Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Social Network Visualization, Methods of

  • Jürgen PfefferEmail author
  • Linton C. Freeman
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_496-2
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Glossary

Adjacent

Two nodes are adjacent to another if there is an edge connecting them.

Arrow

A line with an arrowhead from one node to another representing a directed link.

Binary relation

A two valued yes/no or on/off relation.

Bipartite graph

A graph, B = 〈N, E〉, where N is a finite set of nodes and E is a collection of pairs of nodes in which N is partitioned into two disjoint subsets, N1 and N2, and no edge in E has both end points in the same subset.

Blockmodeling

A procedure for clustering actors such that the actors in each cluster share similar patterns of ties both within and between clusters.

Connected

Any two nodes in a graph are said to be connected if there is a path from one to the other; a graph is connected if there is a path connecting every pair of nodes.

Cycle

Any path that begins and ends at the same node.

Digraph

A directed graph.

Directed graph

A graph D = 〈N, A〉 where N is a finite collection of nodes and A is a set of pairs linked by directed lines or arrows.

Directed line

A line going from a node to another representing a nonreciprocated link.

Edge

A line connecting two nodes representing a reciprocated link.

Edge-labeled graph

A graph in which at least two kinds of connections between nodes are identified.

Formal concept analysis

A method of data analysis based on Galois lattice structure.

Galois lattice

A dual structure that displays the dependencies of both objects and their properties.

Geodesic

The shortest path between two nodes.

Graph

A graph G = 〈N, E〉 where N is a finite set of nodes and E is a collection of pairs of nodes represented as edges.

Hyperedge

An edge in a hypergraph that can enclose more than two nodes.

Hypergraph

A hypergraph, F = 〈N, H〉, consists of a set of nodes N and a collection of hyperedges, H.

Indegree

The indegree of a node is the number of directed lines it receives.

Irreflexive

A relation in which no edge connects any node with itself.

Multidimensional scaling

A search procedure designed to represent an observed set of proximities or distances in a small number of dimensions.

Node

A point in a graph.

One-mode matrix

A data matrix in which the rows and columns both represent the same objects.

Outdegree

The outdegree of a node is the number of directed lines it sends out.

Path

A path is a sequence of nodes and edges beginning with a node that has an edge connecting it to the next node in the sequence and so on.

Path length

The length of a path connecting two nodes is the number of edges it contains.

Permutation

A reordering of the rows, columns, or rows and columns of a matrix.

Principal diagonal

The set of cells in a square matrix that runs from the upper left to the lower right.

Relation

A collection of ordered or unordered pairs of nodes.

Singular value decomposition

An algebraic procedure that decomposes a data matrix into its “basic structure.”

Sociometry

An early version of social network analysis introduced by Jacob Moreno and Helen Jennings.

Spring embedder

A kind of multidimensional scaling based on a model in which it is assumed that nodes are connected by springs that pull and push on them.

Symmetric

A relation in which if a node a is adjacent to another, b, then b is adjacent to a.

Tree

A graph is a tree if it is connected and contains no cycles.

Two-mode matrix

A data matrix in which the rows and columns represent different objects.

Definition of the Subject

Social network visualization refers to the practice of constructing pictorial images of the connections linking social actors. The use of such images provides two benefits. It allows investigators to gain new insights into the patterning of social connections, and it helps investigators to communicate their results to others. Here, readers can find different aspects of network visualization discussed in the context of mostly historic network visualizations. All visualizations here are re-created by the authors (Pfeffer and Freeman 2015), except for Figs. 23, 25, and 27, which are reprinted with permission by their authors.

Introduction

Social network analysis did not emerge as a systematic field of research until early in the twentieth century (Freeman 2004). But visual images of social networks were produced more than a millennium earlier. In this text, we discuss the historic development of different aspects of network visualization. The earliest of these images that we have uncovered were produced in Spain in the middle of the ninth century. They are attributed to the prolific writer and Roman Catholic Saint, Isidore de Séville. His images display relationships based on genealogical descent. From the earliest times, people have been interested in kinship ties – in who is related to whom. This interest is evident in the descent lists found in the Christian bible and in the oral genealogies that were required to be memorized by Hawaiian nobles (Schweitzer 1998).

The fact that Isidore de Séville’s pictures take the form of trees shows that as early as the ninth century, people saw the analogy between the branching structure of descent and that of trees. This notion was captured in a mathematical formalization in 1857 by Arthur Cayley (1857). Cayley defined a tree in mathematical graph theoretic terms. Biggs et al. (1977, p. 38) characterized Cayley’s definition by saying that his “… use of the word ‘tree’ in this context is presumably derived from the diagrammatic form of these graphs, and is akin to the traditional use of the word in describing genealogical or family trees.”

The use of trees to depict descent was, of course, continued. As time passed, however, their form became simplified. Lewis Henry Morgan (1871/1997) was an attorney and an anthropologist. He was interested in comparing how different peoples reckoned kinship, and in 1871 he published a mammoth work containing a collection of kinship trees. Each tree depicted descent as conceived by a society somewhere in the world. Morgan’s trees are quite simple. Figure 1 shows descent as it was reckoned in ancient Rome.
Fig. 1

Descent in ancient Rome

Twelve years later, a mathematician-physicist, Alexander Macfarlane (1883a), produced a different kind of graphic image based still on kinship. Macfarlane set out to examine British marriage prohibitions, and he represented them both algebraically and visually. His visual images depict males using plus signs (+) and females with circles (o). Earlier generations he placed higher on the page. Descent is shown by lines connecting points. A short line crossing a descent line indicates another person, of either sex, in an intermediate generation. And the lowest point is always the prohibited offspring.

The illustration shown in Fig. 2 displays all the two-step marriage relations that are prohibited by British law. The left image shows that a male may not marry his granddaughter. The middle image shows that he may not marry his sister. And the right image shows that he may not marry his grandmother. Or, put the other way, a woman may not marry her grandfather, her brother, or her grandson.
Fig. 2

Macfarlane’s images of two-step marriage prohibitions

Macfarlane’s paper also included algebraic expressions that captured all of the same marriage prohibitions. But Sir Francis Galton, who attended Macfarlane’s presentation, declared that his “diagrammatic form” seemed “the most distinctive and self-explanatory” of the two treatments (Macfarlane 1883b).

Finally, in 1894, John Hobson produced a visual image of a social network that was not based on kinship. He had collected two-mode (corporation by director) data on interlocking corporate directorates (Hobson 1894). He reasoned that, to the degree that corporations shared directors, they could be expected to cooperate and work together.

Hobson’s illustration was designed to show the interlock among, as he put it, “the small inner ring of South African finance.” Corporations are depicted as circles, and interlock is shown by overlapping or by a line connecting two circles. Hobson’s image is reproduced here as Fig. 3.
Fig. 3

Hobson’s image of corporate interlocks

The important feature of this image is that it displays a connection linking more than two corporations. Hobson’s data showed that three corporations, Charter, Rand, and De Beers, all shared directors in common. And, at the same time, Rand and De Beers also both shared directors with coal mines, telegraphs, rails, and others. The overlaps in his image allowed him to display which companies shared with which others.

It is clear, then, that a concern with connections among social actors and the use of visual images have a long history of intimate association. It should come as no surprise therefore that images played an important part in the development of social network analysis when it did emerge as an organized field of research.

Visualization in Social Network Analysis

In the book cited above (Freeman 2004), the modern science of social network analysis is described as possessing four defining properties. They were:
  1. 1.

    It embodies ideas about the importance of social ties linking social actors.

     
  2. 2.

    It collects data reflecting those ties.

     
  3. 3.

    It involves the use of graphic imagery.

     
  4. 4.

    It employs mathematical and/or computational models.

     

Pre-network research often included one or two of those properties, but in the late 1920s, each of two independent research teams came up with efforts that included all four.

One took place in the early 1930s. It involved a psychiatrist, Jacob L. Moreno, and a psychologist, Helen H. Jennings. Together, they developed an approach they called “sociometry.” They reported two huge studies, both focused on examining the structure of social ties. One was conducted among prisoners at Sing Sing Correctional Facility in Ossining, New York (Moreno 1932), and the other among young delinquents at the New York State Training School for Girls in Hudson, New York (Moreno 1934).

Both Moreno-Jennings studies involved the extensive use of graphic images. The image shown in Fig. 4 was included in their report on the research at Sing Sing Correctional Facility (Moreno 1932). In that figure, individuals or other kinds of social actors are represented as points or nodes and links between pairs of social actors are lines or edges connecting pairs of nodes. In Fig. 4 Moreno was concerned with the positions of individuals and the patterning of their ties. As he put it, the individuals at the top and the bottom were “dominant,” and the image showed that those dominant individuals were linked both “directly” and “indirectly.”
Fig. 4

Image of a pattern of linkages

Most of the data collected by Moreno and Jennings involved asking individuals whom they liked or disliked. In data of that sort, choices are seldom reciprocated. So Moreno and Jennings drew lines with arrowheads to reveal who chose whom. Mutual choices were drawn without arrowheads, and they also included a small line bisecting the main line connecting the two nodes.

Moreno and Jennings often required subjects to report both their likes and their dislikes. By using different colors, red for likes and black for dislikes, a single image could display both. The image shown in Fig. 5 was published in the Moreno-Jennings report on The Hudson School (Moreno 1934). It depicts positive and negative choices among 13 members of an American football team. Moreover, it contains another innovation. The various team members are placed in the drawing in approximately the same relative locations that they occupied on the football field. That arrangement shows the players’ positions and permits the viewer to evaluate the impact of physical proximity on the patterning of social linkages.
Fig. 5

Positive and negative choices in a football team

Figures of the sort used by Moreno and Jennings had a major impact on the style of graphic imagery used subsequently in social network analysis. For the most part, social network analysts have represented social actors as nodes and links between actors as edges or as directed lines with arrowheads.

The second introduction of the social network approach also occurred in the early 1930s. An anthropologist, W. Lloyd Warner, and a collection of his colleagues and students at Harvard, conducted three elaborate network analytic projects. One was a study of an industrial factory, the Western Electric plant in Cicero, Illinois (Roethlisberger and Dickson 1939). The other two were studies of communities: one focused on a New England town, Newburyport, Massachusetts (Warner and Lunt 1941), and the other on a southern town, Natchez, Mississippi (Davis et al. 1941).

The image shown in Fig. 6 was produced as part of the factory study. It displays observed friendship ties among pairs of individuals who worked together in the same workroom. It was drawn using nodes and two-headed lines instead of edges, but it is very similar to the images produced by Moreno and Jennings. In addition, as in Fig. 5, the impact of physical space was displayed; workers were placed in the drawing in positions that reflected the locations of their workstations.
Fig. 6

Friendships linking factory workers

In reporting their study of Newburyport, Warner and Lunt used the kind of drawing of overlapping circles that Hobson had used to construct Fig. 3. But here that image was introduced, not to describe data but to propose an idea they had about social structure. The diagram in Fig. 7 represents the investigators’ idealized version of the expected structure of overlaps among subgroups in the presence of social class. The idea is that only subgroups that are close to one another in class ranking are likely to have overlapping memberships.
Fig. 7

An idealized pattern of overlapping “cliques”

In their study of Natchez, Davis et al. (1941) employed the same diagrammatic form to display two-mode data reflecting on the earlier Newburyport hypothesis. Figure 8 shows subgroups of black males and their overlaps. In that image, the men are arranged in terms of both social class and age. Both, it turned out, provided important bases for grouping.
Fig. 8

Stratification, age, and overlapping groups

Finally, in that same report, Davis, Gardner, and Gardner also introduced an entirely different kind of social network image. Like Hobson, they had collected two-mode network data. Eighteen women were designated in the rows of their data matrix, and fourteen social events were depicted in the columns. That matrix is reproduced here as Fig. 9.
Fig. 9

The Davis, Gardner, and Gardner data on women’s attendance at social events

The data shown in Fig. 9 were all collected during a single year. But, by examining the column headings, it is clear that Davis and his colleagues did not arrange the social events according to the dates upon which they took place. Instead, they listed both the events and the women who attended them in such a way that the arrangement itself suggests that these women were organized into two groups. The two groups overlap, but for the most part, they are distinct. Most of the women in the top half of the matrix attended the leftmost five events. And most of the women in the bottom half attended the rightmost five events. The middle four events apparently brought both groups of women together.

This arrangement of women and events was self-consciously produced by the authors. Davis, Gardner, and Gardner were convinced that these women were organized into two groups, and they presented their data matrix in a way that would illustrate that conclusion. The interesting thing is that these authors never commented explicitly about how they had rearranged the columns and rows in their matrix. They simply organized their display in a way that would make the point.

From the outset, then, four kinds of images have played important parts in the development of social network analysis. These first network graphics included drawings displaying (1) one-mode undirected relations, (2) one-mode directed relations, (3) two-mode relations, and (4) one- or two-mode data matrices. A few other kinds of network images have been used since then, but the four originals – particularly those based on one-mode undirected and one-mode directed relations – still dominate the field. In the next four sections, we will examine the four original kinds of images and how their use has evolved in the social network context.

Images Based on One-Mode Undirected Relations

J. Clyde Mitchell collected data on the social ties among the 19 individuals involved in the personal network of a homeless woman in Britain (Mitchell 1994). The top left image of Fig. 10 is a graph representing these individuals and ties. A program called NetDraw (Borgatti 2002) was used to place the nodes representing individuals in this figure in random positions. That calls attention to the importance of the locations of points in graphic displays. Given the locations of the points in this image, it is very difficult for the viewer to see anything interesting in the patterning of this woman’s network.
Fig. 10

Links in the network of a homeless woman; top left, random node positions; top right, spring embedder layout; bottom left, layout based on singular value decomposition; bottom right, spring embedder layout with link weights

Compare the top left image in Fig. 10 with that in the top right of the same figure. This second figure was also produced using NetDraw, but this time, the points were placed using a spring embedder (Eades 1984) layout. A spring embedder is a computer algorithm that, in effect, places a spring of unit length between every pair of adjacent nodes and a much longer spring between nodes that are not adjacent. It starts with a random placement of nodes, and then the whole apparatus is set in motion, and the various springs push and pull until they reach an equilibrium.

The advantage of using a spring embedder is that it does not require the investigator to make ad hoc judgments in locating nodes in a graph. It uses a standard computer algorithm to place the nodes automatically. There are several different spring embedding algorithms. And they are all examples of a more general class of computer algorithms that search for optimal locations for nodes in relatively few dimensions. This general class of search algorithms is called multidimensional scaling (Krempel 1999).

An alternative method for placing nodes automatically is grounded in algebra. It is called singular value decomposition (Weller and Romney 1990). Singular value decomposition is not search based. Instead, it uses matrix operations to produce a linear transformation of the data and thus to position the nodes in one, two, three, or more dimensions. There is no guarantee that it will always be effective, but often singular value decomposition provides very good placements of the nodes in few enough dimensions that visualization is possible (Freeman 2005). A NetDraw image based on singular value decomposition of Mitchell’s data is shown in the lower left image in Fig. 10.

Mitchell’s report, however, included even more details. It included estimates of the strength of the tie linking each of the pairs of individuals. He classified each tie as either strong or weak. We can embody this additional information in our NetDraw image by adding another component to our graph. The final image in Fig. 10, then, was produced using the spring embedder, and, in addition, it is an edge-labeled graph.

These last three images all show that the whole network is organized into three densely connected groups that are only loosely linked to one another. That is interesting, but it does not tell us anything about the bases for the groupings. By adding a little information, and continuing to use NetDraw, we can transform the graphs of Fig. 10 into a node-labeled graph (see Fig. 11). In this figure, the edges are drawn in gray to increase the readability of the labels.
Fig. 11

Node and edge-labeled graph visualization of a homeless woman’s network

Given the labels, we can identify the homeless woman, the “respondent.” We can also see how her network is split up. One division includes her original family, another her friends along with her social worker, and the third contains her estranged husband and his family, her in-laws.

In Fig. 11 strong ties are indicated by wide edges. By examining their patterning, we learn that the individuals within each family are linked together mostly by strong ties, while the homeless woman’s friends have fewer strong ties linking them together. This result is not surprising, but it does provide additional insight about the structural position of the woman in question. Clearly, it would be easier for either family to achieve consensus and provide support than it would be for the respondent’s loosely connected collection of friends (Bott 1957).

It should be clear, then, that the placement of nodes and the labeling of both nodes and edges are critical for the ability of a graph to communicate important information. Good images can provide investigators with new insights about the structural properties of the social networks they are studying. And they can, of course, help to communicate the results of social network research to outsiders.

Images Based on One-Mode Directed Relations

It was obvious from the outset that these simple graphs would not permit many kinds of displays of interest to social network analysts. Even Moreno and Jennings saw the need to display the direction of choice in their sociograms. The direction of connections can be expressed using directed graphs or digraphs.

A digraph D = 〈N, A〉 where N is a finite collection of nodes and A is a set of pairs shown as directed lines or arrows. When an arrow is directed from node a to node b in a digraph, then a is the tail of the arrow and b is the head; a is the immediate predecessor of b and b is the immediate successor of a. The outdegree of a node is the number of arrows for which it is the tail and its indegree is its number for which it is the head.

In any study that involves social links that are not symmetric, digraphs provide a natural representation. Consider Fig. 12 that was produced by a program called Visone (Baur et al. 2001). In preparing a book on the development of social network analysis, interviews with a number of the founders were conducted (Freeman 2004). Each was asked to name others who had influenced them to think in network terms. The result is a data set that obviously lacks symmetry.
Fig. 12

Influences on some founders of social network analysis

The interest, however, was with clusters, or blocs, of influentials and nominees. So nodes were placed using a spring embedder designed by Kamada and Kawai (1989). The resulting figure shows that there seem to be two fairly well-defined subgroups, one on the left and one on the right. The two groups are relatively dense but they are only loosely connected together. The people on the left are almost entirely sociologists and those on the right are mostly from other fields. And from the patterning, one can suspect that there was some kind of split between these two groups.

For some kinds of data, the search for clusters or groups is not appropriate. For example, when we are dealing with data that should embody some sort of ordering, digraph representations are particularly important. To illustrate how digraphs can be used to display ordering, consider the data collected by Forkman and Haskell (2004). They studied several communities, each made up of six domestic hens. In five of these communities, the hens formed strict pecking orders in which the top hen pecked all the others; the second pecked all but the top and so on. Figure 13 shows a visone image of the data from one of those five communities. There the nodes are arranged, top down, in terms of their outdegrees, and the pecking order is obvious.
Fig. 13

Dominance among six hens

Often data approach, but do not achieve, a strict order. David Krackhardt (1996), for example, collected data on who sought advice from among 14 employees in the internal auditing staff of a large company. Krackhardt’s data could not be drawn with all the arrows pointing in one direction. So, in Fig. 14, he arranged the individuals in such a way that as many arrows as possible were pointing up. The viewer, then, can immediately see there is an important hierarchical element displayed by these data. From the image, it appears that Nancy is at the top of the advice chain and Bob, Wynn, Carol, Harold, and Susan are at the bottom.
Fig. 14

Visone image of advice seeking. (From Brandes et al. 2001)

There is, however, an important limitation in this figure. Nancy seeks advice from Donna, Donna seeks advice from Manuel, and Manuel seeks advice from Nancy. Thus, these three form a directed cycle of advice seeking. Given such a circular arrangement, no possible hierarchy among these three individuals can be established. Any order in which they were arranged would be misleading. In addition, Stuart and Charles cannot be ordered because they chose each other. The same is true for Kathy and Tanya.

The apparent ordering of nodes in Krackhardt’s image was imposed by human judgment. There are computer algorithms that can automatically arrange the nodes into a hierarchical form (Brandes et al. 2001).

Images Based on Two-Mode Relations

Any time we deal with a relation that can link more than two social actors, we cannot use graphs or directed graphs. Both graphs and directed graphs can deal only with links between pairs. Two-mode data, however, allow for relations that link three or more actors. So, whenever we have two-mode data, like that collected by Hobson (1894) or Davis et al. (1941), we need another way to construct images.

There are several ways to construct images of two-mode data. We will consider three of them in the present section, hypergraphs, bipartite graphs, and lattices. Then, in the next section, we will discuss the use of matrix representations for both one-mode and two-mode data.

Hobson (1894) collected two-mode data on corporations and their directors. He produced the image shown in Fig. 3 showing corporate interlocks as overlapping areas. Mathematically, images like Hobson’s are hypergraphs. A hypergraph, F = 〈N, H〉, consists of a set of nodes, N, and a collection of hyperedges, H. While an edge in an ordinary graph connects two nodes, a hyperedge in a hypergraph may link any arbitrary subset of the nodes in N. Pictorially, hyperedges are represented as boundaries enclosing sets of nodes.

The use of hypergraphs was demonstrated in a report by Estrada and Rodríguez-Velázquez (2005). They began with one-mode data that showed the patterning of predation among the members of 11 species in a Malaysian rain forest. Their graph, showing who preys on whom, is shown in Fig. 15.
Fig. 15

Who preys on whom in a Malaysian rain forest

Figure 15 shows which species preys on which other species. But if the investigator is interested, as those who study food webs often are, in defining ecological niches in terms of co-predation, Fig. 15 makes the overall pattern less than obvious. As an alternative, we can build a hypergraph.

The matrix shown in Fig. 16 is based on the data in Fig. 15. It was built by considering each of the species in turn as prey. Then all of the species that share each given prey are pooled together. Species 1, 6, and 9 have no prey in the set. And species 4, 1, and 9 are the targets of co-predation. So the new matrix is two modes. It has the three targets of co-predation as columns and the eight predators as rows.
Fig. 16

Two-mode matrix of co-predation

That matrix is captured visually by the hypergraph in Fig. 17. It immediately reveals that there are three niches. The one labeled E1 includes all the species who preyed on species 4, E2 those who preyed on species 1, and E3 those who preyed on 9. Thus, each edge in Fig. 17 encloses a collection of species that compete directly for at least one prey.
Fig. 17

Hypergraph of co-predation

There are, however, other ways to picture two-mode data. In a study of corporate interlocks, Joel Levine (1979) reported data on the board memberships of seven major American corporations. Those corporations turned out to have ten directors who appeared on two or more of their boards. Levine presented his interlock data using a bipartite graph. A bipartite graph B = 〈N, E〉 is a graph where N is partitioned into two disjoint subsets, N1 and N2, and no edge in E has both end points in the same subset. He used singular value decomposition to place the nodes representing both corporations and board members and produced a bipartite image similar to the one displayed in Fig. 18. We prepared that figure using Pajek (Batagelj and Mrvar 1998) and a spring embedder layout algorithm. There, the corporations are shown as green squares, and the board members are yellow circles. Thus, both the colors and the shapes of the nodes stress the bipartite nature of the graph.
Fig. 18

A Pajek image of Levine’s interlock data as a bipartite

There is still another form of graphic display, one that reveals even more structural information about a two-mode data set. It is based on an algebraic procedure called Galois lattice analysis or formal concept analysis (Wille 1982, 1984; Duquenne 1987; Freeman and White 1993). A Galois or formal concept lattice is defined on an object by property matrix. Let O be a set of objects and A be a set of attributes. The binary matrix O × A indicates which objects possess which attributes.

We can define a pair 〈Oi, Ai〉 such that Oi is a subset of O and Ai is a subset of A and every object in Oi has every attribute in Ai. Moreover, both O and A must be maximal. Thus, for every attribute in A that is not in Ai, there is an object in Oi that does not have that attribute. And for every object in O that is not in Oi, there is an attribute in Ai that the object lacks.

These pairs are dual and they can be partially ordered by inclusion. Given two pairs 〈Oi, Ai〉 and 〈Oj, Aj〉, we say that 〈Oi, Ai〉 is less than 〈Oj, Aj〉 when Oi is a subset of Oj or, equivalently, when Aj is a subset of Ai. Since all these pairs have unique least upper bounds and greatest lower bounds, they form a dual (Galois) lattice.

We will illustrate by considering again the woman by event data collected by Davis et al. (1941). Let the women (1 through 18) be the objects and the events (A through N) be the attributes. The data, originally arranged into a Galois lattice by a program called GLAD (Duquenne 1999), are shown in Fig. 19.
Fig. 19

The Davis, Gardner, and Gardner data as a Galois lattice

The lattice displays the same three classes of events that define the same two groups of women that we saw in Fig. 9. But, in addition to the classes of events and groups of women, we can now see the containment structures of both events and women. To begin with, by following lines up from the bottom, we can see which women attended which events. When we get to the top, we hit the set of all events, and at the same time, because no woman attended all 14 events, it is also the null set of women.

The uppermost events (E, F, G, H, I, K, L) involved the largest sets of women. Other events are contained in the lower intersections of these events. Event C, for example, is contained in E; everyone who attended C was present at E. And, at the next lower level, B and D are both contained in C. The events, then, can be seen as varying in their “openness.”

At the same time, the figure shows the upward containment structure of the women in terms of their patterns of attendance. Because no event attracted all 18 women, the lowest point represents the set of all women as well as the null set of events. Then, the lowest set of women (1, 2, 3, 4, 13, 14, and 15) are the “core” attendees, so to speak. The next level contains woman 9 who never attended unless woman 3 was also present and woman 5 whose attendance depended on that of women 4 and 3. Women 6, 7, 8, 10, 11, 12, 17, and 18 are also at this second level. In some sense, these are all secondary or peripheral participants in these events. And, finally, woman 16 turns out to be a third-level participant; she was extremely peripheral. Woman 16 attended events only when secondary attendees 8–12 and core attendees 1, 3, and 13 were all present. All in all, then, the image of the Galois lattice reveals a great deal about the internal structure of attendance.

In this section, we have shown three ways of visualizing two-mode data. All three of them, however, share one important limitation. That limitation stems from the fact that all three of them can only be used for very small data sets. As the number of cases grows, they all produce images that become increasingly difficult to read.

Images Based on One- or Two-Mode Data Matrices

When Davis et al. (1941) first used matrix permutation, they did so without calling attention to the process. But since that first use a number of contributors have suggested procedures explicitly designed to rearrange the rows and columns of matrices. As time has passed, the overall tendency has been to come up with more effective procedures. And, with the introduction of computers, it has become possible to manipulate ever larger matrices. Presently, there is no end in sight.

Matrix permutations, moreover, can be used with either one-mode or two-mode data. Five years after Davis, Gardner, and Gardner introduced matrix permutation in their two-mode data set, Elaine Forsyth and Leo Katz (1946) explicitly proposed permuting matrices as a way to uncover and display social groups in a one-mode data set. They illustrated using data from one of Moreno’s (1934) sociometric studies. The young women in a residence hall had each been asked to name others in their hall for whom they had positive feelings and those for whom their feelings were negative. Positive choices were recorded using plus signs and negative choices were recorded as minus signs.

Forsyth and Katz adopted a brute-force procedure that involved rearranging rows and columns and redrawing the image again and again until as many of the plus signs fell as close to the principal diagonal as possible. At that point, cohesive groups become visible as clusters of plus signs around the diagonal. Their result is shown in Fig. 20.
Fig. 20

The Forsyth and Katz image of sociometric choices

Obviously, the Forsyth and Katz procedure was extremely cumbersome. But Beum and Brundage (1950) soon came up with a systematic iterative procedure for finding groups by rearranging the rows and columns of a one-mode matrix. And, by the late 1950s, when computers emerged on the scene, Coleman and MacRae (1960) developed a series of UNIVAC programs at the Operations Analysis Laboratory at the University of Chicago that were designed to uncover the groups in large networks.

An entirely different kind of matrix permutation procedure was proposed by Harrison White and his students. They introduced the idea of blockmodeling (White et al. 1976). In so doing, they provided a theoretical basis for reordering network data matrices, and they developed a number of algorithms for doing so.

The aim of this new thrust was to reorder the matrix in such a way that it could be partitioned to reveal two or more collections of social actors who were not linked by some social relation of interest. So, instead of arraying actors along the diagonal of a matrix, White et al. sought permutations that would define zero blocks – sets of actors between which there were no social links. They used their approach to examine a great many network data sets. One example is shown in Fig. 21.
Fig. 21

Left, Sampson’s data on who was a negative influence on whom; right, White, Boorman, and Breiger’s partitioning of the negative influence data matrix from Sampson

The data in Fig. 21 were collected by Sampson (1969) in his study of a monastery. Sampson asked each of a collection of 18 novices to report their relationships with each of the others. Figure 21 shows on the left an 18 by 18 matrix of their responses to a question asking the novices about which others had negative influences on them. A response of 3 indicated a first choice. A 2 was a second choice and a 1 was a third choice. White et al. reasoned that only first and second choices represented strong responses, so they ignored the third choices and treated the entries of 1 as if they did not exist.

One of the several procedures for working with matrix data is called CONCOR (Breiger et al. 1975). CONCOR is a recursive procedure that begins by calculating correlations between the rows (or columns) of a network data matrix. Then correlations are calculated between the rows of the resulting correlation matrix. That procedure continues until it produces a matrix of correlations that uniformly displays values of +1 and −1. Those positive and negative values are used to partition the individuals into two subsets. The CONCOR procedure can be repeated using the data contained within each of the partitions. Thus, the original matrix can thus be refined to any desired degree.

White and his students used CONCOR on Sampson’s data in an attempt to uncover blocks that contained only 0s. They could then use these zero blocks to reduce the complexity of the data matrix. That matrix produced three zero blocks. They are shown on the right side in Fig. 21.

In the reduced model in Fig. 22, each cell represents one of the blocks in Fig. 21. Thus, the 18 by 18 matrix is reduced to a 3 by 3 array. The reduction is consistent with Sampson’s original ethnographic description of subgroups among the novices. Moreover, its pattern of zero blocks in the principal diagonal indicates that no block member saw any fellow block member as having a negative influence. But the members of each block saw at least some of the members of both of the other blocks as negative influences. This makes sense in the light of the ongoing conflict that Sampson described in his report.
Fig. 22

The CONCOR reduction of the Sampson matrix

Since that time, displays based on matrix permutations have grown in size, complexity, and sophistication. One particularly striking example was produced by Richards and Seary (2000). Their data were drawn from a study of participants in a needle exchange program in Baltimore, Maryland (Valente et al. 1998). Richards and Seary examined data on 4259 individuals who picked up and returned needles at each of 4 exchange sites over a 30-month period. Each cell in the matrix is a record of the number of needles picked up by the individual in that row and returned by the individual in that column. About a third of all needles fall in the principal diagonal of the matrix.

Richards and Seary used the data from the largest weak component in the data set. That component involved 100,000 needle exchanges among 36,000 individuals. Richards and Seary used their program MultiNet (Seary 1995) and scaled the data using a form of singular value decomposition called correspondence analysis (Weller and Romney 1990). They used the coordinates provided by the first eigenvector to reorder the rows and columns of the matrix. They then colored the entries in terms of frequencies. The color scale is logarithmic: gray is 1 needle, blue 2–3, green 4–7, red 8–15, magenta 16–31, and yellow 32 and above. Their image is shown in Fig. 23.
Fig. 23

The largest component in the Baltimore needle exchange data

Figure 23 dramatically illustrates the utility of images based on matrix permutation. It shows that there was not a single community of needle users in Baltimore. Instead, there were two distinct communities of individuals who regularly obtain, return, and exchange needles with one another. These two relatively large communities were centered around two of the four needle exchange sites.

Recent Developments

Overall, the trend in visualizing social networks has been to rely on computers to do more and more of the job. First, computers used a version of singular value decomposition to locate nodes in two-dimensional images (Bock and Husain 1952). Then, soon thereafter, Coleman and MacRae (1960) programmed a computer both to permute rows and columns of a matrix and to print out an image of the result. And, in the early 1970s, Alba (1972) wrote a program that performed calculations to place nodes and then went on to draw node and edge images of the results (Kadushin 1974).

Since the 1970s, then, network analysts have increasingly used computers both for calculations and to draw images. And increasingly, multidimensional scaling and singular value decomposition have been used to determine locations for nodes. Moreover, when two dimensions are not enough to display network structure, three-dimensional images are being produced.

When microcomputers became available, it quickly became possible to produce images that gave the appearance of being three-dimensional. Figure 24 represents data collected by Kirke (1996) on social links among teenagers in a suburb of Dublin. Nodes were first located in three dimensions using multidimensional scaling. And then the Virtual Reality Modeling Language (VRML) was used to produce the appearance of three dimensions. The image was created using a program called KiNG, which builds on Mage and was written by Richardson and Richardson (1992). The image is designed as a display on computer screens, and it allows the viewer to move into the picture as well as to spin and rotate it. It is useful, then, for exploring the patterning of structural data in three apparent dimensions. Richards and Seary’s (Seary 1995) program, MultiNet, produces a wide range of graphic images. Included are images that actually can be viewed in three dimensions using anaglyphic glasses in which one lens is red and one is blue.
Fig. 24

VRML image of friendship among teens in a Dublin suburb

The most recent development in visualizing social networks involves the production of animated graphics. As more and more process data are collected and as more process models are constructed, animated images are a natural development. A group at Stanford University has written a Java program, SoNIA, that makes it quite simple to produce animated node and edge and node and directed line images (Moody et al. 2005; Bender-deMoll and McFarland 2006). These images allow users to explore the changing structural forms generated by process data.

Moreover, with the advent of color screens, color images began to be produced. Colors can be used to enhance the ability of an image to communicate important information (Pfeffer 2017). In Fig. 25 Höpner and Krempel (2003) used a spring embedder and Krempel’s own programs to arrange the nodes in two dimensions. The nodes represent the 100 largest German corporations in the year 2000. They used color to label both nodes and directed lines. In their image, each company is represented as a node, and an arrow pointing from one node to another means that the first node holds shares in the second. The size of a node indicates the number of connections to other nodes it has. Financial companies are shown as yellow nodes and industrial companies are red. Links between financial companies are yellow, those between industrial companies are red, and links between financial and industrial companies are orange. By using color, this directed graph reveals a great deal of information about the organization of German industry and finance.
Fig. 25

Shareholding among German corporations in 2000

Future challenges for network visualization will be driven by the needs that future data will bring. Especially in the context of computer-assisted data generation (e.g., emails, data from social media), we can observe that network data have been becoming richer in terms of size, temporal information, geographic information, and contextual information. Consequently, we will see more approaches tackling these challenges. Figure 26 shows the network of about one million people based on their telephone communication. In this figure just nodes are drawn to increase readability. The layout was calculated by an algorithm developed by Ulrik Brandes and Christian Pich (2007) that is implemented in Pajek (Batagelj and Mrvar 1998). The result shows geographically different areas that dominate call behavior.
Fig. 26

Visualizing a communication network with one million nodes

Incorporating geographical and temporal information of data requires new algorithms and interactive tools that support visual reasoning. Figure 27 shows the Cultural Heritage Cube from Windhager and Mayr (2012). They created an interactive visual tool that incorporates approaches from time geography. The tool can be used in museums for interactive data exploration.
Fig. 27

Networks in time and space visualized with the Cultural Heritage Cube

Overall then, in the period between Moreno’s hand drawn ad hoc images and the latest animations of dynamic network processes, there has been a dramatic growth in our ability to visualize social network structure. The major contribution has come from computers. Today we can use a wide variety of readily available computer programs to both design images and to produce screen images and/or printed output.

But, as the job of producing images becomes easier, we must be careful not to lose our sense of why we are producing network visualization in the first place. From the very beginning, the important point has always been that the visual images of social networks are not produced simply to be decorative. In every case, the early images were drawn in order to dramatize some feature of social structure. Moreno produced Fig. 4 to illustrate the importance of considering the number of connections in evaluating the structural position of an individual. In Fig. 5, the number of negative ties received by one of the running backs showed, as Moreno (1934, p. 213) put it, “It is easy to see that when 5/RB is running with the ball he is not apt to get the maximum of cooperation in interference and blocking.”

Figure 7 was a pictorial statement by Warner and Lunt that when cohesive subgroups overlap, they should not be expected to bridge wide differences in social class. Figure 8, from Davis, Gardner, and Gardner, demonstrated that the Warner-Lunt hypothesis was supported by data with respect to both social class and age. And, finally, Fig. 9 illustrated the presence of cohesive groups and of the variation of different individuals in their involvement in those groups. In every case, each of these early authors had a point to make, and in every case, the image helped to make that point. That is the key to the effective use of visual materials in social network analysis.

In the future, we can expect to see continued development of computer programs designed to aid in visualizing social networks. We can look forward to continued refinement of algorithms for displaying group structure that are based on multidimensional scaling, particularly spring embedding. We can anticipate better algorithms for displaying hierarchies and approximate hierarchies. We can expect to have more powerful programs for animation. And, at the same time, we can expect to be able to produce higher-quality and more refined visual displays of all sorts.

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Copyright information

© Springer Science+Business Media LLC 2019

Authors and Affiliations

  1. 1.Bavarian School of Public PolicyTechnical University of MunichMunichGermany
  2. 2.Department of Sociology, School of Social SciencesUniversity of CaliforniaIrvineUSA