Core-periphery Structures: Operationalizing patterns of dependence and dominance in binary and valued networks

Carl Nordlund Center for Network Science; Department of Political Science, CEU, Budapest nordlundc@ceu.hu

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Abstract:

With origins in the postwar development discourse, the core-periphery concept has spread all over the social, and increasingly the natural, sciences. Used initially to denote two broad regional categories distinguished by differences in socio-economic parameters, its structural connotations paved way for more relational, and less attributional, specifications. As reflected in the blockmodeling tradition and as implemented in the well-established index of Borgatti and Everett (1999), contemporary network scholars view a core-periphery as a structural template, where cores are depicted as internally cohesive and peripheries as disconnected from each other.

Through an extensive review of postwar literature utilizing the concept, this article finds support for the intra-categorical density differential characteristic of core-periphery structures. However, beyond the occasionally specified density of inter-categorical ties midway between the intra-categorical extremes, the literature review lends equal, possibly more, support to a set of inter-categorical features that characterize core-periphery structures: dominated and dependent peripheries, and dominating cores.

This paper proposes new core-periphery metrics that operationalize dependency and dominance. Expressed as ideal block types found in generalized blockmodeling, the proposed inter-categorical indices supplement the index proposed by Borgatti and Everett, resulting in a composite index that captures the characteristic features of core-periphery structures as found in the postwar literature. Whereas the heuristic for binary data is fairly rudimentary, its extension to valued networks exemplifies the application of a novel approach to generalized blockmodeling of valued networks that is more sensitive to patterns, rather than strengths, of ties.

Testing the proposed metrics and heuristics with the binary and valued example datasets used by Borgatti and Everett (1999), new insights and details about core-periphery structures can be obtained. This is particularly evident when, circling back to the original domains of the concept, applying the heuristic to valued inter-continental trade data and data on bilateral trade between 18 countries.

Keywords:

core-periphery, valued networks, regular blockmodeling

Core-periphery Structures: Operationalizing patterns of dependence and dominance in binary and valued networks

Introduction

The terminological origin of ‖center/core‖ and ‖periphery‖, and their coupling into the conjoint concept of repute, is rightly attributed to the work of Prebisch (1950) (de Janvry, 1975; Love, 1980). Originally designated to represent two types of regions distinguishable by their differences in actual and potential socioeconomic development, the concept became integral to the more heterodox strands of postwar development thinking (Chase-Dunn and Hall, 1991; de Janvry, 1975; dos Santos, 1970; Frank, 1970; Galtung, 1971; Meier and Baldwin, 1957; Wallerstein, 1974). Prior to its recent entry into mainstream economics of today (Krugman, 1998, 1991, 1990; Hojman and Szeidl, 2008), the core-periphery concept was anything but dormant:whether as a descriptive, explanatory, or analytical device, as a model, structure, or process, or as something spatial, metaphorical, or something in-between, the core-periphery concept is all over the social, and increasingly also the natural, sciences.

Within political science, the core-periphery concept appears in studies ranging from political participation (Galtung, 1964; Langholm, 1971) to international relations (Berman, 1974; Chan, 1982; Dominguez, 1971; e.g. Galtung, 1966; Gochman and Ray, 1979; Snyder and Kick, 1979; Thompson, 1981). Within human geography, core-peripheries are found in regional studies (Friedmann, 1966; Hanink, 2000; Kauppila, 2011), in transport geography (Gleditsch, 1967; Goetz and Sutton, 1997), in communication studies (Sun and Barnett, 1994), and urban studies at the intra-city (Uzzi and Spiro, 2005), inter-city (Wellhofer, 1989), and world city level (Alderson and Beckfield, 2004). Within sociology (plus friends), the core-periphery concept describes organizations and workgroups (Crowston et al., 2006; Cummings and Cross, 2003), exchange among Papua New Guinea hunters (Healey, 1990), community prevention programs (Feinberg et al., 2005), stakeholder networks (Boutilier, 2011), eco-system-related advice networks in Chile (Giuliani and Bell, 2004), Ghana (Isaac et al., 2007), and Stockholm (Ernstson et al., 2008) – and how accounting and auditing standards are regulated in Canada (Richardson, 2009). Its spread within sociology might relate to the migration patterns of sociology PhD students, exchanges that resemblance a core-periphery structure (Burris, 2004). Within biology, the human metabolic network has been perceived as a core-periphery structure (Zhao et al., 2007), and so have the citation patterns of the biology scholars themselves (Mullins et al., 1977) and co-citation behavior of academic scholars in general (White, 1990; Zuccala, 2006). Apparently an infectious concept – and if its spread shares similarities with medical infections (Christley, 2005), this epistemological diffusion could very well be described in – ta-dam – core-periphery terms.

Its relational connotations has given the core-periphery concept recognition, specifications, and article keyword prominence among network scholars. Stripped of any would-be discipline-specific substance, core-periphery in network analysis is a structural template – a network-topological type – that contains two ideal types of actors, reflecting different structural properties, and where the relevance of such a distinction, similar to its raison d'être in social sciences at large (McKenzie, 1977, p. 55), rests on the idea that the general relationship between core and periphery is of importance for understanding the system at large.

Borgatti and Everett (1999) have introduced a heuristic for the categorical partitioning of relational data into core and peripheral actors. Based on a suggested index of core-peripheralness that percevies the subset of core actors as being densely internally connected and where ties between peripheral actors are perceived as absent, the heuristic searches for an optimal partition that, in various ways, reflects this difference in intra-categorical tie frequencies. Although the heuristic (and its implementation) allows for specifying the density of inter-categorical ties, i.e. the ties that tie core and peripheral actors to each other, Borgatti and Everett recommend treating such ties as missing data in their core-periphery fitting function (1999, p. 383).

Viewing their paper ‖as a starting point in a methodological debate on what constitutes a core/periphery structure‖ (Borgatti and Everett, 1999, p. 376), this article continues this debate,

arguing that densities within each of the two ideal actor categories only captures one aspect of core-periphery structures. Based on a cross-disciplinary literature review where the concept is specified and used, this paper argued that an equally, or possibly even more, significant characteristic of core-periphery structures is to be found in the inter-categorical ties between core and periphery. Such inter-categorical ties should thus not be ignored, nor captured as an imperfect 1-block; rather, core-periphery structures are characterized by a specific pattern of ties that connect core and peripheral actors to each other.

This paper (re)introduces and operationalizes a set of inter-categorical criteria that supplement the intra-categorical density differential characteristic: peripheral dependency, dominated peripheries (connectivity), and dominating cores. Operationalized as combinations of ideal block types used in blockmodeling (Doreian et al., 2005), this paper proposes novel metrics that capture how well such ideal inter-categorical patterns exist in a given 2-position blockmodel. Two heuristics1 are proposed: whereas the version for binary networks is relatively simple, the version for continuous data exemplifies a novel approach to generalized blockmodeling of valued networks that is more sensitive to patterns, rather than mere stengths, of ties.

Paper structure

The remainder of this paper is divided into four parts. The first part contains a review of previous literature where the core-periphery concept has been used, described, and/or specified. Traversing multiple disciplines and decades, the review is divided into the network-analytical separation between attributional and relational specifications, the latter divided into micro-, meso- and macro-level specifications.

Based on the characteristic core-periphery features derived from the literature review, the second part models such features as specific combinations of the ideal block types used in generalized blockmodeling. Based on deviations from such ideal patterns, this part operationalize a metric for binary networks that capture to what extent the pattern of ties between a core and a peripheral subset reflect patterns of dependency and dominance. Combining this metric with the intra-categorical index proposed by Borgatti and Everett (1999), this section reexamines the binary examples of the latter study.

The third part extends the heuristic to valued networks. A novel approach to blockmodeling of valued networks is introduced, applied here to identify the particular block types of core-periphery patterns. The proposed heuristic – and the approach to blockmodeling of valued networks that it represents – differs from existing approaches (cf. Žiberna, 2007a), arguably being more sensitive to patterns, rather than strength, of ties. Operationalized as two indices for intra- and inter-categorical blocks, respectively, the suggested composite core-periphery index is tested on the valued example data in Borgatti and Everett (1999). Also, thematically circling back to the developmental discourse from which the core-periphery concept stems, the heuristic is demonstrated using inter-continental trade data, as well as international trade between 18 sample countries.

A summary of the findings and the suggested approach concludes this paper. This part also discusses a possible incorporation of a semi-peripheral position and ongoing work to extend the valued core-periphery heuristic to generalized blockmodeling of valued networks.

Core-periphery as specified and defined in the literature

Disciplinary contextual and non-relational, actor attributes are of scant use when deriving generalized structural properties. In the case of core-periphery structures, it is nevertheless instructive to examine a handful of such definitions since its original formulation actually was specified in terms of attributes. This initial lack of formal relational definitions underlines that any topological perceptions about core-periphery structures indeed is debatable. ―Conceptions‖,

1 Implementing the two heuristics, two Windows software clients – CorePeripheryBinary and

CorePeripheryValued – are available for download at http://cnslabs.ceu.hu/

Wallerstein argues, ―precede and govern measurements‖ (1979, p. 36): motivating the somewhat extensive literature review in this paper, such a debate should be if not based on but at least informed by previous conceptualizations of cores, peripheries, and core-periphery structures, including its attributional, non-topological genesis and the disciplinary and substantive context from which the concept stems.

Core and periphery as attributes

Using the center-periphery terminology in lectures and presentations in the mid-1940's (Love, 1980, pp. 52–54), Raul Prebisch's 1950 report on the postwar development prospects of Latin American countries is typically regarded as the terminological origin. In this report, the center-periphery concept was part of a broader critique towards the neoclassical modernization school: Prebisch argued that theories and models stemming from the developed world, termed the center, were not applicable to the situation and historical experiences facing the so-far non-developed world (termed the periphery) (Prebisch, 1950, p. 7 note 1). Building on a previous UN report on the diminishing terms of trade for countries exporting primary goods vis-a-vis countries exporting manufactured goods, Prebisch found that productivity increases, wherever they occur, tend to benefit the manufacturing center more than the agricultural periphery. This unequal sharing of productivity gains was primarily due to differences in labor institutions: strong labor organizations in the center led to wage increases in economic upswings and prevented wages from dropping in downswings, whereas disorganized labor in the periphery implied stagnant wages. The solution, according to Prebisch, was in domestic policy: industrial fostering, import-substitution, and a sound financial policy were seen as suitable tools for fixing the flawed production structures in the periphery, argued to be the root cause of the peripheral condition.

Under the heading of Latin American structuralism, the ideas of Prebisch and his colleagues at the Economic Commission for Latin America had a tremendous impact on both economic policy and subsequent strands of development thinking. Abandoning the ‖whole nation bias‖ of existing approaches (e.g. Wellhofer, 1989, p. 341, 1988, p. 282ff), the center-periphery concept facilitated development studies at intra- and international systemic levels. Still, Prebisch did treat center and periphery as two broad regional categories, indeed connected but nevertheless defined by internal properties such as wage levels, production structures, export composition, and other similar attributes. The ‖structuralism‖ of this Latin American scholarly tradition thus did not refer to international structures per se; rather, socio-economic problems and remedies were domestic (Kay, 2009) where the center-periphery connections only were seen as conditioning – not causing (Oman and Wignaraja, 1991, p. 142) – the peripheral situation.

Within geography, the regional scholar John Friedmann described center-periphery as the spatial manifestation of a presumed transitional phase between non-industrial and industrial society (Friedmann, 1966, p. 7). Although describing center-periphery relations as colonial (1966, pp. 8, 12), the periphery as ‖imperfectly related to this [singular] center‖ (1966, p. 9) and peripherality as spatial distance from the center (1966, p. 11), Friedmann nevertheless distinguishes the categories by the levels of investment, type of export products, modes of production etc (see also de Janvry, 1975; Kauppila, 2011), i.e. regional attributes. However, viewing center-periphery as ‖[a] dualistic structure...imprinted upon the space economy‖ where the non-industrial territory ‖becomes locationally obsolete‖ (Friedmann, 1966, p. 9), Friedmann‘s usage overlaps that of dualism (Boeke, 1953; Lewis, 1954; cf. Frank, 1970, p. 6), i.e where the modern and traditional sectors are deemed as separated from each other. Similar to the growth pole literature (e.g. Perroux, 1950; see also Rościszewski, 1977, p. 13), Friedmann optimistically argued for a continued focus on center growth that eventually would absorb the peripheral areas, reflecting a functional separation between centers and peripheries that is in line with his attributional, non-relational categorical specifications.

Whereas the modernization school (and perhaps also the Latin American structuralists) saw development as a function of time, the world-system perspective (and dependency school – see below) deemed it as space-functional: development in certain parts of the world are intrinsically tied to the under-development in other parts. In this scholarly tradition, the two categories are complemented by a semi-peripheral category, representing a unique functional position between core and periphery (Wallerstein, 1974, p. 349), and an ‖external area‖ representing the nations and

geographic regions not (yet) integrated into the grander world-system. Inter-national and -regional relations – particularly economic (global commodity chains, monopolistic trade, and unequal exchange) – are seen as fundamental for understanding the different developmental trajectories of various national states and regions in the world-system (Wallerstein, 1974, p. xi). Still, despite this explicit focus on relations that tie component parts into a coherent whole and the role of such ties for change at all levels of the system, the trichotomy of the world-system perspective is nevertheless typically described in attributional terms (Bousquet, 2012; Chase-Dunn, 1998, p. 77; Goldfrank, 2012, p. 100; Kentor, 2000, pp. 36–38; Wallerstein, 1974, pp. 102, 349), particularly the international division of labor, a categorical specification that has, it is claimed, ‖raised little debate‖ (Bousquet, 2012, p. 124). However, such attributional definitions has been contested (Duvall, 1978, p. 59; Vanolo, 2010, p. 30), for instance in the series of network-analytical world-system studies (e.g. Nemeth and Smith, 1985, p. 521ff; Smith and White, 1992, p. 859; Snyder and Kick, 1979, p. 1102): although correlations might exist, ‖[country attributes] do not represent such position any more than an individual's income or education measures his or her (discrete) class position‖ (Snyder and Kick, 1979, p. 1102). A relational approach, it was argued, implies that ‖the focus of the analysis is no longer on characteristics of individual countries, but on the relationships between countries.‖ (Nemeth and Smith, 1985, p. 522).

Borrowing the categorical terminology from Wallerstein and the other ‖radicals‖ (Krugman, 1981, p. 149), Paul Krugman views it ‖as nearly scandalous that economists have ignored [the core-periphery model] until now‖ (Krugman, 1998, p. 13). By combining Dixit-Stiglitz economies of scale with spatial distance, Krugman conjured the so-called New Economic Geography (Krugman, 1998, 1991, 1990), bypassing what he deemed to be an economic geography obsessed with ‖geometric tricks involving triangles and hexagons‖ (Krugman, 1991, p. x). Deriving his market equilibrium model from a two-regional example (Krugman, 1991, p. 18), he subsequently expands it to multiple discrete locations laid out and connected in a circle (Krugman, 1998, p. 13, 1991, p. 24). The model includes two economic sectors: agricultural production employing immobile labor producing at constant returns to scale, and a mobile industrial labor producing at economies of scale, where the two types of labor also constitute the market for both goods. Depending on transport cost and consumer preferences, his model results in manufacturing production being concentrated in relatively few locations. It is this concentration of industry that defines a core, and the lack thereof a periphery, i.e. a purely attributional definition (see also Lange and Quaas, 2010).

Attribute-based core-periphery specifications are also found in other disciplines – such as Johan Galtung's 1964 study on how social position of individuals is related to their foreign policy orientations. Supplementing the center and periphery categories with a ‖decision-making nucleus‖ and ‖extreme periphery‖ (Galtung, 1964, p. 207), his measure of social position, interpreted as distance to the nucleus, was an aggregated index of attributes such as sex, age, education, income, occupation etc (Galtung, 1964, p. 217ff). Similarly, Langholm's study of political participation defines the center as where political decisions are made. Although described in relational terms, discussing peripherality in terms of accessibility and closeness to the center ‖defined as positions on the communications networks of society in the broadest sense, including spatial closeness‖ (Langholm, 1971, p. 276), he finds that ‖the underlying principle of this concept of 'distance' would be similarity-dissimilarity with center‖, operationalized as a comparison between individual attributes ‖such as education, occupation, income, property‖ (ibid.).

The above attributional core-periphery specifications are, similar to its original usage, tightly knit to particular scientific disciplines, substantial contexts, and research question, as such being of scant relevance when specifying core-periphery structures in topological terms. As such, they do underline that would-be characteristics of core-periphery structures are, and indeed should be, debatable.

Core and periphery as relational properties

Despite the initial attributional descriptions, the dual-categorical concept did lead to specifications that were more relational. Seven years after Prebisch's original formulation, Meier and Baldwin described the centers and peripheries of the world economy in more relational terms:

A country can be termed a center of the world economy if it plays a dominant, active role in world trade. […] Foreign trade revolves around it: it is a large exporter and

importer, and the international movement of capital normally occurs from it to other countries. In contrast, a country can be considered on the periphery of the world economy if it plays a secondary or passive role in world trade. […] The common feature of a peripheral economy is its external dependence on the center as the source of a large proportion of imports, as the destination for a large proportion of exports, and as the lender of capital. (Meier and Baldwin, 1957, p. 146ff)

This proto-topological description of centers and peripheries was followed by studies that similarly, at various degrees of formality, characterized center-periphery structures in relational terms, at the micro-, meso- and macro-levels of networks.

Micro-level: core and periphery in terms of centrality

The notion of a difference with regards to centrality is ingrained in the concept itself, where peripherality often translates to the distance to a preconceived centers (Friedmann, 1966, p. 10; Uzzi and Spiro, 2005, p. 476). As noted above, Galtung (1964) and Langholm (1971) described peripheralness as such distances, even though operationalizing this in terms of actor attributes (see also Heinz, 2011, p. 458ff).

The network-topological centrality of actors has occasionally been directly associated with the notions of core and periphery (e.g. Galtung, 1971, p. 103; see also Scott, 2000, p. 88ff). Using centrality measures for identifying core and peripheral actors could seem reasonable as it would rest on the centrality differences among actors in a star-network as well as differences in actor centrality between dense vis-à-vis sparse subgroups; however, as noted by Borgatti and Everett (1999), even though core actors ‖are necessarily highly central as measured by virtually an measure...the converse is not true‖ (1999, p. 393).

Studies do exist that apply centrality indices to confirm the existence of assumed, pre-determined core-periphery structures. In Gochman and Ray (1979) study of international power relations between 1950-1970, the authors compare two political subsystems: the US and USSR spheres of influence in, respectively, Latin America and East Europe. They depict these two subsystems as star-shaped networks where the hub in each network is respective hegemonic power, assumptions that their study sets out to investigate by looking at a degree-based measure of received diplomatic ties and a share-of-trade index. In the substantive context of their study, degree centrality seems viable for verifying the existence of a pre-determined hypothetical star-shaped network.

The partner concentration of a country's export vector, formalizing the perception that peripheries typically depend on relatively few trade partners, is a micro-level approach even though it typically focuses on inter-categorical ties (e.g. Gidengil, 1978, p. 56). This metric, similar to measures of centrality, could indicate the existence of a core-periphery structure, but rather than being characteristic features per se, they instead reflect core-periphery characteristics specified at the meso- and macro-level of networks, the former to which we now turn.

Meso-level: core and periphery as subgroups and dyads

The perceived density differential of intra-categorical ties that Borgatti and Everett (1999) base their metric and heuristic on finds ample support in the literature. This meso-level characterization depicts core actors as tightly connected to each other and peripheral actors as ideally devoid of ties (Berman, 1974, p. 4ff; Chan, 1982, p. 315; Dominguez, 1971, p. 176; Galtung, 1971, p. 89, 1966, p. 146; Gleditsch, 1967, p. 369; Mullins et al., 1977, pp. 49–56; Nemeth and Smith, 1985, p. 538) – or simply corresponding to the remaining non-core actors (e.g. Holme, 2005).

In the blockmodel tradition (Breiger, 1976; Breiger et al., 1975; Doreian et al., 2005; Mullins et al., 1977; White et al., 1976), the block image type that corresponds to a core-periphery (and centralized2) structure similarly specify core-core interaction as an ideal 1-block (i.e. a total subgraph) and intra-peripheral ties as an ideal 0-block (i.e. with no dyads between peripheral

2 The difference between the core-periphery and centralized block images is related to the directionality of inter-categorical ties: the ideal centralized block images depict ties between core and peripheral actors as uni-directional.

actors). Somewhat peculiar, Wasserman and Faust state that ‖peripheral blocks may or may not be internally cohesive‖ (1994, p. 419). According to Galtung (1966), a periphery gone cohesive is no longer part of a center-periphery structure: rather, with Marxian-Engelsian undertones, a center-periphery system ‖can be destroyed if the underdogs unite‖, transforming the system into a ‖class system‖(1966, p. 147).

The density differential characteristic is often combined with macro-level characteristics concerned with properties of inter-categorical ties. Leaving the macro-level characteristics of Galtung's feudal interaction structure for the subsequent section, it is worth noting that the density of inter-categorical ties is often depicted as midway between the densities of the two intra-categorical ties of core and periphery (e.g. Galtung, 1966, p. 146; Gleditsch, 1967, p. 369), which indeed corresponds to the non-diagonal imperfect 1-blocks as specified by Borgatti and Everett (1999, p. 378).

The studies of Galtung (1966) and Gleditsch (1967) exemplify how density differentials are used to verify predetermined center-periphery structures. Using a ‖topdog-underdog‖ terminology rather than ‖center-periphery‖, both studies examine the rank-ordered density differential hypothesis, i.e. that the frequency of intra-center ties exceeds that of inter-categorical ties, followed by the lowest tie frequency among peripheral actors (Galtung, 1966, p. 146; Gleditsch, 1967, p. 369). Their respective focus are however slightly different: whereas Gleditsch analyze air routes of a singular (world) system, Galtung focuses on the intra- and inter-categorical ties between the two, analytically separated sets of countries found in NATO and the Warsaw pact in the 1960's. Galtung's study is thus an attempt to examine whether the hypothesis on rank-ordered densities holds true for an international system that is, evidently, separated into two political subsystems (cf. Gochman and Ray, 1979). Both studies conform the density differential characteristic of core-periphery structures.

There are two aspects of the studies of Galtung (1966) and Gleditsch (1967) that make them particularly interesting from a network-analytical perspective. First, both use complete relational datasets in their analysis: Gleditsch uses air route data for four time periods between 1930-1965, and Galtung uses survey data from 23 foreign embassies in Oslo covering 15 types of ties – diplomatic, political, economic, cultural and travel relations – between these nations. Secondly: even though topdog and underdog actors are pre-determined based on attributional data, their subsequent sorting of actors, calculation of intra- and inter-categorical densities, and interpreting by comparing these densities to an ideal model (Galtung, 1966, p. 163; Gleditsch, 1967, p. 377), is in essence a blockmodeling procedure, conducted several years before formally labeled as such (Breiger et al., 1975; White, 1974a, 1974b; White et al., 1976).

Studies in transport geography also lend support to the rank-ordered density differentials hypothesis. Employing 14 different indices when studying 18 domestic railway networks, Kansky (1963) finds a relationship between Beta-indices (the ratio between number of edges and nodes) and per capita energy consumption (1963, p. 42). His Pi-index, indicating whether a network is circular or elongated, also points to a relationship between density of railway links and per capita GDP (ibid.). Kansky's conclusion, i.e. that ‖less developed countries are served by transportation systems which look more like disconnected graphs or trees [whereas] highly developed countries benefit from highly connected transport networks‖ (1963, p. 12), not only reflects the density differential hypothesis in postwar human geography (e.g. Haggett, 1965, p. 71; Taaffe et al., 1963, p. 504), but it also characterizes core regions in the New Economic Geography (Krugman, 1991, p. 23ff).

There are also approaches that identify core-periphery structures at the dyadic level. Similar to the studies above, such approaches typically deal with pre-determined categorical partitions of actors, but rather than looking at intra-categorical ties, dyadic sampling typically focuses on ties from periphery to core (e.g. Dominguez, 1971; Gidengil, 1978; Thompson, 1981). An example of such dyadic sampling in the development context is the study by Dominguez (1971). Analyzing bilateral trade flows between 15 former colonies and their respective hosts – France, Netherlands, United Kingdom, and USA (with respect to the Philippines) – for three years (1938, 1954 and 1964), Domínguez uses two dyadic indices of ‖preferentiality‖: relative acceptance, and economic importance (1971, p. 181ff). These indices reflect to what extent directional valued dyads are above or below what is expected based on total in- and outdegrees. Despite a slight overall decline

in peripheral economic dependency to their former hosts over the time period, the dyadic analysis of Domínguez does point to the existence of star-shaped exchange patterns within former colonial spheres (1971, p. 183).

Although conducted at the meso-level of network analysis, dyadic analysis that look at inter-categorical ties rests on structural hypotheses at the macro-level, the level to which we now turn to.

Macro-level: patterns of ties between core and periphery

Whether treated as two broad categories, as subsets of actors, or as a particular spatial configuration, the utility of the coupled concept hinges on core and periphery somehow being connected and interrelated to each other. Although there are exceptions (e.g. Holme, 2005), the connectivity of core-periphery structures is emphasized in both classical and contemporary usages (e.g. Borgatti and Everett, 1999, p. 382; Dominguez, 1971, p. 176; Frank, 1970, p. 7; Galtung, 1971, p. 82, 1966, p. 146ff; McKenzie, 1977, pp. 55, 59; Meier and Baldwin, 1957, pp. 144, 146; Snyder and Kick, 1979, p. 1102; Wallerstein, 1974, p. 63). This assumed interrelatedness differs from the two analytically separated categories of the dualist discourse (Boeke, 1953; see also Friedmann, 1966, p. 9; Rościszewski, 1977, p. 20), the latter in which the non-developed ‖traditional‖ sector, at best seen as an unlimited supply of labor (Lewis, 1954), eventually will be absorbed into the ―modern‖ sector. In the world-system tradition, regions and countries (still) outside the world-system are categorized as the ‖external area‖ (Wallerstein, 1974, p. 300ff), a seemingly suitable label to denote actors that are neither core nor periphery.

The classical ideal block images for core-periphery (and centralized) structures depict inter-categorical ties as 1-blocks (White et al., 1976, pp. 742, 744). Initial blockmodel studies preferred the zeroblock (lean fit) criteria for classifying positional ties (Wasserman and Faust, 1994, p. 399; White et al., 1976), a criteria that treats all non-empty block as 1-blocks. Thus, based on the rank-ordered density differential postulate, i.e. where the density of inter-categorical ties lies between respective density of the intra-categorical ties (e.g. Galtung, 1966; Gleditsch, 1967), the non-diagonal blocks of core-periphery block images are better viewed as imperfect 1-blocks (Borgatti and Everett, 1999, p. 378). Combined with the meso-level characteristics of a fully internally connected core and an internally disconnected periphery results in the block image in Figure 1.

Core Periphery

Core 1-block Imperfect 1-block

Periphery Imperfect 1-block

0-block

Figure 1: Classical ideal core-periphery block image

Although an imperfect 1-block implies an inter-categorical density between the two intra-categorical ones, the density per se does not capture what I argue to be fundamental properties of core-periphery structures, but additional inter-categorical criteria are needed. The first criteria is concerned with connectivity. Assuming an internally connected core and an absence of intra-peripheral ties, the overall connectivity of the core-periphery structure would only prevail if the periphery-to-core block is row-regular, i.e. that the block in question has at least one tie on each row3. If not, the peripheral actors corresponding to each such row will lack any connection to the core (and other peripheral actors), as such being part of the ‖external area‖. Inter-categorical block densities are thus not sufficient to ensure that the connectivity criteria is fulfilled.

Two additional criteria for the inter-categorical blocks in the classical core-periphery block image is provided by the postwar development literature. The dependency scholars' description of global monopolies and dendritic interaction patterns did address the patterns of ties between core and periphery (see below), but it is Galtung's work on imperialism and the feudal interaction structure (Galtung, 1971) that provides the most comprehensive and formalized topological description of

3 In the case of symmetric data, or directional data where ties go from core to periphery, the core-to-periphery block has to be column-regular, i.e. at least one tie must exist in each column of the top-right block.

center-periphery structures, a description that shaped conceptual frameworks in many subsequent studies.

Galtung treats imperialism as a specific type of dominance system, primarily but not exclusively between nations, ―that splits up collectivities and relates some of the parts to each other in relations of harmony of interest, and other parts in relations of disharmony of interests, or conflict of interest.‖ (1971, p. 81). Starting off with a simple Prebischian two-country center-periphery model, each subsequently separated into internal centers and peripheries, Galtung outlines the basic principles of conflict and cooperation within and between nations, and between centers and peripheries. Galtung identifies two mechanisms that maintain and reinforce such relations: vertical interaction relations, and the so-called 'feudal interaction structure'. Vertical interaction is concerned with asymmetrical relations between centers and peripheries, whether manifested as differences in terms of trade and division of labor, unequal exchange, exogenous political influence, or unequal sharing of environmental burdens (1971, p. 89). Similar to the conclusions drawn by most dependency and world-system scholars (e.g. dos Santos, 1970; Frank, 1970; Wallerstein, 1974), it is such vertical interaction that causes the differences in socio-economic attributes.

The second mechanism of imperialism – the feudal interaction structure – facilitates the first mechanism (Galtung, 1971, p. 89). Mentioned in his earlier writings (Galtung, 1966), the 1971 paper specifies the role and characteristics of feudal interaction structures in greater detail. Expressed as a set of rules on the interaction between central and peripheral actors, the feudal interaction structure is in essence a topological meso- and macro-level specification of an ideal center-periphery structure, Quoting Galtung (1971, p. 89), these rules are as follows:

1. interaction between Center and Periphery is vertical

2. interaction between Periphery and Periphery is missing

3. multilateral interaction involving all three is missing

4. interaction with the outside world is monopolized by the Center, with two implications:

a. Periphery interaction with other Center nations is missing

b. Center as well as Periphery interaction with Periphery nations belonging to other Center nations is missing.

Reproduced in Figure 2, Galtung provides a visual example of a feudal interaction structure containing four centers and nine peripheries.

Whereas the first rule reiterates the first mechanism of imperialism, the subsequent three rules constitute a formal topological specification of what Galtung sees as characteristic of systemic interaction between centers and peripheries. At the meso-level, the internally non-connectivity of the periphery is given by the second rule and although a corresponding rule for intra-core properties is missing, as such reflecting the genre's overall focus on the peripheral situation, the visual example does depict a core that is connected and relatively dense (0.67). Block-image-wise, the inter-categorical blocks have densities of 0.25, i.e, in accordance with the rank-ordered density differential idea from previous writings (Galtung, 1966; Gleditsch, 1967).

Of particular interest is the fourth rule and its implications for patterns of core-periphery relations. Whereas the connectivity characteristic implies that each peripheral actor is connected to at least one core actor, Galtung's fourth rule states that a periphery is connected to no more than one core actor. Core actors can indeed be connected to several peripheral actors, but peripheral actors are here depicted as only having a singular tie to a singular core actor. Relations between peripheries and other parts of the structure are thus monopolized by core actors, under the implicit agreement among core actors that ―'if you stay off my satellites, I will stay off yours'‖ (Galtung, 1971, p. 89).

P1

11

P1

22

C1

C2

C3

C4

P2

1

P2

2

P3

1 P3

2

P3

3

P4

1

P4

2

Figure 2: Galtung's example of feudal

interaction structure

According to Galtung, it is this monopolization of periphery-core ties and the enforced inability of peripheries to interact with other actors beside ―their own‖ centers that constitute the raison d'être of the feudal interaction structure:

The feudal interaction structure is in social science language nothing but an expression of the old political maxim divide et impera, divide and rule, as a strategy used systematically by the Center relative to the Periphery nations. How could – for example – a small foggy island in the North Sea rule over one quarter of the world? By isolating the Periphery parts from each other, by having them geographically at sufficient distance from each other to impede any real alliance formation, by having separate deals with them so as to tie them to the Center in particularistic ways, by reducing multilateralism to a minimum with all kinds of graded membership, and by having the Mother country assume the role of window to the world. (Galtung, 1971, p. 90)

The ―rules‖ above and the interaction patterns they give rise to were not something Galtung simply took out of the blue to suit his specific theory of imperialism. Rather, they were mere formalizations of patterns already perceived by other scholars in the heterodox development tradition. Of particular relevance here is the dependency tradition, where the mixing of Latin American structuralism with neo-Marxism led to theories where the prospects of peripheral development were somewhat dire (dos Santos, 1970; Frank, 1970). This school typically saw development and underdevelopment as two sides of the same capitalistic coin: peripheries are peripheries because cores are cores. As a remnant of colonial relations, Frank depicted contemporary interaction between developed and underdeveloped regions and countries as a hierarchical series of monopolistic metropole-satellite relations, in which each satellite was confined to dealing only with their respective metropole (dos Santos, 1970, p. 235; Frank, 1970, pp. 7, 15). Similar to how Galtung viewed the feudal interaction structure as a facilitator of vertical interaction between center and periphery, dependency scholars tended to view this global dendritic structure, channelling profits from the many 3rd world peasants towards the few European industrialist, as the root cause for the development of underdevelopment.

One can indeed disagree with the general dependency narrative and the specific dominance system described by Galtung, but their claims regarding the dendricity and, through this, the monopolistic patterns of ties between the developed center and the non-developed periphery find significant support in economic-historical accounts. In Baran's study of post-colonial trade in West Africa, the economic interface between peripheries and the world market was often represented by a handful of foreign (Western) firms acting both as buyers of local produce and sellers of Western goods (Bauer, 1954, p. 99; Meier and Baldwin, 1957, p. 313), monopolistic-oligopsonistic gateways that in form and function shared many similarities with previous colonial powers:

[These] leading firms are engaged in a remarkably wide range of activities geographically and functionally. Those of the United Africa Company range from the operation of an ocean shipping line to the maintenance of small stores and produce-buying stations in remote villages, and from the management of huge estates in Central Africa and elsewhere to the purchase of dates in Iraq, the operation of a department store in Istanbul, and the maintenance of buying offices in numerous cities all over the world. (Bauer, 1954, p. 103)

The dendritic nature of core-periphery exchange structures has also found support within peripheral countries. In the Gold Coast, trade in cocoa and other local produce was conducted by 1,500 brokers connected to approximately 37,000 sub-brokers (Meier and Baldwin, 1957, p. 313), a structural setup that arguably affects relative bargaining powers of core and peripheral actors:

The peasant producers have frequently had to face a small group of exporting and processing firms who have monopsonistic powers in buying the crop. And as the consumers of imported commodities, the peasants have confronted the same group of firms who are the monopolistic sellers or distributors of these commodities. […] To this extent, it may be said that the native's real income has not risen as much as it would have if he had sold and bought in more competitive markets. (Meier and Baldwin, 1957, p. 332)

A large buyer may often squeese a dependant supplier, but as long as the supplier has alternative outlets there are limits to the extent of the squeese. […] The real problem for the small country is to maintain the possibility of alternative markets. (Condliffe, 1950, p. 816)

Between each pair of successive levels in the hierarchy perceived by Frank et al, the dendritic and monopolistic patterns of relations between center and periphery are manifestations of Galtung's last rule, and its structural implications, of feudal interaction structures. Separated as a subsystem consisting of a singular center (metropolis) with its uniquely attached peripheries (satellites), we have in effect a star network identical to the network types analyzed by Galtung (1966), Gleditsch (1967), and several others (e.g. Berman, 1974; Chan, 1982; de Janvry, 1975; Dominguez, 1971; Gochman and Ray, 1979; Thompson, 1981), i.e. where each peripheral actor has a singular connection to a core actor but where cores can be, and typically are, connected to several peripheral actors.

The formal specification of Galtung (1971) and the more descriptive roles and structural situations of dependency and dominance as found in the postwar heterodox development discourse form the basis for the novel index of core-periphery structures that we now turn to.

Dependency and dominance: operationalizing core-periphery structures

As reflected in the ideal block images of the blockmodeling tradition and the metric of Borgatti and Everett (1999), contemporary network-analytical conceptualizations of cores and peripheries are overwhelmingly specified in terms of intra-categorical density differentials. This meso-level specification, where cores are deemed as internally cohesive and peripheries as disconnected from each other, finds significant support in the historical literature.

Core-periphery structures have equally been described and specified at the macro-level. Based on the literature, this paper argues for the inclusion of three additional characteristics at the macro-level – connectivity, dependence, and dominance – characteristics that, it is argued, are at least as significant as the meso-level intra-categorical characteristic.

As a conjoint concept where the two categories are somehow related to each other, a baseline criteria for a topological representation of core-periphery structures is connectivity. If we assume an internally disconnected periphery and a connected core, the connectivity of the overall structure would only prevail if each periphery were connected to at least one core. This topological necessity is very much in line with the various usages of the concept: whether conditioning or causing the relative detrimental properties of the periphery, it is the tie(s) between core and periphery that mutually defines and presupposes each category.

As the post-ECLA neo-Marxist school is labeled, peripheral dependency is a central tenet in these heterodox development schools. Facing monopolistic-oligopsonic (neo-)colonial structures in their relations to the outside world, the perceived dependency of peripheries implies that each periphery has at most one tie to a (singular) core actor. Combined with the connectivity criteria, a peripheral actor is thus connected to exactly one core actor.

Mirroring peripheral dependency, core actors are typically described in terms of dominance: those actors that peripheries are dependent upon. In Galtung‘s feudal interaction structure, each core actor has ties to a unique set of peripheral actors. It is nevertheless debatable whether being part of the cohesiveness of core actors per se qualifies as coreness or whether it also should imply ―owning‖ one or more peripheries.Thus, although I include dominating cores as an optional characteristic in the suggested operationalization, it could in certain context be relevant to separate dominating from non-dominating cores.

Expressed in terms of ideal blocks (Doreian et al., 2005, p. 212), an ideal core-periphery structure is given in Figure 3 below. Connectivity and dependency is manifested as the row- (and column-)functional ideal blocks, implying that there is exactly one tie in each row (column) of the P-to-C (C-to-P) block. Core dominance is depicted as the column- (and row-) regular blocks, implying that there is at least one tie in each column (and row) of the P-to-C (C-to-P) block.

Core Periphery

Core 1-block Row-regular & Column-functional

Periphery Row-functional & Column-regular

0-block

Figure 3: Proposed ideal core-periphery block image

With three characteristics of core-periphery patterns identified, adding would-be directionality of relations, Table 1 below summarizes possible deviations from an ideal core-periphery pattern with corresponding penalty types.

Periphery-to-Core:

Connectivity Peripheral

dependency Core dominance

Criteria

Each periphery has at least one tie to a core actor

Each periphery has at most one tie to the core

Each core has at least one tie from a peripheral actor (→ Each periphery has one tie to a singular core)

Ideal block type Row-functional Column-regular

Penalty score Number of peripheries without ties to the core (PCnonConn)

Number of peripheries that have ties to many core actors (PCnonDep)

Number of core actors without ties from the periphery (PCnonDom)

Core-to-Periphery:

Connectivity Peripheral

dependency Core dominance

Criteria

Each periphery has at least one tie from a core actor

Each periphery has at most one tie from the core

Each core has at least one tie to a peripheral actor

(→ Each periphery has one tie from a singular core)

Ideal block type Column-functional Row-regular

Penalty score Number of peripheries without ties from the core (CPnonConn)

Number of peripheroies that have ties from many core actors (CPnonDep)

Number of core actors without ties to the periphery (CPnonDom)

Table 1: Inter-categorical characteristics and their respective penalty type

The penalty types to include in an analysis may vary, and the deviations from the ideal blocks can either be the raw penalty scores or normalized. The suggested (normalized) deviation measure that accepts non-dominating cores is as follows:

dev=PCnonConn+PCnonDep+CPnonConn+CPnonDep

2⋅Np (1)

(where Np is the number of peripheral actors in the given partition)

If coreness should imply having peripheries, the metric is as follows:

dev=PCnonConn+PCnonDep+PCnonDom+CPnonDep+CPnonConn+CPnonDom

2⋅N (2) (where N is total number of actors)

The complement of the normalized deviation constitute the proposed measure of fit for inter-categorical core-periphery patterns:

cpinter

=1−dev (3)

The above index should be combined with the correlation measure of Borgatti and Everett (1999) – or any other suitable measure that captures intra-categorical density differentials. As implemented

in the CorePeripheryBinary software, a two-step search algorithm first calculates meso-level indices (cpintra) for different partitions, subsequently calculating the inter-categorical index (cpinter) for those partitions that pass a specified meso-level threshold. Multiplying4 these two indices with each other, a composite index of core-peripherality is obtained:

cpcomp

=cpintra⋅cp

inter (4) (where cpintra is a (normalized) meso-level core-periphery index, such as the one suggested by Borgatti and Everett

(1999))

The cpinter for Galtung‘s visual example (Figure 2) is, not surprisingly, at unity, but the non-perfect 1-block of intra-core ties yields a cpintra (BEcorr) of 0.79. Alternative metrics for the density differential characteristic are feasible, but as most scholars (including pre-1971 Galtung) depict such density differentials as a defining feature of core-peripheralness, the BEcorr is used here as the cpintra measure when re-examining the example binary networks in Borgatti and Everett (1999) below.

Described as an intuitive core-periphery structure by Borgatti and Everett (1999, p. 337), the network in Figure 4 has a cpintra of unity. All cores dominate and all peripheries are connected, but two of the latter – actor 5 and 8 – are non-dependent, having ties to two cores. This results in an cpinter (and cpcomp) value of 0.8, a maximum among all alternative partitions.

Figure 4: Intuitive core-periphery structure according to Borgatti and Everett (1999)

Using dichotomized and symmetrized co-citation data among social work journals (Baker, 1992), Borgatti and Everett finds the optimal partition (cpintra=BEcorr=0.86) as given in Figure 5. The non-dependency of 10 (out of 13) peripheries results in a cpinter of 0.5 for this partition, yielding a composite core-periphery index of 0.435, i.e. hardly the core-periphery structure given by the BEcorr value alone.

4 Although not explored in this paper, cpcomp could be tweaked further, for instance by attaching weights to

respective index or by combining them in different ways (e.g. square-root of product, mean value, etc). The same goes for the cpinter index where the three inbound penalty scores could be weighted and/or combined differently.

5 The maximum cpcomp (0.48) for this network is obtained by moving PW and SWG to the core.

Figure 5: Co-citation data (dichotomized and symmetrized), optimal BEcorr partition

For the non-symmetrized co-citation data, the maximum cpintra value (0.83) is obtained at the partition given in Figure 6 below. Including all penalties for both direction results in a cpinter at 0.475 and a cpcomp at 0.39 for this partition. The maximum cpcomp (0.41; cpintra=0.79, cpinter=0.525) is obtained by moving SWG to the core.

Figure 6: Co-citation data (dichotomized, directional), optimal BEcorr partition (clockwise-directional arcs)

Incorporating connectivity, dependence and dominance when operationalizing core-peripheralness, the above heuristic for binary networks does not add much novelty to the block deviation measures suggested by Doreian et al (2005). However, although the below extension to core-periphery structures in valued networks introduces and exemplifies a novel approach to blockmodeling of valued data, the same heuristic is still used, i.e. where intra- and inter-categorical indices are combined into a composite index that reflects the characteristic traits of the core-periphery concept as found in the historical literature.

Core-periphery structures in valued networks

A would-be dirty little secret of network analysis is that its methods are primarily designed for binary networks, not valued. This is evident in blockmodeling: although several methods for partitioning valued networks into role-equivalent positions exists, the ideal blocks for subsequent comparisons are nevertheless binary, such as the regular and functional blocks used in this paper to conceptualize core-periphery patterns.

To conform to binary methods, valued data is often dichotomized. Using a statistically, theoretically, or arbitrarily determined cutoff value, finer details are lost, with interpretations dependent on chosen cutoffs. Furthermore, the feasibility of dichotomization is ties to assumptions of ‗relational capacity‘. If we map interaction time among school children during a 45-minute lunch break, we should be able to determine significant ties as a minimum time of interaction, applicable across all dyads. However, when relational capacities differ between actors, e.g. when bullies spend half their lunch breaks in the principal‘s office, the use of a system-wide dichotomization cutoff will emphasize strengths, rather than patterns, of ties.

Addressing the dilemmas with blockmodeling of valued networks (Žiberna, 2009, 2008, 2007a, 2007b), Žiberna‘s suggestions excludes dichotomization of the raw data. Redefing the ideal blocks of generalized blockmodeling using a system-wide threshold parameter (Žiberna, 2007a, pp. 108, 111), possibly including censoring/pruning of exceptionally large values (Žiberna, 2007a, p. 125), such thresholds do still imply specifying significance across all actors based on absolute values. The second suggestion for blockmodeling of valued networks is for homogeneity blockmodeling (see Borgatti and Everett, 1992), i.e. where the blockmodel is optimized for minimum intra-block variance. Suggesting ways to identify ideal block types in such blocks (Žiberna, 2007a, p. 115), optimal partitions in homogeneity blockmodeling are nevertheless based on similar strengths, rather than patterns, of ties.

Tackling these issues when expanding the core-periphery heuristic to valued networks, a novel approach to valued blockmodeling is introduced where significance is determined on a per-actor-to-actor basis. Here applied in search of the regular and functional ideal blocks that characterize dependence, dominated peripheries, and dominating core, the heuristic could be equally applicable to valued blockmodeling in general. Similar to the binary heuristic, a meso-level index captures the density differential characteristic, possibly followed by a penalty-based measure for inter-categorical ties and a composite core-periphery index for valued networks. A sample 7-actor dataset of inter-continental trade exemplifies the approach (Table 2), followed by a re-examination of the monkey interaction and citation networks in Borgatti and Everett (1999, pp. 380, 386). A study of international trade between 18 countries concludes this section.

AFR ASI EUD EUE LAT NAM AUS

Africa (AFR)

15 38 1 3 13 0

Asia (ASI) 16

263 16 31 358 27

Europe, West (EUD) 38 239

109 50 201 19

Europe, East (EUE) 2 20 100

2 10 0

Latin America (LAT) 2 26 41 3

141 1

North America (NAM) 10 226 185 9 133

17

Australasia (AUS) 1 40 10 1 1 8

Table 2: Inter-continental trade (bn USD), annual means, 1995-1999 (Source: Comtrade)

Intra-categorical characteristics: The ‘density differential’ of valued datasets

Borgatti and Everett deem their core-periphery correlation index equally applicable to valued data (1999, p. 384), i.e. by correlating the values of intra-categorical ties to 1 (core) and 0 (periphery). An optimal BEcorr for valued data thus depends on intra-categorical variance.

The feasibility of using the Borgatti-Everett correlation index (‗BEcorr‘) for valued networks is questioned by the examples in Figure 7. The zero variance of the dense core and sparse periphery of Figure 7:A yields a perfect BEcorr value6. Equally so, arguably contra-intuitive, does Figure 7:B. By removing two of those intra-peripheral ties (Figure 7:C), peripheral variance increase, lowering the BEcorr value. Similarly, the intra-core variance in Figure 7:D lowers the BEcorr index further. Noteworthy, the partition in Figure 7:D is not optimal: by classifying either A, B, or C as peripheral would lead to zero intra-categorical variance, yielding a BEcorr at unity for such non-intuitive core-periphery partitions.

Rather than defining core and peripheries in terms of variance, the proposed cpintra index for valued data builds on the simple idea that cores prefer cores and peripheries do not prefer peripheries.

We begin by creating two (zero-sum) normalized versions of the original valued sociomatrix, normalized with respect to rows (RN) and columns (CN), respectively. The row (column) vectors in RN (CN) thus depict the distribution of actor outdegrees (indegrees) among alters.

rna , b

=( xa , b

/∑i

N

xa , i

)−(1

N−1) , a≠b

(5)

cna , b

=( xa , b

/∑i

N

xi , b

)−(1

N−1) , a≠b

(6) (where xa,b is the raw value, N is the number of actors)

In the suggested metric, an ideal core-periphery partition {C, P} implies that intra-core values in RN and CN are above zero and corresponding intra-peripheral values below zero. Counting deviations from such, a normalized cpintra index for valued network could look as follows:

cpintra

=(cardGTZ (RN (C ,C))+cardGTZ (CN (C ,C ))

NC⋅(N

C−1)

−cardGTZ (RN (P , P))+cardGTZ (CN (P , P))

NP⋅(N

P−1)

)/2

(7)

(where the function cardGTZ(B) is the number of values above zero in block B)

The cpintra for the examples in Figure 7 are 1 (A), 0 (B), 0.67 (C), and 1 (D). For A and D, equally perfect are partitions where one of the cores are classified as periphery, and cpintra for C actually increases (to 0.83) when reclassifying one of its core as a periphery.

Table 3 and Table 4 below depict the RN and CN matrices for the inter-continental trade data, sorted according to the partition where cpintra reaches unity, highlighting above-zero values. This

6 ...and so do partitions where either A, B, or C is deemed peripheral.

B C

A

F

D E

100 100

100

90 90

90

B C

A

F

D E

100 100

100 B C

A

F

D E

100 100

100

90

B C

A

F

D E

100 100

350

A

BEcorr = 1.0

B

BEcorr = 1.0

C

BEcorr = 0.76

D

BEcorr = 0.74

PE

RIP

HE

RY

C

OR

E

Figure 7: Using the Borgatti-Everett correlation for valued networks: examples

partition differs from the one that maximizes BEcorr (0.98), when only EUD and NAM are identified as cores.

Core Periphery

ASI EUD NAM AFR EUE LAT AUS

Co

re ASI

0.20 0.34 -0.14 -0.14 -0.12 -0.13

EUD 0.20

0.14 -0.11 0.00 -0.09 -0.14

NAM 0.22 0.15

-0.15 -0.15 0.06 -0.14

Pe

rip

he

ry

AFR 0.05 0.38 0.02

-0.15 -0.12 -0.17

EUE -0.02 0.58 -0.09 -0.15

-0.15 -0.17

LAT -0.05 0.02 0.49 -0.16 -0.15

-0.16

AUS 0.49 0.00 -0.04 -0.15 -0.15 -0.15

Table 3: Row-normalized (zero-marginal) matrix RN of inter-continental trade flows

Core Periphery

ASI EUD NAM AFR EUE LAT AUS

Co

re ASI

0.25 0.32 0.07 -0.05 -0.03 0.26

EUD 0.26

0.11 0.38 0.62 0.06 0.13

NAM 0.23 0.12

-0.02 -0.10 0.44 0.10

Pe

rip

he

ry

AFR -0.14 -0.11 -0.15

-0.16 -0.15 -0.17

EUE -0.13 -0.01 -0.15 -0.14

-0.16 -0.17

LAT -0.12 -0.10 0.03 -0.14 -0.15

-0.15

AUS -0.10 -0.15 -0.16 -0.15 -0.16 -0.16

Table 4: Column-normalized (zero-sum) matrix CN of inter-continental trade flows

Inter-categorical characteristics: ideal blocks in valued blockmodels

The inter-continental trade data exemplifies unequal relational capacities and the dilemmas with dichotomizing valued data. For instance, the trade flow from EUD to its eastern counterpart (EUE), valued at 109 billion USD, constitute 17 percent of the former‘s export, but the very same flow constitute 78 percent of the latter‘s import. Thus, rather than specifying significance on a global level, it is preferable to specify it on a per-actor basis or, which lies at the core of the herein suggested approach to blockmodeling of valued data, on a per-actor-to-actor basis. To demonstrate the procedure, we focus exclusively on the Periphery-to-Core block in the partition obtained above (Table 4).

To map dependency and dominance, we have to identify significant periphery-to-core ties. As core actors differ with respect to total indegree, the significance of such ties depend both on the distribution of aggregate peripheral ties to each core actor, as well as the distribution of outflows from each periphery to the core.

We begin by calculating how total peripheral flows to the core are distributed among core actors – see Table 5. A periphery whose distribution of core ties is similar to the ―expected‖ distribution in Table 5 could thus be deemed as balanced.

ASI EUD NAM Total:

Periphery 101 189 172 462

Marginal-normalized 0.22 0.41 0.37 1

Marginal-norm (zero-sum) -0.11 0.08 0.04 0 Table 5: Peripheral flows to core as distributed among core actors

Using total flows to the core for each peripheral actor, we do a block-internal row-based (zero-sum) normalization of the C-to-P block, where the resulting row vectors reflect each periphery‘s distribution of flows among receiving core actors – see Table 6.

Core

ASI EUD NAM Total:

Pe

rip

he

ry

AFR -0.11 0.24 -0.14 0

EUE -0.18 0.44 -0.26 0

LAT -0.21 -0.14 0.34 0

AUS 0.36 -0.16 -0.20 0

Table 6: Row-based (zero-sum) block-internal normalization of P-to-C block

Comparing these distributions (Table 6) with the expected periphery-to-core distribution (Table 5), the discrepancies between these are given in Table 7 below. Expressed as percentages above and below expected values in Table 8, we note that EUE exports to EUD is 88 percent higher than ―expected‖, and that African flows to North America is 47 percent lower. For both tables, values above zero indicate a higher-than-expected value, which combined with a suitable threshold allow us to identify those ties that are significant for the periphery.

ASI EUD NAM

AFR 0.01 0.17 -0.18

EUE -0.06 0.36 -0.30

LAT -0.09 -0.21 0.31

AUS 0.47 -0.24 -0.23

Table 7: Row-based deviations between actual and expected (balanced) periphery-to-core flows

ASI EUD NAM

AFR 0.04 0.41 -0.47

EUE -0.30 0.88 -0.79

LAT -0.43 -0.52 0.82

AUS 2.15 -0.58 -0.63

Table 8: Percentual difference between actual and expected (balanced) periphery-to-core flows

Percentual difference between actual and expected (balanced) periphery-to-core flows.

Repeating the procedure for the Core-to-Periphery block, this time normalizing with respect to columns, we arrive at the deviations in Table 9, with the alternative percentage-based metric in Table 10.

AFR EUE LAT AUS

ASI 0.06 -0.07 -0.04 0.24

EUD 0.14 0.36 -0.22 -0.15

NAM -0.20 -0.29 0.27 -0.09

Table 9: Column-based deviations between actual and expected (balanced) core-to-periphery flows

AFR EUE LAT AUS

ASI 0.32 -0.37 -0.24 1.26

EUD 0.31 0.79 -0.49 -0.34

NAM -0.56 -0.81 0.75 -0.24

Table 10: Percentual difference between actual and expected (balanced) core-to-periphery flows

The blockmodel for this particular partition is given in Table 11, highlighting ties deemed significant by being at least 33 percent7 larger than expected (see Table 8 and Table 10).

7 I.e. the inverse of the number of cores for this partition.

ASI EUD NAM AFR EUE LAT AUS

ASI 263 358 16 16 31 27

EUD 239 201 38 109 50 19

NAM 226 185 10 9 133 17

AFR 15 38 13 1 3 0

EUE 20 100 10 2 2 0

LAT 26 41 141 2 3 1

AUS 40 10 8 1 1 1

Table 11: Blockmodel of inter-continental trade flows (bn USD), highlighting significant flows

Mapping ―connectivity‖8, dependence and dominance of significant ties in both directions, the cpinter (and cpcomp) for the above partition is 0.93, strongly indicating a core-periphery structure as herein defined. Of particular interest is the singular penalty identified – Africa‘s apparent lack of significant core inflows – but consulting Table 10, we note that trade flows from ASI and EUD are both very close to the stipulated cutoff. Lowering the threshold for significance of ties, AFR would have two significant sources of its imports, possibly reflecting a contemporary struggle between Asia and West Europe for the African market. Noticeably, the same phenomena is not evident in African exports, where EUD is a distinct destination of African exports and where flows from Africa to Asia are only 4 percent higher than what would be expected (see Table 8).

The heuristic and the substantive findings above raises three additional points. First, the ties identified as significant are determined from the peripheral point of view. Whereas reasonable for determining connectivity and dependence, the domination of peripheries is possibly better seen as a core property. Altering the normalization directions and instead interpreting core-based deviations could indicate that ties deemed significant for a periphery might not be deemed as such by core actors. As it is debatable whether the domination of peripheries is a defining core characteristic or not, I have here chosen to determine significance of ties from the peripheral viewpoint, as such reflecting the traditional emphasis in the postwar developmentalist literature.

Secondly, returning to the African vector of core inflows (Table 10 and Table 11), we note that the 16 bn USD inflow from Asia is 32 percent higher than expected, and that the 38 bn USD inflow from Western Europe is 31 percent higher than expected. Thus, even though the former is less than half that of the latter, the heuristic actually deems the former to be slightly more significant than the latter. This underlines that the heuristic not only deems significance on a per-actor basis, but on a per-actor-to-actor basis, a phenomena which the last example will demonstrate further.

Lastly, whereas this heuristic explicitly searches for the characteristics of a 2-positional core-periphery blockmodel, a project is currently underway to expand this approach to generalized blockmodeling of valued networks. Comparing individual values with expected values as determined by series of normalized block-to-actor (and/or actor-to-block) vectors, combining the analysis of rows and columns of blocks that are deemed to be non-zero-blocks as determined by similar normalization procedures at the macro-level of the valued networks, it should be possible to determine the significance of flows on a per-block level. Once identified, such ties could subsequently be compared with the ideal block types found in generalized blockmodeling, with similar penalty scores and goodness-of-fit measures as above.

8 For valued data where there indeed may exist non-zero yet non-significant ties, the notion of ―connected

periphery‖ is better seen as ―dominated periphery‖. For consistency with the binary heuristic, I use the former notion here as well.

Example valued datasets

Monkey interaction

Using (valued, symmetrical) data on interaction among 20 Macaca fuscata monkeys, Borgatti and Everett (1999, p. 380ff) refutes (BEcorr=0.21) the hypothesis that the males (monkey 1-5) constitute a core and the females (6-20) a periphery. At an equally low cpintra (0.23) and a cpinter at 0.55, the resulting cpcomp of 0.13 indeed refutes this hypothesis.

Tracking connectivity, dependency, and dominance, with a 0.10 significance threshold, the maximum cpcomp (0.45) is obtained when three females – 7, 12, and 14 – constitute the core (cpintra = 0.64; cpinter = 0.7). Although exceeding the cpcomp index of the male-only core partition, the conclusion is nevertheless that this dataset lacks a core-periphery structure. The C-to-P deviations between actual and expected monkey interactions, highlighting significant ties, are given in Table 12 below. Peripheral ties to the three female core monkeys are balanced to a, possibly surprisingly, high degree.

1 2 3 4 5 6 8 9 10 11 13 15 16 17 18 19 20

7 0.18 0.08 -0.04 -0.02 0.02 0.10 0.18 0.05 0.02 -0.13 -0.07 -0.12 0.02 0.02 0.02 -0.12 0.02

12 0.00 0.01 0.05 -0.09 0.01 -0.06 0.01 -0.21 0.00 0.17 -0.04 0.01 0.11 0.00 -0.17 0.01 0.11

14 -0.18 -0.09 -0.01 0.11 -0.02 -0.04 -0.19 0.16 -0.01 -0.04 0.11 0.11 -0.12 -0.01 0.15 0.11 -0.12

Table 12: Deviations between actual and expected monkey interactions

Co-citation data

For the dichotomized co-citation data, Borgatti and Everett found an optimal 7-actor-core at a BEcorr at 0.86. This differs from their analysis of the valued data, where BEcorr is maximized (0.81) for a 3-actor core partition. With only two (non-connected) penalties, this partition results in a cpcomp of 0.80406 (cpintra=0.8934; cpinter=0.9), a partition that maximizes cpcomp and indeed reflects a core-periphery structure. Evidently (Table 13), peripheral social work journals have less balanced core ties than do peripheral monkeys.

CAN BJSW AMH ASW CSWJ FR IJSW JGSW JSP JSWE PW CCQ CW CYSR SWG SWHC SWRA

SSR -0.18 -0.18 -0.18 0.03 0.00 -0.18 -0.18 -0.18 0.83 -0.01 -0.18 -0.18 -0.02 0.11 -0.05 -0.18 0.21

SW 0.00 0.43 0.43 0.15 -0.17 -0.24 0.43 -0.04 -0.57 0.04 0.43 -0.57 -0.03 -0.01 0.15 0.11 -0.13

SCW 0.17 -0.26 -0.26 -0.18 0.16 0.41 -0.26 0.22 -0.26 -0.04 -0.26 0.74 0.04 -0.10 -0.10 0.06 -0.08

Table 13: Social work journals: deviations from expected core-to-periphery ties

International commodity trade (18 countries)

The final example dataset in this article constitute international commodity trade data between 18 countries (see Table 14). As a ‗semi-random‘ sample of countries from three regions – North America, West Europe, and South-east Asia – countries were also chosen to represent a significant span in relational capacities (Figure 8). Still, without China, Russia, Middle East, East Europe, Brazil and Latin America, more substantive interpretations of the state of the world economy has to wait for subsequent studies.

USA DEU CAN JPN GBR FRA MEX NLD SGP MYS IRL IDN IND PHL FIN CRI EST GHA

USA

49,749 177,636 65,404 41,356 28,101 118,973 23,987 23,443 14,776 9,941 3,886 8,307 9,340 2,426 3,767 248 346

DEU 86,934

8,481 17,823 68,551 81,704 8,670 49,811 5,953 5,087 5,359 1,781 5,552 1,169 8,711 159 1,408 283

CAN 291,866 3,317

8,924 6,654 2,477 6,169 1,333 812 563 353 698 976 256 289 79 25 102

JPN 141,950 26,816 12,220

16,160 12,890 13,078 7,358 19,233 16,579 2,612 6,906 3,679 8,464 1,916 532 486 119

GBR 52,369 47,060 8,602 6,703

27,867 1,866 16,923 3,935 1,684 21,935 645 4,300 418 2,598 56 281 369

FRA 34,772 66,282 4,126 8,528 37,328

2,570 13,789 3,811 1,489 2,388 708 2,212 418 2,083 157 295 159

MEX 172,481 2,520 12,050 2,535 1,841 869

661 451 134 168 46 99 35 136 430 6 1

NLD 15,503 58,951 1,262 2,130 34,785 19,664 925

1,805 885 2,803 369 1,047 421 2,333 91 240 180

SGP 15,388 4,849 804 6,695 6,384 4,274 2,226 3,578

13,356 1,078 9,471 3,159 3,865 160 19 19 24

MYS 34,676 4,635 2,156 14,669 3,310 1,844 3,658 4,957 27,335

699 2,149 2,436 1,851 377 46 60 42

IRL 28,770 18,623 1,696 3,762 18,226 8,951 774 4,258 1,046 711

81 214 534 543 428 35 24

IDN 12,947 3,017 789 20,817 1,775 1,429 654 1,832 10,447 4,372 132

3,019 1,093 205 10 18 50

IND 19,875 4,195 1,475 3,192 5,138 2,618 959 1,665 4,076 1,101 289 1,052

371 164 27 21 164

PHL 9,694 2,381 761 7,700 1,346 553 1,323 2,610 4,648 3,220 173 322 203

140 39 7 2

FIN 4,532 9,670 1,008 1,231 4,313 2,929 306 2,781 350 218 337 329 474 89

15 1,291 6

CRI 3,602 483 292 221 1,035 123 883 1,664 176 231 44 3 32 48 75

2 0

EST 547 456 46 27 638 185 28 316 0 3 23 2 6 0 1,836 0

3

GHA 173 116 24 76 197 155 6 298 9 49 40 4 74 0 5 0 25

Table 14: Total commodity trade, selected countries, 2012 (Source: Comtrade)

Figure 8: Gross (total imports and exports) for selected countries

We begin with the hypothesis that coreness implies large gross trade flows, picking a core with the eight left-hand countries in Figure 8. Using the inverse of the number of cores as deviation threshold for significant ties, including all penalty types for both directions, we arrive at a cpinter at 0.64. Multiplying this with the relatively low cpintra (0.34), the resulting cpcomp of 0.22 seemingly refutes this hypothesis.

Swapping the positions of Mexico and Singapore, cpintra increases to 0.41. With the same inter-categorical parameters, we get a cpinter of 0.72, with inter-categorical deviations and significant ties as given in Table 15 below. Even though a meagre cpintra index yields an equally meagre cpcomp at 0.29, the patterns of core-periphery ties reveal interesting findings. Mexico is the only periphery with a significant tie to USA, despite the latter‘s overall dominance. Irish imports are quite dependent on Britain, whereas Ireland‘ export seemingly have an alternative German market (cf. Condliffe 1950). India has no ―significant‖ (i.e. non-balanced) trade ties, making it a possible

0

500

1,000

1,500

2,000

USA

DEU

CAN

JPN

GBR

FRA

MEX

NLD

SGP

MYS IRL

IDN

IND

PHL

FIN

CRI

EST

GHA

Gross trade (bn USD), 2012

candidate for the ―external area‖ (cf. Wallerstein 1974). In East- and South-east Asia, Indonesia and the Philippines have significant ties from and to Japan, where Malaysia‘s exports primarily go its southern neighbor Singapore.

USA DEU CAN JPN GBR FRA NLD SGP MEX MYS IRL IDN IND PHL FIN CRI EST GHA

USA

0.30 -0.20 -0.26 -0.32 -0.19 -0.09 -0.36 0.30 -0.39 -0.26

DEU

-0.05 -0.01 0.01 -0.03 0.09 -0.06 0.32 -0.07 0.36 0.07

CAN

0.01 -0.02 -0.02 0.00 0.01 -0.02 -0.01 -0.01 -0.02 0.04

JPN

-0.07 0.16 -0.09 0.13 -0.02 0.20 -0.06 -0.04 0.01 -0.07

GBR

-0.08 -0.06 0.38 -0.07 0.05 -0.08 0.03 -0.08 0.00 0.14

FRA

-0.02 -0.01 0.02 -0.01 0.04 -0.02 0.07 -0.00 0.06 0.07

NLD

-0.02 -0.01 0.04 -0.01 0.01 -0.01 0.09 -0.01 0.05 0.09

SGP

-0.08 0.15 -0.07 0.30 0.02 0.07 -0.08 -0.09 -0.09 -0.08

MEX 0.36 -0.07 0.02 -0.09 -0.06 -0.03 -0.04 -0.09

MYS -0.17 -0.037 -0.01 0.06 -0.04 -0.02 0.01 0.20

IRL -0.20 0.13 -0.02 -0.06 0.14 0.07 0.01 -0.08

IDN -0.29 -0.03 -0.02 0.29 -0.04 -0.01 -0.01 0.11

IND -0.07 0.01 0.00 -0.03 0.05 0.03 0.00 0.01

PHL -0.21 -0.01 -0.01 0.16 -0.03 -0.02 0.05 0.07

FIN -0.37 0.27 0.000 -0.06 0.09 0.07 0.06 -0.08

CRI -0.06 -0.02 0.00 -0.07 0.07 -0.02 0.18 -0.07

EST -0.29 0.12 -0.02 -0.09 0.22 0.05 0.10 -0.09

GHA -0.37 0.03 -0.02 -0.03 0.12 0.11 0.25 -0.08

Table 15: Deviations between actual and expected inter-categorical trade flows

In the above blockmodel, Canada and France are are non-dominating, i.e. no periphery deem either as a significant partner. Moving these two to the periphery as an alternative hypothesis, reducing the deviation threshold to 0.17 (1/6), cpintra and cpinter increase to 0.51 and 0.83 respectively, resulting in a cpcomp of 0.43. In this partition, Canada is a periphery to USA, and France is a periphery to Germany. The increased threshold though implies that Ireland looses both its significant export ties.

The optimal 5-actor core, with a deviation threshold of 0.15, contains USA, Germany, Japan, Great Britain and the Netherlands in the core – see Table 16. With a cpinter of 0.89, the patterns of core-periphery interaction are almost ideal: Singapore and India lacks significant ties to the core, and no periphery deems the Netherlands as a significant source. Despite these near-ideal patterns, a cpintra of 0.61 results in a composite index of 0.54, i.e. barely indicative of a core-periphery structure. As the core indeed is intuitive for this particular dataset, this could motivate the design of an alternative index to capture the intra-categorical property of coreness.

USA DEU JPN GBR NLD CAN FRA MEX SGP MYS IRL IDN IND PHL FIN CRI EST GHA

USA

49,749 65,404 41,356 23,987 177,636 28,101 118,973 23,443 14,776 9,941 3,886 8,307 9,340 2,426 3,767 248 346

DEU 86,934

17,823 68,551 49,811 8,481 81,704 8,670 5,953 5,087 5,359 1,781 5,552 1,169 8,711 159 1,408 283

JPN 141,950 26,816

16,160 7,358 12,220 12,890 13,078 19,233 16,579 2,612 6,906 3,679 8,464 1,916 532 486 119

GBR 52,369 47,060 6,703

16,923 8,602 27,867 1,866 3,935 1,684 21,935 645 4,300 418 2,598 56 281 369

NLD 15,503 58,951 2,130 34,785

1,262 19,664 925 1,805 885 2,803 369 1,047 421 2,333 91 240 180

CAN 291,866 3,317 8,924 6,654 1,333

2,477 6,169 812 563 353 698 976 256 289 79 25 102

FRA 34,772 66,282 8,528 37,328 13,789 4,126

2,570 3,811 1,489 2,388 708 2,212 418 2,083 157 295 159

MEX 172,481 2,520 2,535 1,841 661 12,050 869

451 134 168 46 99 35 136 430 6 1

SGP 15,388 4,849 6,695 6,384 3,578 804 4,274 2,226

13,356 1,078 9,471 3,159 3,865 160 19 19 24

MYS 34,676 4,635 14,669 3,310 4,957 2,156 1,844 3,658 27,335

699 2,149 2,436 1,851 377 46 60 42

IRL 28,770 18,623 3,762 18,226 4,258 1,696 8,951 774 1,046 711

81 214 534 543 428 35 24

IDN 12,947 3,017 20,817 1,775 1,832 789 1,429 654 10,447 4,372 132

3,019 1,093 205 10 18 50

IND 19,875 4,195 3,192 5,138 1,665 1,475 2,618 959 4,076 1,101 289 1,052

371 164 27 21 164

PHL 9,694 2,381 7,700 1,346 2,610 761 553 1,323 4,648 3,220 173 322 203

140 39 7 2

FIN 4,532 9,670 1,231 4,313 2,781 1,008 2,929 306 350 218 337 329 474 89

15 1,291 6

CRI 3,602 483 221 1,035 1,664 292 123 883 176 231 44 3 32 48 75

2 0

EST 547 456 27 638 316 46 185 28 0 3 23 2 6 0 1,836 0

3

GHA 173 116 76 197 298 24 155 6 9 49 40 4 74 0 5 0 25

Table 16: Optimal core-periphery blockmodel, international trade data example

An alternative to the ideal 1-block of intra-core ties is to treat it as a regular block, possibly at a certain level of overfitting. Examining the intra-core blocks in RN and CN, respectively, the criteria for regular blocks are fulfilled in both of these. A meso-level index allowing for ―regular cores‖ would imply that the core actors very well could be internally fragmented: even though contradicting the notion of a ‗dense‘ and internally connected core, such an index would capture cores with non-connected actors, such as the two separate hegemonic systems in Gochman and Ray (1979), thus defining cores more by their inter-categorical than the density of their intra-categorical ties.

Conclusion

―Conceptions‖, Wallerstein argues (1974, p. 36), ―precedes and govern measurements‖, and indeed, the intra-categorical density differential of the core-periphery index of Borgatti and Everett is in line with previous conceptions. However, such conceptions also include the pattern of ties that connect core and peripheral entities into the conjoint concept of repute. Supplementing the density differential characteristic, this paper proposes metrics that capture connectivity, dependence, and dominance for both binary and valued networks.

Whereas the heuristic for binary data is relatively rudimentary, using the correlation index suggested by Borgatti and Everett to capture density differentials, the heuristic for valued data implies confronting the inherent dilemmas of valued networks and their would-be dichotomization. Using a series of block-internal normalizations, the identification of inter-categorical core-periphery patterns exemplifies a novel approach for generalized blockmodeling of valued networks that is more sensitive to patterns, rather than strengths, of ties, while still using the same repertoir of ideal block types found in binary blockmodeling.

Testing the approach on the example data in Borgatti and Everett (1999) as well as two datasets on inter-continental and inter-national trade, the proposed indices seems better at capturing the characteristic features of core-periphery structures as found in the postwar literature on the subject. The trade flow analyses yield particularly interesting findings: evidently, the identified core-periphery patterns in these data sets do not only conform to the feudal interaction structures as specified by Galtung (1971, 1966), but the ties identified as significant reflect historically intuitive anomalies that a dichotomization of the data would not reveal. However, the findings underline a

potential need for an alternative to modeling intra-core relations as a 1-block: although not explored in this paper, it is possible that such ties are better seen as a regular block, possibly at a certain level of overfitting.

Apart from alternative ways to operationalize the density differential characteristic for valued networks with inherently large degree spans, the proposed heuristic has to be tested on larger datasets – such as more complete data on international trade. Alternative ways to combine the inter-categorical penalties should also be explored, as well as possibly adding positions for, respectively, non-dominating cores and non-dependent peripheries (external area). It would also be interesting to explore the structural properties of a would-be semi-peripheral position. As a theoretically distinct stratum in the world-system tradition, it could be possible to explore its structural features and role patterns as reflected in its intra- and inter-positional blocks through empirical analysis of international relations, where a pre-analytical selection of semi-peripheral countries would be based on the more qualitative findings in the relevant literature.

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