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Meyers: Encyclopedia of Complexity and Systems Science — Entry 205 — 2008/10/10 — 16:28 — page 1 — le-tex��

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Social Network Visualization, Methods of 1

Social Network Visualization,Methods ofLINTON C. FREEMAN1Department of Sociology and Institute for Mathematical2Behavioral Science, School of Social Sciences,3University of California, Irvine, USA4

Article Outline5

Glossary6Definition of the Subject7Introduction8Visualization in Social Network Analysis9Images Based on one Mode Undirected Relations10Images Based on one Mode Directed Relations11Images Based on two Mode Relations12Images Based on one or two Mode Data Matrices13Future Directions14Bibliography15

Glossary16

Adjacent A node is adjacent to another if there is an edge17connecting them.18

Arrow A line directed from one node to another.19Binary relation A two valued yes/no or on/off relation.20Bipartite graph A graph, B D hN; Ei where N is a finite21

set of nodes and E is a collection of pairs of nodes in22which N is partitioned into two disjoint subsets, N123and N2, and no edge in E has both end points in the24same subset.25

Blockmodeling A procedure for clustering actors such26that the actors in each cluster share similar patterns of27ties both within and between clusters.28

Connected Any two nodes in a graph are said to be con-29nected if there is a path from one to the other; a graph30is connected if there is a path connecting every pair of31nodes.32

Cycle Any path that begins and ends at the same node.33Digraph A directed graph.34Directed graph A graph D D hN;Ai where N is a finite35

collection of nodes and A is a set of pairs linked by di-36rected lines or arrows.37

Directed line A line going from a node to another repre-38senting a non-reciprocated link.39

Edge A line connecting two nodes representing a recipro-40cated link.41

Edge labeled graph A graph in which at least two kinds42of connections between nodes are identified.43

Formal concept analysis Amethod of data analysis based 44on Galois lattice structure. 45

Galois lattice A dual structure that displays the depen- 46dencies of both objects and their properties. 47

Geodesic The shortest path between two nodes. 48Graph A graph G D hN; Ei where N is a finite set of 49

nodes and E is a collection of pairs of nodes repre- 50sented as edges. 51

Hyperedge An edge in a hypergraph that can enclose 52more than two nodes. 53

Hypergraph A hypergraph, F D hN;Hi, consists of a set 54of nodes N and a collection of hyperedges,H. 55

Indegree The indegree of a node is the number of directed 56lines it receives. 57

Irreflexive A relation in which no edge connects any node 58with itself. 59

Multidimensional scaling A search procedure designed 60to represent an observed set of proximities or distances 61in a small number of dimensions. 62

Node A point in a graph. 63One mode matrix A data matrix in which the rows and 64

columns both represent the same objects. 65Outdegree The outdegree of a node is the number of di- 66

rected lines it sends out. 67Path A path is a sequence of nodes and edges beginning 68

with a node that has an edge connecting it to the next 69node in the sequence and so on. 70

Path length The length of a path connecting two nodes is 71the number of edges it contains. 72

Permutation A reordering of the rows, columns, or rows 73and columns of a matrix. 74

Principle diagonal The set of cells in a square matrix that 75runs from the upper left to the lower right. 76

Relation A collection of ordered or unordered pairs of 77nodes. 78

Singular value decomposition an algebraic procedure 79that decomposes a data matrix into its “basic struc- 80ture”. 81

Sociometry An early version of social network analysis 82introduced by Jacob Moreno and Helen Jennings. 83

Spring embedder A kind of multidimensional scaling 84based on a model in which it is assumed that nodes 85are connected by springs that pull and push on them. 86

Symmetric A relation in which if a node a is adjacent to 87another, b, then b is adjacent to a. 88

Tree A graph is a tree if it is connected and contains no 89cycles. 90

Twomode matrix A data matrix in which the rows and 91columns represent different objects. 92

Please note that the pagination is not final; in the print version an entry will in general not start on a new page.

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2 Social Network Visualization, Methods of

Definition of the Subject93

Social network visualization refers to the practice of con-94structing pictorial images of the connections linking social95actors. The use of such images provides two benefits. It al-96lows investigators to gain new insights into the patterning97of social connections, and it helps investigators to commu-98nicate their results to others.99

Introduction100

Social network analysis did not emerge as a systematic field101of research until early in the twentieth century [1]. But vi-102sual images of social networks were produced more than103a millennium earlier. The earliest of these images that I104have uncovered was produced in Spain in the middle of105the ninth century. That image is attributed to the prolific106writer and Roman Catholic Saint, Isidore de Séville. It is107reproduced here as Fig. 1.108

The image shown in Fig. 1 displays relationships based109on genealogical descent. From the earliest times, people110have been interested in kinship ties – in who is related to111whom. This interest is evident in the descent lists found112in the Christian bible and in the oral genealogies that were113required to be memorized by Hawaiian nobles [2].114

The fact that Fig. 1 takes the form of a tree shows that115as early as the ninth century people saw the analogy be-116tween the branching structure of descent and that of trees.117This notion was captured in a mathematical formaliza-118tion in 1857 by Arthur Cayley [3]. Cayley defined a tree119in mathematical graph theoretic terms. Biggs, Lloyd and120Wilson (p. 38 in [4]) characterized Cayley’s definition by121saying that his “. . . use of the word ‘tree’ in this context is122presumably derived from the diagrammatic form of these123graphs, and is akin to the traditional use of the word in124describing genealogical of ‘family’ trees.”125

The use of trees to depict descent was, of course, con-126tinued. As time passed, however, their form became sim-127plified. Lewis Henry Morgan [5] was an attorney and an128anthropologist. He was interested in comparing how dif-129ferent peoples reckoned kinship and in 1871 he published130a mammoth work containing a collection of kinship trees.131Each tree depicted descent as conceived by a society some-132where in the world. Morgan’s trees are quite simple. Fig-133ure 2 shows descent as it was reckoned in ancient Rome.134

Twelve years later a mathematician-physicist, Alexan-135der Macfarlane [6], produced a different kind of graphic136image based still on kinship. Macfarlane set out to exam-137ine British marriage prohibitions and he represented them138both algebraically and visually. His visual images depict139males using plus signs (+) and females with circles (o).140Earlier generations he placed higher on the page. Descent141

is shown by lines connecting points. A short line crossing 142a descent line indicates another person, of either sex, in 143an intermediate generation. And the lowest point is always 144the prohibited offspring. 145

The illustration shown in Fig. 3 displays all the two- 146step marriage relations that are prohibited by British law. 147The left image shows that a malemay not marry his grand- 148daughter. The middle image shows that he may not marry 149his sister. And the right image shows that he may not 150marry his grandmother. Or, put the other way, a woman 151may not marry her grandfather, her brother or her grand- 152son. 153

Macfarlane’s paper also included algebraic expressions 154that captured all of the same marriage prohibitions. But 155Sir Francis Galton [7], who attendedMacfarlane’s presen- 156tation, declared that his “diagrammatic form” seemed “the 157most distinctive and self-explanatory” of the two treat- 158ments. 159

Finally, in 1894, John Hobson [8] produced a visual 160image of a social network that was not based on kin- 161ship. He had collected twomode (corporation by director) 162data on interlocking corporate directorates. He reasoned 163that, to the degree that corporations shared directors, they 164could be expected to cooperate and work together. 165

Hobson’s illustration was designed to show the inter- 166lock among, as he put it, “the small inner ring of South 167African finance.” Corporations are depicted as circles, and 168interlock is shown by overlapping or by a line connecting 169two circles. Hobson’s image is reproduced here as Fig. 4. 170

The important feature of this image is that it displays 171a connection linking more than two corporations. Hob- 172son’s data showed that three corporations, Charter, Rand 173and De Beers, all shared directors in common. And, at 174the same time, Rand and De Beers also both shared di- 175rectors with coalmines, telegraphs, rails, and others. The 176overlaps in his image allowed him to display which com- 177panies shared with which others. 178

It is clear, then, that a concern with connections 179among social actors and the use of visual images have 180a long history of intimate association. It should come as 181no surprise therefore that images played an important part 182in the development of social network analysis when it did 183emerge as an organized field of research. 184

Visualization in Social Network Analysis 185

In the book cited above (p. 3 in [1]), I described the mod- 186ern science of social network analysis as possessing four 187defining properties. They were: 188

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Social Network Visualization, Methods of, Figure 1Tree of consanguinity with six degrees of relationship

1. It embodies ideas about the importance of social ties189linking social actors.190

2. It collects data reflecting those ties.1913. It involves the use of graphic imagery.1924. It employs mathematical and/or computational mod-193

els.194

Pre-network research often included one or two of those195properties, but in the late 1920s each of two independent196research teams came up with efforts that included all four.197

One took place in the early 1930s. It involved a psychi- 198atrist, Jacob L. Moreno, and a psychologist, Helen H. Jen- 199nings. Together, they developed an approach they called 200“sociometry.” They reported two huge studies, both fo- 201cused on examining the structure of social ties. One was 202conducted among prisoners at Sing Sing Prison in Ossin- 203ing, NewYork [9] and the other among young delinquents 204at the New York State Training School for Girls in Hud- 205son, New York [10]. 206

Both Moreno–Jennings studies involved the extensive 207use of graphic images. The image shown in Fig. 2.1 TS2 was 208

TS2 Please check figure, there is no figure with label 2.1.


linNote2.1 comes from an earlier draft; it should read "5", not "2.1".

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Social Network Visualization, Methods of, Figure 2Descent in ancient Rome

Social Network Visualization, Methods of, Figure 3Macfarlane’s images of two-step marriage prohibitions

included in their report on the research at Sing Sing prison209(Moreno, 1932). In that figure, individuals or other kinds210of social actors are represented as points or nodes and links211between pairs of social actors are lines or edges connecting212pairs of nodes. In Fig. 5 Moreno was concerned with the213positions of individuals and the patterning of their ties. As214he put it, the individuals at the top and the bottom were215“dominant” and the image showed that those dominant216individuals were linked both “directly” and “indirectly”.

Social Network Visualization, Methods of, Figure 4Hobson’s image of corporate interlocks

Social Network Visualization, Methods of, Figure 5Image of a Pattern of Linkages

Most of the data collected by Moreno and Jennings 217involved asking individuals whom they liked or disliked. 218In data of that sort, choices are seldom reciprocated. So 219Moreno and Jennings drew lines with arrowheads to reveal 220who chose whom. Mutual choices were drawn without ar- 221rowheads and they also included a small line bisecting the 222main line connecting the two nodes. 223

Moreno and Jennings often required subjects to report 224both their likes and their dislikes. By using different colors, 225red for likes and black for dislikes, a single image could dis- 226play both. The image shown in Fig. 6 was published in the 227

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Social Network Visualization, Methods of, Figure 6Positive and Negative Choices in a Football Team

Moreno–Jennings report on the Hudson School (Moreno,2281934). It depicts positive and negative choices among 13229members of an American football team. Moreover, it con-230tains another innovation. The various team members are231placed in the drawing in approximately the same relative232locations that they occupied on the football field. That ar-233rangement shows the players’ positions and permits the234viewer to evaluate the impact of physical proximity on the235patterning of social linkages.236

Figures of the sort used by Moreno and Jennings had237a major impact on the style of graphic imagery used subse-238quently in social network analysis. For themost part, social239network analysts have represented social actors as nodes240and links between actors as edges or as directed lines with241arrowheads.242

The second introduction of the social network ap-243proach also occurred in the early 1930s. An anthropol-244ogist, W. Lloyd Warner, and a collection of his col-245leagues and students at Harvard, conducted three elab-246orate network analytic projects. One was a study of an247industrial factory, the Western Electric plant in Cicero,248Illinois [11]. The other two were studies of communi-249ties: one focused on a New England town, Newburyport,250Massachusetts [12], and the other on a southern town,251Natchez, Mississippi [13].252

The image shown in Fig. 7 was produced as part of the253factory study. It displays observed friendship ties among254pairs of individuals who worked together in the same255workroom. It was drawn using nodes and two-headed256lines instead of edges, but it is very similar to the images257

Social Network Visualization, Methods of, Figure 7Friendships linking factory workers

Social Network Visualization, Methods of, Figure 8An idealized pattern of overlapping ‘cliques’

produced byMoreno and Jennings. In addition, as in Fig. 6 258above, the impact of physical space was displayed; workers 259were placed in the drawing in positions that reflected the 260locations of their workstations. 261

In reporting their study of Newburyport, Warner and 262Lunt used the kind of drawing of overlapping circles that 263Hobson had used to construct Fig. 4. But here that image 264was introduced, not to describe data, but to propose an 265idea they had about social structure. The diagram in Fig. 8 266represents the investigators’ idealized version of the ex- 267pected structure of overlaps among subgroups in the pres- 268ence of social class. The idea is that only subgroups that 269are close to one another in class ranking are likely to have 270overlapping memberships. 271

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Social Network Visualization, Methods of, Figure 9Stratification, age and overlapping groups

In their study of Natchez Davis, Gardner and Gard-272ner [13] employed the same diagrammatic form to dis-273play two mode data reflecting on the earlier Newburyport274hypothesis. Figure 9 shows subgroups of black males and275their overlaps. In that image themen are arranged in terms276of both social class and age. Both, it turned out, provided277important bases for grouping.278

Finally, in that same report, Davis, Gardner and Gard-279ner also introduced an entirely different kind of social net-280work image. Like Hobson, they had collected two mode281data. Eighteen women were designated in the rows of their282datamatrix and fourteen social events were depicted in the283columns. That matrix is reproduced here as Fig. 10.284

The data shown in Fig. 10 were all collected during285a single year. But, by examining the column headings, it286is clear that Davis and his colleagues did not arrange the287social events according to the dates upon which they took288place. Instead, they listed both the events and the women289who attended them in such a way that the arrangement290itself suggests that these women were organized into two291groups. The two groups overlap, but for the most part292they are distinct. Most of the women in the top half of293the matrix attended the leftmost five events. And most of294the women in the bottom half attended the rightmost five295events. The middle four events apparently brought both296groups of women together.297

This arrangement of women and events was self-con-298sciously produced by the authors. Davis, Gardner and299Gardner were convinced that these womenwere organized300into two groups and they presented their data matrix in301a way that would illustrate that conclusion. The interest-302

ing thing is that these authors never commented explicitly 303about how they had rearranged the columns and rows in 304their matrix. They simply organized their display in a way 305that would make the point. 306

From the outset, then, four kinds of images have 307played important parts in the development of social net- 308work analysis. These first network graphics included draw- 309ings displaying (1) one mode undirected relations, (2) one 310mode directed relations, (3) two mode relations and (4) 311one or two mode data matrices. A few other kinds of net- 312work images have been used since then, but the four orig- 313inals – particularly those based on one mode undirected 314and one mode directed relations – still dominate the field. 315In the next four sections we will examine the four original 316kinds of images and how their use has evolved in the social 317network context. 318

Images Based on oneMode Undirected Relations 319

Mathematically speaking, the node and edge images in- 320troduced by Moreno and Jennings in Fig. 5 are graphs. 321A graph G D hN; Ei where N is a finite set of nodes and E 322is a collection of pairs of nodes. In graph visualizations, 323a pair of nodes in E is presented as a line connecting the 324two nodes in question. Two nodes are called adjacent if 325there is an edge directly connecting them to each other. 326A graph embodies a binary (yes/no or on/off) relation that 327is irreflexive (no node is adjacent to itself) and symmetric 328(if a node a is adjacent to another, b, then b is adjacent 329to a). 330

A path is a sequence of nodes and edges, beginning 331with a node that has an edge connecting it to the next node 332in the sequence. The length of a path between two nodes 333is the number of edges it contains. And the shortest path 334connecting two nodes is called the geodesic. 335

Any two nodes in N are connected if there is a path 336from one to the other. And a whole graphG is connected if 337every pair of nodes in N is connected. If a path begins and 338ends at the same node, that path is a cycle. Finally, a graph 339is a tree if it is both connected and it contains no cycles. 340

The image in Fig. 11 is a graph. It is based on data 341recorded by J. Clyde Mitchell [14] on the social ties among 342the 19 individuals involved in the personal network of 343a homeless woman in Britain. I used a program called Net- 344Draw to place the nodes representing individuals in Fig. 11 345in random positions. That calls attention to the impor- 346tance of the locations of points in graphic displays. Given 347the locations of the points in Fig. 11 it is very difficult for 348the viewer to see anything interesting in the patterning of 349this woman’s network. 350

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Social Network Visualization, Methods of, Figure 10The Davis, Gardner and Gardner data on women’s attendance at social events

Social Network Visualization, Methods of, Figure 11Links in the network of a homeless woman I

Compare the image in Fig. 11 with that in Fig. 12. Fig-351ure 12 was also produced using NetDraw, but this time the352points were placed using a spring embedder [16]. A spring353embedder is a computer algorithm that, in effect, places354a spring of unit length between every pair of adjacent355nodes and a much longer spring between nodes that are356not adjacent. It starts with a random placement of nodes,357then the whole apparatus is set in motion and the various358springs push and pull until they reach an equilibrium.359

The advantage of using a spring embedder is that it360does not require the investigator to make ad hoc judg-361ments in locating nodes in a graph. It uses a standard com-362puter algorithm to place the nodes automatically. Thus,363every user will get the same result. There are several dif-364

Social Network Visualization, Methods of, Figure 12Links in the network of a homeless woman II

ferent spring embedding algorithms. And they are all ex- 365amples of amore general class of computer algorithms that 366search for optimal locations for nodes in relatively few di- 367mensions. This general class of search algorithms is called 368multidimensional scaling [17]. 369

An alternativemethod for placing nodes automatically 370is grounded in algebra. It is called singular value decom- 371position [18]. Singular value decomposition is not search 372based. Instead, it uses matrix operations to produce a lin- 373

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Social Network Visualization, Methods of, Figure 13Links in the network of a homeless woman III

ear transformation of the data, and thus to position the374nodes in one, two or three or more dimensions. There is375no guarantee that it will always be effective, but often sin-376gular value decomposition provides very good placements377of the nodes in few enough dimensions that visualization378is possible [19]. A NetDraw image based on singular value379decomposition of Mitchell’s data is shown in Fig. 13.380

The images in Figs. 12 and 13 both show that the whole381network is organized into three densely connected groups382that are only loosely linked to one another. That is inter-383esting, but it does not tell us anything about the bases for384the groupings. By adding a little information, and continu-385ing to use NetDraw, we can transform the graph of Fig. 12386into a node labeled graph. See Fig. 14.387

Given the labels, we can identify the homeless woman,388the “respondent.” We can also see how her network is split389up. One division includes her original family, another her390friends along with her social worker, and the third con-391tains her estranged husband and his family, her in-laws.392

Mitchell’s report, however, included evenmore details.393It included estimates of the strength of the tie linking each394of the pairs of individuals. He classified each tie as either395strong or weak. We can embody this additional informa-396tion in our NetDraw image by an adding another compo-397nent to our graph. Figure 15, then, was produced using the398spring embedder, it is node labeled, and, in addition, it is399an edge labeled graph.400

In Fig. 15 strong ties are indicated by wide edges. By401examining their patterning, we learn that the individuals402within each family are linked together mostly by strong403ties, while the homeless woman’s friends have fewer strong404ties linking them together. This result is not surprising, but405it does provide additional insight about the structural po-406

Social Network Visualization, Methods of, Figure 14Links in the network of a homeless woman IV

Social Network Visualization, Methods of, Figure 15Links in the network of a homeless woman V

sition of the woman in question. Clearly, it would be easier 407for either family to achieve consensus and provide support 408than it would be for the respondent’s loosely connected 409collection of friends [20]. 410

It should be clear, then, that the placement of nodes 411and the labeling of both nodes and edges are critical for the 412ability of a graph to communicate important information. 413Good images can provide investigators with new insights 414about the structural properties of the social networks they 415

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are studying. And they can, of course, help to communi-416cate the results of social network research to outsiders.417

Images Based on oneMode Directed Relations418

It was obvious from the outset that these simple graphs419would not permit many kinds of displays of interest to so-420cial network analysts. Even Moreno and Jennings saw the421need to display the direction of choice in their sociograms.422The direction of connections can be expressed using di-423rected graphs or digraphs.424

A digraph D D hN;Ai whereN is a finite collection of425nodes and A is a set of pairs shown as directed lines or ar-426rows. When an arrow is directed from node a to node b in427a digraph, then a is the tail of the arrow and b is the head;428a is the immediate predecessor of b and b is the immediate429successor of a. The outdegree of a node is the number of430arrows for which it is the tail and its indegree is its number431for which it is the head.432

In any study that involves social links that are not sym-433metric, digraphs provide a natural representation. Con-434sider Fig. 16 that was produced by a program called Pa-435jek [21]. In preparing a book on the development of social436network analysis, I interviewed a number of the founders.437Each was asked to name others who had influenced them438to think in network terms. The result is a data set that ob-439viously lacks symmetry.440

My interest, however, was with clusters, or blocs, of441influentials and nominees. So I placed the nodes using442a spring embedder designed by Kamada and Kawai [22].443The resulting figure shows that there seem to be two fairly444well defined subgroups, one on the left and one on the445right. The two groups are relatively dense but they are only446loosely connected together. The people on the left are al-447most entirely sociologists and those on the right aremostly448from other fields. And from the patterning one can sus-449pect that there was some kind of split between these two450groups.451

For some kinds of data the search for clusters or groups452is not appropriate. For example, when we are dealing with453data that should embody some sort of ordering, digraph454representations are particularly important. To illustrate455how digraphs can be used to display ordering, consider the456data collected by Forkman and Haskell [23]. They studied457several communities, each made up of six domestic hens.458In five of these communities the hens formed strict peck-459ing orders in which the top hen pecked all the others; the460second pecked all but the top, and so on. Figure 17 shows461a visone [24] image of the data from one of those five462communities. There the nodes are arranged, top down, in463terms of their outdegrees and the pecking order is obvious.464

Often data approach, but do not achieve, a strict order. 465DavidKrackhardt [25], for example, collected data on who 466sought advice from whom among 14 employees in the in- 467ternal auditing staff of a large company. Krackhardt’s data 468could not be drawn with all the arrows pointing in one di- 469rection. So, in Fig. 18 he arranged the individuals in such 470a way that as many arrows as possible were pointing up. 471The viewer, then, can immediately see there is an impor- 472tant hierarchical element displayed by these data. From 473the image, it appears that Nancy is at the top of the ad- 474vice chain and Bob, Wynn, Carol, Harold and Susan are at 475the bottom. 476

There is, however, an important limitation in this fig- 477ure. Nancy seeks advice from Donna, Donna seeks advice 478from Manuel and Manuel seeks advice from Nancy. Thus, 479these three form a directed cycle of advice seeking. Given 480such a circular arrangement, no possible hierarchy among 481these three individuals can be established. Any order in 482which they were arranged would be misleading. In addi- 483tion, Stuart and Charles cannot be ordered because they 484chose each other. The same is true for Kathy and Tanya. 485

The apparent ordering of nodes in Krackhardt’s im- 486age was imposed by human judgment. There are computer 487algorithms that can automatically arrange the nodes into 488a hierarchical form [26]. They are, however, not as well 489grounded or reliable as multidimensional scaling and sin- 490gular value decomposition. 491

Images Based on twoMode Relations 492

Any time we deal with a relation that can link more than 493two social actors, we cannot use graphs or directed graphs. 494Both graphs and directed graphs can deal only with links 495between pairs. Two mode data, however, allow for rela- 496tions that link three or more actors. So, whenever we have 497two mode data, like that collected by Hobson [8] or Davis, 498Gardner and Gardner [13] we need another way to con- 499struct images. 500

There are several ways to construct images of two 501mode data. I will consider three of them in the present 502section, hypergraphs, bipartite graphs and lattices. Then, 503in the next section, I will discuss the use of matrix repre- 504sentations for both one mode and two mode data. 505

Hobson [8] collected two mode data on corporations 506and their directors. He produced the image shown in Fig. 4 507showing corporate interlocks as overlapping areas. Math- 508ematically, images like Hobson’s are hypergraphs. A hy- 509pergraph, F D hN;Hi, consists of a set of nodes N and 510a collection of hyperedges,H. While an edge in an ordinary 511graph connects two nodes, a hyperedge in a hypergraph 512may link any arbitrary subset of the nodes in N. Pictori- 513

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Social Network Visualization, Methods of, Figure 16Influences on some founders of social network analysis

Social Network Visualization, Methods of, Figure 17Dominance among six hens

ally, hyperedges are represented as boundaries enclosing514sets of nodes.515

The use of hypergraphs was demonstrated in a recent516report by Estrada and Rodríguez-Velázquez [27]. They517

Social Network Visualization, Methods of, Figure 18Visone image of advice seeking (from Brandes, Raab and Wag-ner [26])

began with one mode data that showed the patterning 518of predation among the members of eleven species in 519a Malaysian rain forest. Their graph, showing who preys 520on whom, is shown in Fig. 19. 521

Figure 19 shows which species preys on which other 522species. But if the investigator is interested, as those who 523study food webs often are, in defining ecological niches in 524

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Social Network Visualization, Methods of, Figure 19Who preys on whom in aMalaysian Rain Forest

Social Network Visualization, Methods of, Figure 20Twomode matrix of co-predation

terms of co-predation, Fig. 19 makes the overall pattern525less than obvious. As an alternative we can build a hyper-526graph.527

The matrix shown in Fig. 20 is based on the data in528Fig. 19. It was built by considering each of the species in529turn as prey. Then all of the species that share each given530prey are pooled together. Species 1, 6 and 9 have no prey531in the set. And species 4, 1 and 9 are the targets of co-pre-532dation. So the new matrix is two mode. It has the three533targets of co-predation as columns and the eight predators534as rows.535

Prey on TS3536That matrix is captured visually by the hypergraph in537

Fig. 19. It immediately reveals that there are three niches.538The one labeled E1 includes all the species who preyed on539species 4, E2 those who preyed on species 1 and E3 those540

Social Network Visualization, Methods of, Figure 21Hypergraph of co-predation

who preyed on 9. Thus, each edge in Fig. 21 encloses a col- 541lection of species that compete directly for at least one 542prey. 543

There are, however, other ways to picture two mode 544data. In a more recent study of corporate interlocks, Joel 545Levine [29] reported data on the board memberships of 546seven major American corporations. Those corporations 547turned out to have ten directors who appeared on two 548or more of their boards. Levine presented his interlock 549data using a bipartite graph. A bipartite graph, B D hN; Ei 550where N is partitioned into two disjoint subsets, N1 and 551N2, and no edge in E has both end points in the same 552subset. He used singular valued decomposition to place 553the nodes representing both corporations and boardmem- 554bers and produced a bipartite image similar to the one dis- 555played in Fig. 22. I prepared that figure using NetDraw. 556There, the corporations are shown as red circles and the 557board members are blue squares. Thus, both the colors 558and the shapes of the nodes stress the bipartite nature of 559the graph. 560

There is still another form of graphic display, one 561that reveals even more structural information about a two 562mode data set. It is based on an algebraic procedure 563called Galois lattice analysis or formal concept analy- 564sis [30,31,32,33]. A Galois or formal concept lattice is de- 565fined on an object by property, matrix. Let O be a set 566of objects and A be a set of attributes. The binary ma- 567trix O � ATS4 indicates which objects possess which at- 568tributes. 569

We can define a pair hOi ;Ai i such that Oi is a subset 570ofO andAi is a subset ofA and every object inOi has every 571attribute in Ai. Moreover, both O and Amust be maximal. 572Thus, for every attribute in A that is not in Ai, there is an 573

TS3 Please check.TS4 Please confirm notation.


linNoteThis is correct

linNoteThis is simply a typo; please delete it.

linNoteMain entry:bipartite graph

linNoteMain entry:latticesub entry:GaloisMain entry:formal concept analysis

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Social Network Visualization, Methods of, Figure 22A NetDraw Image of Levine’s interlock data as a bipartite

object inOi that does not have that attribute. And for every574object in O that is not in Oi, there is an attribute in Ai that575the object lacks.576

These pairs are dual and they can be partially ordered577by inclusion. Given two pairs hOi ;Ai i and hOj;Aji we say578that hOi ;Ai i is less than hOj;Aji whenOi is a subset ofOj579or, equivalently, when Aj is a subset of Ai. Since all these580pairs have unique least upper bounds and greatest lower581bounds they form a dual (Galois) lattice.582

I will illustrate by considering again the woman by583event data collected by Davis, Gardner and Gardner [13].584Let the women (1 through 18) be the objects and the events585(A through N) be the attributes. The data, arranged into586a Galois lattice by a program called GLAD, are shown in587Figure into a lattice in Fig. 23.588

The lattice displays the same three classes of events589that define the same two groups of women that we saw in590Fig. 10. But, in addition to the classes of events and groups591of women, we can now see the containment structures of592both events and women. To begin with, by following lines593up from the bottom we can see which women attended594which events. When we get to the top we hit the set of all595events and, at the same time, because no woman attended596all 14 events, it is also the null set of women.597

The uppermost events (E–L) involved the largest sets598of women. Other events are contained in the lower inter-599sections of these events. Event C, for example, is contained600

in E; everyone who attended C was present at E. And, at 601the next lower level, B and D are both contained in C. The 602events, then, can be seen as varying in their “openness”. 603

At the same time, the figure shows the upward con- 604tainment structure of the women in terms of their patterns 605of attendance. Because no event attracted all 18 women, 606the lowest point represents the set of all women as well as 607the null set of events. Then, the lowest set of women (1, 2, 6083, 4, 13, 14 and 15) are the “core” attendees, so to speak. 609The next level contains woman 9 who never attended un- 610less woman 3 was also present, and woman 5 whose atten- 611dance depended on that of women 4 and 3. Women 6, 7, 8, 61210, 11, 12, 17 and 18 are also at this second level. In some 613sense, these are all secondary or peripheral participants in 614these events. And, finally, woman 16 turns out to be a third 615level participant; she was extremely peripheral. Woman 16 616attended events only when secondary attendees 8–12 and 617core attendees 1, 3 and 13 were all present. All in all, then, 618the image of the Galois lattice reveals a great deal about the 619internal structure of attendance. 620

In this section I have shown three ways of visualizing 621two mode data. All three of them, however, share one im- 622portant limitation. That limitation stems from the fact that 623all three of them can only be used for very small data sets. 624As the number of cases grows, they all produce images that 625become increasingly difficult to read. 626

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Social Network Visualization, Methods of, Figure 23The Davis, Gardner and Gardner data as a Galois lattice

Images Based on one or twoMode DataMatrices627

WhenDavis, Davis andGardner [13] first usedmatrix per-628mutation, they did so without calling attention to the pro-629cess. But since that first use a number of contributors have630suggested procedures explicitly designed to rearrange the631rows and columns of matrices. As time has passed, the632overall tendency has been to come up with more effective633procedures. And, with the introduction of computers, it634has become possible to manipulate ever larger matrices.635Presently, there is no end in sight.636

Matrix permutations, moreover can be used with ei-637ther one mode or two mode data. Five years after Davis,638Gardner and Gardner introduced matrix permutation in639their two mode data set, Elaine Forsyth and Leo Katz [34]640explicitly proposed permuting matrices as a way to un-641cover and display social groups in a one mode data set.642They illustrated using data from one of Moreno’s [10] so-643ciometric studies. The young women in a residence hall644had each been asked to name others in their hall for whom645they had positive feelings and those for whom their feel-646ings were negative. Positive choices were recorded using647plus signs and negative choices were recorded as minus648signs.649

Forsyth and Katz adopted a brute-force procedure that650involved rearranging rows and columns and redrawing the651

image again and again until as many of the plus signs fell 652as close to the principle diagonal as possible. At that point, 653cohesive groups become visible as clusters of plus signs 654around the diagonal. Their result is shown in Fig. 24. 655

Obviously, the Forsyth and Katz procedure was ex- 656tremely cumbersome. But Beum and Brundage [35] soon 657came up with a systematic iterative procedure for find- 658ing groups by rearranging the rows and columns of a one 659mode matrix. And, by the late 1950s, when computers 660emerged on the scene, Coleman and MacRae [36] devel- 661oped a series of Univac programs at the Operations Anal- 662ysis Laboratory at the University of Chicago that were de- 663signed to uncover the groups in large networks. 664

An entirely different kind matrix permutation proce- 665dure was proposed by Harrison White and his students. 666They introduced the idea of blockmodeling [37]. In so do- 667ing, they provided a theoretical basis for reordering net- 668work data matrices, and they developed a number of algo- 669rithms for doing so. 670

The aim of this new thrust was to reorder the matrix 671in such a way that it could be partitioned to reveal two 672or more collections of social actors who were not linked 673by some social relation of interest. So, instead of arraying 674actors along the diagonal of a matrix, White et al. sought 675permutations that would define zero blocks – sets of actors 676between which there were no social links. They used their 677

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Social Network Visualization, Methods of, Figure 24The Forsyth and Katz image of sociometric choices

Social Network Visualization, Methods of, Figure 25Sampson’s data on who was a negative influence on whom

approach to examine a great many network data sets. One678example is shown in Fig. 25.679

The data in Fig. 25 were collected by Sampson [38] in680his study of a monastery. Sampson asked each of a collec-681tion of 18 novices to report their relationships with each of682the others. Figure 25 shows an 18 by 18 matrix of their re-683sponses to a question asking the novices about which oth-684ers had negative influences on them. A response of 3 in-685dicated a first choice. A 2 was a second choice and a 1 was686a third choice. White et al. reasoned that only first and sec-687ond choices represented strong responses, so they ignored

Social Network Visualization, Methods of, Figure 26White, Boorman and Breiger’s partitioning of the negative influ-ence data matrix from Sampson

the third choices and treated the entries of 1 as if they did 688not exist. 689

One of the several procedures procedure White et al. 690introduced was called CONCOR. CONCOR is a recursive 691procedure that begins by calculating correlations between 692the rows (or columns) of a network data matrix. Then 693correlations are calculated between the rows of the result- 694ing correlation matrix. That procedure continues until it 695produces a matrix of correlations that uniformly displays 696values of + 1 and � 1. Those positive and negative values 697are used to partition the individuals into two subsets. The 698CONCOR procedure can be repeated using the data con- 699tained within each of the partitions. Thus, the original ma- 700trix can thus be refined to any desired degree. 701

White and his students used CONCOR on the data 702shown in Fig. 25 in an attempt to uncover blocks that con- 703tained only 0s. They could then use these zero blocks to 704reduce the complexity of the data matrix. That matrix pro- 705duced three zero blocks. They are shown in Fig. 26. 706

In the reduced model in Fig. 27 each cell represents 707one of the blocks in Fig. 26. Thus, the 18 by 18 matrix is 708reduced to a 3 by 3 array. The reduction is consistent with 709Sampson’s original ethnographic description of subgroups 710among the novices. Moreover, its pattern of zero blocks in 711the principle diagonal indicates that no block member saw 712any fellow block member as having a negative influence. 713But the members of each block saw at least some of the 714members of both of the other blocks as negative influences. 715This makes sense in the light of the ongoing conflict that 716Sampson described in his report. 717

Since that time, displays based on matrix permutations 718have grown in size, complexity and sophistication. One 719particularly striking example was produced by Richards 720

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Social Network Visualization, Methods of, Figure 27The CONCOR reduction of the Sampsonmatrix

and Seary [39]. Their data were drawn from a study of721participants in a needle exchange program in Baltimore,722Maryland [40]. Richards and Seary examined data on 4259723individuals who picked up and returned needles at each of724four exchange sites over a 30 month period. Each cell in725the matrix is a record of the number of needles picked up726by the individual in that row and returned by the individ-727ual in that column. About a third of all needles fall in the728principle diagonal of the matrix.729

Richards and Seary used the data from the largest730weak component in the data set. That component in-731volved 100 000 needle exchanges among 36 000 individu-732als. Richards and Seary used their program MultiNet [41]733scaled the data using a form of singular value decomposi-734tion called correspondence analysis [18]. They used the co-735ordinates provided by the first Eigenvector to reorder the736rows and columns of the matrix. They then colored the737entries in terms of frequencies. The color scale is logarith-738mic: gray is 1 needle, blue 2–3, green 4–7, red 8–15, ma-739genta 16–31, yellow 32 and above. Their image is shown in740Fig. 28.741

Figure 28 dramatically illustrates the utility of images742based on matrix permutation. It shows that there was not743a single community of needle users in Baltimore. Instead,744there were two distinct communities of individuals who745regularly obtain, return and exchange needleswith one an-746other. These two relatively large communities were cen-747tered around two of the four needle exchange sites.748

Future Directions749

Overall, the long range the trend in visualizing social net-750works has been to rely on computers to do more and751more of the job. First, computers used a version of sin-752gular value decomposition to locate nodes in two dimen-753sional images [42]. Then, soon thereafter, Coleman and754MacRae [35] programmed a computer both to permute755

Social Network Visualization, Methods of, Figure 28The largest component in the Baltimore needle exchange data

Social Network Visualization, Methods of, Figure 29VRML image of friendship among teens in a Dublin suburb

rows and columns of a matrix and to print out an image of 756the result. And, in the early 1970s, Alba [43] wrote a pro- 757gram that performed calculations to place nodes and then 758went on to draw node and edge images of the results [44]. 759

Since the 1970s, then, network analysts have increas- 760ingly used computers both for calculations and to draw 761images. And increasingly, multidimensional scaling and 762singular value decomposition have been used to determine 763locations for nodes. Moreover, when two dimensions are 764not enough to display network structure, three dimen- 765sional images are being produced. 766

Whenmicrocomputers became available it quickly be- 767came possible to produce images that gave the appear- 768ance of being three dimensional. Moreover, with the ad- 769vent of color screens, color images began to be produced. 770

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Social Network Visualization, Methods of, Figure 30Links among German corporations in 2000

Figure 29 represents data collected by Kirke [45] on social771links among teenagers in a suburb of Dublin. Nodes were772first located in three dimensions using multidimensional773scaling. And then the virtual reality modeling language774(VRML) was used to produce the appearance of three di-775mensions. Figure 27 was produced as a cover design for776a book [46] and the colors were used simply make the im-777age more attractive.778

Colors can, however, be used to enhance the ability779of an image to communicate important information. In780Fig. 30 Höpner and Krempel [47] used a spring embed-781der and Krempel’s own programs to arrange the nodes in782two dimensions. The nodes represent the 100 largest Ger-783man corporations in the year 2000. They used color to la-784bel both nodes and directed lines. In their image each com-785pany is represented as a node and an arrow pointing from786one node to another means that the first node holds shares787in the second. The size of a node indicates the number788of connections to other nodes it has. Financial companies789are shown as yellow nodes and industrial companies are790red. Links between financial companies are yellow, those791between industrial companies are red and links between792financial and industrial companies are orange. Bu using793color, then, his directed graph reveals a great deal of in-794formation about the organization of German industry and795finance.796

The image in Fig. 31 was made using a program797called MAGE [48]. MAGE was written by Richardson and798

Richardson [49]. It is designed as a display on computer 799screens, and it allows the viewer to move into the picture 800as well as to spin and rotate it. It is useful, then, for ex- 801ploring the patterning of structural data in three apparent 802dimensions. 803

In Fig. 31 students in a university professional school 804program reported who their friends in their class were. 805Nodes were placed using a multidimensional scaling pro- 806gram and then they were colored according to their pro- 807gram in the school. It turned out that most of their friend- 808ship choices linked those that shared a program. 809

Richards and Seary’s [50] program, Multinet, pro- 810duces a wide range of graphic images. Included are im- 811ages that actually can be viewed in three dimensions using 812anaglyphic glasses in which one lens is red and one is blue. 813Obviously, I cannot illustrate their program here, but any- 814one who wants to see real 3D images should exploreMulti- 815net. 816

Themost recent development in visualizing social net- 817works involves the production of animated graphics. As 818more andmore process data are collected and asmore pro- 819cess models are constructed, animated images are a natu- 820ral development. A group at Stanford University has writ- 821ten a Java program, SoNIA [51], that makes it quite sim- 822ple to produce animated node and edge and node and di- 823rected line images [52,53]. These images allow users to ex- 824plore the changing structural forms generated by process 825data. 826

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Social Network Visualization, Methods of, Figure 31MAGE image of friendship among classmates

Overall then, in the period between Moreno’s hand827drawn ad hoc images and the latest animations of dynamic828network processes, there has been a dramatic growth in829our ability to visualize social network structure. The major830contribution has come from computers. Today we can use831a wide variety of readily available computer programs to832both design images and to produce screen images and/or833printed output.834

But, as the job of producing images becomes easier, we835must be careful not to lose our sense of why we are pro-836ducing then in the first place. From the very beginning,837the important point has always been that the visual images838of social networks are not produced simply to be decora-839tive. In every case, the early images were drawn in order840to dramatize some feature of social structure. Moreno pro-841duced Fig. 5 to illustrate the importance of considering the842number of connections in evaluating the structural posi-843tion of an individual. In Fig. 6 the number of negative ties844received by one of the running backs showed, as Moreno845(p. 213 in [10]) put it, “It is easy to see that when 5/RB is846running with the ball he is not apt to get the maximum of847cooperation in interference and blocking.”848

Figure 7 was a pictorial statement byWarner and Lunt849that when cohesive subgroups overlap, they should not be850expected to bridge wide differences in social class. Fig-851ure 8, from Davis, Gardner and Gardner, demonstrated852that the Warner–Lunt hypothesis was supported by data853with respect to both social class and age. And, finally,854

Fig. 9 illustrated the presence of cohesive groups and of 855the variation of different individuals in their involvement 856in those groups. In every case each of these early authors 857had a point to make, and in every case the image helped 858to make that point. That is the key to the effective use of 859visual materials in social network analysis. 860

In future we can expect to see continued development 861of computer programs designed to aid in visualizing social 862network networks. We can look forward to continued re- 863finement of algorithms for displaying group structure that 864are based on multidimensional scaling, particularly spring 865embedding. We can anticipate better algorithms for dis- 866playing hierarchies and approximate hierarchies. We can 867expect to have more powerful programs for animation. 868And, at the same time, we can expect to be able to pro- 869duce higher quality and more refined visual displays of all 870sorts. 871

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TS5 Please provide the publisher and location of publishing.


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