A new look at clustering coefficients with generalization to weighted and multi-faction networks

In this paper we propose a new method for studying local and global clustering in networks employing random walk pairs. The method is intuitive and directly generalizes standard local and global clustering coefficients to weighted networks and networks containing nodes of multiple types. In the case of two-mode networks the values obtained for commonly considered social networks are in sharp contrast to those obtained, for instance, by the method of Opsahl (2013), and provide a different viewpoint for clustering. The approach is also applicable in questions related to the general study of segregation and homophily. Applications to existent data sets are considered.     

Stochastic weighted graphs: Flexible model specification and simulation

In most domains of network analysis researchers consider networks that arise in nature with weighted edges. Such networks are routinely dichotomized in the interest of using available methods for statistical inference with networks. The generalized exponential random graph model (GERGM) is a recently proposed method used to simulate and model the edges of a weighted graph. The GERGM specifies a joint distribution for an exponential family of graphs with continuous-valued edge weights. However, current estimation algorithms for the GERGM only allow inference on a restricted family of model specifications. To address this issue, we develop a Metropolis–Hastings method that can be used to estimate any GERGM specification, thereby significantly extending the family of weighted graphs that can be modeled with the GERGM. We show that new flexible model specifications are capable of avoiding likelihood degeneracy and efficiently capturing network structure in applications where such models were not previously available. We demonstrate the utility of this new class of GERGMs through application to two real network data sets, and we further assess the effectiveness of our proposed methodology by simulating non-degenerate model specifications from the well-studied two-stars model. A working R version of the GERGM code is available in the supplement and is incorporated in the GERGM CRAN package.     

The sequence analysis of hospitals that established social work department under fuzzy environment

This paper reviews the existing literature on hospital social work and discusses intervention strategies for improving social work practice in hospital. The objective of this study was to improve the quality of medical care. But few studies have compared social work services between different hospitals. This study describes qualitative analysis under fuzzy environment, extracts the main influencing factors and establishes a comprehensive evaluation index system. It provides comprehensive evaluation for alternative hospitals by the fuzzy clustering method. This paper proposes a new mixed fuzzy clustering algorithm on the basis of analysing the axiomatic fuzzy set (AFS) and K-means algorithm, which is not affected by some complicated parameter issues and has higher statistical validity. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is applied for selecting the best option for each cluster and a comparative analysis is done. Results from a case study in Shanghai, China, confirm that the proposed approach is effective by using information entropy to test. By comparing AFS, K-means and C-means algorithms, the hybrid algorithm can find the two closest attributes of evaluation index of hospital social work, and the proposed approach can be easily help raise the level of hospital social work service.     

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.     

Modelling a disease-relevant contact network of people who inject drugs

This study uses social network analysis to model a contact network of people who inject drugs (PWID) relevant for investigating the spread of an infectious disease (hepatitis C). Using snowball sample data, parameters for an exponential random graph model (ERGM) including social circuit dependence and four attributes (location, age, injecting frequency, gender) are estimated using a conditional estimation approach that respects the structure of snowball sample designs. Those network parameter estimates are then used to create a novel, model-dependent estimate of network size. Simulated PWID contact networks are created and compared with Bernoulli graphs. Location, age and injecting frequency are shown to be statistically significant attribute parameters in the ERGM. Simulated ERGM networks are shown to fit the collected data very well across a number of metrics. In comparison with Bernoulli graphs, simulated networks are shown to have longer paths and more clustering. Results from this study make possible simulation of realistic networks for investigating treatment and intervention strategies for reducing hepatitis C prevalence.     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time is the class of exponential random graph models (ERGMs), also known as p * models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, Markov chain Monte Carlo (MCMC) algorithms have been developed that produce approximate maximum likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data.
This paper proposes new specifications of exponential random graph models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistics are proposed: geometrically weighted degree distributions, alternating k -triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.     

A comparison of email networks and off-line social networks: A study of a medium-sized bank

Recently there has been a surge in the availability of online data concerning the connections between people, and these online data are now widely used to map the social structure of communities. There has been little research, however, on how these new types of relational data correspond to classical measures of social networks. To fill this gap, we contrast the structure of an email network with the underlying friendship, communication, and advice seeking networks. Our study is an explorative case study of a bank, and our data contains emails among employees and a survey of the ego networks of the employees. Through calculating correlations with QAP standard errors and estimating exponential random graph (ERG) models, we find that although the email network is related to the survey-based social networks, email networks are also significantly different: while off-line social networks are strongly shaped by gender, tenure, and hierarchical boundaries, the role of these boundaries are much weaker in the email network.     

Emergence of social support networks among breast cancer survivors through a community-based physical activity program in Colombia

Physical activity (PA), social networks, and social support have been associated with decreased mortality and improved quality of life among breast cancer survivors (BCS). This study used social network analysis to understand the social co-benefits of a community-based PA program for BCS in Colombia. Two types of social support networks emerged from the program: friendship (the number of edges increased by 90 %) and PA support (35 % of participants practiced PA together after the program). Using egocentric and socio-centric analysis we show the presence of homophily for friendship and PA support relations and the BCS’s roles in their networks.     

An agent-based random-utility-maximization model to generate social networks with transitivity in geographic space

Stochastic actor-based approaches receive increasing interest in the generation of social networks for simulation in time and space. Existing models however cannot be readily integrated in agent-based models that assume random-utility-maximizing behavior of agents. We propose an agent-based model to generate social networks explicitly in geographic space which is formulated in the random-utility-maximizing (RUM) framework. The proposed model consists of a friendship formation mechanism and a component to simulate social encounters in a population. We show how transitivity can be incorporated in both components and how the model can be estimated based on data of personal networks using likelihood estimation. In an application to the Swiss context, we demonstrate the estimation and ability of the model to reproduce relevant characteristics of networks, such as geographic proximity, attribute similarity (homophily), size of personal networks (degree distribution) and clustering (transitivity). We conclude that the proposed social-network model fits seamlessly in existing large-scale micro-simulation systems which assume RUM behavior of agents.     

Considering clustering measures: Third ties,means, and triplets

Measures that estimate the clustering coefficients of ego and overall social networks are important to social network studies. Existing measures differ in how they define and estimate triplet clustering with implications for how network theoretic properties are reflected. In this paper, we propose a novel definition of triplet clustering for weighted and undirected social networks that explicitly considers the relative strength of the tie connecting the two alters of the ego in the triplet. We argue that our proposed definition better reflects theorized effects of the important third tie in the social network literature. We also develop new methods for estimating triplet, local and global clustering. Three different types of mathematical means, i.e. arithmetic, geometric, and quadratic, are used to reflect alternative theoretical assumptions concerning the marginal effect of tie substitution.     

Balance in stochastic signed graphs

Signed graphs provide models for investigating balance in connection with various kinds of social relations. Since empirical social networks always involve uncertainty because of errors due to measurement, imperfect observation or sampling, it is desirable to incorporate uncertainty into signed graph models. We introduce a stochastic signed graph and investigate the properties of some indices of balance involving triads. In particular we consider the balance properties of a graph which is randomly signed and of one which has been randomly sampled from a large population graph.     

Triadic closure in two-mode networks: Redefining the global and local clustering coefficients

As the vast majority of network measures are defined for one-mode networks, two-mode networks often have to be projected onto one-mode networks to be analyzed. A number of issues arise in this transformation process, especially when analyzing ties among nodes’ contacts. For example, the values attained by the global and local clustering coefficients on projected random two-mode networks deviate from the expected values in corresponding classical one-mode networks. Moreover, both the local clustering coefficient and constraint (structural holes) are inversely associated to nodes’ two-mode degree. To overcome these issues, this paper proposes redefinitions of the clustering coefficients for two-mode networks.     

Cluster analysis of multiplex networks: Defining composite network measures

Social relations are multiplex by nature: actors in a group are tied together by various types of relationships. To understand and explain group processes it is, therefore, important to study multiple social networks simultaneously in a given group. However, with multiplexity the complexity of data also increases. Although some multivariate network methods (e.g. Exponential Random Graph Models, Stochastic Actor-oriented Models) allow to jointly analyze multiple networks, modeling becomes complicated when it focuses on more than a few (2–4) network dimensions. In such cases, dimension reduction methods are called for to obtain a manageable set of variables. Drawing on existing statistical methods and measures, we propose a procedure to reduce the dimensions of multiplex network data measured in multiple groups. We achieve this by clustering the networks using their pairwise similarities, and constructing composite network measures as combinations of the networks in each resulting cluster. The procedure is demonstrated on a dataset of 21 interpersonal network dimensions in 18 Hungarian high-school classrooms. The results indicate that the network items organize into three well-interpretable clusters: positive, negative, and social role attributions. We show that the composite networks defined on these three relationship groups overlap but do not fully coincide with the network measures most often used in adolescent research, such as friendship and dislike.     

Analysing networks of networks

We consider data with multiple observations or reports on a network in the case when these networks themselves are connected through some form of network ties. We could take the example of a cognitive social structure where there is another type of tie connecting the actors that provide the reports; or the study of interpersonal spillover effects from one cultural domain to another facilitated by the social ties. Another example is when the individual semantic structures are represented as semantic networks of a group of actors and connected through these actors’ social ties to constitute knowledge of a social group. How to jointly represent the two types of networks is not trivial as the layers and not the nodes of the layers of the reported networks are coupled through a network on the reports. We propose to transform the different multiple networks using line graphs, where actors are affiliated with ties represented as nodes, and represent the totality of the different types of ties as a multilevel network. This affords studying the associations between the social network and the reports as well as the alignment of the reports to a criterion graph. We illustrate how the procedure can be applied to studying the social construction of knowledge in local flood management groups. Here we use multilevel exponential random graph models but the representation also lends itself to stochastic actor-oriented models, multilevel blockmodels, and any model capable of handling multilevel networks.     

Effects of missing data in social networks

We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three principal missing data mechanisms: network boundary specification (non-inclusion of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos E-print Archive as well as random bipartite graphs. The simulation results show that network boundary specification and fixed choice designs can dramatically alter estimates of network-level statistics. The observed clustering and assortativity coefficients are overestimated via omission of affiliations or fixed choice thereof, and underestimated via actor non-response, which results in inflated measurement error. We also find that social networks with multiple interaction contexts may have certain interesting properties due to the presence of overlapping cliques. In particular, assortativity by degree does not necessarily improve network robustness to random omission of nodes as predicted by current theory.     

Clearing the FOG: Fuzzy,overlapping groups for social networks

Humans are well known to belong to many associative groups simultaneously, with various levels of affiliation. However, most group detection algorithms for social networks impose a strict partitioning on nodes, forcing entities to belong to a single group. Link analysis research has produced several methods which detect multiple memberships but force equal membership. This paper extends these approaches by introducing the FOG framework, a stochastic model and group detection algorithm for fuzzy, overlapping groups. We apply our algorithm to both link data and network data, where we use a random walk approach to generate rich links from networks. The results demonstrate that not only can fuzzy groups be located, but also that the strength of membership in a group and the fraction of individuals with exclusive membership are highly informative of emerging group dynamics.     

A comparative study of social network models: Network evolution models and nodal attribute models

This paper reviews, classifies and compares recent models for social networks that have mainly been published within the physics-oriented complex networks literature. The models fall into two categories: those in which the addition of new links is dependent on the (typically local) network structure (network evolution models, NEMs), and those in which links are generated based only on nodal attributes (nodal attribute models, NAMs). An exponential random graph model (ERGM) with structural dependencies is included for comparison. We fit models from each of these categories to two empirical acquaintance networks with respect to basic network properties. We compare higher order structures in the resulting networks with those in the data, with the aim of determining which models produce the most realistic network structure with respect to degree distributions, assortativity, clustering spectra, geodesic path distributions, and community structure (subgroups with dense internal connections). We find that the nodal attribute models successfully produce assortative networks and very clear community structure. However, they generate unrealistic clustering spectra and peaked degree distributions that do not match empirical data on large social networks. On the other hand, many of the network evolution models produce degree distributions and clustering spectra that agree more closely with data. They also generate assortative networks and community structure, although often not to the same extent as in the data. The ERGM model, which turned out to be near-degenerate in the parameter region best fitting our data, produces the weakest community structure.     

Longitudinal analysis of personal networks. The case of Argentinean migrants in Spain

This paper discusses and illustrates various approaches for the longitudinal analysis of personal networks (multilevel analysis, regression analysis, and SIENA). We combined the different types of analyses in a study of the changing personal networks of immigrants. Data were obtained from 25 Argentineans in Spain, who were interviewed twice in a 2-year interval. Qualitative interviews were used to estimate the amount of measurement error and to isolate important predictors. Quantitative analyses showed that the persistence of ties was explained by tie strength, network density, and alters’ country of origin and residence. Furthermore, transitivity appeared to be an important tendency, both for acquiring new contacts and for the relationships among alters. At the network level, immigrants’ networks were remarkably stable in composition and structure despite the high turnover. Clustered graphs have been used to illustrate the results. The results are discussed in light of adaptation to the host society.     

The use of sparsest cuts to reveal the hierarchical community structure of social networks

We show that good community structures can be obtained by partitioning a social network in a succession of divisive sparsest cuts. A network flow algorithm based on fundamental principles of graph theory is introduced to identify the sparsest cuts and an underlying hierarchical community structure of the network via maximum concurrent flow. Matula [Matula, David W., 1985. Concurrent flow and concurrent connectivity in graphs. In: Alavi, Y., et al. (Eds.), Graph Theory and its Applications to Algorithms and Computer Science. Wiley, New York, NY, pp. 543–559.] established the maximum concurrent flow problem (MCFP), and papers on divisive vs. agglomerative average-linkage hierarchical clustering [e.g., Matula, David W., 1983. Cluster validity by concurrent chaining. In: Felsenstein, J. (Ed.), Numerical Taxonomy: Proc. of the NATO Adv. Study Inst., vol. 1. Springer-Verlag, New York, pp. 156–166 (Proceedings of NATO ASI Series G); Matula, David W., 1986. Divisive vs. agglomerative average linkage hierarchical clustering. In: Gaul, W., and Schader, M. (Eds.), Classification as a Tool of Research. Elsevier, North-Holland, Amsterdam, pp. 289–301; Thompson, Byron J., 1985. A flow rerouting algorithm for the maximum concurrent flow problem with variable capacities and demands, and its application to cluster analysis. Master Thesis. School of Engineering and Applied Science, Southern Methodist University] provide the basis for partitioning a social network by way of sparsest cuts and/or maximum concurrent flow.