3.6 Normalised matrices
3.6.2 The Normalised adjacency matrix
Definition 3.40
LetGbe a graph such that every vertex has at least one outgoing edge. Then the normal matrix N (G) is defined as:
N(G) :=−→
D(G)−1A(G), where −→
D(G) denotes the out–degree matrix of G, this is a diagonal matrix which has the entries deg→(·). The normal spectrum of G is defined as:
ΛN (G) := Λ (N(G)).
The normal matrix is also called thenormalised adjacency matrix. SinceN is in general not symmetric its spectrum need not be real. We have:
Lemma 3.41
Let G be an graph on n vertices such that every vertex has at least one outgoing edge. Then its normal matrix is stochastic. It has1 as eigenvalue.
IfG is undirected we gain additionally:
N(G) = In− D(G)−12 LN(G)D(G)12 =In− D(G)−1L(G) (3.17) Proof:
The normal matrix is nonnegative since it is the product of nonnegative matrices (proposition 3.5). Let v be a vertex. The adjacency matrix A(G) has exactly deg→(v) entries with value 1 and all other entries are 0 in the v–th row. So every row sum ofN(G) equals 1 and therefore it is stochastic.
SinceN is stochastic 1 is an eigenvalue.
IfG is undirected we have L(G) =D(G)− A(G), therefore we have N(G) = D(G)−1
D(G)− L(G)
= In− D(G)−1L(G)
= In− D(G)−1
D(G)12 LN(G)D(G)12
= In− D(G)−12 LN(G)D(G)12
Corollary 3.42
LetGbe an undirected graph such that every vertex has at least one incident edge. Then its normal spectrum is real and (µ, w) is an eigenpair of N (G) iff LN(G)has (1−µ,D(G)−12 w) as an eigenpair.
Proof:
It is sufficient to show the equivalence, sinceLN(G) has a real spectrum. Let B =D(G)12. The matrix B−1LN(G)B has the same eigenvalues asLN(G) and v is an eigenvector of LN(G) iffB−1v is an eigenvector of B−1LN(G)B.
Therefore equation (3.17) implies the equivalence.
Since every stochastic has spectral radius one, every eigenvalue is located in the unit disc in the complex plane. We use the notation
µ1, . . . , µn−1, µn= 1
for the eigenvalues the µs are sorted non descending with respect to their real parts. If G has no bipartite connected component we have |µj| <1 for all j = 1, . . . , n−1.
Corollary 3.43
Let G= (V, E)be an undirected graph with δ(G)≥1. Two eigenvectors to different eigenvalues of N(G) are orthogonal with respect to the following scalar product:
hx, yiN :=X
v∈V
(degv)·[x]v[y]v (3.18) Proof:
Since LN (G) is symmetric we know by theorem 3.21 that eigenvectors to different eigenvalues are orthogonal (with respect to h,i1). By corollary 3.42 we know that ifvis an eigenvector ofLN(G) thenD(G)−12 vis an eigenvector of N (G). Let x0, y0 be two eigenvectors of N(G) to different eigenvalues and x:=D(G)−12 x0 and y:=D(G)−12 y0. Then we have:
Sincexand y are eigenvectors ofLN(G) to different eigenvaluesx andy are
orthogonal and thereforehx, yi1 = 0.
The normal matrix can be used as a model for random walks in a graph. In section5.1 we describe this model in an application background.
Cluster analysis
In this chapter we give a brief overview of cluster analysis. We are espe-cially interested in graph clustering respectively clustering with methods from graph theory. The first section considers problem settings and aims of clustering. Then we present some selected methods and discuss advan-tages and disadvanadvan-tages. In the last section we deal with a graph clustering method using spectral methods.
4.1 Problem settings and purpose
What is clustering? Giving an informal definition we could say:
Cluster analysis is the study of recognizing natural groups within a class of entities with the help of mathematical tools.
The weak point of this definition is of course the term natural groups. In the past many different approaches have been made to formalise this term.
This is one reason why today there exists a toolbox of different procedures rather than one method. Summarising we give the following general problem definition:
Problem 4.1 (general clustering)
Input: a set C of entities, a weighted binary relation on C and a cost function for partitions on C
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Output: a partitionP ofC such that P has minimal cost with respect to all partitions of C
The partition P is also called a cluster and its components thecluster com-ponents. One detail which is hidden in this kind of problem description is the role of the binary relation. Its purpose is measuring the degree of rela-tion between two entities. This can be done in a positive way, for example expressing similarity, or in a negative one, for example modeling dissimilarity or differences. Throughout the rest of this paper we consider only similar-ity as weighted binary relation between the entities. The binary relation is usually a complete relation. Problem 4.1 can also be seen as a maximising problem, if the cost function is replaced by a weight function or something similar.
Before we describe certain cost or weight functions in detail we state why we think that cluster analysis is a fundamental principle in science. Many scien-tific disciplines deal with the problem of exploring huge data sets. A general interest is to identify structures within the entities. Consider for example the classification of animals and plants in biology. Using clustering tech-niques one can reduce the complexity of the exploration space and identify local relations between the entities. In disciplines like biology, psychology, sociology, linguistics, geology and many more there is a general principle to identify new classifications:
1. Examine a small data space and identify certain structures or relations.
2. Formulate a hypothesis. The thesis consists of two parts: first, variables which influence the result, and second, the connection between result and variables.
3. Try to prove or disprove the hypothesis. This can be done for example by searching for a counter–example or probabilistically by considering a larger data space.
4. If the hypothesis was correct, then a new law was found, else reenter step 1or 2and modify current results.
In this scheme cluster analysis is often used in step 3and sometimes also in step 1. For a more detailed introduction into cluster analysis consider for example [JD88], [Zup82], [And73] or [Har37].