Occupational Network Structure and Vector Assortativity for illustrating patterns of social mobility
OOccupational Network Structure and Vector Assortativity for illustratingpatterns of social mobility
Vinay Reddy VenumuddalaDoctoral Student, Public PolicyIndian Institute of Management Bangalore
Abstract
In this study we arrive at a closed form expression for measuring vector assortativity in networksmotivated by our use-case which is to observe patterns of social mobility in a society. Based on existingworks on social mobility within economics literature, and social reproduction within sociology literature,we motivate the construction of an occupational network structure to observe mobility patterns. Basingon existing literature, over this structure, we define mobility as assortativity of occupations attributedby the representation of categories such as gender, geography or social groups. We compare the resultsfrom our vector assortativity measure and averaged scalar assortativity in the Indian context, relying onNSSO 68th round on employment and unemployment. Our findings indicate that the trends indicated byour vector assortativity measure is very similar to what is indicated by the averaged scalar assortativityindex. We discuss some implications of this work and suggest future directions.
In this study, we devise a framework to depict patterns of social mobility with occupations as units ofanalysis, where we define social mobility as the opposite of durable inequality introduced by Tilly (1998).In his study, Tilly argues that, insofar as the structuration of occupational boundaries within organizationshappen along categorical lines, categorical inequalities remain durable. A category for instance can bedefined on the basis of gender, religion, or social group, and categorical groups correspond to groupswithin a given category. In this work, we frame an occupational network structure, with relation betweenoccupations defined by other forces of structuration such as the industry and educational requirements.We define strength of connection between occupations as the similarity of occupations along these non-categorical factors. We also define categorical attributes for each occupation as a vector capturing therepresentation of distinct categorical groups in that occupation, relative to their representation in theoverall work force. We then devise a vector assortativity measure to depict the stratification of occupationalnetwork along categorical lines and proxy it for categorical inequality following Tilly (1998). Observing thisassortativity measure over consecutive birth cohorts, allows us to illustrate the patterns of social mobilitygiven a particular category. Our findings indicate that assortativity along sector is more or less stagnant,along gender it’s falling in the recent cohorts, but for social groups it has been consistently increasing withslight stagnation observed in recent cohorts. Further we believe that looking at the changing contribution tothe overall assortativity by different industry-education combinations (set of non-categorical structurationforces) through time, can allows us to comment on those structuration forces which are amicable to socialmobility and those which perhaps are not. 1 a r X i v : . [ ec on . GN ] N ov Social Mobility and Reproduction
Studies on social mobility often characterize its measurement as a relative or absolute improvement in socio-economic status of individuals within any society, seen either inter-generationally or intra-generationally(Narayan et al., 2018). Here the socio-economic status of individuals are indicated either by income (Beckerand Tomes, 1979; Black and Devereux, 2010; Chetty et al., 2014; Solon, 1999), or education (Asher et al.,2018; Azam and Bhatt, 2015), or class positions determined by employment relations and defined as aggre-gate occupational groupings (Erikson and Goldthorpe, 2002). The one commonality among these studiesis that the measurement of social mobility is pegged at a macro or societal level. Each of these studiesalso base their analysis on plausible pre-suppositions and justifications about why they had to choose anyparticular indicator as an individual’s socio-economic status and how does it help in the measurementof social mobility. Further these ontological pre-suppositions also help researchers to provide plausibleexplanations for the processes/mechanisms that cause variation in social mobility across societies.For instance, rational choice based social mobility models within economics literature rely on individualattributes, both ascriptive and non-ascriptive, to theorize the processes/mechanisms underlying mobilityas well as to explain the variation in mobility across societies. For example, the seminal rational choicebased theoretical model on intergenerational mobility proposed by Becker and Tomes (1979), describethese processes through a dependence between parent’s earnings and child’s earnings. Such dependenceaccording to them is mediated by factors such as human and non-human capital investments of parents ontheir children, individual ‘endowments’ determined by ascriptive characteristics such as ethnicity, familyconnections, and so on. Much of the literature on social mobility within economics rely on empiricallyevaluating the relative influence of each of these processes/mechanisms across different societies (Blackand Devereux, 2010; Solon, 1999).Within sociology literature pre-suppositions are often made about the structure of society, in particularin terms of the class positions that constitute it. The class schema proposed by Erikson and Goldthorpe(2002) defined using ‘employment status and occupation as indicator of employment relations’ has mo-tivated several empirical works on social mobility within this strand of literature (Azam, 2015; Iversenet al., 2016; Motiram and Singh, 2012). In so far as mechanisms underlying social mobility are concerned,many macro sociological theories, shed light instead into the mechanisms underlying reproduction of so-cial inequalities (Tilly, 1998). Bourdieu (2013), for example, illustrates the mechanism underlying socialreproduction through the notion of habitus generating action. According to Bourdieu, the processes as-sociated with social stratification and its reproduction can be depicted within a field or a social spacein which individuals or groups are closely linked to different class/social positions. Each such position isbound up with systems of dispositions called habitus that is bound up with a particular set of culturaltastes or capital. This habitus in turn, dictates the practices (or actions) of individuals occupying theirrespective class positions. These individual actions, affected by a misrecognition of historically contingentsocial relations within the field, tend to perpetuate the very influence (and location in the hierarchy) of aclass position ad-infinitium, reproducing social inequality and the observed patterns of social stratificationwith time (Burawoy, 2018; Riley, 2017). For Tilly (1998), the mechanisms underlying reproduction ofsocial or categorical inequalities predominantly situate within organizations in any society. Tilly (1998)proposes mechanisms that contribute to installation of the widely recognized categories (such as gender,religion, caste, ethnicity and so on) internally, their maintenance, and their percolation into many organi-zational forms within society. By explicitly focusing on the persistence of structures that influence world ofwork within and across organizations, Tilly (1998) provides explanations for the mechanisms that underliedurable categorical inequalities. 2mpirical works that test much of the macro sociological theories, do so by illustrating the stabilityof stratification or class structures within societies by making plausible assumptions about the basis ofstratification. Bourdieu (2013) for example, treat cultural capital of individuals as constituting the basisfor stratification. Empirically highlighting an association of cultural tastes of individuals with the occupa-tions they are situated in, Bourdieu (2013) plots occupations onto social space and depict a hierarchy ofoccupations or the stratification structure in terms of differences in their cultural capital. Recent workswithin CAMSIS (Cambridge Social Interaction and Stratification) tradition (Bottero, 2004; Griffiths andLambert, 2012; Lambert and Griffiths, 2018), base their stratification structure on social relations betweenindividuals across occupations to determine whether occupations are socially close or distant in a socialspace. They empirically map occupations onto a two-dimensional space or as a network in order to identifyemergent classes or strata based on closely spaced occupations. More recent work by Toubøl and Larsen(2017) assume occupational mobility as the basis for class formation based on the works of Max Weber.They map occupations onto a network with edge weights determined by the extent of intra-generationalmobility between any two occupations, to explore class formations based on clustering patterns within theconstructed occupational network. In all these studies occupations are treated as the units of analysiswhile observing stratification or class structure. “Again, the founder of a small manufacturing firm, following models already established in thetrade, divides the firm’s work into clusters of jobs viewed as distinct in character and qualifica-tions and then recruits workers for those jobs within well-marked categories. As turnover occursand the firm expands, established workers pass word of available jobs among friends and rela-tives, collaborating with and supporting them once they join the work force. Those new workerstherefore prove more reliable and effective than others hired off the street, and all concernedcome to associate job with category, so much so that owner and workers come to believe in thesuperior fitness of that category’s members for the particular line of work.” (Tilly, 1998)Charles Tilly in his seminal work the ‘Durable Inequality’ (Tilly, 1998), proposes mechanisms thatoperate within and across organizations and which are at the root of persistent inequalities along categoriessuch as gender, race, caste, ethnicity and so on. For him, causes for social inequality and its reproductionwithin society can be understood in terms of mechanisms that sustain inequalities along categorical groups.The central argument of his thesis is that: “Large, significant inequalities in advantages among humanbeings correspond mainly to categorical differences such as black/white, male/female, citizen/foreigner,or Muslim/Jew rather than to individual differences in attributes, propensities, or performances.” (Tilly,1998). According to him, ‘durable inequality depends heavily on institutionalization of categorical pairs’,and this occurs when organizations at large, match individuals from external unequal categories withinternal work roles for the purpose of efficiency and maintenance. Four mechanisms are key to this matchingof unequal categorical structures internally. First, the installation of categorical pairs occurs when peoplecommanding resources at the helm of organizations although draw returns from the work of others, theynevertheless exclude others from the full value added by their effort. This is the mechanism of exploitation.Second, when individuals or groups belong to a ‘categorically bounded network’, any resources acquired bysuch members are supported and often enhanced by the ‘network’s modus operandi’. This is the mechanismof opportunity hoarding. Third, the mechanism of emulation, happens when established organizationalmodels are copied or replicated by many organizations across the society. Fourth, the elaboration of dailyroutines within organizations across its internal work boundaries, often happens on the basis of the unequalcategorical structures, which is the mechanism of adaptation. While exploitation and opportunity hoarding3acilitate the installation of categorical boundaries into organizations, emulation and adaptation generalizetheir influence across society (Tilly, 1998).“
The notion of upward mobility contains the idea that it represents the triumph of individualachievement over structural constraints ” (Bottero, 2004).So long as categorical boundaries continue to match work boundaries within and across organizations,individuals can also be constrained by such unequal categorical structures within the world of work. SocialMobility through the lens of Tilly, is therefore possible only when the substantive work boundaries be-come independent of the categorical boundaries. According to Tilly (1998), introduction of organizationalforms and with work structures that foreclose possibility of categorical matching, is the only way out forovercoming ‘durable inequality’. According to him, “reduction or intensification of racist, sexist, or xeno-phobic attitudes will have relatively little impact on durable inequality, whereas the introduction of certainnew organizational forms - for example, installing different categories or changing the relation betweencategories and rewards - will have great impact” (pg. 19). In essence, the structure of social inequalityand mobility, is constituted by persistent inequalities along categories such as gender, caste, religion andso on, that are institutionalized within organizations. And the mechanisms that persist such inequalitieswithin society are also the mechanisms that maintain boundaries between work roles within and acrossorganizations by matching them with categorical boundaries. Approximately equating work roles withoccupations, in so far as the occupational boundaries continue to match with categorical boundaries in anysociety, it indicates the persistent influence of the aforementioned inequality reproducing mechanisms.
Institutions are the resilient social structures, that comprise “ regulative, normative, and cultural-cognitive elements that, together with associated activities and resources, provide stability andmeaning to social life ” (Scott, 2013).In describing the mechanisms underlying persistence of categorical inequalities, Tilly (1998) extensivelyrelies on the stability of organizational forms and the institutionalization of categorical pairs within them.This institutionalization occurs through scripts, practices, established hierarchies, and other forms of net-worked relationships between work roles, that structure the actions of incoming actors within organizationsin such a way that it reinforces the internal work boundaries along categorical lines.However, it is hard to deny that several other forces of structuration do exist, which act upon organi-zations and alter their forms through time. These forces do not necessarily depend upon institutionalizedpractices, scirpts, and relationships built around categorical pairs. Instead, they depend upon the partic-ular organizational field in which organizations are located and the corresponding rules, norms, practicesthat govern such fields. According to DiMaggio and Powell (1983) organizations that produce similar ser-vices or products, or say, belong to a particular industry, while in aggregate constitute an organizationalfield, they are simultaneously subject to similar institutional pressures that make them isomorphic to oneanother. In their seminal work on ‘institutional isomorphism’, which is at the foundation of institutionaltheory literature, DiMaggio and Powell (1983) highlight three important mechanisms that capture themechanisms or the forces of structuration which influence organizations to become similar to one anotherin an organizational field. These mechanisms are broadly effected by the state (coercive), uncertainties ofthe market (mimetic), and lastly the professions (normative). In so far as the role of state is concerned, themechanism of coercive isomorphism explains for example, how and why organizational changes are a re-sponse to ‘government mandates’, or other ‘legal and technical requirements of the state’. The mechanism4f mimetic isomorphism explains as to how and why, in response to uncertainties of the market, organiza-tions tend to morph themselves similarly with other organizations that are perceived to be legitimate orsuccessful within their field. Lastly, professionalization makes organizations similar within a field throughthe mechanism of normative isomorphism. Following Sarfatti Larson (1977), DiMaggio and Powell (1983)define the idea of professionalization as the “collective struggle of members of an occupation to define theconditions and methods of their work, to control, ‘the production of producers’ (Sarfatti Larson (1977):49-52), and to establish a cognitive base and legitimation for their occupational autonomy”. Formal edu-cation, and professional networks spanning organizations, according to them are the two important aspectsof professionalization that contribute to homogeneity of organizational structures within a given field andthe consequent variation across fields.From the above discussion we note that the changes in political and economic structure of the state, themarket environment, and changes around professionalization, together can alter the organizational formswithin a given field largely independent of the institutionalization of categorical pairs. Therefore, suchchanges in organizational forms could also contain in them the seeds of social mobility (Tilly, 1998), whereother forces of structuration discussed above, can overpower the institutionalization of categorical pairs.In the subsequent section, we propose occupational network structure as a framework that allows us tolook at these two kinds of forces separately, and 1. Identify the patterns of social mobility and 2. Lookout for the probable mechanisms or forces of structuration that could have contributed to social mobility.
According to Tilly (1998), matching of external categories (such as religion, gender, language, caste and soon) onto internal work roles within organizations is critical to the reproduction of categorical inequalities.Insofar as work roles or occupations are structured and therefore segregated or stratified along categoricallines, it is indicative that the mechanisms of durable inequality are at play. That is, in such case, theforces of structuration operating along categorical pairs, also cast their influence over the proximity ordistance between occupations in the substantive world of work. In the earlier section we have seen thatorganizations are structured by the institutional influences of the corresponding organizational field. Theseinstitutional influences therefore also tend to structure the occupations that comprise such organizations.Further we have also seen that occupations by themselves are structured by forces beyond organizationalfield such as the formal education of its occupants and the professional networks that undergrid such occu-pations (Sarfatti Larson, 1977). Infact Grusky and Weeden (2001) emphasize that the substantive forces ofstructuration happen locally at the level of disaggregated occupations, rather than aggregate classes. Sucha structuration, according to them, could manifest as ‘tangible and symbolic control over the supply oflabour’ in the form of stipulated educational requirements, instituted licensing systems, organized unions,occupational associations and so on (Grusky and Weeden, 2001).Above we have seen above, there are two sets of structuration forces that are at play operating overoccupations within organizations. One as per Tilly (1998), that says institutionalized categorical pairsdetermine occupational boundaries. Second, which say that occupations within organizations are structuredby the institutional forces operating on the corresponding organizational fields and also by the stipulatededucational requirements, licensing systems, unions, associations and so on (DiMaggio and Powell, 1983;Grusky and Weeden, 2001). In order to evaluate social mobility we therefore attempt to look at theoperation of these different sets of structuration forces over occupations, separately in order to check forsocial mobility patterns. We conceive of an occupational network structure that is constructed based on theforces of stratification that are non-categorical in nature, and look for the extent to which such a network5s also stratified along attributes defined by categorical representations in each occupation. We describethis network construction below, and discuss the mathematical aspects associated with measurement ofsocial mobility and mechanisms in the subsequent section.
For us, given a category (such as gender), each occupation is defined by 1. Representation of individualsacross groups within such category (such as male/female in case of gender), relative to their representationin the overall workforce, 2. Distribution of workforce in this occupation across different industries, and 3.Distribution of workforce in this occupation across different education levels. We proxy (2) for the struc-turation forces operating on an occupation that come from its association with multiple organizationalfields, and (3) for the stipulated educational requirements specific to an occupation. Note that, owing todata availability we limit to only (2) and (3) for proxying the structuration forces that are specific to anoccupation, however in reality one could include other factors such as licencing requirements, associationsand so on. We consider (1) to be the label attributes of occupations that are important while measuringstratification along categorical lines using graph assortativity measures as we shall see in subsequent sec-tions.Before going into the details one could ask at this point, as to why we do not look at each occupationas a separate entity and depict the structure of world of work as constituted by bundle of independentoccupations? Or, one might also ask, could you not impose a hierarchy of some sort to the occupations ? We acknowledge that both these ways of looking at the occupational structure are plausible, where inthe former case there is no notion of distance between occupations, and in the latter, there is a clear-cuthierarchy imposed to define the occupational structure. However, given our conceptualization about thestructure of the world of work we find occupational network as lying somewhere in between these twoextremes. In addition it serves as a framework to observe social mobility patterns and simultaneously lookfor changes in the occupational network that could possibly have answers for mechanisms or structurationforces that contribute to social mobility. Definition 1
Occupational Network:
Let E = { E , E , .., E m } be set of education levels, I = { I , I , ..., I n } be set of industry sectors. Consider a distribution defined over support S := E × I . We denote the non-categorical structuration forces operating on an occupation at a given point in time (or for a given birthcohort) by the distribution of individuals associated with it over the support S . Now the distance betweenoccupation O i and O j can be given by the distributional distance D between O i and O j defined over thesupport S . This means, if two occupations are closer that indicates that the non-categorical structurationforces operating on them are similar. We use total variation distance as our measure of distributional dis-tance as it is always bounded in the range [0 , , with indicating exact same distributions, while indicatescompletely disjoint distributions. We transform distributional distance into edge weight, where edge weightis given by w i,j = 1 − D ( O i , O j ) . After constructing an adjacency matrix with these edge-weights, we re-move the edge-weights that connect the same occupations to avoid self-loops, and subsequently normalize itsvalues by the sum total of all the remaining edge-weights. Our resultant network is therefore an undirectedweighted network with weights given by corresponding values in the normalized adjacency matrix A . In the following section, we first introduce scalar assortativity measure defined over undirected weightedgraphs, where node labels are given by scalar real valued attributes. In our case, however, with nodes as For example, bundling occupations along hierarchically situated class locations following established class schemas such asthe one proposed by Erikson and Goldthorpe (2002), or rank occupations on the basis of average education or average incomeof the individuals.
Here we first discuss the assortativity measure over graphs with nodes having scalar attributes. Newman(2003) defines this measure over scalar attribute graphs as Pearson’s correlation coefficient across edges.We first adjust this correlation coefficient to weighted graphs, and make explicit, the notion of attributes ofnodes on an edge as random variables, and edge-weights as corresponding probabilities, following Peel et al.(2018). Applying similar intuition, we then extend the scalar assortativity measure to vector attributes byreplacing Pearson’s correlation coefficient with distance correlation (Lyons, 2013; Sz´ekely et al., 2007). Wetrade-off the interpretation of linear independence as offered by assortativity measure based on Pearson’scorrelation coefficient with the interpretation of independence. Since our attribute space is comprised ofvectors of a chosen dimension, we consider this assumption as a plausible one to make, given our problem.
Assortativity defines the property of a network or a graph where nodes with similar attributes have atendency to be strongly connected than those with dissimilar attributes (Newman, 2003). In the contextof scalar node attributes (such as age) Newman (2003) indicates that assortative mixing over a socialnetwork could suggest stratification of society along such attributes. He defines assortativity measure asPearson’s correlation coefficient measured between attributes of adjacent nodes in the network. In thefollowing proposition we adjust this measure to the case of a weighted graph following Peel et al. (2018).We also explicate the assumptions that underpin this approach of measurement, which as we will showsubsequently, will help us to extend this measure to graphs with nodes having vector attributes.
Proposition 1
Consider an undirected weighted graph G = ( V, E ) having n nodes and m edges. Everyedge in the graph is defined by pair of nodes ( i, j ) , with strength of the connection given by a weight w ij ,and attributes of the nodes given by x i and x j such that x i , x j ∈ R . Assortativity measure r on this networkfollowing Newman (2003) and Peel et al. (2018) is given by r = (cid:80) ij A ij ( x i − ¯ x )( x j − ¯ x ) (cid:80) i k i ( x i − ¯ x ) where A denotes normalized adjacency matrix such that A ij = w ij (cid:0) (cid:80) i ≤ j w ij (cid:1) , if i = j w ij (cid:0) (cid:80) i ≤ j w ij (cid:1) , if i (cid:54) = j , k i = (cid:80) j A ij , and ¯ x = (cid:80) i x i k i .Proof: Consider X and Y to be random variables denoting the attributes of start and end nodes corre-sponding to any randomly chosen edge in the graph (Note that every undirected edge between two differentnodes is treated as two directed edges). X and Y follow the same distribution with support defined overall the scalar attribute values over the undirected graph. Joint distribution of X, Y , is defined in terms ofedge-weights as follows, 7 ( X = x, Y = y ) = (cid:88) i,j ∈ Vx i = x,x j = y A ij Assortativity index according to Newman (2003) is given by Pearson’s correlation coefficient, r = E [( X − µ x )( Y − µ y )] (cid:0) E [( X − µ x ) ] E [( Y − µ y ) ] (cid:1) / = E [( X − µ )( Y − µ )] E [( X − µ ) ] (Since X and Y have identical marginal distributions with mean µ )= (cid:80) x,y ( x − µ )( y − µ ) P ( X = x, Y = y ) (cid:80) x ( x − µ ) P ( X = x )= (cid:80) ij A ij ( x i − µ )( x j − µ ) (cid:80) i k i ( x i − µ ) as P ( X = x, Y = y ) = (cid:88) i,j ∈ Vx i = x,x j = y A ij , P ( X = x ) = (cid:88) y P ( X = x, Y = y ) Which is same as Equation (B2) in Peel et al. (2018) (cid:3)
In the above formulation of assortativity index, the probability P ( X = x, Y = y ) indicates the fractionof edges that connect nodes having the attribute x with nodes having attribute y . A ij on the other handindicates the strength of the connection between two nodes i and j . Our idea behind illustrating the aboveformulation is to bring out a distributional assumption that the strength of an edge is proportional to theprobability of selecting adjacent nodes (as the start and end nodes) constituting that edge. A higher assor-tativity therefore indicates that strongly connected nodes, whose consequent edges are also more probableof selection, are also close in terms of their corresponding node attributes. In our case where the networkis constituted by occupations and their interconnections, the aforementioned index reflects the stratifica-tion of occupations along any scalar real-valued attribute defined over each of the occupations. A higherassortativity indicates that occupations are largely stratified along this attribute.However, in so far as our study is concerned, we are interested in categorical attributes of occupationswhich can be multi-dimensional. Since we are interested in stratification of occupations along categoricalattributes, we associate a vector made up of representations of different groups within a given category toeach occupation relative to their corresponding representation in the total workforce. For example, if weare interested in stratification based on sector, which is a category constituted by two groups rural andurban, then the vector constituted by representations of rural and urban workforce in given occupation,relative to the rural and urban workforce in the total population, defines the corresponding occupation’sattribute. Although in categories defined by only two groups, representation of any one category can betreated as scalar real-valued attribute to compute assortativity index like above, but for those defined bymore than two groups (such as language, religion, caste, ethnicity and so on) we will have to accommodatevector attributes as well. We can also treat parent’s occupations as categorical groups, which is usuallythe case in contingency table based mobility measurement approaches.8his motivates us to define a vector assortativity index to measure stratification of occupations alongcategorical attributes, which captures the essence of categorical inequality in Tilly’s terms. Observinghow inequalities along various categories such as religion, gender, caste and so on, are changing over time,will tell us whether or not the substantive work boundaries are becoming independent of given categoricalboundaries. Higher the assortativity, higher is the extent of categorical inequality, and persistence of ahigher assortativity through time relates to absence of social mobility. In other words, vector assortativityhelps us infer about whether the changes in the occupational network structure through time is reflectssocial mobility or the reproduction of categorical inequalities.To construct a vector assortativity index, we build on the notion of linear dependence (or correlation)of real-valued scalar random variables (denoting node attributes), and extend it to multi-dimensional realvalued random vectors (also denoting node attributes). Since linear dependence of random vectors has littlemeaning (as signified by the usage of Pearson’s correlation coefficient), we instead consider independenceof random vectors. We therefore replace Pearson’s correlation coefficient with distance correlation (Sz´ekelyet al., 2007; Lyons, 2013) between two random vectors to extend the above proposition to vector attributesof nodes. We illustrate it in the following proposition. Proposition 2
Consider an undirected weighted graph G = ( V, E ) having n nodes and m edges. Everyedge in the graph is defined by pair of nodes ( i, j ) , with strength of the connection given by a weight w ij ,and attributes of the nodes given by x i and x j such that x i , x j ∈ R d . Vector assortativity measure r onthis network is given by a distance correlation measure, following Sz´ekely et al. (2007), and Lyons (2013),which is the square root of r = f f Where f is given by the following expression (cid:88) i (cid:48) ,j (cid:48) (cid:88) i,j A ij A i (cid:48) j (cid:48) d ( x i , x (cid:48) i ) d ( x j , x (cid:48) j ) − (cid:88) i,j A ij (cid:32)(cid:88) i (cid:48) A i (cid:48) . d ( x i , x (cid:48) i ) (cid:33) (cid:88) j (cid:48) A .j (cid:48) d ( x j , x (cid:48) j ) + (cid:88) i,i (cid:48) A i. A i (cid:48) . d ( x i , x (cid:48) i ) and f is given by the following expression (cid:88) i (cid:48) ,j (cid:48) A i. A i (cid:48) . ( d ( x i , x (cid:48) i )) − (cid:88) i A i. (cid:32)(cid:88) i (cid:48) A i (cid:48) . d ( x i , x (cid:48) i ) (cid:33) + (cid:88) i,i (cid:48) A i. A i (cid:48) . d ( x i , x (cid:48) i ) A here denotes normalized adjacency matrix such that A ij = w ij (cid:0) (cid:80) i ≤ j w ij (cid:1) , if i = j w ij (cid:0) (cid:80) i ≤ j w ij (cid:1) , if i (cid:54) = j , A i. = (cid:80) j A ij ,and d ( x i , x j ) = || x i − x j || , where || . || denotes Euclidean norm.Proof: Consider X and Y to be random vectors denoting the vector attributes of start and end nodescorresponding to a randomly selected edge from the network (Note that every undirected edge betweentwo different nodes is treated as two directed edges). X and Y follow the same distribution with supportdefined over all possible vector attributes over the undirected graph. Joint distribution of X, Y , is definedin terms of edge-weights as follows, 9 ( X = x, Y = y ) = (cid:88) i,j ∈ Vx i = x,x j = y A ij Since we have vector attributes, instead of Pearson’s correlation coefficient here we consider populationdistance correlation which was first defined by Sz´ekely et al. (2007). We follow an equivalent definition ofdistance covariance given by Lyons (2013), in order to arrive at a simplified form given that our populationdistribution is completely determined by the normalized adjacency matrix A .Consider ( X (cid:48) , Y (cid:48) ) be independent and identically distributed copies of ( X, Y ). Then following Lyons(2013), the distance covariance and variance are given by, dCov ( X, Y ) := E [ d µ ( X, X (cid:48) ) d ν ( Y, Y (cid:48) )] ,dCov ( X, X ) := E [ d µ ( X, X (cid:48) ) ] , where, d µ ( X, X (cid:48) ) = d ( X, X (cid:48) ) − a µ ( X ) − a µ ( X (cid:48) ) + D ( µ ) and, d ν ( Y, Y (cid:48) ) = d ( Y, Y (cid:48) ) − a ν ( Y ) − a ν ( Y (cid:48) ) + D ( ν )and d ( X, X (cid:48) ) = || X − X (cid:48) || , a µ ( X ) := E X (cid:48) [ || X − X (cid:48) || ]and D ( ν ) = E [ || X − X (cid:48) || ] µ and ν represent the distributions followed by X and Y respectively. However in our case µ = ν .We define vector assortativity index as distance correlation, which following Sz´ekely et al. (2007), andLyons (2013) is given by square root of, r = dCov ( X, Y )( dCov ( X, X ) × dCov ( Y, Y )) We first simplify the numerator of the above expression, dCov ( X, Y ) = E [ d µ ( X, X (cid:48) ) d ν ( Y, Y (cid:48) )]= (cid:88) x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y d ν ( X, X (cid:48) ) d ν ( Y, Y (cid:48) ) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )Expanding the expression, d ν ( X, X (cid:48) ) d ν ( Y, Y (cid:48) ) =( d ( x, x (cid:48) ) − a µ ( x ) − a µ ( x (cid:48) ) + D ( µ )) ( d ( y, y (cid:48) ) − a ν ( y ) − a ν ( y (cid:48) ) + D ( ν )), we have the following sets of terms.1. d ( x, x (cid:48) ) d ( y, y (cid:48) )2. − a ν ( y ) d ( x, x (cid:48) ) , − a µ ( x ) d ( y, y (cid:48) ) , a µ ( x ) a ν ( y ) , − a µ ( x (cid:48) ) d ( y, y (cid:48) ) , − a ν ( y (cid:48) ) d ( x, x (cid:48) ) , a µ ( x (cid:48) ) a ν ( y (cid:48) )3. D ( µ )( d ( y, y (cid:48) ) − a ν ( y )) , D ( ν )( d ( x, x (cid:48) ) − a µ ( x ))4. D ( µ ) D ( ν ) , − D ( µ ) a ν ( y (cid:48) ) , − a µ ( x (cid:48) ) D ( ν ) , a µ ( x (cid:48) ) a ν ( y ) , a µ ( x ) a ν ( y (cid:48) )10et us consider expanding the first term in (2) over the summations defining covariance (cid:88) x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y − a ν ( y ) d ( x, x (cid:48) ) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= (cid:88) x,y (cid:88) x (cid:48) ,y (cid:48) − a ν ( y ) d ( x, x (cid:48) ) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) ) P ( X = x, Y = y )= (cid:88) x,y − a ν ( y ) (cid:32)(cid:88) x (cid:48) d ( x, x (cid:48) ) P ( X (cid:48) = x (cid:48) ) (cid:33) P ( X = x, Y = y )= (cid:88) x,y − a ν ( y ) a µ ( x ) P ( X = x, Y = y )Since distributions µ and ν are identical and by symmetry it follows that all the terms in (2) without thesign end up with same value. Therefore all the terms in (2) together simplify as − (cid:80) x,y a ν ( y ) a µ ( x ) P ( X = x, Y = y )Now consider the first term in (3), we have (cid:88) x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y D ( µ )( d ( y, y (cid:48) ) − a ν ( y )) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= D ( µ ) (cid:88) x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y ( d ( y, y (cid:48) ) − a ν ( y )) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= D ( µ ) (cid:88) x,y (cid:88) x (cid:48) ,y (cid:48) ( − a ν ( y )) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) ) P ( X = x, Y = y ) + D ( µ ) (cid:88) x,y (cid:88) x (cid:48) ,y (cid:48) d ( y, y (cid:48) ) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) ) P ( X = x, Y = y )= − D ( µ ) (cid:88) x,y a ν ( y ) P ( X = x, Y = y ) + D ( µ ) (cid:88) x,y a ν ( y ) P ( X = x, Y = y )= 0Similarly the other term in (3) also vanishesNow consider the third and fourth terms together in (4)11 x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y (cid:2) − a µ ( x (cid:48) ) D ( ν ) + a µ ( x (cid:48) ) a ν ( y ) (cid:3) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= (cid:88) x (cid:48) ,y (cid:48) − a µ ( x (cid:48) ) (cid:32)(cid:88) x,y [ D ( ν ) − a ν ( y )] P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= (cid:88) x (cid:48) ,y (cid:48) − a µ ( x (cid:48) ) (cid:32) D ( ν ) − (cid:88) x,y a ν ( y ) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= (cid:88) x (cid:48) ,y (cid:48) − a µ ( x (cid:48) ) (cid:32) D ( ν ) − (cid:88) y a ν ( y ) P ( Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= 0 , since (cid:88) y a ν ( y ) P ( Y = y ) = D ( ν )Similarly terms two and five in (4) also cancel each other out, and only the first term remains. Thereforeexpression for dCov ( X, Y ) is eventually simplified as dCov ( X, Y ) = E [ d µ ( X, X (cid:48) ) d ν ( Y, Y (cid:48) )]= (cid:32)(cid:88) x,y d ν ( X, X (cid:48) ) d ν ( Y, Y (cid:48) ) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) )= (cid:88) x (cid:48) ,y (cid:48) (cid:32)(cid:88) x,y d ( x, x (cid:48) ) d ( y, y (cid:48) ) P ( X = x, Y = y ) (cid:33) P ( X (cid:48) = x (cid:48) , Y (cid:48) = y (cid:48) ) − (cid:88) x,y a ν ( y ) a µ ( x ) P ( X = x, Y = y ) + D ( µ ) , since ν , and µ have same support.Now using P ( X = x, Y = y ) = (cid:88) i,j ∈ Vx i = x,x j = y A ij , we get, dCov ( X, Y ) = (cid:88) i (cid:48) ,j (cid:48) (cid:88) i,j A ij A i (cid:48) j (cid:48) d ( x i , x (cid:48) i ) d ( x j , x (cid:48) j ) − (cid:88) i,j A ij (cid:32)(cid:88) i (cid:48) A i (cid:48) . d ( x i , x (cid:48) i ) (cid:33) (cid:88) j (cid:48) A .j (cid:48) d ( x j , x (cid:48) j ) + (cid:88) i,i (cid:48) A i. A i (cid:48) . d ( x i .x i . ) Similarly, we can simplify the denominator of r , given by ( dCov ( X, X ) × dCov ( Y, Y )) = dCov ( X, X )to the following expression, (cid:88) i (cid:48) ,j (cid:48) A i. A i (cid:48) . ( d ( x i , x (cid:48) i )) − (cid:88) i A i. (cid:32)(cid:88) i (cid:48) A i (cid:48) . d ( x i , x (cid:48) i ) (cid:33) + (cid:88) i,i (cid:48) A i. A i (cid:48) . d ( x i , x (cid:48) i ) This is the end of our simplification. (cid:3) Data and Findings
For the purpose of our study we carry out our analysis using NSS 68th round on employment and unem-ployment which captures details of individual occupation, industry and education. We recode educationalong four levels, 1. Below Primary, 2. Below Secondary and above primary, 3. Below Graduation butabove secondary, and 4. Above graduation. Industry information is captured in the survey according toNIC-2008 code structure, and we consider this information at the least disaggregated level captured by 20section codes. Occupation information is captured according to NCO-2004 code structure, and we consider2-digit codes for the purpose of our study.
For a given birth cohort we build network based on our definition of occupational network. From year 1940to year 1980 we consider consider consecutive and overlapping 10 year birth cohorts considering slidingwindows with spacing of one year. Following are the patterns of vector assortativity and averaged scalarassortativity observed across years. We compute averaged scalar assortativity as the average of scalarassortativity computed based on the proportional representation (relative to workforce), for each of thecategorical groups within a given category, which is a scalar label attribute for each occupation. We findthat the trends observed for each category are similar following either of these measures (See Figures 1 and2). Either way we find that while assortativity along sector (rural/urban) is more or less stagnant acrosscohorts, assortativity along gender (male/female) has come down slightly for the recent cohorts, and hasbeen steadily increasing over the years along social group (GEN/OBC/SC/ST) with a slight dip in therecent cohorts.
In so far as occupations are less assortative along ascriptive categorical attributes, it indicates that thestructural constraints determined by such ascriptive characteristics are less important in the world of work.Within the social mobility literature dealing with intergenerational transition of individuals from one occu-pational position to another (Motiram and Singh, 2012; Iversen et al., 2016; Azam, 2015), the categoricalstructures constratining individuals are assumed to be their initial class/occupational positions. Theseinitial class positions are usually indicated by the occupation of their respective parents. Our frameworkalso allows us to define category as constituted by groups of individuals with similar parent occupations.This work addresses key data limitation in developing countries where mobility measurement using incomeindicators is difficult. Our approach treats occupations as units of analysis, and since occupation informa-tion is typically found in most of the sample surveys and even census, it allows for mobility measurementdespite such data limitations.Our study also makes a minor contribution to network science literature. Although there are existingworks that deal with computing vector assortativity (see Pelechrinis and Wei (2016)), here we attempt toprovide a closed form expression for computing it based on distance correlation following Sz´ekely et al.(2007) and Lyons (2013).
Future Work : In order to identify social mobility mechanisms, Tilly (1998) argues that one can findthem only when one deep-dives to understand about the forces of structuration that influence work rolesor occupations within organizations. Since we considered industry education combinations as the set ofnon-categorical structuration forces that operate over the occupations within the world of work, one couldalso attempt to identify or measure the contribution to assortativity by each one of these combinations.13 .10.20.30.40.5 1940 1950 1960 1970 1980 year V e c t o r A ss o r t a t i v i t y category sectorsexsocialgroup across sliding windows of 10 year birth cohorts, slide by a year Trends depicting Vector Assortativity
Figure 1: Vector Assortativity year S c a l a r A ss o r t a t i v i t y A v e r age category sectorsexsocialgroup averaged across sliding windows of 10 year birth cohorts, slide by a year Trends depicting Scalar Assortativity
Figure 2: Scalar Assortativity
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