Exploring the Complex Structure of Labour Mobility Networks: Evidence from Veneto Microdata

This essay investigates the network structure of inter-firm worker mobility in Veneto, an industrial region of Northern Italy, using comprehensive employer-employee matched data. The empirical network reveals a small world pattern that hinges critically upon a few hub firms. Main hubs are found to be: long-established manufacturing companies; wholesale companies; and companies supplying workforce to third parties. The methodology of investigation provides a tool kit for monitoring labour market evolution, and should enable industry policies supporting labour reallocation mechanisms.


Introduction
This paper investigates the network structure of inter-firm worker mobility in a regional labour market in Northern Italy during the 1990s. Worker reallocations are selected and assembled together so as to form a knit texture, formally defined as a binary directed graph where vertices indicate firms and links represent transfers of workers between firms; the characteristics of such structure are examined using the original lens of complex network theory.
The study represents a tentative bridging between the analyses of labour mobility based on linked employeremployee data and the analyses of complex networked systems applied to human behaviours. The main goal is to improve the general understanding of the worker reallocation phenomenon in a well-defined regional labour market, offering policy makers a new tested toolkit for monitoring worker flows that could support precisely targeted interventions directed at improving or preserving the functioning of reallocation mechanisms.
For the first time, a large worker mobility network, consisting of hundreds of thousands of vertices and links and covering the universe of private sector employment in an industrial region of Europe over a full decade, is thoroughly analyzed. In such a context, network analysis allows to explore several aspects of labour mobility that otherwise would remain hidden in the mass of individual work histories contained and intertwined in a linked employer-employee dataset, and which could not be easily mined by standard econometrics. In particular, two are the issues it is possible to appreciate: whether the regional economy under scrutiny proves to be globally integrated into a single labour market and to what extent this labour market is easily practicable by workers; and whether or not the flowing of workers from job to job depends critically on a few pivotal employers.
The investigation is based upon mapping individual reallocations in a region of Italy, Veneto. The key statistical source is the VWH database, a longitudinal dataset that covers the universe of worker histories in Veneto in the private sector for more than twenty years, up to 2001. 1 In respect to the specific research purposes of this essay, the VWH database presents the unique feature of allowing to recover virtually all individual transitions between firms; moreover, the administrative nature of the data ensures that the obtained network is indeed a reliable representation of the true fabric of labour mobility.
Veneto is a highly industrialized Nuts-2 region that ranks among the most developed and densely populated regions of Europe. Most notably, the economy is by far dominated by small and medium size ventures; economic activities are widespread over the entire territory, at the same time showing clear spatial agglomeration pivoted around a plurality of small and medium urban centres and industrial districts (Brusco, 1986;Becattini et al., 1990), but overall there emerges no single organization nor economic concentration that can play as a gravity centre of the system, contrary for instance to regions hosting a big metropolitan area or deeply dependent on very large enterprises (Tattara and Anastasia, 2003).
Throughout the period under scrutiny, the Veneto labour market has been characterized by a positive rate of job creation in both manufacturing and services, and by almost frictional 1 VWH is the acronym for Veneto Worker Histories, a longitudinal linked employeremployee dataset developed at the Department of Economics of the University of Venice Ca' Foscari on the basis of administrative records of the Italian Social Security Institute. unemployment, accompanied by high labour mobility, with worker turnover comparable to Anglo-Saxon countries.
Well understood network models are applied in order to characterize the structure of inter-firm labour mobility. The first focus is on assessing the extent to which the network is pervious to reallocation flows, allowing workers the potentiality to navigate easily across the labour market; to this aim the actual arrangement of inter-firm links is studied and it is compared with the 'small world' model (Milgram, 1967;Watts, 1999;Newman, 2000;Goyal et al., 2006), which is taken as a benchmark for efficient mobility (Latora and Marchiori, 2001). The attention is then turned to the role of employers, examining the statistical distribution of firm links, in order to appreciate how connectivity is distributed within the system; to this aim the empirical connectivity distribution is compared with Pareto-like models (Barabasi and Albert, 1999;Newman, 2005;Gabaix, 2009). Finally, the most connected firms are closely examined, appreciating their importance in integrating different areas of the regional mobility network by means of evaluating the consequences of simulating the failure of such firms, as opposed to the failure of firms chosen at random (Albert et al., 2000).
The paper is organized as follows. Section two briefly discusses the nexus between labour reallocation and networks; Section three formally introduces the network concepts and the empirical strategy; Section four illustrates the data and the main traits of the Veneto economy; Section five contains the small-world analysis; Section six focuses on the distribution of links and on the role of most connected firms; Section seven highlights the characteristics of important employers; finally, Section eight concludes.

Worker reallocation and network approach
The increasing availability of longitudinal linked employeremployee data (Abowd and Kramarz, 1999;Hamermesh, 2008) and the recent development of interpretative models of complex networked systems (Watts, 2003;Newman et al., 2006;Jackson, 2008) offer a unique chance of analyzing in a systematic way the entangled set of tracks naturally arising from the flowing of workers between employers in a given territory.
A distinguishing feature of longitudinal linked employeremployee data is that individuals and their employers are identified and followed over time. Labour economist have taken full advantage of these data to revisit several classic questions, related to employment and wage dynamics, labour turnover, reallocation and productivity growth, just to name a few. More importantly, the endeavour in the construction and analysis of these data has shown its potential in opening a host of new research directions, often at the crossroads between different branches of economics and sociology. In particular, the powerful support of linked employer-employee data allows to scrutinize anew the micro-level behaviour of worker flows originated by individual reallocations; the progress in the science of networks offers the analytical tools to effectively mine information from the data.
The interest in seeking for new aspects of labour mobility finds its root in the key role labour reallocation plays in most industrial economies, with critical implications both on individual working lives, and on the productivity of the economic system. The reallocation of production factors between concurrent uses is widely understood to represent an important ingredient of growth in market economies. 2 In real markets, firms, procedures, and products are continuously replaced by new ones, while capital is diverted between alternative investments, and workers transfer across firms, much in the Schumpeterian spirit of creative destruction.
Most dynamic economies exhibit significant rates of resource reallocation across production units and economic sectors; in particular, the reallocation of workers between jobs in the context of regional economies emerges as a distinctive feature of industrial economies. Not only labour mobility is essential for a system in order to adapt and respond quickly and effectively to competitive and fast-changing international scenarios, but it may significantly contribute to the spread of knowledge, ideas, and know-how within the economic system, possibly resulting in higher productivity and growth. 3 Regional economies which are administratively bounded represent an appropriate setting for empirically investigating labour mobility, which tends to be mostly localized and it is influenced by the institutional environment. In such contexts, the traditional flow approach to labour markets helps us studying the behaviour of aggregated employment flows according to cyclical fluctuations and/or sectoral and employer size classifications, but still it does not reveal the details of the inter-firm arrangement of job to job mobility flows. 4 In this respect, the seminal work of Granovetter (1973) first addressed the problem of job changes at the individual level, using network-based concepts and showing the 2 The positive relationship between reallocation and productivity has been addressed among others by Baily et al. (1992); Bartelsman and Dhrymes (1998);Foster et al. (2001); and more recently by Acemoglu (2009); empirical studies concerning transition Countries are those by Brown and Earle (2008). 3 The empirical regularities of these processes have been extensively investigated and documented, see for instance Davis and Haltiwanger (1999). 4 For a review of the flow approach to labour markets, see for instance Schettkat (1996), and Davis et al. (2006); a recent study of worker mobility in Veneto, using the flow approach, is that of Tattara and Valentini (2010).

Network concepts and empirical strategy
Two are the main issues the present study seeks to quantitatively address by means of a network approach: (1) understanding how much the reallocation market behaves like an integrated system that is pervious to labour flows; (2) identifying which are, if any, the pivotal employers acting as crossroads or brokers for reallocation flows, and what is their role in the global architecture of the network. For the network 5 Network methodologies have been recently spreading in many different fields of economics. Prominent examples can be found for international trade (Fagiolo et al., 2010), knowledge flows and patent citations (Sorenson et al., 2006;, and collaborations in research and innovation (Frenken, 2000;Fleming and Frenken, 2007;Verspagen and Duysters, 2007;Ter Wal, 2010;Kauffeld-Monz and Fritsch, 2010), just to name a few. To the best of our knowledge, the only work dealing with large labour mobility networks is that of Currarini and Feri (2006). nomenclature to be used henceforth, see the Appendix, which provides precise mathematical definitions.
The first line of enquiry is developed in order to primarily appreciate whether the firms in the reallocation market prove to be integrated into a unique web of flows, that is a network with links arranged in such a way that vertices are Hence, hubs appears to be suitable target candidates for interventions aimed for instance at granting operation continuity, should they be deemed essential to a good functioning of the reallocation mechanisms.
The literature on complex network has been explored in search for appropriate ways of appreciating the network properties just discussed. Noticeably, network theory directly relates the notion of integration and permeability to a specific model of network known as the small world (Watts and Strogatz, 1998), whose crucial functional significance is precisely that of making vertices easily reachable one from another, employing a parsimonious assembly of links; importantly, the small-world model has been demonstrated to be highly efficient, in the sense that it grants reliable communication and flow exchange between vertices (Latora and Marchiori, 2001) . 6 The original approach adopted in this work hence uses the small-world properties of the inter-firm network as a proxy indicator of labour market integration and permeability to reallocation flows: the closest the actual network to a small world, the more integrated and permeable the system is to reallocation flows. According to a consolidated literature, a network is said to be a small world when the following four conditions simultaneously apply: 1   (White and Houseman, 2002).
In order to quantitatively appraise the small-world properties, evaluation criteria are derived from the leading network literature and listed in the third column of Table 1.
The combination of such criteria identifies a range of parameters values for which a network shows small-world characteristics. The first goal of the analysis thus becomes establishing whether the empirical labour mobility network is close to a small world or not.
The question to be addressed next is whether firms in the population under scrutiny tend to have similar positions in the network, or they exhibit some degree of heterogeneity. This amounts to check for the existence and to appreciate the role of pivotal employers in the reallocation market, that is employers which are very central in the network, being able of acting as major crossroads for reallocations. The most straightforward way of evaluating how much a firm is central in the reallocation network is to look at the number of links, the degree in network terminology, that the firm has both on entry and exit; this simple statistic is usually referred to as degree centrality (Freeman, 1979).
In particular, a way of grasping at a glance the centreperiphery structure of a network is to examine the statistical distribution of firm links. The presence of pivotal connectors is signalled by a right-skewed distribution, with a long or fat tail that contains vertices having many more connections than average and appearing much more frequently than in a standard Gaussian distribution. In the present study, the empirical strategy devised in order to verify the presence of a hub backbone consists of two steps: first analyzing some moments of the degree distribution, and then moving to the examination of the functional form of the statistical distribution.
The widely acknowledged literature on complex networks shows that the degree distribution of a network with a hub backbone is very often well approximated in the tail by a Paretian distribution, more specifically, by a negative power law Albert, 1999, 2002;Newman, 2003). This essay hence takes the power law distribution as a benchmark for evaluating the presence of a hub backbone; the empirical degree distribution is first fitted to a power law, then the goodness of the fit is rigorously evaluated using the tests proposed by Clauset et al. (2009).

The Veneto economy and the VWH dataset
Inter-firm worker mobility is studied within an industrialized region of Northern Italy, Veneto, representing a well-defined labour market for which comprehensive and detailed information about worker careers are available over a long period of time. The actual analysis covers the period from 1991 to 2000. Basic statistics for the Veneto economy are listed in Table 2.

Small-world analysis
From the assembled data a labour mobility network of 379,391 vertices and 1,899,898 directed links is obtained; this is the main object of study. The basic network statistics are listed in Table 3.
Following the rules of Table 1, the first task in order to verify the consistency of the empirical pattern with the smallword one is to evaluate the level of network integration by looking at the network components. It can be noticed from the data in Table 3   Considering these results altogether, the first finding of the analysis can be therefore stated as follows: the labour mobility network shows a highly integrated configuration, with a centre of gravity in the SCC, which is compatible with the small-world model.
The actual link density of network scores 1.32e-05, that is five orders of magnitude lower than the maximum possible density, that is the density of a complete network, where a link is present between each possible pair of vertices. The empirical value is very low, hence, the second finding of the analysis is that: the labour mobility network exhibits a sparse topology, as required by the small-world model.
A step further in the small-world analysis consists of computing the average network distance between vertices belonging to the SCC, deriving the average path length, and comparing it with the benchmark value ln(n s c c ). Table 3 reports a value of 4.41 for the APL, meaning the whole SCC -hence almost half of the global network, more than 175 000 verticescan be traversed going along less than five links on average, which is a far small number compared to the SCC size, even smaller than the benchmark value ln(n s c c ). Such a result leads to conclude that: the labour mobility network shows the short distance typical of a small world.
Finally, in order to complete the small-world analysis the average clustering coefficient ACC is computed, so as to verify whether the labour mobility network exhibits an assembly of cohesive subgroup of vertices into a broader architecture. The Table 3 is equal to 3.48e-02; in order to evaluate this number, the actual network is compared with two randomized variants taken as null models (Watts and Strogatz, 1998;Newman et al. 2001): when there is no agreement between the actual network and the null models, a pattern in the data exists that is worth exploring; namely, if the actual network exhibits significantly higher clustering than the comparison networks, it can be rightly claim that vertices come into cohesive subgroups.

ACC reported in
As first term of comparison, a network of the same number of vertices and links as the actual network, but with links placed completely at random, is built. Such network exhibits an ACC equal to 5.49e-05, that is three orders of magnitude smaller than the one observed in the real data. Then, a second random network is built, of the same size as the actual network, and the links placed at random under the constraint that the resulting in-degree distribution must be equal to the actual one; such network yields an ACC equal to 3.53e-04, that is two orders of magnitude smaller the actual value. Hence: the real network reveals itself to be appreciably clustered, as required by the small-world model.
The observed clusterization is likely to result from the combination of two factors: first, worker mobility tend to remain mostly local in spatial terms, because of the cost of long-distance commuting; second, economic activities tend to come in spatially localized groups, especially in Veneto, where Marshallian districts represent a major phenomenon. Districts are clusters of firms closely knit by means of a range of different types of relationships, among which the mobility of labour plays an important role (Tattara, 2001). Another distinguishing trait of Veneto is the presence of a plurality of middle-size urban centres, with weak hierarchical relationships, which may deepen the tendency of labour mobility to exhibit clustering, since each conurbation is expected to be a densely interconnected labour market.
The general conclusion of the analysis performed in this Section is that: the labour mobility network in Veneto is a small world. The labour reallocation market consequently appears to be globally integrated and pervious to worker flows.

Hub analysis
The role of the firm populating the labour mobility network is explored by examining the statistical distribution of links. has k>100. Such a pattern points to a very unequal distribution, markedly right-skewed. Similar considerations hold true for both the in-degree and the out-degree distributions.
In Figure 1 the complementary cumulative distributions, CCDDs, for total-degree (a), in-degree (b), and out-degree (c) are plotted on a double logarithmic scale. 9 It can be noticed at a glance that each plot follows a clear-cut negative relation that appears to be almost linear above some threshold, located at k≈100 for the total-degree and the in-degree, and at k≈50 for In what follows, it is rigorously assessed whether the degree distributions characterizing the labour mobility network behave as power laws. It is worth stressing that, in most practical cases, it is extremely difficult to know for certain whether a given quantity is drawn from a power-law distribution; what can be typically done is rather to verify that the data at hand are consistent with a model in which the given quantity is compatible with a power-law distribution.
The probability distribution of a quantity k is said to be power law if it is drawn from a probability distribution of the form given by the following expression where α represents the scaling exponent. Notice that such distribution diverges as k tends to zero, so, provided that α>1, the power-law behaviour must set in only above a certain threshold k mi n >0. Moreover, in the case of network degree, the quantity of interest can take only positive integer values, i.e. it is discrete. Considering these two elements, the expression for the complementary cumulative distribution reads where the function ζ is the generalized, or Hurwitz zeta function. The goal of the analysis is now to fit expression (2) to the empirical data; this is done by closely following the procedure proposed by Clauset et al. (2009). The methodology runs as follows: first, a power law of the form given in (2) is fitted to the empirical data using maximum likelihood, simultaneously estimating the scaling parameter α and the lower bound of the scaling region k mi n ; then, the goodness of the power-law fit is evaluated by calculating an appropriate pvalue.   These last results lead to the conclusion that: hubs are critical in keeping connected different parts of the network, and such feature is precisely mirrored in the high vulnerability of the system to hub closures. Paraphrasing a terminology risen to fame in the context value chain studies, hubs appear to play a role under many respects analogous to system integrator firms.

The main hubs
Making use of the information available in the VWH database, we classify the first 50 hubs according to the sector of activity, and for each sectoral group we display the average workforce, together with the main average network statistics; the results are reported in Table 5.
At a glance, we notice that hubs are mostly large firms.
This result is not surprising, it generally true that a larger workforce generates a larger turnover at the firm level that might result in more network connections either on entry and exit. In the network under scrutiny, the Pearson correlation coefficient between average firm size and total-degree equals 0.57, highly significant, indicating a correlation of moderate intensity. 3. companies specialized in providing workforce and logistics to third parties.
Among the manufacturing hubs, it is easy to recognize several distinctive activities of the Veneto economy. Four companies produce household and professional appliances, either as final makers or as suppliers of dedicated parts; they belong to the so called 'Inox Valley' district, located in the province of Treviso, and they hold prominent positions in their respective market segments at the international level. Two hubs are the undisputed worldwide leaders of the eyeglass industry; they are the major players around which the eyewear district of the Belluno province is organized. Another company is a historical producer of wool fabrics, now mostly involved in the garment sector, which during various decades has fed a vast textile district in the province of Vicenza.
Each of these firms is deeply rooted in a local socioeconomic context, and tightly nested into a local cluster of firms; at the same time, each player has connections spanning different distant areas of the regional and also national economy, indeed, all manufacturing hubs are well-known, historical trade names whose popularity can attract applicant workers from afar. 10 Firms involved in the service sector -comprising hotel trade and tourism, personal and social services, commerce, and catering -share a major trait: they typically carry out business through multiple local units. The ten department stores or supermarkets listed in Table 5  A different condition is that of companies specialized in providing workforce to third parties. Two different categories of such firms can be identified: firms providing logistic services as subcontractors, and labour market intermediaries providing temporary workforce. Especially for the latter firms, acting as hubs in the labour market is somewhat inborn in their statutory activity. In Italy, logistic subcontractors represent a long-standing phenomenon, that has been intensifying over time, hand in hand with the increasing segmentation of production processes, and the rising importance of logistics in manufacturing value chains. In contrast, private firms dealing with temporary workers in a range of different fields represent a novelty.
The rapid settlement of a number of private labour market intermediaries -suddenly emerged in the role of hubs -is the result of the policies adopted in the second half of the 1990s, aimed at rendering the labour market more flexible. 11 Employment agencies -often local branches of companies operating worldwide -act in a natural way as transmission belts for labour reallocation. Moreover, they treat a variety of occupational profiles, and allocate skilled personnel; whereas, companies specialized in logistic services deal mostly with low-skilled personnel, often recruited within the weakest components of the workforce, as for instance immigrants.

Concluding remarks
This essay studies the phenomenon of worker reallocation in the Veneto region by means of a network approach which is new to regional labour market studies.
The empirical investigation is articulated in two parts. In the first one, the paper attempts to assess to what extent the regional labour market is integrated and permeable to worker flows, and it quantifies such properties by comparing the actual network with a small-world one, showing that the interfirm network examined exhibits a clear-cut small-world pattern. In the second part, the paper asks whether there exist employers playing a strategic role in order to guarantee network connectivity; this question is framed in precise quantitative terms by examining the shape of the link distribution, showing the presence of dominant firms playing the role of hubs for labour flows.
The revealed small-world architecture points to a labour market structure that is permeable to worker flows, and that appears to be capable of effectively absorbing localized cyclical shocks; this is a highly desirable characteristic policy makers might want to preserve and promote. But such connectivity pattern turns out to depend crucially on the presence of a small number of highly connected hubs that span the network from side to side, bridging together distinct local clusters of firms, and keeping the network globally integrated.
This latter feature makes the network vulnerable; indeed, the failure of even a small number of hubs most likely results in the split of the labour market into different separated pieces; a more segregated labour market is in turn especially exposed to the harmful effects of asymmetric shocks, which could not be effectively re-absorbed, and hence may cause persistent unemployment in the areas of the economy that remain isolated.
Examining the assembled evidence altogether, some appealing policy implication can be derived; namely, policy makers might want to especially support those firms that guarantee major connectivity to the system, and in this case an effective reallocation policy would hinge critically upon the ability to localize hubs and their connection patterns. Among the most important hubs in Veneto, a few large and longtradition manufacturing firms are detected. Safeguarding such firms, which are nowadays particularly exposed to the competitive pressure stemming from low-wages countries, could be a first-order goal for policy. The identification and characterization of hubs might offer a criterion for assigning subsidies -as well as extraordinary funds aimed at supporting employment -which are in the discretional disposal of the Regional Government or other Public Administrations.
Furthermore, the paper shows that temporary staffing agencies, a new family of actors in the regional economy under scrutiny, enter the reallocation market and rapidly rise to hub positions, calling for further investigation.
More in general, a network analysis of labour mobility of the type proposed in this essay seems to offer a suitable toolkit to be put aside more traditional econometrics in order to achieve an effective appreciation and monitoring of structural traits of the reallocation market. Noticeably, the study also reveals that, even in a regional economy whose development has been based for decades on very small organizational dimensions, a few large organizations still play a central role in keeping integrated different parts of the system and allowing for workers to find reallocation opportunities that otherwise would probably be more difficult or impossible to be seized.  (Brandt et al., 2009;Schweitzer et al., 2009), recognizing the potential of this technique. The stakes seem to be high and so it is worth investing in this direction.
The present essay lies precisely in this wake, illustrating a possible application of network methodologies to labour mobility on the basis of large-scale linked employer-employee data from administrative source.
Several directions for future developments are in sight.
For instance, dynamic network analysis can be performed in order to map the evolution of regional labour markets, again For the sake of simplicity, in the text the word degree is used to refer to total-degree. The average degree of a network is equal to the average degree of its vertices, denoted by k(G). Path and components are exemplified in the Figure A2.

FIGURE A2 AROUND HERE
The length of a path from i to j is equal to the number of links one has to run along to reach j starting from i. The shortest path from i to j is called geodesic, and its length is denoted by d i j . The average path length (APL) of a network is defined to be the average length of the geodesics between all possible pairs of vertices in the SCC, and it is denoted by d(G), The set of vertices with which vertex i is directly connected, both on entry and exit, is called (nearest) neighbourhood of i, and it is defined as    The ACC is at least two orders of magnitude higher than the ACC of a random network of the same size as and with the same in-degree distribution of the actual network