Journal of Mathematical Sociology, 29:265–294, 2005
Copyright # Taylor & Francis Inc.
ISSN: 0022-250X print/1545-5874 online
DOI: 10.1080/00222500590957473
Eigenspectral Analysis of Hermitian Adjacency
Matrices for the Analysis of Group Substructures
Bettina Hoser
Andreas Geyer-Schulz
Information Services and Electronic Markets,
Universität Karlsruhe (TH), Germany
In this paper we propose the use of the eigensystem of complex adjacency matrices
to analyze the structure of asymmetric directed weighted communication.
The use of complex Hermitian adjacency matrices allows to store more data relevant to asymmetric communication, and extends the interpretation of the resulting eigensystem beyond the principal eigenpair. This is based on the fact, that the
adjacency matrix is transformed into a linear self-adjoint operator in Hilbert
space.
Subgroups of members, or nodes of a communication network can be characterised
by the eigensubspaces of the complex Hermitian adjacency matrix. Their relative
‘traffic-level’ is represented by the eigenvalue of the subspace, and their members
are represented by the eigenvector components. Since eigenvectors belonging to
distinct eigenvalues are orthogonal the subgroups can be viewed as independent
with respect to the communication behavior of the relevant members of each
subgroup.
As an example for this kind of analysis the EIES data set is used. The substructures and communication patterns within this data set are described.
Keywords: Hermitian matrix, eigensystem, perturbation, social network analysis,
rank prestige index
1. INTRODUCTION
Today’s trend towards embedded, wearable, and networked devices
leads to an increasing stream of communication traffic data, which
Address correspondence to Bettina Hoser, Institut fur Informationswirtschaft
und-management, Abteilung für Informationsdienste und elektronische Märkte,
Universität Karlsruhe (TH), Kaiserstrasse 12, D-76128 Karlsruhe, Germany. E-mail:
[email protected]
265
266
B. Hoser and A. Geyer-Schulz
can be used to reveal social network structures. Wellman, for example,
envisions the rise of network societies that require new tools for
enhancing social capital (Wellman, 2001).
Eigenvector centrality measurements have become a standard
procedure in the analysis of group structures. Mostly symmetric
(dichotomized) data have been used. Bonacich and Lloyd (Bonacich
and Lloyd, 2001) present an introduction of the use of eigenvector-like
measurements of centrality for asymmetric data. The analysis of directed, valued, asymmetric relationships within a social network poses
some difficulties. In this paper we will propose a method based on
the status (rank prestige) index method described in Wasserman
and Faust (Wasserman, 1994, p. 205–219) but adapted to complex
adjacency matrices. As an example, we discuss the use of this method
in the analysis of part of the full EIES data set as presented in Wasserman and Faust (Wasserman, 1994) as well as an EIES subset as presented by Freeman (Freeman, 1997).
The following constructs from social network analysis are commonly used in the analysis of group structure (Wasserman, 1994):
. Centrality: A member in a newsgroup or a subgroup is viewed as
central, whenever he or she has a high out degree (¼ number of outgoing connections with a high number of different co-members or
other subgroups).
. Prestige: A member in a newsgroup or a subgroup is viewed as prestigious, whenever he or she has a high in degree (¼ number of
inbound connections with a high number of different co-members
or other subgroups).
. Rank prestige or status: Centrality and prestige will be viewed with
respect to the rank of the member within the group. This is the
construct of interest for the rest of this paper.
Wasserman and Faust (Wasserman 1994, p. 205–219) describe to a
great extent the history of the status (or as they define it: rank prestige) index. This index is based on the idea, that the rank of a group
member depends on the rank of the members to whom he or she is
connected. Stated in mathematical terms this yields the eigenvalue
equation (for an eigenvalue equal to 1). The components of the principal eigenvector are the rank prestige indices of each group member.
The highest ranked member, the one with the largest rank prestige
index, is the one who is either chosen by a few but high ranking comembers, or by many relatively low ranked co-members. Until now,
this index has only been calculated for real matrices, based on directed and possibly weighted graphs, which can give rise to the problem
Eigenspectral Analysis of Hermitian Adjacency Matrices
267
of complex eigenvalues (which have, in this context, no interpretation
as of now). As a second problem, this index can only use the principal
eigenvector for the interpretation of the group’s rank structure. This
is due to the fact that the eigenvector system does not construct a
vector space that lends itself easily to interpretation in the field of
social network analysis.
For asymmetric relations, Bonacich and Lloyd (Bonacich and Lloyd,
2001) show that for some networks the standard eigenvector measures
for network centrality do not lead to meaningful results. As a solution
they suggest the notion of a-centrality, which combines an individual’s
own inherent status with a weight a for the relative importance of the
perceived status.
There have been different approaches to the analysis of asymmetric
communication behavior. Freeman (Freeman, 1997) proposed to use
the possibility to split any asymmetric square matrix into its symmetric and skew-symmetric part, perform a singular value decomposition of the skew-symmetric matrix, and showed, that the result
could be interpreted as a ranking of dominance.
Recent research by Tyler et al. (Tyler et al., 2003) showed that
they could identify subgroups in asymmetric email networks by analyzing betweenness centrality in the form of inter-community edges
with a large betweenness value. These edges are then removed until
the graph decomposes into separate communities, thus re-organizing
the graph structure. Barnett and Rice (Barnett and Rice, 1985)
showed, that asymmetrical data in its raw form might have more
information than matrices that have been transformed to avoid negative eigenvalues.
Chino (Chino, 1998) proposed the use of Hermitian matrices and
Hilbert space theory in the analysis of preference data over time in
psychology. He could show that the sign of the eigenvalues yields
information about the orientation of behavior within a given subspace,
and that it can then be shown how people move (behave) towards each
over time (or move apart).
We propose here the use of complex adjacency matrices (Hoser
and Geyer-Schulz, 2003) and Hilbert space theory to be able to keep
the information concerning the communication asymmetry and at
the same time to get more information concerning the structure
of the newsgroup as well as the communication behavior within the
subgroups of the newsgroup.
The remainder of this paper is organized as follows: In section 2 we
introduce the properties of Hilbert space and complex Hermitian
matrices. Also, the characteristics of their eigensystem and their
behavior under perturbation are discussed. We then show in section
268
B. Hoser and A. Geyer-Schulz
3 how we construct a complex-valued sociomatrix that is then
transformed into a Hermitian matrix. The structure and characteristics of the resulting eigenspace is discussed. We further introduce
some relevant constructs from social network analysis needed for
the interpretation of the resulting eigensystem. In section 4 we
present two examples with synthetic data and two applications to real
data. Section 5 presents our conclusions.
2. HILBERT SPACE AND HERMITIAN MATRICES
In this section, we introduce the notation and basic facts about complex numbers, Hilbert space, eigensystems and the perturbation of
eigensystems.
The complex number z can be represented in algebraic form or
equivalently in exponential form:
z ¼ a þ ib ¼ jzjei/
ð1Þ
with the real part of z being denoted as ReðzÞ
¼ a, the
ffi imaginary part
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
as ImðzÞ ¼ b, the absolute value as jzj ¼ a2 þ b2 , and the phase
2
as 0 / ¼ arccos ReðzÞ
jzj p, with i the imaginary unit (i ¼ 1).
z ¼ a ib denotes the complex conjugate of z. The following four equations are of special interest in this paper:
z1 z2 ¼ jz1 jjz2 jeið/1 þ/2 Þ
ð2Þ
z þ z ¼ 2ReðzÞ
z¼z
zz ¼ jzj2
if and only if
ð3Þ
z2R
ð4Þ
ð5Þ
The notation will be as follows: unless otherwise stated all numbers
are complex (2 C). Column vectors will be written in bold face x, with
the components xj ; j ¼ 1; . . . ; n. The vector space will be denoted by
V ¼ Cn . Matrices will be denoted as capital letters A with akl representing the entry in the k-th row and the l-th column. Greek letters
will denote eigenvalues with kk representing the k-th eigenvalue.
The complex conjugate transpose of a vector x will be denoted as x .
The transpose of a vector x will be denoted as xt .
The outer product of two vectors x and y is defined as:
0
1
x1 y1 . . . x1 yn
xy ¼ @ . . . . . . . . . A
ð6Þ
xn y1 . . . xn yn
Eigenspectral Analysis of Hermitian Adjacency Matrices
269
The cross product of two vectors x; y 2 R2 is defined as:
x y ¼ x1 y2 y1 x2
ð7Þ
The inner product of x; y 2 Cn is a semilinear form on a given vector
space V and will be represented as
hxjyi ¼ x y ¼
n
X
ð8Þ
xk yk
k¼1
The norm will be denoted k x k and will be defined as follows:
pffiffiffiffiffiffiffiffiffiffiffiffi
hxjxi ¼ k x k
ð9Þ
For vector space V where the norm is defined as in Equation (9) the
following rules hold:
hxjxi 0 withhxjxi ¼ 0 if and only if x ¼ 0
a2C
haxjyi ¼ ahxjyi; hxjayi ¼ ahxjyi
hx þ yjzi ¼ hxjzi þ hyjyi
hxjyi ¼ hyjxi
ð10Þ
ð11Þ
ð12Þ
ð13Þ
Hilbert space is a complete normed inner product space as defined
by Equations (8)–(9). Completeness is given because the Cauchy condition holds.
The following inequalities will also be helpful:
. Cauchy-Schwarz inequality:
jhxjyij pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi
hxjxi hyjyi
ð14Þ
(with equality if x ¼ ay; a 2 C)
. Bessel inequality: for any vector h and an orthonormal basis xk
n
X
jhhjxk ij2 k h k2
ð15Þ
j¼1
. For any vector h and a complete orthonormal basis xk the Bessel
inequality becomes Parseval’s equality:
n
X
k¼1
jhhjxk ij2 ¼k h k2
ð16Þ
270
B. Hoser and A. Geyer-Schulz
. Triangle inequality:
k x þ y kk x k þ k y k
ð17Þ
(with equality if x ¼ ay as in Eq. (14) and in addition a 2 R).
In Hilbert space the Equations (14) and (17) become equalities and
the Parseval equality Eq. (16) holds.
The adjoint space X of X is the set of all semilinear forms (Equation
(11)) (e.g., inner product) on vector space X (Kato, 1995, p. 11). A selfadjoint linear operator is called Hermitian. A matrix H is called
Hermitian, if and only if
H ¼ H
ð18Þ
with H representing the conjugate complex transpose of H. This
means that the matrix entries can be written as hlk ¼ hkl . Hermitian
matrices are also normal:
HH ¼ H H
ð19Þ
Hx ¼ kx
ð20Þ
The eigenvalue equation
of a complex Hermitian matrix H can be represented due to its complete orthonormal eigenvector system in the spectral (Fourier) representation:
H¼
n
X
kk P k ;
Pk ¼ xk xk
ð21Þ
k¼1
Pn
with Pk as orthogonal projectors. Note, that
k¼1 Pk ¼ I; Pk ¼ Pk ;
2
Pk ¼ Pk . The set of all eigenvalues is called spectrum.
The eigenvector components are interpreted as a generalized rank
prestige index of each subgroup member. In addition, each member
has for each subgroup structure=eigenvector in the spectral representation a different rank prestige index. This index depends on his
relation to the respective anchor of the subgroup.
An orthonormal basis can be chosen such that:
hxk jxl i ¼ dkl
(follows from (18))
ð22Þ
with
dkl ¼ 0
1
if k 6¼ l;
if k ¼ l
ð23Þ
Eigenspectral Analysis of Hermitian Adjacency Matrices
271
This also holds true for arbitrary rotation:
hei/k xk jei/l xl i ¼ ei/k ei/l hxk jxl i
ð24Þ
¼e
ið/l /k Þ
hxk jxl i
ð25Þ
¼e
ið/l /k Þ
dkl
ð26Þ
Hermitian matrices have full rank, and therefore all eigenvalues are
simple and real:
kk 2 R
8k
ð27Þ
because hHxjxi ¼ hkxjxi ¼ khxjxi and hxjHxi ¼ hxjkxi ¼ khxjxi and
H ¼ H imply hHxjxi ¼ hxjHxi and thus k ¼ k which means k 2 R
(see Meyer, 2000, p. 548; Kato, 1995, p. 53).
Since all eigenvalues of a Hermitian matrix are real Equation (27),
the interpretation of the eigenvalues does not pose the difficulty of
interpreting complex eigenvalues of non-symmetric real matrices.
Since the matrix H has full rank and is diagonalizable,
1 ¼ algebraic multiplicity ¼ geometric multiplicity which means that
every one of the n simple eigenvalues has but one eigenvector (see
Meyer, 2000, p. 510).
For a complex Hermitian matrix with trðHÞ ¼ 0 some eigenvalues
have to be negative due to the fact that
trðHÞ ¼
n
X
k¼1
hkk ¼
n
X
kk ¼ 0
ð28Þ
k¼1
As a special case consider a matrix B of order n þ m with
0nn A
B¼
A
0mm
ð29Þ
with A representing a n by m matrix. In the special case of n ¼ 1 and
m ¼ k this matrix represents a directed, weighted star graph. The two
non-zero eigenvalues of that system are described as follows:
rðBÞ ¼ fþk1 ; k2 g
ð30Þ
(see Meyer, 2000, p. 555) with jk1 j ¼ jk2 j.
The interpretation of negative eigenvalues poses a challenge, but
Richards and Seary (Richards, 2000, section F), Chino (Chino, 1998),
and Barnett and Rice (Barnett and Rice, 1985) offer some suggestions
for the interpretation of negative eigenvalues within the social network analysis framework.
272
B. Hoser and A. Geyer-Schulz
Richards and Seary, who use real matrices, point out that while
positive eigenvalues give information about the ‘‘clustering of interconnected nodes,’’ negative eigenvalues emphasize the ‘‘partitioning
of the network into sets of nodes that have similar patterns of connections with nodes in other sets, but few connections amongst themselves’’ (Richards, 2000, section F).
Chino uses Hilbert space theory to explain asymmetric behavior
(Chino, 1998). He constructs a complex Hermitian matrix out of a real
valued sociomatrix by using a well-known representation for square
matrices (Bronstein et al., 2000, p. 263) S ¼ Ss þ Sas which splits S
in a symmetric part Ss and an skew-symmetric part Sas . Chino starts
with a real square asymmetric similarity matrix S. He decomposes
this matrix by S ¼ 12 ðS þ St Þ þ 12 ðS St Þ ¼ Ss þ Sas where St is the
transpose of S. Chino constructs the Hermitian matrix H as
H ¼ Ss þ iSas . He uses the resulting eigenspace to explain behavior
in groups over time. The sign of the eigenvalues in the complex
eigensubspace under consideration defines the positive direction of
movement between members of the group. Formally, Chinos
pffiffi matrix
HC and the matrix H proposed here are related by HC ¼ 22 H as we
show in section 3. The difference is in the better interpretation of
the proposed method, since the orthogonal projectors can easily be
back rotated and thus yield insight into the filtered adjacency
matrices.
By virtue of Equation (16) it follows that because of
n
X
k H k2 ¼
k2k
ð31Þ
k¼1
2
the sum of all eigenvalues definesP
the total variation
in H
Pn r contained
P
n
2
(see Chino, 1998, p. 58). k H k ¼k P i¼1 ki Pi k¼ h i¼1 ki Pi j ni¼1 ki Pi i ¼
hk1 P1 jk1 P1 i þ þ hkn Pn jkn Pn i ¼ ni¼1 k2i because hPi jPj i ¼ dij .
The sign of the eigenvalues helps to identify heterogeneity in the
communication patterns. Another approach to this view was taken
by Barnett and Rice (Barnett and Rice, 1985, p. 293) who defined
the warp to explain heterogeneity. This is in the case of matrices with
trðAÞ ¼ 0 not adaptable. But still the sign gives information about the
behavior of a subgroup towards the rest of the group and the behavior
of the subgroup within itself.
The eigenvalues sorted by their absolute value jk1 j jkn j help
to identify the dominant substructures for interpretation.
For social network analysis, we can identify the most active group
member, because for jkmax j the largest absolute eigenvector component
jxmax;m j belongs to the most active group member. Since there can be
an eigenvalue with the same absolute value but different sign (see
Eigenspectral Analysis of Hermitian Adjacency Matrices
273
Equation (30)), there could also be a second eigenvector with the same
subgroup members but with a phase shift of p.
Next we assume an unperturbed Hermitian matrix A with eigenvalues a1 ; . . . ; an , with a1 an and a Hermitian perturbation matrix
E with eigenvalues E1 ; . . . ; En , with E1 En . For the eigenvalues
k1 ; . . . ; kn , with k1 kn of the resulting Hermitian matrix
H ¼ A þ E a Weyl type perturbation bound is given by
aj þ E1 kj aj þ En
ð32Þ
See (Horn and Johnson, 1990, p. 181–182), (Kato, 1995, p. 61) and
(Meyer, 2000, p. 551–552). This bound can help to analyze mixed patterns of communication.
3. THE CONSTRUCTION OF A COMPLEX ADJACENCY
MATRIX
In the following we consider communication patterns which can
be modeled as a directed, and weighted graph G ¼ fN; Eg with N
denoting the nodes or members and E denoting the edges, links or
communications between different members. Self references are
excluded.
Consider, for example, an email network. Network members
represent nodes in the graph, the links are built from the unique
message IDs and the ‘‘To’’ and ‘‘From’’ fields, whenever two authors
are linked by a message. The direction of the link is given by the
sequence of the message IDs, the weight by the number of messages
sent in this direction.
In the literature, several methods of constructing real adjacency
matrices from such graphs are being used. For example, some authors
consider only one direction in the construction process, others consider
the difference and dichotomize according, e.g., to frequency, . . . In the
process of constructing a real adjacency matrix from such a graph
information is lost.
Therefore, we propose the following construction rules for a complex
adjacency matrix H from a graph G as defined above:
1. We construct a square complex adjacency matrix A with n members
by
akl ¼ m þ ip
ð33Þ
with m the number of outbound messages from node k to node l,
and p the number of inbound messages from node l to node k and
i representing the imaginary unit. As can be seen akl ¼ ialk .
274
B. Hoser and A. Geyer-Schulz
ip
2. We rotate A by multiplying A with e 4 (Eq.2) in order to obtain a
Hermitian matrix H:
p
H ¼ A ei4
ð34Þ
Proof:
akl ¼ rei/
alk ¼ iakl ¼ irei/
akk ¼ 0 because of the exclusion of self references
hkl ¼ akl eiw ¼ rei/ eiw ¼ reið/þwÞ
p
hlk ¼ alk eiw ¼ irei/ eiw ¼ ei2 reiðw/Þ
For Hermitian matrices hkl ¼ hlk must hold, therefore,
p
reið/þwÞ ¼ reið2þw/Þ .
(g) This holds, if / þ w ¼ p2 w þ /.
(h) Solving for w leads to 2w ¼ p2. Thus w ¼ p4 :
qed
(a)
(b)
(c)
(d)
(e)
(f)
3. Under this similarity transformation the coordinate independent
characteristics of the original communication patterns is kept
(Meyer, 2000, p. 256), no information is lost.
Table 1 shows, how the following four qualitatively different types
of communication behavior remain visible after the rotation:
1. No self reference.
2. More outbound than inbound traffic ðk ! l > l ! kÞ leads to a
negative sign of the imaginary part of hkl .
3. More inbound than outbound traffic ðl ! k > k ! lÞ leads to a positive sign of the imaginary part of hkl .
4. Outbound equals inbound traffic ðk ! l ¼ l ! kÞ.
Note that the matrix is invariant under rotation.
This construction of H is related to ChinoÇs construction HC by
pffiffi
HC ¼ 22 H. To show this we set HC ¼ 12 ðB þ Bt Þ þ i 12 ðB Bt Þ and
p
H ¼ Aei2 , and we compute HC ¼ 12 ðB þ Bt Þ þ i 12 ðB Bt Þ ¼ 12 ðBð1 þ iÞ
pffiffi
pffiffi
pffiffi
p
p
p
þBt ð1 iÞÞ ¼ 22 ðBei2 þ Bt ei2 Þ ¼ 22 ðB iBt Þei2 ¼ 22 H.
TABLE 1 Communication Behavior Representation
Communication behavior
no self reference
k!l>l!k
l!k>k!l
k!l¼l!k
akl ¼ m þ ip
hkl ¼ mr þ ipr
akk ¼ 0
m>p
m<p
m¼p
hkk ¼ 0
pr < 0
pr > 0
pr ¼ 0, mr > 0
Eigenspectral Analysis of Hermitian Adjacency Matrices
275
Because of the rotation invariance of a complete orthonormal eigenvector system, we can improve the visibility of special eigenvector components by applying a rotation which makes these components real.
For example, Mathematica (Wolfram Research, 1999) automatically
rotates the eigenvector so that the eigenvector component with the
highest absolute value becomes real positive. This element is regarded
in this paper as the anchor or most influential member of a subgroup.
4. EIGENSPECTRAL ANALYSIS OF NETWORK DATA
To show how the eigensystem of a complex Hermitian adjacency
matrix reflects a communication structure, we present two different
synthetic structures and as an example of real data we take the EIES
data set (Wasserman, 1994) as well as a subset thereof as described by
Freeman (Freeman, 1997, p. 11). The eigensystems have been calculated using the function Eigensystem of Mathematica (Wolfram
Research, 1999).
4.1 Eigenspectral Analysis of a Star Graph
As a basic construct, consider a directed and weighted star graph with
5 members as in Figure 1.
The complex adjacency matrix A1 belonging to the graph in Figure 1
is:
0
1
0
1 þ 2i 2 þ 3i 2 þ i 1 þ i
B2 þ i
0
0
0
0 C
B
C
C
A1 ¼ B
3
þ
2i
0
0
0
0
ð35Þ
B
C
@ 1 þ 2i
0
0
0
0 A
1þi
0
0
0
0
FIGURE 1 Star graph with 5 vertices corresponding to Eq. (35).
276
B. Hoser and A. Geyer-Schulz
TABLE 2 Eigensystem for H1 with z ¼ a þ ib
k
5.0
5.0
0
0
0
x
0.71
0:3 þ 0:1i
0:5 þ 0:1i
0:3 0:1i
0:2
0.71
0:30 0:10i
0:50 0:1i
0:30 þ 0:10i
0.20
0
0:27 þ 0:09i
0:138 þ 0:028i
0:083 0:028i
0.94
0
0:36 þ 0:27i
0:19 þ 0:11i
0.86
0:083 þ 0:028i
0
0:72
0:64 0:08i
0:18 þ 0:13i
0:134 þ 0:045i
which after rotation becomes the Hermitian matrix H1 in Eq. (36):
0
1
0
2:1 þ 0:7i 3:5 þ 0:7i 2:1 0:7i 1:4
B 2:1 0:7i
0
0
0
0 C
B
C
C
H1 ¼ B
3:5
0:7i
0
0
0
0
ð36Þ
B
C
@ 2:1 þ 0:7i
0
0
0
0 A
1:4
0
0
0
0
The corresponding eigensystem is given in two different representations to make certain aspects more visible. As can be seen in both
Tables 2 and 3, there are two eigenvalues of the same absolute value
but with different sign. As was shown in Equation (30) this is indeed
the characteristic of the adjacency matrix of a star graph. As can also
be seen the eigenvectors belonging to the two eigenvalues are the
same in absolute values but differ by p in phase. Note, that the member with ID 1 is the center of the star graph and is indicated as such
by the real eigenvector component that has no shift in phase between
eigenvector one and two. This member can thus be described as the
most central and prestigious member of the group. It can also be
seen, that the member with ID 3 is the one that has the most contact
with the anchor within the group because he has the second-highest
absolute eigenvector component. It can also be seen that the
TABLE 3 Eigensystem for H1 with z ¼ jzjei/ðzÞ
5:0
0
5:0
0
0
k
x
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
0.71
0.32
0.51
0.32
0.20
0
2.82
2.94
2.82
p
0.71
0.32
0.51
0.32
0.20
0
0.32
0.20
0.32
0
0
0.28
0.14
0.087
0.94
undef
2.82
2.94
2.82
0
0
0.45
0.22
0.86
0.087
undef
2.50
2.62
0
2.82
0
0.72
0.64
0.22
0.14
undef
0
3.02
0.64
0.32
Eigenspectral Analysis of Hermitian Adjacency Matrices
277
members 2 and 4 are of same relevance (same absolute value of the
eigenvector component) but show opposite direction in communication (complex conjugate imaginary part), indicating that while
member 2 writes more to member 1, member 4 receives more from
member 1.
4.2 Eigenspectral Analysis of a Perturbed Star Graph
Next, let us now consider a perturbed star graph as presented
in Figure 2. The matrix A2 represents the graph shown in
Figure 2. The perturbation is given by an inbound and outbound
connection between members with ID 2 and 3 of weight 2 in both
directions.
0
1
0
1 þ 2i 2 þ 3i 2 þ i 1 þ i
B2 þ i
0
2 þ 2i
0
0 C
B
C
C
A2 ¼ B
3
þ
2i
2
þ
2i
0
0
0
ð37Þ
B
C
@ 1 þ 2i
0
0
0
0 A
1þi
0
0
0
0
which after rotation becomes the Hermitian matrix H2 in Equation
(38):
1
0
0
2:1 þ 0:7i 3:5 þ 0:7i 2:1 0:7i 1:4
B 2:1 0:7i
0
2:8
0
0 C
C
B
B
ð38Þ
H2 ¼ B 3:5 0:7i
2:8
0
0
0 C
C
@ 2:1 þ 0:7i
0
0
0
0 A
1:4
0
0
0
0
FIGURE 2 Perturbed star graph with 5 vertices as in Eq. (37).
278
B. Hoser and A. Geyer-Schulz
TABLE 4 Eigensystem for H2 with z ¼ a þ ib
4:5
k
6.2
x
0.61
0:46 0:13i
0:56 0:13i
0:21 þ 0:07i
0.14
2:5
0.70
0:26 0:01i
0:027 þ 0:068i
0.78
0:57 þ 0:07i 0:50 0:06i
0:33 0:11i
0:22 þ 0:08i
0:22
0:15 þ 0: 103 i
0.79
0
0:23 0:08i
0:34 þ 0:21i
0:25 þ 0:17i
0.70
0:42 0:14i
0
0
0
0:51 0:17i
0.85
The eigensystem belonging to matrix H2 has the form shown in
Tables 4 and 5.
As predicted by Equation (32) the eigenvalues of matrix H1 are now
disturbed. The consequence of the perturbation can be seen when we
view H2 as the sum of the unperturbed matrix H1 and the perturbation matrix P2 of the form
1
0
0
0
0
0 0
B0
0
2 þ 2i 0 0 C
C
B
B
ð39Þ
P2 ¼ B 0 2 þ 2i
0
0 0C
C
@0
0
0
0 0A
0
0
0
0 0
The eigenvalues of matrix P2 after rotation are k1 ¼ 2:8; k2 ¼ 2:8.
Thus the largest eigenvalue k1 ¼ 6:2 of the perturbed matrix H2
should be inside the Weyl-type bounds given by 5:0 þ 2:8 ¼ 7:8 6:2 2:2 ¼ 5:0 2:8 and the smallest eigenvalue k2 ¼ 4:5 should be inside
the bound given by 5 þ 2:8 ¼ 2:2 4:5 5 2:8 ¼ 7:8. This is
the case as seen in Table 4. Note, that due to the perturbation the symmetry of the eigenvalues in Table 2 is destroyed. In addition, the perturbation leads to a break in symmetry between the eigenvectors. The
eigenvector components affected by the perturbation do no longer
show the regular phase shift of p anymore. For members with ID 2
TABLE 5 Eigensystem for H2 with z ¼ ðjzj; /ðzÞÞ
4:5
6.2
2:5
0
0.79
k
x
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
jzj
/ðzÞ
0.61
0.48
0.57
0.22
0.14
0
0.27
0.22
0.32
0
0.70
0.073
0.58
0.35
0.22
0
1.19
3.02
2.82
p
0.27
0.78
0.50
0.23
0.15
3.10
0
3.03
0.37
0.04
0.25
0.40
0.31
0.70
0.44
0.32
2.60
2.54
0
0.32
0
0
0
0.53
0.85
undef
undef
undef
2.82
0
Eigenspectral Analysis of Hermitian Adjacency Matrices
279
and 3 this is visible in Table 5. However, a perturbed star graph can
still be recognized by the the existence of an anchor, which can be
identified by the eigenvector component with the highest absolute
value in two eigenvectors at the same ID position. In addition the difference in subgroup behavior can be seen. The subspace corresponding
to k1 shows a more equal distribution of absolute values then the subspace corresponding to k2 . This second subspace strengthens the star
like character of the communication.
4.3 Eigenspectral Analysis of Freeman’s EIES Data Set
To show the advantages of the method, we present the analysis of the
EIES data set as given in Wasserman and Faust (Wasserman, 1994,
p. 747). We have deleted the diagonal entries, since self reference is
not discussed.
When this data set is transformed as described in section 3, the distribution of the spectrum is given in Figure 3. For a better visualization of the symmetry inherent in the spectrum we have rearranged
the eigenvalues such that in Figure 4 the positive and negative eigenvalues kj sorted by jkj j.
From Figure 5 it is clear to see that the cumulative variance
PK 2
kk
r2c;K ¼ Pk¼1
with K < N covered by the first 5 eigenvalues is with
N
2
k¼1
kk
98% already close to 1. For this reason we will only consider the five
corresponding eigenspaces in the following analysis.
FIGURE 3 Eigenspectrum of the complete EIES data set sorted by value.
280
B. Hoser and A. Geyer-Schulz
FIGURE 4 Eigenspectrum of the complete EIES data set sorted by absolute
value and sign.
The symmetry of the spectrum suggests a star like pattern with perturbation as described in section 4.2. This is indeed the case as can be
seen in the distribution of the absolute value of the eigenvector components in Table 6. In Table 6 the eigenvectors corresponding to the first
5 eigenvalues and the 10 highest ranked members are given. As can be
seen from this table, the first ranked members of subspaces 1 and 2
FIGURE 5 Cumulative covered variance r2c;K by eigenvalues kk .
281
ID
k1
1.0
ID
k2
0.56
ID
k3
0.31
ID
k4
0.23
ID
k5
0.17
jx5l j
/5l
jx4l j
/4l
jx3l j
/3l
jx2l j
/2l
jx1l j
/1l
1
0.56
0
1
0.74
0
8
0.69
0
29
0.60
0
8
0.49
0
29
0.41
0.13
29
0.44
2.86
32
0.50
3.12
24
0.43
2.92
32
0.39
0.07
8
0.35
0.04
2
0.39
3.00
30
0.34
2.95
2
0.43
2.89
30
0.37
0.23
2
0.33
0.17
8
0.21
2.33
11
0.29
2.96
31
0.29
0.40
29
0.33
3.10
32
0.28
0.015
32
0.11
2.15
29
0.14
2.22
15
0.20
3.02
24
0.30
2.95
31
0.24
0.24
11
0.11
2.78
1
0.12
1.65
32
0.17
2.74
22
0.20
2.61
11
0.21
0.03
15
0.07
2.78
2
0.11
1.30
12
0.15
2.89
11
0.20
0.11
TABLE 6 Ranking of Authors by ID in the First Five Subspaces of the Full Data Set
24
0.16
0.09
18
0.06
3.13
9
0.10
0.45
11
0.13
2.71
2
0.19
2.82
30
0.10
0.23
26
0.06
2.74
31
0.09
0.24
14
0.13
2.82
15
0.19
3.06
27
0.09
0.24
31
0.06
1.10
17
0.06
3.09
10
0.11
2.65
14
0.14
2.73
282
B. Hoser and A. Geyer-Schulz
FIGURE 6 Distribution of jx1l j.
are the same as well as for subspaces 3 and 5, suggesting centers of
star-like patterns.
Figures 6 and 7 show the distribution of the absolute value of the
eigenvector components in the first two subspaces. The distribution
in the first subspace is more similar to a uniform distribution than
the second. This implies a connected communication pattern in the
FIGURE 7 Distribution of jx2l j.
Eigenspectral Analysis of Hermitian Adjacency Matrices
283
FIGURE 8 Distribution of /ðx1l Þ.
first subspace. The rapid decrease in the second distribution on the
other hand indicates a strong star-like pattern (around author with
ID 1) with the relevant authors with ID 29 and 2.
The distribution of the phase in the two subspaces is given in
Figures 8 and 9. As can be seen the distribution in Figure 8 is such
that the phase only varies between 0 / p4. This suggests that the
FIGURE 9 Distribution of /ðx2l Þ.
284
B. Hoser and A. Geyer-Schulz
communication between members in this pattern is balanced with
respect to direction with a little more weight on outbound communication with respect to the central member. Figure 9, on the other
hand, supports the star-like communication pattern because of the
high incidence of phase between þ 34 p and 34 p, which enhances
the relevance of the members of the star. It can also be seen that
while Wellman (ID 29) writes more often to the center, members
with IDs 2 and 8 receive more messages from the center. This is also
suggested by the spectrum and the absolute value distribution in
Figure 7. This second subspace serves as a corrective pattern for
the first subspace to show the star like structure around author
with ID 1.
In the first two subspaces we find that the group is first of all
connected and that the author with ID 1 is the major source and
focus of communication. In addition, it can be seen that one subgroup
(IDs 1, 29, 8 and 2) around the author with ID 1 is is very active and
more connected within itself than with the rest of the group. The
direction within this group is such that while the author with ID
29 is more outbound oriented, authors 2 and 8 are more inbound
oriented with respect to author 1. Members 8 and 32 show a high
similarity with respect to communication direction with the center,
as can be seen from their respective phase being close to 0 (or p).
These two (ID 8 and 32) play the major role in the third and fifth
subspaces.
Let us now consider the third and fifth subspace, since they share
the same most relevant member, namely the author with ID 8.
The distribution of the absolute values of the eigenvector components
is given in Figures 10 and 11. Figure 10 suggests a star like pattern
with one strong center around author 8 and four authors (32, 30, 29
and 24) who are relevant in that pattern. Figure 11 on the other hand
points towards a well connected pattern. This is similar to the subspaces 2 and 1.
The phase distributions given in Figures 12 and 13 support this
view. In Figure 13 it can be seen that the phase varies mainly between
0 / p4 and 3p
4 / p, which implies a balanced pattern regarding direction. Figure 12 on the other hand, shows a distribution almost
across the entire range.
In subspace 4 Wellman (ID 29) is the most relevant member.
Mullins (ID 24) and White (ID 2) show a strong similarity in pattern
behavior, which is obvious because of the same absolute value and
the same phase in comparison with the central member. The star-like
behavior can again be deduced from the distribution of the absolute
values in Figure 14 and the phase distribution in Figure 15.
Eigenspectral Analysis of Hermitian Adjacency Matrices
285
FIGURE 10 Distribution of jx3l j.
To summarize, the analysis of the EIES data set yields the following
information:
. The strongest pattern (eigenvalue k1 ) is that of an interconnected
group, with a lot of traffic centered on member with ID 1 (Freeman).
He communicates much with IDs 29, 8 and 2. This pattern states
FIGURE 11 Distribution of jx5l j.
286
B. Hoser and A. Geyer-Schulz
FIGURE 12 Distribution of /ðx3l Þ.
that a strong subgroup is connected to the complete group. This
strong pattern that covers about 60% of the total variance is most
probably because Freeman was the main communicator due to his
role as administrator in that exchange.
. The second-strongest pattern (eigenvalue k2 ) shows the behavior
within the subgroup. The subgroup is centered around ID 1 with
FIGURE 13 Distribution of /ðx5l Þ.
Eigenspectral Analysis of Hermitian Adjacency Matrices
287
FIGURE 14 Distribution of jx4l j.
IDs 29 and 8 being of next highest relevance in that subgroup. This
subgroup has a star-like communication pattern. The direction of
communication within that subgroup is also given.
. The third subspace reveals the pattern and subgroup hidden within
the main buzz of communication in the first subspaces. This pattern
now states that Bernard (ID 8) is the center of a star with members
FIGURE 15 Distribution of /ðx4l Þ.
288
B. Hoser and A. Geyer-Schulz
32, 30 and 11. Since Bernard is also part of the star around
Freeman, it may well be concluded that he serves as a link between
those subgroups.
. The fifth subspace shows that the subgroup around Bernard is again
well connected with the rest of the group.
. The fourth subspace shows that Wellman (ID 29) is the center of the
star with Mullins (ID 24) and White (ID 2) as strong members. This
again suggests the idea that Wellman is the link between the subgroup around Freeman and the people connected to himself in his
subgroup. The direction of communication is from Wellman to his
subgroup members.
The members with IDs 1 (Freeman), 29 (Wellman), 8 (Bernard),
2 (White) and 24 (Mullins) also play a major role in the subset analysis
of the next section.
4.4 Eigenspectral Analysis of a Subset of Freeman’s
EIES Data Set
As a smaller data example, consider the subset of EIES data given in
(Freeman, 1997, p.11). The difference here in comparison to the full
data set is that the one-to-all communication has been eliminated
(Freeman, 2004). According to (Freeman 2004), we give for each member of the subset his ID in the subset and the complete set to link the
two data sets: Freeman (ID 1,1), White (ID 2,2), Alba (ID 3,4), Bernard
(ID 4,8), Doreian (ID 5,11), Mullins (ID 6,24), and, last but not least,
Wellman (ID 7,29). In the rest of this section only names will be used
when referring to members of the subset. In Table 7 the outbound
numbers of messages between the member of the group are given.
TABLE 7 Number of Message Exchanged between Members in the
EIES Sub Set
Author
ID
Freeman
1
White
2
Alba
4
Bernard
8
Doreian
11
Mullins
24
Wellman
29
Freeman
White
Alba
Bernard
Doreian
Mullins
Wellman
1
2
4
8
11
24
29
0
84
16
127
57
23
118
115
0
10
22
9
4
24
17
4
0
17
4
3
5
93
5
15
0
57
9
35
53
5
3
57
0
8
15
33
0
3
12
8
0
45
84
15
4
34
10
33
0
Eigenspectral Analysis of Hermitian Adjacency Matrices
289
The resulting complex adjacency matrix F is given in Equation (40):
0
0
B 84 þ 115i
B
B
B 16 þ 17i
B
B
F ¼ B 127 þ 93i
B
B 57 þ 53i
B
B
@ 23 þ 33i
118 þ 84i
115 þ 84i 17 þ 16i
93 þ 127i 53 þ 57i
4 þ 10i
5 þ 22i
10 þ 4i
0
15 þ 17i
3 þ 4i
3 þ 3i
22 þ 5i
17 þ 15i
0
57 þ 57i
12 þ 9i
9 þ 5i
4 þ 3i
57 þ 57i
0
8 þ 8i
4
3 þ 3i
9 þ 12i
8 þ 8i
0
24 þ 15i
5 þ 4i
35 þ 34i
15 þ 10i
45 þ 33i
0
5 þ 9i
33 þ 23i 84 þ 118i
4i
1
15 þ 24i C
C
C
4 þ 5i C
C
C
34 þ 35i C
C
10 þ 15i C
C
C
33 þ 45i A
0
ð40Þ
The eigensystem of matrix F after rotation is given in Tables 8 and 9.
The results suggest the following interpretation:
. Freeman is the center of a very active star. About 93% of the total
communication level variance (Equation 31) is explained by this
structure. This corresponds to the results in section 4.3.
To see this we identify ID 1 as the anchor of eigenvectors x1 and x2
in Table 9 and compute the explained variance of this substructure
P
with k21 þ k22 = 7i¼1 k2i approximately 93%. Main corresponding partners are White, Bernard and Wellman, which also corresponds to
the previous results in Table 6. Only the ranking differs slightly.
To check for a star pattern we decomposed the matrix F into
the unperturbed ‘‘Freeman-star’’-matrix G, with g1l ¼ f1l and
gk1 ¼ fk1 for all k; l and gk;l ¼ 0 for all others, and a perturbation
matrix D with d1l ¼ 0; 8l and dk1 ¼ 0; ; 8k and dkl ¼ fkl ; 8k ¼
f2; . . . ; 7g; l ¼ f2; . . . ; 7g. We showed, that for the eigenvalues of
the F ¼ G þ D the Weyl-type bounds hold, which is an indication
that we really have correctly identified a star-shaped substructure.
. The first eigenvalue k1 ¼ 340 suggests, that this represents the communication behavior that Freeman and his subgroup show with
respect to the whole group. It points towards a strong connected
group which is again connected to the rest. Whereas the sign of
the second eigenvalue k2 ¼ 230 seems to point towards the fact
that the subgroup forms a star graph. These results are suggested
by the distribution of the absolute value of the eigenvector component. In addition from the phase information it can be seen that
the the direction of communication is different between White on
the one hand and Wellman and Bernard on the other in regard to
Freeman. While Bernard and Wellman are more outbound oriented,
White is more inbound oriented.
. In the eigenvector x3 the anchor is Bernard. His corresponding
partners are Doreian and Wellman. Bernard is at the center of a
link between Wellman and Doreian with almost equal amounts of
290
230
0.76
0.39 0.05i
0.011 0.012i
0.38 þ 0.06i
0.079 0.011i
0.013 0.032i
0.33 þ 0.07i
340
0.62
0.33 þ 0.06i
0.098 0.008i
0.45 0.07i
0.29 0.02i
0.17 þ 0.01i
0.40 0.06i
kk
xkl
p
55
0.073 0.021i
0.55 þ 0.07i
0.015 þ 0.004i
0.20 þ 0.03i
0.39 0.02i
0.55
0.53 þ 0.13i
90
0.056 0.005i
0.018 0.057i
0.121 þ 0.003i
0.65
0.55 þ 0.02i
0.22 þ 0.001i
0.45 þ 0.03i
TABLE 8 Eigensystem for Fei4 with z ¼ a þ ib
0.027 0.039i
0.12 0.16i
0.16 þ 0.03i
0.35 þ 0.15i
0.47 þ 0.25i
0.56
0.42 0.11i
25
0.15 0.001i
0.69
0.118 0.037i
0.18 þ 0.1i
0.32 þ 0.21i
0.51 0.11i
0.16 0.04i
13
0.059 0.026i
0.106 0.059i
0.97
0.001 þ 0.015i
0.14 þ 0.02i
0.15 þ 0.05i
0.002 þ 0.014i
2.9
291
p
/ðzÞ
0
0.18
0.09
0.15
0.07
0.05
0.15
jzj
0.62
0.34
0.098
0.45
0.29
0.17
0.41
kk
xkl
340
0.76
0.39
0.016
0.38
0.080
0.034
0.34
jzj
0
3.00
2.32
3.00
3.00
1.97
2.93
/ðzÞ
230
0.056
0.06
0.121
0.65
0.56
0.22
0.45
jzj
jzj
0.076
0.44
0.016
0.21
0.39
0.55
0.55
/ðzÞ
0.09
1.88
3.12
0
3.10
0.02
3.07
90
TABLE 9 Eigensystem for Fei4 with z ¼ ðjzj; /ðzÞÞ
/ðzÞ
2.86
0.15
0.29
2.99
0.04
0
2.91
55
0.047
0.20
0.16
0.38
0.53
0.56
0.44
jzj
25
0.96
0.92
2.93
2.75
2.64
0
0.25
/ðzÞ
0.15
0.69
0.12
0.20
0.38
0.52
0.16
jzj
/ðzÞ
0.02
0
0.30
2.64
2.56
2.94
2.89
13
0.064
0.12
0.97
0.015
0.14
0.16
0.014
jzj
/ðzÞ
2.73
2.63
0
1.67
3.03
0.34
1.73
2.9
292
B. Hoser and A. Geyer-Schulz
communication between Bernard and his two partners. They cover
about 4%.
This result differs slightly from the previous results, since authors
with ID 32 and 30 are not in the subset data. Apart from that
Doreian and Wellman also are strong partners in the full data set.
. In eigenvector x4 and x5 Mullins is the center of a star. His star
though explains only about 2% of the total communication level.
In addition he communicates mostly with Wellman, which can be
seen in the fact that the absolute value of the corresponding eigenvector components are very close together.
To summarize, these examples show:
1. The eigenvalues of F show the perturbation behavior of a matrix as
in Equation (29).
2. The eigenvalues react to the perturbation in that they split asymmetrically according to Equation (32).
3. The eigenvalues can be used to show how much of the total variation is covered by one subgroup Equation (31).
4. The eigenvectors belonging to two distinct, but coordinated eigenvalues show the same subgroup members, maybe slightly in different order of relevance depending on the kind of perturbation. See
Table 9 columns x4 and x5 for the ‘‘Mullins-star.’’
5. The largest absolute value eigenvector component relates to the
most relevant subgroup member. See last paragraph in section 3.
6. Since for dominant star graph-like structures the eigenvalues come
in pairs of the same absolute value but different sign, which means
their subspaces are rotated by p, the eigenvector components also
show this rotation and if the matrix is perturbed the components
will also have a perturbation of their phase.
5. CONCLUSIONS
We showed that the proposed eigensystem analysis of Hermitian
adjacency matrices is an improvement in comparison to the existing
analysis tools based on the rank prestige index as defined in
Wasserman and Faust (Wasserman, 1994, p. 205-219) in that the proposed method describes the complete substructure of a group by virtue
of the use of all eigenpairs based on their relevance as defined by their
explanatory strength in terms of variance and in that no arbitrary
assumptions about the symmetrization of real adjacency matrices
are used. It also seems to support the notion that not only group
structure but also core=periphery questions might be solved with this
Eigenspectral Analysis of Hermitian Adjacency Matrices
293
method. The additional phase information might further help to define
in which direction the communication flows.
We presume, that the behavior of the eigenvalues and their
respective eigenvector components is comparable to the behavior of
a perturbed matrix of the form H ¼ A þ E, even if we do not know
the exact form of the unperturbed matrix. We tentatively decompose
the matrix in its subspaces according to the identified dominant substructure from the spectral decomposition of the perturbed matrix
and check if the Weyl-type bounds still hold for this decomposition.
If this is true, we take this as an indication that we correctly identified a substructure. We do not yet know whether this can be done
repeatedly.
The analysis of the data sets showed that the proposed method
offers consistent interpretation of a well-known data set. In addition,
if offers more information regarding direction of communication with
groups and between subgroups. To investigate this further we plan
to design and analyze communication networks in such a way as to
gain a deeper understanding of the possibilities. One promising communication network is a market, where the trading behavior can be
modeled in such a way that it resembles an asymmetric communication network. We have set up two types of markets, namely election
markets (http://psm.em.uni-karlsruhe.de/psm/) and technology forecast markets (http://tfm.em.uni-karlsruhe.de/psm/) and we plan to
analyze the data resulting from these markets with the method proposed in this market. Further application might be e-mail traffic
within an organization. Thus there is a high potential in the method
to gain further insight into the communication behavior and its
relevance in respect to group structure.
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