Direct blockmodeling of valued and binary networks: a dichotomization-free approach

A long-standing open problem with direct blockmodeling is that it is explicitly intended for binary, not valued, networks. The underlying dilemma is how empirical valued blocks can be compared with ideal binary blocks, an intrinsic problem in the direct approach where partitions are solely determined through such comparisons. Addressing this dilemma, valued networks have either been dichotomized into binary versions, or novel types of ideal valued blocks have been introduced. Both these workarounds are problematic in terms of interpretability, unwanted data reduction, and the often arbitrary setting of model parameters.This paper proposes a direct blockmodeling approach that effectively bypasses the dilemma with blockmodeling of valued networks. By introducing an adaptive weighted correlation-based criteria function, the proposed approach is directly applicable to both binary and valued networks, without any form of dichotomization or transformation of the valued (or binary) data at any point in the analysis, while still using the conventional set of ideal binary blocks from structural, regular and generalized blockmodeling.The proposed approach seemingly solves two other open problems with direct blockmodeling. First, its standardized goodness-of-fit measure allows for direct comparisons across solutions, within and between networks of different sizes, value types, and notions of equivalence. Secondly, through an inherent bias of point-biserial correlations, the approach puts a premium on solutions that are closer to the mid-point density of blockmodels. This, it is argued, translates into solutions that are more intuitive and easier to interpret.The approach is demonstrated by structural, regular and generalized blockmodeling applications of six classical binary and valued networks. Finding feasible and intuitive optimal solutions in both the binary and valued examples, the approach is proposed not only as a practical, dichotomization-free heuristic for blockmodeling of valued networks but also, through its additional benefits, as an alternative to the conventional direct approach to blockmodeling.     

A deviational approach to blockmodeling of valued networks

This article proposes a novel approach to blockmodeling of valued (one-mode) networks where the identification of (binary) block patterns in the valued relations differ from existing approaches. Rather than looking at the absolute values of relations, or examining valued ties on a per-actor basis (cf. Nordlund, 2007), the approach identifies prominent (binary) ties on the basis of deviations from expected values. By comparing the distribution of each actor's valued relations to its alters with the macro-level distributions of total in- and outdegrees, prominent (1) and non-prominent (0) ties are determined both on a per-actor-to-actor and a per-actor-from-actor basis. This allows for a direct interpretation of the underlying functional anatomy of a non-dichotomized valued network using the standard set of ideal blocks as found in generalized blockmodeling of binary networks.In addition to its applicability for direct blockmodeling, the article also suggests a novel indirect measure of deviational structural equivalence on the basis of such deviations from expected values.Exemplified with the note-sharing data in Žiberna (2007a), citations among social work journals (Baker, 1992), and total commodity trade among EU/EFTA countries as of 2010, both the direct and indirect approach produce results that are more sensitive to variations at the dyadic level than existing approaches. This is particularly evident in the case of the EU/EFTA trade network, where the indirect approach yields partitions and blockmodels in support of theories of regional trade, despite the significantly skewed valued degree distribution of the dataset.     

k-means-based algorithm for blockmodeling linked networks

The paper presents a k-means-based algorithm for blockmodeling linked networks where linked networks are defined as a collection of one-mode and two-mode networks in which units from different one-mode networks are connected through two-mode networks. The reason for this is that a faster algorithm is needed for blockmodeling linked networks that can better scale to larger networks. Examples of linked networks include multilevel networks, dynamic networks, dynamic multilevel networks, and meta-networks. Generalized blockmodeling has been developed for linked/multilevel networks, yet the generalized blockmodeling approach is too slow for analyzing larger networks. Therefore, the flexibility of generalized blockmodeling is sacrificed for the speed of k-means-based approaches, thus allowing the analysis of larger networks. The presented algorithm is based on the two-mode k-means (or KL-means) algorithm for two-mode networks or matrices. As a side product, an algorithm for one-mode blockmodeling of one-mode networks is presented. The algorithm’s use on a dynamic multilevel network with more than 400 units is presented. A situation study is also conducted which shows that k-means based algorithms are superior to relocation algorithm-based methods for larger networks (e.g. larger than 800 units) and never much worse.     

Notes on blockmodeling

We attempt to develop further the blockmodeling of networks, so as better to capture the network structure. For this purpose a richer structure than ordinary (valued) graphs has to be used for a model. Such structures are valued graphs with typified (complete, dominant, regular, etc.) connections. Based on the proposed formalization, the blockmodeling is cast as an optimization problem.     

Actor non-response in valued social networks: The impact of different non-response treatments on the stability of blockmodels

Social network data usually contain different types of errors. One of them is missing data due to actor non-response. This can seriously jeopardize the results of analyses if not appropriately treated. The impact of missing data may be more severe in valued networks where not only the presence of a tie is recorded, but also its magnitude or strength. Blockmodeling is a technique for delineating network structure. We focus on an indirect approach suitable for valued networks. Little is known about the sensitivity of valued networks to different types of measurement errors. As it is reasonable to expect that blockmodeling, with its positional outcomes, could be vulnerable to the presence of non-respondents, such errors require treatment. We examine the impacts of seven actor non-response treatments on the positions obtained when indirect blockmodeling is used. The start point for our simulation are networks whose structure is known. Three structures were considered: cohesive subgroups, core-periphery, and hierarchy. The results show that the number of non-respondents, the type of underlying blockmodel structure, and the employed treatment all have an impact on the determined partitions of actors in complex ways. Recommendations for best practices are provided.     

A multiple indicator approach to blockmodeling signed networks

Regardless of whether the focus is on algebraic structures, elaborating role structures or the simple delineation of concrete social structures, generalized blockmodeling faces a pair of vulnerabilities. One is sensitivity to poor quality of the relational data and the other is a risk of over fitting blockmodels to the details of specific networks. Over fitting blockmodels can lead to multiple equally well fitting partitions where choices cannot be made between them on a principled basis. This paper presents a method of tackling these problems by viewing (when possible) observed social relations as multiple indicators of an underlying affect dimension. Quadratic assignment methods using matching coefficients, product moment correlations and Goodman and Kruskal's gamma are used to assess the appropriateness of using the sum of observed relations as input for applying generalized blockmodeling. Data for four groups are used to show the value of this approach within which multiple equally well fitting blockmodels for single relations are replaced by unique (or near-unique) partitions of the summed data. This strategy is located also within a broader problem of blockmodeling three-dimensional networks data and suggestions are made for future work.     

Approaches to blockmodeling dynamic networks: A Monte Carlo simulation study

Blockmodeling refers to a variety of statistical methods for reducing and simplifying large and complex networks. While methods for blockmodeling networks observed at one time point are well established, it is only recently that researchers have proposed several methods for analysing dynamic networks (i.e., networks observed at multiple time points). The considered approaches are based on k-means or stochastic blockmodeling, with different ways being used to model time dependency among time points. Their novelty means they have yet to be extensively compared and evaluated and the paper therefore aims to compare and evaluate them using Monte Carlo simulations. Different network characteristics are considered, including whether tie formation is random or governed by local network mechanisms. The results show the Dynamic Stochastic Blockmodel (Matias and Miele 2017) performs best if the blockmodel does not change; otherwise, the Stochastic Blockmodel for Multipartite Networks (Bar-Hen et al. 2020) does.     

A new criterion function for exploratory blockmodeling for structural and regular equivalence

We present a new criterion function for blockmodeling two-way two-mode relation matrices when the number of blocks as well as the equivalence relation are unknown. For this, we specify a measure of fit based on data compression theory, which allows for the comparison of blockmodels of different sizes and block types from different equivalence relations. We arm an alternating optimization algorithm with this criterion and demonstrate that the method reproduces consensual blockings of three datasets without any pre-specification. We perform a simulation study where we compare our compression-based criterion to the commonly used criterion that measures the number of inconsistencies with an ideal blockmodel.     

Networks containing negative ties

Social network analysts have often collected data on negative relations such as dislike, avoidance, and conflict. Most often, the ties are analyzed in such a way that the fact that they are negative is of no consequence. For example, they have often been used in blockmodeling analyses where many different kinds of ties are used together and all ties are treated the same, regardless of meaning. However, sometimes we may wish to apply other network analysis concepts, such as centrality or cohesive subgroups. The question arises whether all extant techniques are applicable to negative tie data. In this paper, we consider in a systematic way which standard techniques are applicable to negative ties and what changes in interpretation have to be made because of the nature of the ties. We also introduce some new techniques specifically designed for negative ties. Finally we show how one of these techniques for centrality can be extended to networks with both positive and negative ties to give a new centrality measure (PN centrality) that is applicable to directed valued data with both positive and negative ties.     

Identifying regular blocks in valued networks: A heuristic applied to the St. Marks carbon flow data,and international trade in cereal products

While the concept of regular equivalence is equally applicable to dichotomous as well as valued networks, the identification of regular blocks in regular blockmodels is somewhat problematic when dealing with valued networks. Applying the standard procedure for identifying ties in such blockmodels, a procedure perhaps most suited for dichotomous networks, does tend to generate block images and reduced graphs that differ from intuitive notions of such structures.     

Social context,spatial structure and social network structure

Frequently, social networks are studied in their own right with analyses devoid of contextual details. Yet contextual features – both social and spatial – can have impacts on the networks formed within them. This idea is explored with five empirical networks representing different contexts and the use of distinct modeling strategies. These strategies include network visualizations, QAP regression, exponential random graph models, blockmodeling and a combination of blockmodels with exponential random graph models within a single framework. We start with two empirical examples of networks inside organizations. The familiar Bank Wiring Room data show that the social organization (social context) and spatial arrangement of the room help account for the social relations formed there. The second example comes from a police academy where two designed arrangements, one social and one spatial, powerfully determine the relational social structures formed by recruits. The next example is an inter-organizational network that emerged as part of a response to a natural disaster where features of the improvised context helped account for the relations that formed between organizations participating in the search and rescue mission. We then consider an anthropological example of signed relations among sub-tribes in the New Guinea highlands where the physical geography is fixed. This is followed by a trading network off the Dalmatian coast where geography and physical conditions matter. Through these examples, we show that context matters by shaping the structure of networks that form and that a variety of network analytic tools can be mobilized to reveal how networks are shaped, in part, by social and spatial contexts. Implications for studying social networks are suggested.     

Block structures of automorphism groups of social relations

This paper expands the concept of “block”, as in “blockmodeling”, by relating it to the “blocks” of permutation groups. Crucial to this development is the idea of graph automorphism, which captures the essence of “regular equivalence” in a way that allows the flexibility of the group block concept. Blocks, unlike regular or structural equivalence classes, are not disjoint, allowing overlapping and hierarchical structures to be described. One of the main application of these techniques is to “crack” a disappointingly small number of orbits (say one or two) found in highly symmetric graphs into a richer block structure.     

Parameter estimation procedures for exponential-family random graph models on count-valued networks: A comparative simulation study

The exponential-family random graph models (ERGMs) have emerged as an important framework for modeling social networks for a wide variety of relational types. ERGMs for valued networks are less well-developed than their unvalued counterparts, and pose particular computational challenges. Network data with edge values on the non-negative integers (count-valued networks) is an important such case, with examples ranging from the magnitude of migration and trade flows between places to the frequency of interactions and encounters between individuals. Here, we propose an efficient parallelizable subsampled maximum pseudo-likelihood estimation (MPLE) scheme for count-valued ERGMs, and compare its performance with existing Contrastive Divergence (CD) and Monte Carlo Maximum Likelihood Estimation (MCMLE) approaches via a simulation study based on migration flow networks in two U.S. states. Our results suggest that edge value variance is a key factor in method performance, while network size mainly influences their relative merits in computational time. For small-variance networks, all methods perform well in point estimations while CD greatly overestimates uncertainties, and MPLE underestimates them for dependence terms; all methods have fast estimation for small networks, but CD and subsampled multi-core MPLE provides speed advantages as network size increases. For large-variance networks, both MPLE and MCMLE offer high-quality estimates of coefficients and their uncertainty, but MPLE is significantly faster than MCMLE; MPLE is also a better seeding method for MCMLE than CD, as the latter makes MCMLE more prone to convergence failure. The study suggests that MCMLE and MPLE should be the default approach to estimate ERGMs for small-variance and large-variance valued networks, respectively. We also offer further suggestions regarding choice of computational method for valued ERGMs based on data structure, available computational resources and analytical goals.     

Connectedness in a general model for valued networks

Social networks have been closely identified with graph theoretical models, which constitute their most familiar mode of representation. There are a number of such models which may embody symmetric, directed, or valued relationships. But the study of networks with valued linkages, using the natural formalization provided by the valued graph or digraph, has been impeded by a traditional lack of analytical machinery for dealing with valued structures. In this paper, we demonstrate the development and elaboration of formalizations for the central network concepts of reachability, joining, and connectedness through graph theoretical models of increasing complexity, culminating in their expression within a general model for valued structures. This model for valued (symmetric or directed) graphs, or vigraphs, provides a unified representation and matrix methodology for dealing with qualitative and quantitative structures, incorporates many existing methods as special cases, and suggests new applications. Some of the most interesting of these follow the recognition, consistent with the model, that the “values” assigned to network linkages may be sorts of entities other than numbers.     

On the core of characteristic function games associated with exchange networks

An exchange network is a social system in which the actors gain valued resources from bilateral transactions, but an opportunity to negotiate a deal is given only to those pairs of actors whose positions are tied with each other in a fixed communication network. A transaction consists in a mutually agreed-on division of a resource pool assigned to a network line. An additional constraint imposed on such a network restricts the range of transaction sets which may happen in a single negotiation round to those consistent with a given “exchange regime.” Under the one-exchange regime every actor is permitted to make no more than one deal per round. Bienenstock and Bonacich [Bienenstock, E.J., Bonacich, P., 1992. The core as a solution to exclusionary networks. Social Networks 14, 231–243] proposed to represent a one-exchange network with an n-person game in characteristic function form. The aim of this paper is to develop a mathematical theory of games associated with homogenous one-exchange networks (network homogeneity means that all lines are assigned resource pools of the same size). The focus is on the core, the type of solution considered most important in game theory. In particular, all earlier results obtained by Bonacich are re-examined and there is given a new graph-theoretic necessary and sufficient condition for the existence of nonempty core for the game representing a homogenous one-exchange network.     

Uncovering CSR meaning networks: A cross-national comparison of Turkey and Slovenia

This study addressed an important question about the meaning of corporate social responsibility (CSR), and how it is measured. Based on a comparison of the meaning networks of CSR in two countries with fundamentally different cultural and socioeconomic backgrounds, we argue that there is a need for an institutional perspective when studying CSR associations and expectations in a particular society. Thus empirical study involved the use of three methods the word-association technique, social network analysis, and blockmodeling using Pajek software; to provide deep insight into the structure of CSR associations. The findings suggest that the two societies have diverse collective cognitive structures regarding CSR. In Turkey, the philanthropic understanding of CSR is highly dominant, while the Slovenian social meaning of CSR is multidimensional. The findings point to the social construction of the concept of CSR with implications both for academic research and practice.     

Environmental stewardship footprint research: linking human agency and ecosystem health in the Puget Sound region

Urbanization processes challenge ecosystem health in many metropolitan areas. New policy and program approaches are needed to restore and sustain natural systems as public agencies and organizations face greater demands and declining budgets. Environmental stewardship is an often overlooked intervention strategy, and the full potential of civic engagement by citizens on behalf of ecosystem health is little understood. Using a coupled systems approach, integrated analysis of social and ecological footprints can lead to greater theoretical understanding and more effective programs at the landscape scale. Here we outline two pilot studies as part of an emergent research program that is investigating the extent and impact of environmental stewardship. Qualitative interviews of stewardship managers revealed multiple dimensions of motivations and purposes for stewardship, ranging from the practical to the conceptual. A regional organization census yielded a surprisingly large number of organizations that conduct stewardship, with social and ecological values being of comparable emphasis. The initial research is based in the Puget Sound area of Washington State, U.S., but results have relevance to other urban areas. Pilot study findings now guide additional research effort about motivations, organizational networks, and theory of integrated socio-ecological systems to be derived from comprehensive footprint analysis of stewardship activity.     

Masculinity,Gender, and Rural Policy*

Abstract This paper explores the operation of gender relations in the context of rural policy. Framed by debates on new rural governance, it considers how both the content and the culture of recent rural regeneration policy reflect highly masculine values and the maintenance of traditional power relations. New forms of decision making in rural areas promote a style of policy making that values and grants priority to male networks in the construction of elite groups and styles of management, and devalues community participation. We use examples from the United Kingdom to demonstrate the implications of shifts in the mechanisms and practice of policy making and implementation for men's and women's differential involvement and experience with rural regeneration. We go on to show how gender relations are also reflected in the content of contemporary rural regeneration policy. Decisions concerning the most appropriate types of initiative are predicated on a male‐oriented view of previous economic activity and local labor markets, and represent a highly masculinist approach to regeneration.     

An introduction to exponential random graph (p) models for social networks

This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement.     

Graph and semigroup homomorphisms on networks of relations

The algebraic definitions presented here are motivated by our search for an adequate formalization of the concepts of social roles as regularities in social network patterns. The theorems represent significant homomorphic reductions of social networks which are possible using these definitions to capture the role structure of a network. The concepts build directly on the pioneering work of S.F. Nadel (1957) and the pathbreaking approach to blockmodeling introduced by Lorrain and White (1971) and refined in subsequent years (White, Boorman and Breiger 1976;Boorman and White 1976; Arabie, Boorman and Levitt, 1978; Sailer, 1978).Blockmodeling is one of the predominant techniques for deriving structural models of social networks. When a network is represented by a directed multigraph, a blockmodel of the multigraph can be characterized as mapping points and edges onto their images in a reduced multigraph. The relations in a network or multigraph can also be composed to form a semigroup.In the first part of the paper we examine “graph” homomorphisms, or homomorphic mappings of the points or actors in a network. A family of basic concepts of role equivalence are introduced, and theorems presented to show the structure preserving properties of their various induced homomorphisms. This extends the “classic” approach to blockmodeling via the equivalence of positions.Lorrain and White (1971), Pattison (1980), Boyd, 1980, Boyd, 1982, and most recently Bonacich (1982) have explored the topic taken up in the second part of this paper, namely the homomorphic reduction of the semigroup of relations on a network, and the relation between semigroup and graph homomorphisms. Our approach allows us a significant beginning in reducing the complexity of a multigraph by collapsing relations which play a similar “role” in the network.