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
[email protected]
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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 intracategorical extremes, the literature review lends equal, possibly more, support to a set of intercategorical 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 macrolevel 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 intracategorical 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 intercategorical 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 centerperiphery 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 coreperiphery 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 centerperiphery 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 intercategorical 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 coreperiphery 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 intercategorical 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 intercategorical 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 rankordered 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 nondiagonal 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
Core
1-block
Periphery
Imperfect 1block
Periphery
Imperfect 1block
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 coreperiphery structures, but additional inter-categorical criteria are needed. The first criteria is
concerned with connectivity. Assuming an internally connected core and an absence of intraperipheral 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-toperiphery 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).
P1
P1
11
22
C1
P4
1
P2
C4
C2
P4
1
P2
C3
2
P3
P3
1
2
P3
3
2
Figure 2: Galtung's example of feudal
interaction structure
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).
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 producebuying 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 macrolevel – 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 (Cto-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
Row-regular
& Column-functional
Core
1-block
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
Criteria
Each periphery has at least
one tie to a core actor
Peripheral
dependency
Each periphery has at
most one tie to the core
(→ Each periphery has one tie to a singular core)
Ideal block type
Row-functional
Penalty score
Number of peripheries
without ties to the core
(PCnonConn)
Number of peripheries
that have ties to many
core actors
(PCnonDep)
Core dominance
Each core has at least
one tie from a
peripheral actor
Column-regular
Number of core actors
without ties from the
periphery
(PCnonDom)
Core-to-Periphery:
Connectivity
Criteria
Each periphery has at least
one tie from a core actor
Peripheral
dependency
Each periphery has at
most one tie from the
core
(→ Each periphery has one tie from a singular core)
Ideal block type
Column-functional
Penalty score
Number of peripheries
without ties from the core
(CPnonConn)
Number of peripheroies
that have ties from
many core actors
(CPnonDep)
Core dominance
Each core has at least
one tie to a peripheral
actor
Row-regular
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⋅N p
(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 intercategorical 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 =cp intra⋅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 nondependency 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
5
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.
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 coreperipheralness, 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 coreperiphery 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-actorto-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 intercategorical 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
Africa (AFR)
Asia (ASI)
Europe, West (EUD)
Europe, East (EUE)
Latin America (LAT)
North America (NAM)
Australasia (AUS)
16
38
2
2
10
1
ASI
15
239
20
26
226
40
EUD
38
263
100
41
185
10
EUE
LAT
1
16
109
3
9
1
3
31
50
2
133
1
Table 2: Inter-continental trade (bn USD), annual means, 1995-1999 (Source: Comtrade)
NAM
13
358
201
10
141
8
AUS
0
27
19
0
1
17
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.
PERIPHERY
CORE
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 coreperiphery partitions.
100
A
100
B
D
100
100
C
B
E
D
A
100
90
90
100
100
C
B
E
D
A
100
100
90
100
C
B
E
D
A
350
100
C
E
90
F
F
F
F
A
B
C
D
BEcorr = 1.0
BEcorr = 1.0
BEcorr = 0.76
BEcorr = 0.74
Figure 7: Using the Borgatti-Everett correlation for valued networks: examples
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.
N
rna , b =( x a , b /∑ x a , i)−(
i
N
cna , b =( x a , b / ∑ x i , b )−(
i
1
) , a≠b
N −1
(5)
1
) , a≠b
N −1
(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 )) cardGTZ ( RN (P , P))+cardGTZ (CN ( P , P))
−
)/2
N C⋅(N C −1)
N P⋅( N P −1)
(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.
(7)
partition differs from the one that maximizes BEcorr (0.98), when only EUD and NAM are identified
as cores.
Core
ASI
EUD
NAM
0.20
0.22
0.15
Periphery
ASI
Core
EUD
0.20
AFR
EUE
LAT
AUS
0.05
-0.02
-0.05
0.49
0.38
0.58
0.02
0.00
NAM
0.34
0.14
AFR
-0.14
-0.11
-0.15
0.02
-0.09
0.49
-0.04
Periphery
EUE
LAT
-0.14
-0.12
0.00
-0.09
0.06
-0.15
-0.15
-0.15
-0.16
-0.15
-0.15
-0.15
-0.12
-0.15
AUS
-0.13
-0.14
-0.14
-0.17
-0.17
-0.16
-0.15
Table 3: Row-normalized (zero-marginal) matrix RN of inter-continental trade flows
Core
ASI
EUD
NAM
0.26
0.23
0.12
Periphery
ASI
Core
EUD
0.25
AFR
EUE
LAT
AUS
-0.14
-0.13
-0.12
-0.10
-0.11
-0.01
-0.10
-0.15
NAM
0.32
0.11
Periphery
EUE
LAT
-0.05
-0.03
0.62
0.06
0.44
-0.10
AFR
0.07
0.38
-0.02
-0.15
-0.15
0.03
-0.16
-0.16
-0.14
-0.14
-0.15
-0.15
-0.16
-0.15
-0.16
AUS
0.26
0.13
0.10
-0.17
-0.17
-0.15
-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
Periphery
Marginal-normalized
Marginal-norm (zero-sum)
EUD
NAM
Total:
101
189
172
462
0.22
-0.11
0.41
0.08
0.37
0.04
1
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.
Periphery
AFR
EUE
LAT
AUS
ASI
-0.11
-0.18
-0.21
0.36
Core
EUD
0.24
0.44
-0.14
-0.16
NAM
-0.14
-0.26
0.34
-0.20
Total:
0
0
0
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.
AFR
EUE
LAT
AUS
ASI
0.01
-0.06
-0.09
0.47
EUD
0.17
0.36
-0.21
-0.24
NAM
-0.18
-0.30
0.31
-0.23
Table 7: Row-based deviations between actual and expected (balanced) periphery-to-core flows
AFR
EUE
LAT
AUS
ASI
0.04
-0.30
-0.43
2.15
EUD
0.41
0.88
-0.52
-0.58
NAM
-0.47
-0.79
0.82
-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
ASI
EUD
NAM
0.06
0.14
-0.20
EUE
-0.07
0.36
-0.29
LAT
-0.04
-0.22
0.27
AUS
0.24
-0.15
-0.09
Table 9: Column-based deviations between actual and expected (balanced) core-to-periphery flows
AFR
ASI
EUD
NAM
0.32
0.31
-0.56
EUE
-0.37
0.79
-0.81
LAT
-0.24
-0.49
0.75
AUS
1.26
-0.34
-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
ASI
EUD
NAM
263
AFR
EUE
LAT
AUS
358
16
16
31
27
201
38
109
50
19
10
9
133
17
1
3
0
2
0
EUD
239
NAM
226
185
AFR
15
38
13
EUE
20
100
10
2
LAT
26
41
141
2
3
AUS
40
10
8
1
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 coreperiphery 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.
7
1
2
3
4
5
6
8
9
10
11
13
15
16
17
18
19
20
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
SSR
-0.18
-0.18
-0.18
0.03
0.00
-0.18
-0.18
SW
0.00
0.43
0.43
0.15
-0.17
-0.24
0.43
SCW
0.17
-0.26
-0.26
-0.18
0.16
0.41
-0.26
JSP
JSWE
PW
CCQ
CW
CYSR SWG SWHC SWRA
-0.18
0.83
-0.01
-0.18
-0.18
-0.02
0.11
-0.05
-0.18
0.21
-0.04
-0.57
0.04
0.43
-0.57
-0.03
-0.01
0.15
0.11
-0.13
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
USA
DEU
DEU
CAN
JPN
GBR
FRA
MEX
NLD
SGP
MYS
49,749 177,636 65,404 41,356 28,101 118,973 23,987 23,443 14,776
86,934
CAN 291,866
8,481
3,317
IRL
IDN
IND
PHL
3,886 8,307 9,340 2,426 3,767
1,781 5,552 1,169 8,711
8,670
49,811
5,953
5,087
5,359
8,924
2,477
6,169
1,333
812
563
353
16,160 12,890
13,078
7,358
19,233 16,579
2,612
27,867
1,866
16,923
3,935
1,684
21,935
645
4,300
418
2,570
13,789
3,811
1,489
2,388
708
2,212
418
661
451
134
168
46
99
1,805
885
2,803
369
1,047
13,356
1,078
699
JPN 141,950 26,816
12,220
GBR
52,369
47,060
8,602
6,703
FRA
34,772
66,282
4,126
8,528
37,328
MEX 172,481
2,520
12,050
2,535
1,841
NLD
15,503
58,951
1,262
2,130
34,785 19,664
SGP
15,388
4,849
804
6,695
6,384
4,274
2,226
3,578
MYS
34,676
4,635
2,156
14,669
3,310
1,844
3,658
4,957
27,335
IRL
28,770
18,623
1,696
3,762
18,226
8,951
774
4,258
1,046
711
IDN
12,947
3,017
789
20,817
1,775
1,429
654
1,832
10,447
4,372
132
869
925
CRI
9,941
17,823 68,551 81,704
6,654
FIN
698
976
256
289
159
EST GHA
248
346
1,408 283
79
25
102
532
486
119
2,598
56
281
369
2,083
157
295
159
35
136
430
6
1
421
2,333
91
240
180
9,471 3,159 3,865
160
19
19
24
2,149 2,436 1,851
377
46
60
42
543
428
35
24
205
10
18
50
164
27
21
164
140
39
7
2
15
1,291
6
2
0
6,906 3,679 8,464 1,916
81
214
534
3,019 1,093
IND
19,875
4,195
1,475
3,192
5,138
2,618
959
1,665
4,076
1,101
289
1,052
PHL
9,694
2,381
761
7,700
1,346
553
1,323
2,610
4,648
3,220
173
322
203
371
FIN
4,532
9,670
1,008
1,231
4,313
2,929
306
2,781
350
218
337
329
474
89
CRI
3,602
483
292
221
1,035
123
883
1,664
176
231
44
3
32
48
75
EST
547
456
46
27
638
185
28
316
0
3
23
2
6
0
1,836
0
GHA
173
116
24
76
197
155
6
298
9
49
40
4
74
0
5
0
3
25
Table 14: Total commodity trade, selected countries, 2012 (Source: Comtrade)
Gross trade (bn USD), 2012
2,000
1,500
1,000
500
GHA
EST
CRI
FIN
PHL
IND
IDN
IRL
MYS
SGP
NLD
MEX
FRA
GBR
JPN
CAN
DEU
USA
0
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 intercategorical 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
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
USA
MEX
MYS
IRL
IDN
IND
PHL
FIN
CRI
EST
GHA
0.30
-0.20
-0.26
-0.32
-0.19
-0.09
-0.36
0.30
-0.39
-0.26
0.07
DEU
-0.05
-0.01
0.01
-0.03
0.09
-0.06
0.32
-0.07
0.36
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
MYS -0.17 -0.037
-0.09
-0.06
-0.03
-0.04
-0.09
-0.01
0.06
-0.04
-0.02
0.01
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
IRL -0.20
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
FIN -0.37
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 coreperiphery 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
JPN
GBR
86,934
17,823 68,551 49,811
JPN 141,950 26,816
16,160 7,358
GBR
52,369 47,060 6,703
NLD
15,503 58,951 2,130 34,785
CAN 291,866 3,317
FRA
NLD
CAN
FRA
MEX
SGP
MYS
IRL
49,749 65,404 41,356 23,987 177,636 28,101 118,973 23,443 14,776 9,941
USA
DEU
DEU
8,924
16,923
6,654
IND
PHL
FIN
CRI
EST
GHA
248
346
3,886 8,307
9,340 2,426 3,767
5,359
1,781 5,552
1,169 8,711
159
1,408
283
12,220 12,890 13,078 19,233 16,579 2,612
6,906 3,679
8,464 1,916
532
486
119
8,481
81,704
8,670
5,953
5,087
1,684 21,935
645
4,300
418
2,598
56
281
369
2,803
369
1,047
421
2,333
91
240
180
563
353
698
976
256
289
79
25
102
1,489
2,388
708
2,212
418
2,083
157
295
159
168
46
99
35
136
430
6
1
9,471 3,159
3,865
160
19
19
24
2,149 2,436
8,602
27,867
1,866
3,935
1,262
19,664
925
1,805
2,477
6,169
812
2,570
3,811
451
134
1,333
34,772 66,282 8,528 37,328 13,789
IDN
4,126
885
MEX 172,481 2,520
2,535
1,841
661
12,050
869
SGP
15,388
4,849
6,695
6,384
3,578
804
4,274
2,226
MYS
34,676
4,635 14,669 3,310
4,957
2,156
1,844
3,658
27,335
IRL
28,770 18,623 3,762 18,226 4,258
1,696
8,951
774
1,046
IDN
12,947
3,017 20,817 1,775
1,832
789
1,429
654
10,447 4,372
132
IND
19,875
4,195
3,192
5,138
1,665
1,475
2,618
959
4,076
1,101
289
1,052
PHL
9,694
2,381
7,700
1,346
2,610
761
553
1,323
4,648
3,220
173
322
203
FIN
4,532
9,670
1,231
4,313
2,781
1,008
2,929
306
350
218
337
329
474
89
CRI
3,602
483
221
1,035
1,664
292
123
883
176
231
44
3
32
48
75
EST
547
456
27
638
316
46
185
28
0
3
23
2
6
0
1,836
0
GHA
173
116
76
197
298
24
155
6
9
49
40
4
74
0
5
0
13,356 1,078
699
711
81
1,851
377
46
60
42
214
534
543
428
35
24
3,019
1,093
205
10
18
50
371
164
27
21
164
140
39
7
2
15
1,291
6
2
0
3
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 intracategorical 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 coreperiphery 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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