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## Abstract

This research paper reviews key ideas in social network analysis methods. Key concepts for describing the connections of individuals and groups into networks of social relations are introduced and defined. The paper identifies the terminology of points and lines for understanding sociograms and reviews concepts of adjacency, degree, centrality, and forms of cohesion and subrouping. Developments in statistical methodology of hypothesis testing are examined and views of the evolution of social networks over time are considered. It is argued that social network analysis is to be considered as a methodological approach rather than a substantive body of theory, although it is shown that there are close links between social networks analysis and theories of social structure and social capital. It is argued that social network analysis is an essential adjunct to any approach to relational sociology.

## Outline

- Introduction
- Concepts and Techniques of Network Analysis
- Statistical Models for Social Network Analysis
- Evolution and Network Dynamics
- Method or Theory?
- Conclusion
- Bibliography

## Introduction

Social network analysis is the study of the patterns of social relations that comprise social structures, treating these relations as networks of connections among the individuals and groups that enter into them. While these relations may be formed by particular individuals, social network analysis is not limited to microlevel interactions. Individuals form social relations as the occupants of institutionally defined positions in social organizations, and the social relations enter into the constitution of patterns of macrolevel relations that may equally be treated as social networks. A large number of concepts have been developed to characterize, measure, and compare network structures and positions in networks. These include the relative centrality of individuals, groups, and positions within networks; their clustering into subgroups; the overall cohesion or density of a network; and the centralization of networks around focal points. A number of statistical methods have been developed to estimate their values and to assess the significance of these measures for observed outcomes.

There are a number of introductions and overviews of social network analysis. A nontechnical introduction is Scott (2013), and there are a number of introductory surveys (Degenne and Forse, 1994; Knoke and Song, 2008; Prell, 2012; Kadushin, 2012; Scott, 2012). The best advanced text is that of Wasserman and Faust (1994). A comprehensive collection of discussions and applications has been brought together by Carrington and Scott (Carrington and Scott, 2011).

## Concepts and Techniques of Network Analysis

In social network analysis, social structures are represented as patterns of points (or vertices) and lines (or edges). The points represent individuals, organizations, or positions, while the lines represent the relations that connect them. Any type of dyadic relationship can be represented, including communication, friendship choices, advice, trust, influence, and exchange relationships. The lines depicting the relationship can be characterized by various attributes. Relations may first be described as directed or undirected. A line can be said to be directed from one point to another, for example, when one person chooses another as a friend but this choice is not reciprocated. The direction assigned to the line represents the likely flow of information or resources between the two participants. These directed lines are often termed ‘arcs.’ Undirected lines are those that represent reciprocated links or common memberships, as in the case of two people who meet as members of the same group or through participation at the same event. Lines can also be distinguished by signs attached to them to indicate the type of relationship. Positive or negative signs on directed lines, for example, enable the representation of positive and negative relations such as cooperation and conflict. The strength of a relation may be represented by a number, either an actual number (such as the size of a shareholding) or a scaled representation of strength. Such lines are said to be ‘valued.’

This language of points and lines, with directions, signs, and values, provides an intuitive form of representation that can help in the drawing of simple sociograms. The sociogram, largely invented by Jacob Moreno (1934), is a simple visual mapping of the pattern of connections and is most easily used in the study of small groups. Indeed, the area of group dynamics (Cartwright and Zander, 1953) largely developed as a reflection on the consequences of different communication patterns in small groups. An example of this kind of work might be Heider’s development of balance theory to study social influence within groups. Heider represented cognitive balance as a signed graph with three points. An individual under study can have a positive or negative relationship to another person. Both persons can have a positive or negative attitude toward another object (e.g., a third person or a certain activity). The cognitive system of the individual under study is in balance if and only if all three relationships are positive or two of them are negative. Harary et al. (1965) generalized the idea of balance to a whole social system.

Developments beyond simple representations in sociograms have involved the use of matrix algebra and graph theory (see particularly Harary et al., 1965). Major computational possibilities became possible by the representation of a social network in a matrix. The rows and columns represent the points, and the cells the relationships from the row to the column point. In an adjacency matrix, only ones (for a directed line from the row point to the column point) or zeros (no such line) are used. In a signed graph, positive ties are represented by ?1 and negative ones by [1]1. In a valued graph, the values of the ties can be given. Graphs with different types of relationships (denoted multigraphs) can be represented by stacked matrices. Graph theory is a branch of formal mathematics that has been used in the modeling of networks – or ‘graphs’ – in such diverse areas as electrical networks, computer networks, and networks of rivers. The use of graph theory allows the formal properties of large networks to be explored and summarized in numerical form. The basic concepts used to describe a pattern of points and lines in graph theory are adjacency, neighborhood, and distance, and these can be used to describe more complex relational structures (Bonacich and Lu, 2012).

Adjacency refers to the simple fact of connection: two connected points are said to be adjacent to each other. The ‘degree’ of a point is the number of other points to which it is adjacent, and the neighborhood of a point comprises the set of other points to which it is adjacent. When account is taken of the direction assigned to a line, it is possible to distinguish the ‘indegree’ and the ‘outdegree’ of a point and so to differentiate its neighborhood. Points in a network are connected by ‘paths’ – sequences of lines – of varying length, the number of lines in a path. The length of the shortest path indicates the distance between two points. Adjacent points are connected by a path comprising a single line and are said to be connected at a distance of one. Indirect links through, for example, a common friend are at a distance of two, and so on. The famous research of Stanley Milgram (1967) showed the significance of six degrees of separation: the fact that randomly chosen people, even if located in different continents, are typically connected to each other through intermediaries by a path of maximally length six.

These basic concepts allow the construction of measures of the relative centrality of points within a network. Freeman (1978) associated the many centrality measures with three important centrality dimensions in communication networks. These are local centrality, global centrality, and betweenness. Local centrality is measured by the degree, the number of points with which a point is directly connected, which is assumed to indicate the communication activity of a point. It is a measure of how well connected a point is in its local environment. In directed networks, centrality in terms of outdegree and indegree should usually be distinguished. In friendship choice networks, for example, the number of choices received (indegree) generally indicates centrality, which may be seen as a measure of likability. In influence networks, local centrality may be based on the number of outgoing relationships (outdegree). When the overall network is considered, rather than its local structure, one is dealing with global centrality. This involves the use of distance-based measures to indicate the relative proximity of points to other points in the network and the extent to which a point can communicate with other points independently of others. In the literature, various statistical measures on the distances of a point to other points are used to measure this dimension of centrality, such as the mean distance to other points or the maximal distance to any of the other points. Betweenness, or rush, is the third type of centrality and measures how important a point is for the transmission of information between other points. Betweenness measures assume that information is mainly transmitted through the shortest paths. A point with a high betweenness might be regarded as a key ‘broker’ or intermediary in a network.

The measures considered so far can all be considered to be attributes of particular points within a network, based on their relations with others. They are often referred to as measures for ‘egonets.’ However, it is also possible to consider the global structure of the network itself and to characterize its overall pattern. The most important are density, centralization, and clustering. Density is the number of distance-one relationships that actually exist relative to the total number that might possibly exist. It measures how ‘complete’ a web of connections may be. Dense networks are more important for control and sanctioning than for information. They also tend to generate a lot of redundant information and so result in many constraints and are inefficient for creative new solutions (Burt, 1992). Comparing densities of networks of quite different sizes is difficult, as large networks tend to be sparse and the measure of density varies directly with the typical neighborhood size (which increases much more slowly than the size of the whole network). The centralization of a network is a measure of the extent to which its connections coalesce around a small subset of globally central points. In connected networks, high centralization corresponds with a high variance of the degrees of the points. Snijders (1981) derived the maximal possible values of the variance, given the number of points or the number of points and lines. He also derived the expected variance for different null models under the same two conditions. This makes a good comparison of centralization possible in networks of different sizes and densities. The third global measure is the clustering or segmentation of the network. A simple summary measure of segmentation is the number of pairs of points at distance two or higher divided by the number of pairs of points at distance three or higher (Baerveldt and Snijders, 1994). More structural measures, however, employ forms of cluster analysis and subdivision to identify clusters, cliques, and components within a network. Such measures of the social circles within which individuals and groups interact provide a picture of the overall shape or landscape of a network. Structurally equivalent points do not have the same connections, but they do have the same types of connections. This kind of analysis has been seen as identifying typical roles or positions within networks.

A large number of other network studies examine the effects of the network structure on the behavior and attributes of the network members. In these studies, the network is considered as given and constant, and the ways in which this network influences processes or individuals in the network is examined. In many of these studies, only relationships of the individuals under investigation are collected (at most combined with their perception of the relationships among their network members). These ego-centered network studies examine the effects of differences in size and composition of the personal networks and the multiplexity of an individual’s personal relations (Wellman and Berkowitz, 1997). Examples of effects studied are an individual’s social well-being, social support, health, labor market position, and career.

Many of the central properties of social networks studied by graph theory have been shown to depend on what have been called small-world conditions (Watts, 1999, 2003). That is, they apply when the pattern of connections is such that Milgram’s suggestion of six degrees of separation holds. Fundamentally different processes will be in operation in networks where the number of connections is so low that this does not happen or where the connections are so high that almost all individuals are closely connected. Thus, a network that changes incrementally may reach a point at which the pattern of connections changes catastrophically and with radical consequences for the members of the network.

The bulk of the measures in social network analysis are based on so-called one-mode data. That is, when individuals are related to each other by common organizational membership, analysis has focused on either the individuals or the organizations. Recent developments have undertaken systematic simultaneous analyses of both individuals and organizations, representing them all as points within the same network. Measures of centrality, for example, are difficult to conceptualize and understand for two-mode data, but much work is progressing to clarify this. An early usage of two-mode data was block modeling, in which both sets of points were simultaneously partitioned into subsets that could be treated as ‘equivalent’ to each other (Everett and Borgatti, 1994).

An important area of development in social network analysis has been the use of multidimensional scaling, principal components analysis, and similar techniques to present graphical visualizations of social networks. These techniques allow a more rigorous extension of the principle of the sociogram and embed images of social networks in a multidimensional space in which distances and directions are depicted as they would be in a geographical map. Thus, distance in this social space differs from the path distance of graph theory as it takes account of the relative closeness of all points within the network. These developments have led to an impressive range of visualization techniques, many of which are accessible through the Pajek software (de Nooy et al., 2005). General network concepts can be usefully explored through the UCINET software developed by Freeman, Borgatti, and Everett (http://www.analytictech.com/).

## Statistical Models for Social Network Analysis

Much research in social network analysis has been descriptive and has prompted questions about the significance of the structures described for actual behavioral outcomes. Recent developments have produced a range of statistical techniques that can allow the use of social network analysis in explanatory studies by assessing the statistical significance of the results.

Standard statistical models are more difficult to apply to social networks because the relationships between the points cannot be treated as independent observations. The most useful alternative to have been developed involves exponential random graph models in which observed patterns are compared with those found in large numbers of random computer simulations given certain characteristics of the overall network to assess the probability that the observations are the result of chance (Wasserman and Pattison, 1996; Snijders et al., 2006; Robins et al., 2007).

## Evolution and Network Dynamics

Along with the move from descriptive to explanatory concerns, there has been a move from static to dynamic analyses. Existing concepts have been largely concerned with the static, crosssectional features of social networks and fail to grasp the complexities of social change. Recent work has taken more seriously the fact that social networks change over time and that these patterns and processes of change must be explained. Like all social structures, social networks are produced and reproduced through individual and collective action, but their structures are generally the unanticipated and unintended consequences of these actions. Agent-based computational models of action (Axelrod, 1997) have been particularly useful in understanding this process and in allowing connections to be made with Markov models of stochastic change.

Holland and Leinhardt (1977) introduced Markov processes as the general framework for stochastic models of network evolution. The basic idea of Markov models is to conceive the social network structure as changing from one state into another over time. The parameters that govern the process concern the likelihood of transition from one of these four states into another. The original Markov models assume that the parameters are stationary over the whole process and that the population is homogeneous. Recent models have considerably increased the analytic possibilities of Markov models by eliminating these strongly limiting assumptions and recognizing that change parameters may well be dependent on the stage of network development and will be different for pairs within and between subgroups (Leenders, 1996). Snijders powerfully integrated Markov models with random utility models to allow a much stronger link between theory and statistical testing (Snijders, 1996), where actors are seen as making and breaking links on the basis of local knowledge and decision rules that generate unplanned but structured long-term changes. Many of the general issues in this area are considered in Doreian and Stokman (1997).

One of the most important theoretical topics in social network analysis concerns the relative importance of selection and influence processes: do, for example, friends seek and select friends with similar characteristics or become friends more similar because of their repeated interactions? As both networks and characteristics like attitudes change over time, the relative importance of the two processes is one of the key questions social networks researchers seek to answer. Snijders’ dynamic approach to network analysis (Snijders, 2011) makes it possible to estimate the relative importance of the two processes in designs where we have data at three or more time points.

## Method or Theory?

Social network analysis is primarily to be understood as a method of analysis and not as a substantive theoretical approach. It is, however, more compatible with certain theoretical approaches and can be considered as an essential adjunct of any relational sociology (Emirbayer, 1997). Early arguments emphasized the links between social network analysis and exchange theory.

The fact that social networks create conditions for cooperation through information and sanctioning is due to exchange processes that create win–win situations (Homans, 1950; Blau, 1964). A fruitful and very promising approach is the study of the effects of social networks in noncooperative game theory (Raub and Weesie, 1990; Flache and Macy, 1996; Bienenstock and Bonacich, 1992). Network exchange theory specifically investigates the effects of network structures on the choice between alternative exchanges and on exchange rates (Willer, 1999). Major effects are particularly due to possibilities for social actors to exclude others. Exchange network theory illustrates again that effects of network structures are context sensitive and cannot be generalized without taking the context and substance into account. The integration of exchange theory and social networks has also been proved to be very successful in the field of policy networks. Most of these models build on Coleman’s social exchange model and confine exchanges to influence network relationships (Laumann et al., 1987). These models make the step from microbehavior to macroeffects explicit and are able to predict outcomes of decisions, to derive the power of social actors and the value of decisions. Later models try to solve a number of remaining theoretical issues (Stokman and Van den Bos, 1992; Pappi and Henning, 1998). Stokman et al. (2013) give a more general theory of bargaining in which three fundamental bargaining processes (persuasion, logrolling, and enforcement) are related to three types of social networks (information, exchange, and hierarchical power networks). They argue that in any bargaining situation all three are present, but only one is likely to be dominant.

Social network analysis has recently, however, been shown to be especially fruitful for discussions of social capital. Social capital gives individuals access to resources of others that can be exploited for the realization of their goals and so constitutes an opportunity structure generated by social relationships (Lin, 1982; Coleman, 1990; Burt, 1992). The amount of social capital depends on the amount of these resources, their value for the goal realization of the individual, and the willingness of others to mobilize them (Flap, 1999). The value of the resources for the individual strongly depends on functional interdependence, the willingness of others to mobilize resources for the individual on their perception of the interdependence (i.e., the cognitive dependence). Social capital emphasizes the relationship between social capital and success of actors, whereas Lindenberg and Foss (2011) change the perspective to success of group production and derive the conditions for joint production motivation. Important ideas developed by Putnam (2000) on bridging and bonding capital can be directly modeled in network terms. Large and dense networks create shared information, high visibility, and common norms in a community. Burt (1992), on the other hand, stresses the importance of unique and nonoverlapping relationships for acquiring unique information in organizations, giving individuals a better chance to find creative solutions for problems and thus providing them better opportunities for career. Vedres and Stark (2010) disputed Burt’s emphasis on broker positions for performance and introduced the concept of structural folds, where success is related to overlapping memberships in different groups rather than isolated positions between groups.

Functional and cognitive interdependencies differ particularly between what Granovetter (1973) has termed strong and weak ties. Strong ties are valued in themselves. The ties are not primarily instrumental for the attainment of other goals. Their value is based on the other individual as a person and the quality of the relationship with that individual. Family and friendship ties are typical examples of such relationships. Strong ties tend to be reciprocal, transitive, and clustered. Strong ties give a sense of belonging to a group and the group often has priority above the individual and individual relationships. Sharing is often based on need and norms tend to promote equality (Lindenberg, 1998). Creating negative attitudes toward other groups often helps to strengthen the predominance of the group, which may give strong negative externalities for society as a whole (think of gangs and other criminal organizations). Weak ties are valuable long-term relationships but their value is primarily instrumental, related to higher ordered goals, goals not primarily located in the relationship or individuals themselves. Weak ties tend to be less clustered; the group is less dominant and often only vaguely delineated. Reciprocity based on equity norms prevails.

## Conclusion

Since the early 1970s, there has been impressive cumulative progress in social network analysis and research. From a rather isolated field, strongly oriented to descriptive structural and static analysis, social network analysis has grown into a wellembedded field, widely accepted as highly important for solving central theoretical problems of cooperation and coordination. With the growing importance of social networks in the information society with virtual communities developing in many segments of society, its importance and contributions to theoretical solutions can only grow. Issues of scope are challenging. Objects of study will vary from small group networks to social networks of billions of points. New techniques to visualize networks are also challenging. These allow certain structural characteristics to become visible and the effects of changes on these characteristics to become transparent. These developments can be followed by linking with the international virtual community of the International Network for Social Network Analysis and the links to be found there (http://www.insna.org/).

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