Our first task is to identify and quantify interactions among occupational specializations across MSAs. This task is made difficult by the absence of readily available data on material, personnel, financial, or informational flows among work places of the sort that would directly signify interaction among workers in different work places. How, then, can one infer from the presence of specialized occupations in an MSA that their co-location is not merely accidental but indicative of possible causal interaction? Here we employ conditional probability: specifically in this context, if the presence of one specialized occupation in an MSA is partly determined by the presence of another specialized occupation, one would expect conditional probabilities to differ from marginal ones. Accordingly, we define the “interdependency” between two occupations and , , as:(2)where and denote a randomly selected MSA (for brevity, the superscripts will hereinafter be omitted); see Methods for detailed discussion on . This metric measures how an MSA's specialization in one occupation may enhance or hinder its specialization in another. The emphasis on “may” acknowledges that – as is the case for many statistical analyses – without additional information or experiments, our analysis cannot imply direct causality; at best, it identifies structural relationships and points to potential places where one may search for such causality. With that caveat in mind, we now proceed to interpret . Positive means that occupations and are more likely to be specialized in the same MSAs than if they are independently distributed across MSAs. The opposite is true for , while means that occupations and are never specialized in the same MSA. Clearly, is closely tied to the more conventional way of representing conditional probability: . Note that is intrinsically symmetric (i.e., ) and thus a suitable metric for building an occupation space (see Methods), while the above relationship preserves the directionality of the conditional probabilities between two occupations, which will be important in our later analysis.

We investigate the structure of occupational specializations in MSAs using 2010 data, the latest year for which both employment and GDP data were available at the MSA level. The histogram (Fig. 1A) indicates that most occupations have positive interdependencies with one another, with 31.2% being negative and 3.8% equal to . The figure also shows that most interdependencies are relatively weak, as indicated by the peak around zero; somewhat surprisingly, and notwithstanding the fact that cities are agglomerations of individuals and businesses, many occupational specializations within urban areas do not strongly interact with one another. Fig. 1B shows the between all 787 distinct occupations considered in a matrix format. Occupations are simply ordered in accordance with BLS occupation grouping codes: the first 33 rows/columns of the matrix correspond to occupations whose codes start with 11 (management occupations); the next 30 rows/columns those starting with 13 (business and financial operations occupations); and so on. Consequently, the presence of some dark green areas (representing strong positive interdependencies) along the diagonal is to be expected: some occupations in the same classification group are closely related and, if an MSA specializes in one, it likely specializes in another. More interesting, however, are the white and red bands (weak or negative interdependencies) along the diagonal, and the off-diagonal dark green areas, indicating that strong positive interdependencies exist among occupations belonging to different occupational groups. These strong interdependencies define the structure of the occupation space.

Deeper insights about the occupation space are gained by revealing its structure through the use of a network representation, shown in Fig. 1C (not all links are shown). Here the nodes represent occupations and the length of the edges between them represent their interdependencies. Several patterns emerge from this network. A number of occupations are specialized in only a few MSAs (small nodes). Many of these rarer specialties are specialized in the same MSAs, resulting in strong positive interdependencies and forming the “core” of the occupation space (Fig. 1C and also Fig. S6 in Supplementary Information S1). We also find that common occupations (larger nodes) interact relatively weakly with other occupations (i.e., they are specialized in many places regardless of what happens with other occupations (Fig. S2 in Supplementary Information S1)) and consequently are located along the periphery of the occupation space. Overall, these occupational specializations form a rather dense network of strong positive interdependencies (see Figs. S3, S4, and S5 in Supplementary Information S1): even when one considers only edges associated with , occupations still have, on average, hundreds of such strong interdependencies with other occupations with a high degree of clustering (Table S1 in Supplementary Information S1). It is worth noting that educational occupations are among those with the largest numbers of such strong positive links, highlighting their importance in urban economies (Table S1 in Supplementary Information S1). Additional topological features of the occupation space are also reported in SI.

What region of occupation space do metropolitan areas with different levels of productivity and wealth inhabit? Answering this question is critical to understanding how an MSA's specialization portfolio is related to its economic performance. We categorize MSAs into quartiles according to their 2010 GDP per capita (Fig. S1 in Supplementary Information S1), a measure of both productivity and wealth, and consider their specialized occupations set (), the set of occupations in which MSAs are specialized (i.e., ). As one moves from the bottom to the top quartile, the gravitate toward the occupation space's core region – characterized by uncommon occupations and strong positive interdependencies (Figs. 1C and 2). Wealthier MSAs specialize in more unique occupations than their poorer counterparts.

No effect of the MSA size has so far been discussed. Size – in terms of population – has been shown to have strong relationships with a city's productivity, diversity, and specialization profile. How do the interdependencies among occupational specializations fit in these relationships? Fig. 3 shows the three-way relationship between size (for which the total number of employees being used as a proxy), GDP per capita, and fraction of interdependencies among specialized occupations that are negative. The partial correlation coefficient between the fraction of negative and GDP per capita with the MSA size held constant is : this represents the relationship between interdependencies and productivity with the effect of size filtered out. Together, Figs. 1, 2 and 3 suggest that on average larger cities are more productive, and their specialized occupations are more unique and have less negative interdependencies. Among cities of similar sizes, those with less negative interdependencies tend to have higher GDP per capita. An MSA's productivity thus depends not only on how many and what jobs are included in its but also on the interdependencies among them.

In addition to its present productivity, the interdependency network within its constrains an MSA's future economic trajectory. Intuitively, one might expect that an MSA is more likely to develop new specializations in occupations that have many positive interdependencies with occupations in its current . To quantify this notion, while capturing the effects of different signs and magnitudes of these interdependencies (cp. Ref. [10]), we introduce the transitional potential that a non-specialized occupation will become specialized in a later year:(3)where is a parameter (see Methods for additional discussion on Eq. 3). Fig. 4 shows that occupations with higher are indeed more likely to become specialized. (Keep in mind, though, that while an occupation with higher may potentially be specialized more easily, whether its specialization is efficient, desired by, or beneficial to the MSA is a different story (see Table S2 in SI); this issue is addressed in our next analysis.) Interestingly, the effects of such constraints seem to saturate after 3 years; this 3-year saturation pattern is robust even when different starting years are used (see SI).

How easy or difficult is it for an MSA to transform its ? Before proceeding, let us note for clarity that hereinafter “transition” refers to a change at the single-occupation level, whereas “transformation” refers to a change of an MSA's entire ; a transformation thus consists of many transitions. We define transitional occupations as those occupations with in a possible new but with in the current (or original) set. Letting denote the set of transitional occupations, we write , where subscripts 1 and 2 represent the original and the new , respectively, and the prime sign denotes a complement. We then measure the ease of transformation from one into another as the average of transitional potentials to all transitional occupations. Letting denote the ease of transformation for occupational portfolios to , we write:(4)where is the total number of transitional occupations. Note that is asymmetric as it depends on the direction of transformation. Fig. 5A indicates that some tradeoff exists between the ease of transformation and the improvement of productivity that might result from it: more difficult transformations are generally associated with greater increases in metropolitan GDP per capita. As one might expect, there is significant uncertainty around this trend.

Given a set of transitional occupations associated with a transformation, what would constitute efficient pathways to achieve the transformation? What is the “best” first transitional occupation to pursue? Candidates for such an occupation should meet the following two properties: easy to transition to and conducive to further transitions. The first property is captured by the transitional potential . We propose to capture the second property by the average transitional probability from a given transitional occupation to all other transitional occupations, denoted by ; that is, for a transitional occupation ,(5)

We apply our method to a specific example: transformation from the of the upper-middle quartile of MSAs (Fig. 2C), to that of the top quartile (Fig. 2D). The result is shown in Fig. 5B, which captures another type of tradeoff – between and . A Pareto frontier of the “optimal” transition occupations is observed: easier first transitions appear to be accompanied by more difficult subsequent ones. Equally useful is the analysis' ability to identify suboptimal transitions that may not be worth pursuing.

The values of used in our analysis are calculated by applying Eq. 1 to the 364 MSAs considered (see Supporting Information for more details). Some reported values of exist that may be based on employment data from both metropolitan and micropolitan statistical areas; these are not to be confused with the 's in the present study.

Our project was originally inspired by the work by Hidalgo et al. [10]. The key idea there was that what product an economy can specialize in is constrained by the economy's location in a “product space,” which represents how different products are related to one another. To Hidalgo et al., this suggests the notion of proximate and distant products, e.g., apples and pears have high proximity, as opposed to apples and copper wires [10]. This structural perspective is relevant to occupations in urban economies as well.

At the heart of their work is a measure that was introduced to capture the relationship between different products. The measure is called ‘proximity.’ Now if we were to apply proximity to our occupational data, the proximity between two occupations and would be defined as:

However, we found that there are a number of unsatisfactory properties associated with .

First and most importantly, there are circumstances under which , as a measure of relatedness between occupations, is ambiguous and misleading. The following counter-example illustrates this point. Consider two occupations and that are specialized in many MSAs (i.e., they are “common”) and are statistically independent of each other. The proximity between them is:

The second equality is obtained because, when considering two statistically independent events, the conditional probability is the same as the marginal probability. Then, because both are specialized in many MSAs, both and would be large. Thus, is large. These lead us to the following result: two statistically independent occupations can have large proximity – which implies a strong positive relationship – between them, and accordingly would appear close to each other in the network representation. Such an outcome is misleading.

Second, the symmetry of the proximity, i.e., , is not intrinsic, but is “forced” by the minimum operator. The symmetry is desired for building the network representation of the occupation space: the distance between two occupations should be based on only one number.

Third, there is no objective threshold of to determine which relationship is beneficial and which is not: it seems that a link, regardless of its corresponding , is always beneficial (see, e.g., the so-called “density” in Ref. [10]). We do not subscribe to this view, and believe that a relationship between two occupations can be either supportive or conflictive in nature.

It turns out that all these unsatisfactory properties can be eliminated by recognizing that conditional probability by itself does not completely capture the relationship between two occupations: it must be compared with the marginal probabilities of the two occupations. This recognition leads us to propose the ‘interdependency’ between occupations and as follows:

This new metric eliminates the ambiguity and misleadingness and logically captures the relationship between two occupations. It is intrinsically symmetric and can therefore be appropriately used to construct the network representation of the occupation space. The supportive or conflictive nature of the relationship can be easily identified by the sign of .

To construct the network representation of the occupation space, we use the forced-spring-and-repulsion scheme, in which the springs pull the nodes (occupations) closer together and the repulsion pushes them away. At the start, we assign very weak repulsion and very weak springs between all nodes. Then, for positive 's, we add strength proportional to the magnitude of 's to the springs. Similarly for negative 's, we add strength propositional to the magnitude of 's to the repulsion. From the random initial conditions, the nodes are allowed to equilibrate with these pulls and pushes and eventually settle into a more-or-less steady-state configuration. This is what is shown in Fig. 1C. Note that it would take a very long time for the occupation space to reach the actual steady state, i.e., no node movements. What we present is the snapshot of the occupation space where the nodes are still moving, but at very, very low speeds, and the occupation space's topology and appearance shows essentially no change.

The expression of , Eq. 3, is motivated by considering a successful transition of a previously non-specialized occupation into a specialized one as a result of a decisive transition from one of the existing occupations in the MSA's current specialized occupation set () into occupation . The probability of success of transitioning from occupation into occupation is assumed to be proportional to the corresponding conditional probability or, in terms of , . It is further assumed that each of these possible transitions is independent of one another. These considerations and assumptions lead the above expression, which is simply the probability that the transition from one or more currently specialized occupations are successful.

Note the emphasis on the term “motivated” in the previous paragraph. It serves as a reminder that we are not claiming that such independent transitions constitute the actual mechanism of transition from non-specialization to specialization of an occupation. As this work indicates, occupational specializations in urban economies form a complex network of interdependencies, and thus the actual mechanism of such a transition will surely be influenced by this network. Rather, the above expression of should be viewed as a kind of approximation in which higher-order interactions are excluded. Indeed, it is not at all uncommon that neglecting higher-order interactions allows one to construct useful and simple models or explanation of some complex phenomena (see, e.g., Ref. [24] for an example in neuroscience and Ref. [25] for another in ecology). In the present case, Figs. 4 and S7 in Supplementary Information S1 indicate that the formulation of fulfills its purpose: a useful, theoretically-based measure of how an MSA's constrains its future occupational specializations.

In all the results presented in this paper, is used to calculate . This value is simply chosen to result in a useful range of values of : too high values of would yield 's that are always close to 1, while too low values of would yield 's that are always close to 0, both of which are not useful for our analysis. Finally, note also that for a very small , can be approximated by