXiv. Retrieved July 20, 2025. https://arxiv.org/abs/2311.00777.
27.
FortunatoSanto. 2010. “Community Detection in Graphs.”Physics Reports486(1):75–174.
28.
GlassDavid Victor. [1954] 2013. Social Mobility in Britain. New York: Routledge.
29.
GoodmanLeo A.1968. “The Analysis of Cross-Classified Data: Independence, Quasi-independence, and Interactions in Contingency Tables with or without Missing Entries: Ra Fisher Memorial Lecture.”Journal of the American Statistical Association63(324):1091–1131.
30.
GoodmanLeo A.1981. “Criteria for Determining Whether Certain Categories in a Cross-Classification Table Should Be Combined, with Special Reference to Occupational Categories in an Occupational Mobility Table.”American Journal of Sociology87(3):612–50.
31.
GrünwaldPeter D.2007. The Minimum Description Length Principle. Cambridge, MA: MIT Press.
32.
GuennebaudGaëlJacobBenoît, et al. 2010. “Eigen v3.” Retrieved July 20, 2025. http://eigen.tuxfamily.org.
33.
HauserRobert M.1978. “A Structural Model of the Mobility Table.”Social Forces56(3):919–53.
34.
HollandPaul W.LaskeyKathryn BlackmondLeinhardtSamuel. 1983. “Stochastic Blockmodels: First Steps.”Social Networks5(2):109–37.
35.
HoutMichael. 1983. Analysing Mobility Tables. Beverly Hills, CA: Sage.
36.
JasraAjayHolmesChris C.StephensDavid A.2005. “Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling.”Statistical Science20(1):50–67.
37.
KarrerBrianNewmanMark E. J.2011. “Stochastic Blockmodels and Community Structure in Networks.”Physical Review E83:016107.
38.
KimYoungdoSonSeung-WooJeongHawoong. 2010. “Finding Communities in Directed Networks.”Physical Review E81:016103.
39.
LancichinettiAndreaFortunatoSanto. 2009. “Benchmarks for Testing Community Detection Algorithms on Directed and Weighted Graphs with Overlapping Communities.”Physical Review E80:016118.
40.
LeeClementWilkinsonDarren J.2019. “A Review of Stochastic Block Models and Extensions for Graph Clustering.”Applied Network Science4(1):1–50.
41.
LegerJean-Benoist. 2016. “Blockmodels: A R-Package for Estimating in Latent Block Model and Stochastic Block Model, with Various Probability Functions, with or without Covariates.” arXiv. Retrieved July 20, 2025. https://arxiv.org/abs/1602.07587.
42.
LinKen-HouHungKoit. 2022. “The Network Structure of Occupations: Fragmentation, Differentiation, and Contagion.”American Journal of Sociology127(5):1551–1601.
43.
LiuDong C.NocedalJorge. 1989. “On the Limited Memory BFGS Method for Large Scale Optimization.”Mathematical Programming45:503–28.
44.
LorrainFrançoisWhiteHarrison C.1971. “Structural Equivalence of Individuals in Social Networks.”Journal of Mathematical Sociology1(1):67–98.
45.
MariadassouMahendraRobinStéphaneVacherCorinne. 2010. “Uncovering Latent Structure in Valued Graphs: A Variational Approach.”Annals of Applied Statistics4(2):715–42.
46.
MarsdenPeter V.1985. “Latent Structure Models for Relationally Defined Social Classes.”American Journal of Sociology90(5):1002–21.
47.
MelamedDavid. 2015. “Communities of Classes: A Network Approach to Social Mobility.”Research in Social Stratification and Mobility41(1):56–65.
48.
MillerSeymour Michael. 1960. “Comparative Social Mobility.”Current Sociology9(1):1–80.
49.
NewmanMark E.J.GirvanMichelle. 2004. “Finding and Evaluating Community Structure in Networks.”Physical Review E69:026113.
50.
NimczikJan Sebastian.2017. “Job Mobility Networks and Endogenous Labor Markets.” Number E05-V3 in Beiträge zur Jahrestagung des Vereins für Socialpolitik 2017: Alternative Geld- und Finanzarchitekturen—Session: Labor Contracts and Markets I, Kiel, Hamburg. ZBW—Deutsche Zentralbibliothek für Wirtschaftswissenschaften, Leibniz-Informationszentrum Wirtschaft.
51.
Norris KeillerAgnes. 2020. “Detecting Labour Submarkets from Worker-Mobility Networks: A Preliminary Study.” IFS Working Paper W20/30. London: Institute for Fiscal Studies.
52.
NowickiKrzysztofSnijdersTom A. B.2001. “Estimation and Prediction for Stochastic Blockstructures.”Journal of the American Statistical Association96(455):1077–87.
53.
PadgettJohn F.1990. “Mobility as Control: Congressmen through Committees.” Pp. 27–58 in Social Mobility and Social Structure, edited by BreigerR. L. Cambridge, UK: Cambridge University Press Cambridge.
54.
PonsPascalLatapyMatthieu. 2006. “Computing Communities in Large Networks Using Random Walks.”Journal of Graph Algorithms and Applications10(2):191–218.
55.
RafteryAdrian E.1995. “Bayesian Model Selection in Social Research.”Sociological Methodology25:111–63.
56.
RandWilliam M.1971. “Objective Criteria for the Evaluation of Clustering Methods.”Journal of the American Statistical Association66(336):846–50.
57.
RosvallMartinAxelssonDanielBergstromCarl T.2009. “The Map Equation.”European Physical Journal of Special Topics178(1):13–23.
58.
RosvallMartinBergstromCarl T.2008. “Maps of Random Walks on Complex Networks Reveal Community Structure.”Proceedings of the National Academy of Sciences105(4):1118–23.
59.
SandersonConradCurtinRyan. 2016. “Armadillo: A Template-Based C++ Library for Linear Algebra.”Journal of Open Source Software1:26.
60.
SatuluriVenuParthasarathySrinivasan. 2011. “Symmetrizations for Clustering Directed Graphs.” Pp. 343–54 in Proceedings of the 14th International Conference on Extending Database Technology. New York: Association for Computing Machinery.
61.
SchmutteIan M.2014. “Free to Move? A Network Analytic Approach for Learning the Limits to Job Mobility.”Labour Economics29(1):49–61.
62.
SnijdersTom A.B.NowickiKrzysztof. 1997. “Estimation and Prediction for Stochastic Blockmodels for Graphs with Latent Block Structure.”Journal of Classification14(1):75–100.
63.
SobelMichael E.HoutMichaelDuncanOtis Dudley. 1985. “Exchange, Structure, and Symmetry in Occupational Mobility.”American Journal of Sociology91(2):359–72.
64.
StephensMatthew. 2000. “Dealing with Label Switching in Mixture Models.”Journal of the Royal Statistical Society: Series B (Statistical Methodology)62(4):795–809.
65.
ToubølJonasLarsenAnton Grau. 2017. “Mapping the Social Class Structure: From Occupational Mobility to Social Class Categories Using Network Analysis.”Sociology51(6):1257–76.
VannemanReeve. 1977. “The Occupational Composition of American Classes: Results from Cluster Analysis.”American Journal of Sociology82(4):783–807.
68.
WangYuchung J.WongGeorge Y.1987. “Stochastic Blockmodels for Directed Graphs.”Journal of the American Statistical Association82(1):8–19.
69.
WassermanStanleyAndersonCarolyn. 1987. “Stochastic A Posteriori Blockmodels: Construction and Assessment.”Social Networks9(1):1–36.
70.
WassermanStanleyFaustKatherine. 1994. Social Network Analysis: Methods and Applications. Cambridge, UK: Cambridge University Press.
71.
WeberMax. [1922] 1978. Economy and Society, Vol. 1. Berkeley: University of California Press.
72.
WhiteHarrison C.BoormanScott A.BreigerRonald L.1976. “Social Structure from Multiple Networks. I. Blockmodels of Roles and Positions.”American Journal of Sociology81(4):730–80.
73.
WuC. F. Jeff. 1983. “On the Convergence Properties of the EM Algorithm.”Annals of Statistics11(1):95–103.
74.
YangZhaoAlgesheimerRenéTessoneClaudio J.2016. “A Comparative Analysis of Community Detection Algorithms on Artificial Networks.”Scientific Reports6:30750.
75.
YuLishaWoodallWilliam H.TsuiKwok-Leung. 2018. “Detecting Node Propensity Changes in the Dynamic Degree Corrected Stochastic Block Model.”Social Networks54:209–27.
76.
ZhangXiaoMartinTravisNewmanMark E. J.2015. “Identification of Core-Periphery Structure in Networks.”Physical Review E91:032803.
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