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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
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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 3
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
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4 Social Network Visualization, Methods of
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 5
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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6 Social Network Visualization, Methods of
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 7
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
linNoteMain entry:spring embedder
linNoteMain entry:multidimensional scaling
linNotesingular value decomposition
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8 Social Network Visualization, Methods of
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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Social Network Visualization, Methods of 9
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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10 Social Network Visualization, Methods of
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 11
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.
Editor’s or typesetter’s annotations (will be removed before the
final TEX run)
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concept analysis
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12 Social Network Visualization, Methods of
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 13
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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14 Social Network Visualization, Methods of
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 15
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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16 Social Network Visualization, Methods of
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 17
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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Social Network Visualization, Methods of 19
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