DIRECTED GRAPH LIMITS AND DIRECTED ...

DIRECTED GRAPH LIMITS AND DIRECTED THRESHOLD GRAPHS

by

Derek Boeckner

A DISSERTATION

Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy

Major: Mathematics

Under the Supervision of Professor Jamie Radcliffe

Lincoln, Nebraska December, 2012

DIRECTED GRAPH LIMITS AND DIRECTED THRESHOLD GRAPHS Derek Boeckner, Ph. D. University of Nebraska, 2012 Adviser: Jamie Radcliffe

iii

Contents

Contents

iii

List of Figures

v

1 Introduction

1

2 Introduction to graph limits

2

2.1

Basic Definitions and Background . . . . . . . . . . . . . . . . . . . . . . . .

2

2.2

Convergence and Correspondence . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

W -random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3 Directed Graph Limits 3.1

3.2

9

Basic Definitions and Background for Main Proof . . . . . . . . . . . . . . .

10

3.1.1

Digraphs and Digraph Homorphisms . . . . . . . . . . . . . . . . . .

10

3.1.2

The Regularity Lemma . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.1.3

The Intricacies of the Main Theorem . . . . . . . . . . . . . . . . . .

15

3.1.4

W -random Graphs and Concentrations of Homomorphism Densities .

20

Convergence and Correspondence . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2.1

Relationship Between Directed Graph Limits and Graph Limits . . .

28

3.2.2

Induced, Injective, and Specified Homomorphism Densities . . . . . .

29

4 Threshold and Directed Threshold Graphs

31

4.1

Threshold Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv 31

4.2

Directed Threshold Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.2.1

Background and Directed Threshold Equivalence . . . . . . . . . . .

35

4.2.2

Adjacency Matrices of Directed Threshold Graphs . . . . . . . . . . .

39

4.2.3

Canonical Form of Defining Sequences for Directed Threshold Graphs

40

5 Limits of Threshold and Directed Threshold Graphs

43

5.1

Limits of Threshold Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

5.2

Probing for Information in Adjacency Functions . . . . . . . . . . . . . . . .

44

5.3

Limits of Directed Threshold graphs

46

. . . . . . . . . . . . . . . . . . . . . .

6 Population Projection Matrices

53

6.1

Mathematical Biology Background . . . . . . . . . . . . . . . . . . . . . . .

53

6.2

The Leading Eigenvalue of a PPM . . . . . . . . . . . . . . . . . . . . . . . .

56

Bibliography

70

v

List of Figures 2.1

W (3) (x, y): Black = 1; White = 0 . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.1

The threshold graph corresponding to the sequence (1, 1, 0, 1, ?) . . . . . . . . .

32

4.2

The limit function W(x,y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.3

Adjacency matrix of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ?) . . . . . . . .

34

4.4

Directed graph from the sequence (1, −1, 0, −1, ?) . . . . . . . . . . . . . . . . .

39

4.5

The adjacency matrix of the directed threshold graph given by the sequence (+ − 0 − 0 + − + 00 − 000 − − − + − ?)

. . . . . . . . . . . . . . . . . . . . .

40

5.1

Uniformly randomly chosen sequence s30 ∈ {0, +, −}30 . . . . . . . . . . . . . .

48

5.2

Uniformly randomly chosen sequence s300 ∈ {0, +, −}300

. . . . . . . . . . . . .

48

5.3

Uniformly randomly chosen sequence s3000 ∈ {0, +, −}3000 . . . . . . . . . . . . .

48

5.4

The limit object W (x, y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

6.1

Perturbations leading to specified maximum eigenvalues. . . . . . . . . . . . . .

65

1

Chapter 1 Introduction A 2004 paper by Lov´asz and Szegedy [5], introduced the concept of graph limits. We’ll describe them here in the introduction before generalizing them to directed graphs in Chapter 2. These limits put a topology on the set of finite graphs and their limit points. Thinking about this topological space, we can look at the closure of a class of graphs. Diaconis, Holmes, and Janson do exactly this with the class of threshold graphs. The limit points of these graphs have a very nice form, they are characteristic functions of increasing sets. In Chapter 4, we develop a class of directed graphs, which naturally corresponds to the class of threshold graphs. In Chapter 5, we look at the limits of this class of directed graphs, and develop a few examples. In Chapter 6, we diverge from graph theory and look at a little bit of math biology. Here we give a mathematical definition of a population projection matrix and examine how perturbations to the growth, survival, and fecundity rates affect the long term growth rates of the population. This has applications to the survival of endagered species and to the prevention of invasive species. There are several examples within this chapter.

2

Chapter 2 Introduction to graph limits Here we start by developing the topology we’re dealing with for undirected graph limits. Lov´asz and Szegedy introduce a notion of what it mean for a sequence of grahps, (Gi )∞ i=1 to converge in [5]. The essence of the notion is that a sequence converges if all structural densities of the Gi converge. For instance, the edge densities must converge, but also the densities of triangles, paths of length 137, cliques of size 74, etc converge. This chapter makes these notions precise and gives us a better feel for graph limits.

2.1

Basic Definitions and Background

To fully use the power of graph limits, we need to define edge weighted graphs, where the weights on the edges correspond to probabilities. This then lets us think about homomorophism probabilities in a more general sense. Definition 2.1.1. An edge weighted graph is a triple, G = (V, E, w) where V is the set of vertices of G, E ⊂ V 2 / (where {x, y} {y, x}) is the set of edges of G, and w : E → [0, 1] is the edge weight function of G. This definition allows for loops but not multiple edges.

Generally, we can think of the edge set being all 2-sets of V with edges we don’t wish3 to consider getting weight 0. Any unweighted graph can be thought of as a weighted graph where edges in the graph have weight 1 and edges not in the graph have weight 0. The weights on the edges can correspond to probabilities that the edge is included in the graph. With this in mind we can think of homomorphism densities in the following manner. Let’s begin by defining the homomorphism density sequence associated with a finite edge weighted graph. Definition 2.1.2. Let G be an edge weighted graph with weight function w. For a finite simple graph F , let ϕ : V (F ) → V (G). The probability that ϕ is a homomorphism is Y

homϕ (F, G) :=

w(ϕ(u), ϕ(v)).

uv∈E(F )

The homomorphism function is defined to be X

hom(F, G) =

homϕ (F, G).

ϕ:V (F )→V (G)

and the homomorphism density of F in G is defined as

t(F, G) =

hom(F, G) . |V (G)||V (F )|

Definition 2.1.3. Let F be the set of all finite graphs and let G be a finite edge weighted graph. The homomorphism density sequence of G is the sequence (t(F, G))F ∈F . For short we just say the density sequence of G. When we want to talk about the sequence without referencing the graphs F we will denote the sequence t(G), and the coordinate t(F, G) will be denoted tF (G). Notice that this sequence lives in the space X = [0, 1]F . Before we continue to define what it means for a graph sequence to converge, let’s first look at a clever way to calculate t(G) by looking at the adjacency matrix of G.

Definition 2.1.4. The adjacency matrix of a finite edge weighted graph G on n vertices is4 the (n × n) square matrix with rows and columns indexed by the vertices of G and entries defined by A(u, v) = w(uv). Notice that this makes A symmetric. We can think of this in another way as well. Labeling the vertices as 1, 2, 3, . . . , |V (G)| and scaling the matrix to fit in the unit square we can think of the adjacency matrix as a function from [0, 1]2 to{0, 1}. This function, WG : [0, 1]2 → [0, 1] is defined as follows:

WG (x, y) := A(dnxe, dnye). We call this the adjacency function of G. With this in mind, let’s revisit he definition of homomorphism density and see how it relates to the adjacency function. First, we have the basic relationship

homϕ (F, G) =

Y

A(ϕ(u), ϕ(v)).

uv∈E(F )

Now extending this to the density definition and doing a little rearrangement and thinking about the adjacency function and product geometrically we get the following sequence of equalities: X

t(F, G) =

ϕ:V (F )→V (G)

X

=

homϕ (F, G) |V (G)||V (F )| Y

ϕ:V (F )→V (G) uv∈E(F )

Z =

Y

A(ϕ(u), ϕ(v)) |V (G)|

WG (xu , xv )d¯ x

(1.1)

[0,1]|V (F )| uv∈E(F )

This now gives us enough background to move to the next stage and state what it means

5

for a graph sequence to converge.

2.2

Convergence and Correspondence

Definition 2.2.1. A graph sequence (Gi )∞ n=1 then is said to converge if the homomorphism density sequences, (t(F, Gi ))F ∈F converge pointwise as i goes to infinity. That is limi→∞ tF (Gi ) exists. Succinctly, a graph sequence converges if the homomorphism densities in Gi of all finite graphs F converge. This is to say, the graphs converge in a structural sense. Topologically, t : F → X, and we say a graph sequence, Gi converges if t(Gi ) converges in the box topology on X. The natural question then, is ‘what is this limiting object?’ It’s certainly no longer a graph. It is the collection of limiting densities, t∞ (F ) = limi→∞ t(F, Gi ). But, we had a nice convergence between the densities of finite graphs and the adjacency function; it would be nice if there were an adjacency function for the graph limit as well. Lov´as and Szegedy prove that there is an adjacency function associated to the graph limit. Theorem 2.2.2. [5] If (Gi )i≥1 is a sequence of graphs, then limi→∞ t(F, Gi ) converges for all F if and only if there exists a symmetric measurable W : [0, 1]2 → [0, 1] such that Z lim t(F, Gi ) =

i→∞

Y

W (xs , xt )dx1 dx2 . . . dxk .

[0,1]k st∈E(F )

With this in mind we have the following definition: Definition 2.2.3. Let W : [0, 1]2 → [0, 1] be a symmetric measureable function. For a finite graph F , let k = |V (F )|. Define Z t(F, W ) =

Y

[0,1]k −−→ xi xj ∈E(F )

W (xi , xj )dx1 dx2 . . . dxk .

This extends the concept of the adjacency matrix to the limit. A quick example might6 help with understanding. Example 2.2.4. Let Gn,p be the finite weighted graph on n vertices without loops where each edge is given weight p. The adjacency matrix for Gn,p has all entries of p except along the diagonal, where the entries are 0. The limiting function then is nearly constant; WGn,p (x, y) = p except for a narrow band a long the line x = y (which only gets smaller as n increases.) We expect then that the limiting function for the limit object is the constant function W (x, y) = p. (Perhaps we would have liked W (x, x) = 0 but the line x = y has measure 0, this doesn’t change the integral and thus is not necessary.) This example illuminates a slight inconvenience. The function W from Theorem 2.2.2 is not unique. The example and theorem make it clear that is not unique up to equivalence almost everywhere. However, even equivalence almost everywhere is not strong enough to gain uniquness. Let’s look again at the adjacency matrix of a finite graph. Permuting the index of the rows and columns (with the same permutation) of the underlying adjacency matrix gives a new adjacency matrix but does not change the calculations of the homomorphism densities. This idea of a measure preserving transformation carries through to the limiting function as well. With these ideas in mind, it will be convenient to name a limiting function of a convergent graph sequence. Definition 2.2.5. An adjacency function of a convergent sequence, (Gn ), is a function W : [0, 1]2 → [0, 1] with the property that t(F, W ) = limn→∞ t(F, Gn ). Lov´asz and Szegedy do not say much more other than W and U are essencially the same limit objects if they satisfy t(F, W ) = t(F, U ) for all finite graphs F . Theorem 2.2.2 gives an alternate means of testing conversion via finding a W such that t(F, W ) is the limit of the density sequence of t(F, Gi ). What is more is that from a symmetric W we can find a convergent sequence of graphs that has W as its limit. To construct this

sequence we can look to W itself and probabilistically construct Gi . This construction is7 explored in the next section.

2.3

W -random Graphs

The proof that a convergent sequence exists for any measureable W uses the idea of a W random graph. Definition 2.3.1. [5] Let W : [0, 1]2 → [0, 1] be a symmetric measureable function. A W -random graph on n vertices, Gn,W , is formed in the following manner. Let (Xi )ni=1 be chosen uniformly at random from [0, 1]. Set V (Gn,W ) = [n]. The edge (s, t) is in E(Gn,W ) with probability W (Xs , Xt ) independent from all other edges. This can be extended to produce random graphs with specified proportions of vertices with specified densities. A more straightforward example would be that constant functions, say W (x, y) = p correspond to the random graphs with edge density p, i.e., Gn,p . (See Example 2.2.4.) Let’s look at another example: Example 2.3.2. Fix k ∈ N. Let

W (k) (x, y) =

   1

if dxke = 6 dyke,

  0 else. Then for k = 3 we get the following graph. Now, for n >> 0 the vertices of Gn,W are partitioned into 3 classes of approximately n/3 vertices each corresponding to A = {xi < 1/3} , B = {1/3 < xi < 2/3} , and C = {2/3 < x} . There are no edges within the sets, and all edges between the sets. This means Gn,W is approximately the Tur´an graph with 3 classes on n vertices, T (n, 3). It is not difficult to show on the other hand that T (n, k) converges to W (k) as n → ∞.

1

y

8

2/3 1/3 (0,0)

1/3 2/3

1

x

Figure 2.1: W (3) (x, y): Black = 1; White = 0

It should be pointed out that given a W , the sequence of W -random graphs, Gn,W converge to W almost surely. This is how Lov´asz and Szegedy find a sequence that converges to W . This is the essence of Corollary 2.6 in [5]. Theorem 2.3.3. [5] The graph sequence Gn,W is convergent with probability 1, and its limit is the function W . Remark 2.3.4. The graphs Gn,W are non-weighted graphs. The graph convergence to W then comes from a sequence of non-weighted graphs. Since this is true in any case, we can always think of graph limits as coming from a sequence of non-weighted graphs. Having the generality of being able to use weighted graphs is nice, but not necessary to get the correspondence theorem from [5].

9

Chapter 3 Directed Graph Limits Now that we thoroughly understand graph limits of undirected graphs, we can generalize them to the directed case. We start again by building up the basic definitions and background and then show that we have similar results in the undirected case as Lov´asz and Szegedy showed in the directed case. We get to use the same topology of pointwise convergence on X but we need to take directed graph homomorphism densities, and use adjacency matrices that now are not necessarily symmetric. The exact statement of the our main theorem is here. We will build up the machinery to prove it in the following section. The proof will come in Section 3.2. Theorem 3.0.5. Let Gn be a sequence of digraphs such that t(F, Gn ) converges for every finite digraph F . Then, there is a measureable W : [0, 1]2 → [0, 1] such that Z lim t(F, Gn ) =

n→∞

Y

W (xi ,j ) dx1 dx2 . . . dx|V (F )| .

[0,1]|V (F )| −−→ xi xj ∈E(F )

Conversely, given a measureable W : [0, 1]2 → [0, 1] there exists a sequence of digraphs Gn such that, Z lim t(F, Gn ) =

n→∞

Y

[0,1]|V (F )| −−→ xi xj ∈E(F )

W (xi , xj ) dx1 dx2 . . . dx|V (F )| .

3.1

10 Basic Definitions and Background for Main Proof

We should start with some basic preliminaries. We give the definition of a directed graph and digraph, and then the definition of the adjacency matrix that will suit our purposes for finding a limit object. Recalling Remark 2.3.4, we could define weighted versions of these graphs, and develop the weighted convergence theorems, but in the end, we have that we may as well have started with unweighted graphs and used the more familiar basic notation. We develop the basic notation and definitions in the first subsection. In the next subsection we give a version of Szemer´edi’s Regularity Lemma. The follow subsection will contain several lemmas used in proving one direction of the main theorem. In the subsequent subsection, we develop the idea of W -random digraphs and prove some concentration results that let us prove the other direction of the main theorem.

3.1.1

Digraphs and Digraph Homorphisms

Definition 3.1.1. A digraph is a pair (V, E) where V is the vertex set and E ⊂ V 2 is a subset of ordered pairs. We think of the pair (x, y) as the directed edge x → y. Definition 3.1.2. The adjacency matrix of a digraph G = (V, E) is the |V | × |V | {0, 1}matrix Aij =

   1 if (i, j) ∈ E   0 if (i, j) ∈ /E

The results of this section take sequences of digraphs and show equivalences for them to converge in the same sense of Lov´asz and Szegedy. Definition 3.1.3. Given two digraphs G = (V (G), E(G)) and H = (V (H), E(H)) we say a function ϕ : V (H) → V (G) is a (digraph) homomorphism if for all (x, y) ∈ E(H) we have that (ϕ(x), ϕ(y)) ∈ E(G).

11 Since we are mostly concerned with homomorphism densities, we define a few density functions next. Definition 3.1.4. Let G, H be two digraphs on finite vertex sets. Say |V (G)| = m and |V (H)| = n. Let Hom(H, G) = {ϕ : V (H) → V (G) : ϕ is a homomorphism} . Similarly, let Hominj (H, G) = {ϕ : V (H) → V (G) : ϕ is an injective homomorphism} . Define

t(H, G) :=

| Hom(H, G)| . mn

Define tinj (H, G) :=

| Hom0 (H, G)| . (m)n

I.e., t(H, G) is the probability that a uniformly randomly chosen map from V (H) → V (G) is a homomorphism, and tinj (H, G) is the probability that a uniformly randomly chosen injective map is a homomorphism. The following relationship between t(H, G) and tinj (H, G) is taken directly from Lemma 2.1 in [5] and needs no translation to the directed case, it is based purely on counting and the proof is identical: Lemma 3.1.5. [5] For digraph H, G we have   |V (H)| 1 . |t(H, G) − tinj (H, G)| < 2 |V (G)| Now, just as in the undirected case,Definition 2.2.1, we state the definition of convergence of directed graphs: Definition 3.1.6. Let (Gn )∞ n=1 be a sequence of directed graphs. We say the sequence converges if (t(H, Gn ))∞ n=1 converges for every finite directed graph H. We have that the definition of the adjacency function transfers by removing symmetry as well.

12

Definition 3.1.7. WG (x, y) := A(dnxe, dnye). We call this the adjacency function of G.

And the same calculation of homomorphism density from the adjacency function transfers, (Equation 1.1.)

X

t(F, G) =

ϕ:V (F )→V (G)

X

=

homϕ (F, G) |V (G)||V (F )| Y

ϕ:V (F )→V (G) uv∈E(F )

Z

Y

=

A(ϕ(u), ϕ(v)) |V (G)|

WG (xu , xv )d¯ x

[0,1]|V (F )| uv∈E(F )

3.1.2

The Regularity Lemma

The original proofs of Lov´asz and Szegedy use the basic Szemer´edi regularity lemma, [10], for dense graphs. Since we are working with digraphs, we will need slightly different notions of regularity and the regularity lemma. We use Scott’s version provided in [8]. We need some notation for the regularity lemma we will be using. Given a matrix A = (aij ) (not necessarily square), with rows indexed by V and columns indexed by W . We write X

kAk =

|aij |.

i∈V, j∈W

For X ⊂ V, Y ⊂ W , we write

wA (X, Y ) =

X v∈X, w∈Y

|avw |.

We say that the density of the submatrix AX,Y (with rows X and columns Y ) is

dA (X, Y ) =

13

wA (X, Y ) . |X||Y |

Definition 3.1.8. We say a submatrix AX,Y is -regular if for all X 0 ⊂ X and Y 0 ⊂ Y with |X 0 | ≥ |X| and |Y 0 | ≥ |Y |, we have

|dA (X 0 , Y 0 ) − dA (X, Y )| ≤ .

This means that the block has near uniform density throughout. In terms of a graph, we’re selecting two sets of vertices and looking at the edge density between them. Taking large enough subsets (more than a proportion  of the total vertices in the sets), we have nearly the same density between the subsets as we do between the original sets. We now define some vocabulary so that we may understand the statement of the regularity lemma we will use. Definition 3.1.9. A block partition (P, Q) of A is a partition P of V together with a partition Q of W ; the blocks, (X, Y ), of the partition are the submatrices AX,Y (these submatrices, in fact, partition the matrix A into blocks.) Definition 3.1.10. We call a block partition balanced if for all X, Y ∈ P we have ||X|−|Y || ≤ 1 and for all S, T ∈ Q we have ||S| − |T || ≤ 1. If V = W and P = Q (i.e., rows and columns are index by the same set and the partitions are equivalent) we say that the block partition is symmetric. We may want to allow for exceptional sets in a block partition. Definition 3.1.11. We we say a block partition (P, Q) has exceptional sets (V0 , W0 ) and refer to the blocks {(V0 , Q) : Q ∈ Q}∪{(P, W0 ) : P ∈ P} as the exceptional blocks. If the partition is symmetric and V0 = W0 we say the partition with exceptional sets is also symmetric. We

14 say a block partition (P, Q) with exception sets (V0 , W0 ) is balanced if it is balanced without the exceptional sets. We now have enough vocabulary about partitions to define what it means for a partition to be -regular. Definition 3.1.12. A block partition (P, Q) (of the V × W matrix A) with exceptional sets (V0 , W0 ) is -regular if the partition is balanced, |V0 | < |V |, |W0 | < |W | and all but at most |P||Q| of the blocks are -regular in the sense of Definition 3.1.8. We say it is ()-regular if it is an -regular partion of the normalized matrix A∗ =

A . dA (V,W )

That is kA∗ k = |V ||W | so that the average modulus of an entry of A is 1. Now, before I state Scott’s version of the regularity lemma I want to review the proof of the original lemma and make an additional statement that is proven while proving the lemma but isn’t directly a consequence. As Scott states, the usual proof of the regularity lemma follows three steps: 1. Define a bounded function f on the partitions of the vertices. 2. Show that for any partition, P, that is not epsilon regular, there is a balanced refinement of it, Q which has f (Q) ≥ f (P) + α where α is a function of . 3. Iterate until, by boundedness, we cannot and thus must have found an -regular refinement of the original partition. Remark 3.1.13. Scott’s proof also follows these basic steps, and so in addition to the statement of his version of the regularity lemma, we get that we can start with any partition and find a refinement of it which is ()-regular. We need this in addition to the general statement of the lemma for the proof of convergence of digraphs. Lemma 3.1.14. [8] For every  > 0 and every positive integer L there is a positive integer M such that for all m, n ≥ M, every real m × n matrix A has an ()-regular partition (P, P) with |P| ∈ [L, M ].

3.1.3

The Intricacies of the Main Theorem

15

Lov´asz and Szegedy’s proof of convergence hinges on a somewhat technical lemma, the proof for digraphs needs a corresponding lemma for convergence. Before stating the technical lemma however, we need to define a norm that will be used in the lemma and in showing convergence. It is used in [5] and they have a few proofs that carry over to the non-symmetric (digraph) case. Definition 3.1.15. For a function f : [0, 1]2 → R we define the square norm, Z kf k =

|f (x, y)|dxdy.

sup A,B⊂[0,1]

A×B

It may not be obvious at first that this is a norm (it’s not if we don’t identify functions that are almost everywhere equivalent), but let’s go through a couple calculations. Lemma 3.1.16. The functional above, k · k , is a norm. Proof. First let’s show that kf k = 0 if and only if f ≡ 0 almost everywhere. Certainly if f ≡ 0 a.e. then kf k = 0. Conversely, if f 6≡ 0 a.e. then there is a measureable set A × B R with positive measure on which f 6= 0. Since then kf k ≥ A×B f dxdy > 0 we have what we needed. To see the triangle inequality, note that ρA,B (f ) =

R A×B

f dxdy satisfies that ρA,B (f +g) =

ρA,B (f ) + ρA,B (g) and ρA,B (cf ) = |c|ρA,B (f ) simply by linearity of integration. This means ρA,B is a seminorm. Now, the supremum of seminorms is again a seminorm. But we showed first that this supremum satisfies separation of points, so that k · k is a norm. We use this norm to help our understanding of how adjacency functions will converge. The first lemma, lemma 4.1 from [5], shows that if we have a sequence Wn : [0, 1]2 → [0, 1] convergencent in k · k then t(H, Wn ) converges for all finite digraphs H.

16 Lemma 3.1.17. [5] Let U, W : [0, 1]2 → [0, 1] be two integrable functions. Then for every finite directed graph H,

|t(H, U ) − t(H, W )| ≤ |E(H)| · kU − W k .

The proof of the preceding lemma was based in counting arguments and algebraic manipulation and did not depend on the graphs being undirected at all. This next lemma will be used for showing examples. It states that pointwise almost everywhere convergence implies convergence in k · k . The proof uses a basic result from an elementary measure theory course. I’ll state Egorov’s Theorem here. Theorem 3.1.18 (Egorov’s Theorem). Let (X, µ) be a measure space with µ(X) < ∞. If fn : X → R converges pointwise almost everywhere to a function f , then for every  > 0 there exists E with µ(E) <  so that fn converges uniformly on X\E. Lemma 3.1.19. If fn : [0, 1]2 → [0, 1] converge pointwise convergence almost everywhere to f , fn converges to f in k · k . Proof. Having pointwise convergence let’s us apply Egorov’s theorem. So, let  > 0 and E be the set of small measure from Egorov’s theorem. Then we have that Z kfn (x, y) − f k =

|fn − f |dxdy

sup A×B⊂[0,1]2

A×B

< sup µ((A × B)\E) + 2 A,B

≤ 3.

This gives convergence in k · k .

The next lemma is the crux of the proof. It will take a convergent graph sequence and create a sequence of functions, similar to adjacency functions, that will converge to the

17 function W in the square norm. This lemma is rather technical, but the proof though long, is fairly straightforward. Lemma 3.1.20. Every digraph sequence, Gn , with |V (G)| → ∞, and lim sup ρ(Gn ) 6= 0, has a subsequence, G0m , for which there exist corresponding sequences, ρm , km and Qm of real numbers, integers, and matrices respectively, with the following properties: (i) Qm is a km × km matrix whose entries are in [0, 1]. (ii) If i < j then ki |kj and the matrix Qi is such that by partitioning the rows and columns of Qj into kj /ki sized square blocks and averaging the entries of the (s, t)th block, we k

k

k

k

get the (s, t) entry of Qi . I.e., for Is = [s kji + 1, (s + 1) kji ] and It = [t kji + 1, (t + 1) kji ] Qj (s, t) =

ki2 kj2

X

Qi (s0 , t0 ).

(s0 ,t0 )∈Is ×It

(iii) lim ρm > 0 and for all j < m the adjacency matrix for G0m , Am , has a ()-regular m→∞

partition Pm,j (in the sense of 3.1.14) with density matrix Qm,j such that

kQm,j − Qj k < 1/j and for 1 ≤ j ≤ m the partition Pm,j is a refinement of Pm,i . Proof. We’ll proceed by induction. First, let An be the adjacency matrix of Gn . Let ρn be the density of Gn , i.e., the average of the entries of An . Since ρn is bounded and lim sup ρn 6= 0, we can find a convergent subsequence, ρ0n → ρ > 0. The first culling of Gn is to look only at the Gn corresponding to the convergent sequence ρ0n . For now we’ll call this sequence (G1n )n≥1 . We take the trivial partition to start making k1 = 1 and Q1 = ρ. Now, for given m suppose we have constructed (Gm n )n≥1 and km × km matrices Qm . For 1 m each digraph Gm n look at its adjacency matrix An and consider an ( m+1 )-regular partition m+1 Pn,m+1 of Am . By Remark 3.1.13, the partition can be a balanced n with density matrix Qn

18 refinement of the partition given by Qm n such that each class of Pn,m is split into rn,m+1 classes. Since rn,m+1 is bounded by the regularity lemma, r ≤ M from Theorem 3.1.14, we can choose an r which is used infinitely often. Now let km+1 = km · r. Further we can cull the sequence ∈ [0, 1]km+1 ×km+1 and noting that again by looking at the (now same sized) matrices Qm+1 n each entry is bounded, so there is a convergent subsequence, say lim Qm+1 → Qm+1 . This is n n→∞

not quite enough, we need to remove the first few terms so that all the terms in the sequence satisfy kQm+1 − Qm+1 k < 1/(2m + 2). n Let G0m = Gm 1 , the first element in each of our successively culled sequences. We have now inductively constructed our sequences G0m , km and Qm . So we just need to look at the statements (i), (ii), and (iii). Statement (i) is clearly true. For statement (ii), we need to note that the divisibility was built into the construction, and that the averaging comes from the fact that each of the matrices Qm n was a refinement of an earlier one. For (iii) we certainly have that limm→∞ ρm exists and is greater than 0. Also, by construction the ()-regular partition for G0m came as a successive refinements of the original ()-regular refinement of the graph G0m in the original sequence. These successive refinements and their associated density matrices are the required Pm,j and Qm,j . That they satisfy ()-regularity and kQm,j − Qj k < 1/j is by construction. The use of ()-regularity in this lemma increases M , the maximum size we can look at for a partition. In some sense, we’re blowing up a matrix by a factor of

1 ρ

to get the average

entry to be 1. When doing this we’ve increased all distances by a rather large factor, and then found an epsilon partition. Shrinking the matrices back down to their original size then makes this partition ρ·-regular, which is (when ρ is small), a much stronger regularity result. Thus the convergence is only helped by doing this, whereas to get the stronger convergence, our partition sizes might increase drastically.

19 Now that we have this sequence of neatly meshed matrices, we have to show the step functions they correspond to converge nicely and exhibit the properties we would like the function W : [0, 1]2 → [0, 1] to have with respect to homomorphism densities. The lemma without symmetry that is stated here is virtually identical to the one in [5]. Lemma 3.1.21. Let (Qm ) be a sequence of matrices satisfying (i) and (ii) from Lemma 3.1.20. Then there exists a measureable function W : [0, 1]2 → [0, 1] such that (a) WQm → W as (m → ∞) almost everywhere (b) for all m and 1 ≤ i, j, ≤ km , we have

(Qm )ij =

2 km

Z

i/km

Z

j/km

W (x, y) dx dy. (i−1)/km

(j−1)/km

Proof. The proof is very similar to the proof of Lemma 5.2 in [5]. Let ϕm : [0, 1) → [km ] be defined by ϕm (x) = dxkm e. This maps the interval [(i − 1)/km , i/km ) to i. We show that if X, Y are independent uniformly chosen elements of [0, 1] then Zm = (Qm )ϕm (X),ϕm (Y ) for m = 1, 2, · · · is a martingale. That is, we want to show

E[Zm+1 |Zm , Zm−1 , . . . , Z1 ] = Zm .

But as ϕm (X), ϕm (Y ) determine Zk for k ≤ m showing that Zm = E[Zm+1 |ϕm (X), ϕm (Y )] suffices. That is (Qm )a,b = E[Zm+1 |a = ϕm (X), b = ϕm (Y )]. This expected value calculation is exactly condition (ii) from Lemma 3.1.20. That means Zm is a martingale as we wanted to show. Now, as Zm is bounded,the martingale convergence theorem states that the pointwise limit exists for almost every (x, y). Define this limit to be

W (x, y) := lim (Qm )ϕm (x),ϕm (y) m→∞

where the limit exists. Define W (x, y) := 0 for those pairs which the limit does not exist.20 This shows W indeed maps [0, 1]2 → [0, 1] and that (a) is satisfied. To show (b), note that

2 km

Z

i/km

Z

j/km

W (x, y) dx dy = (i−1)/km

2 km

(j−1)/km

Z

i/km

Z

j/km

lim (Qn )ϕn (x),ϕn (y) dx dy

(i−1)/km

k2 = lim m2 n→∞ kn

(j−1)/km n→∞ ikn /km

jkn /km

X

X

(Qn )α,β

α=1+(i−1)(kn /km ) β=1+(j−1)(kn /km )

= (Qm )i,j

This gives us enough to show the first part of the main theorem, Theorem 3.0.5, so with the preceding fresh in your mind, you might skip ahead to the next section. However, we continue into the next section by developing W -random digraphs, and some concentration results which will then give us the converse part of Theorem 3.0.5.

3.1.4

W -random Graphs and Concentrations of Homomorphism Densities

We need to generalize the definition of W -random graphs, Definition 2.3.1, to W -random digraphs. This is easy enough; we simply remove symmetry: Definition 3.1.22. Let W : [0, 1]2 → [0, 1] be a measureable function. A W -random graph on n vertices, Gn,W , is formed in the following manner. Let (Xi )ni=1 be chosen uniformly at random from [0, 1]. Set V (Gn,W ) = [n]. The edge (s, t) is in E(Gn,W ) with probability W (Xs , Xt ) independent of all other edges. This lets us form a sequence (Gn,W )n≥1 from a function W . We can then examine, for a fixed digraph H, E[t(H, Gn,W )] and Var(t(H, Gn,W )) in the same way lemma 2.4 does in [5].

21 Lemma 3.1.23. Let W : [0, 1]2 → [0, 1] then for every digraph H and n > |V (H)| the following hold: (i) E[t0 (H, Gn,W )] = t(H, W ). (ii) |E[t(H, Gn,W )] − t(H, W )| ≤

1 |V (H)| n 2


.

(iii) Var[t(H, Gn,W )] ≤ n3 |V (H)|2 . Proof. The proof closely follows the proof of Lemma 2.4 in [5] and note that symmetry wasn’t involved. Let ϕ : V (H) → V (G) be an injective map chosen uniformly at random from the set of injective maps from V (H) to V (G). Then the probability that ϕ is a homomorphism is Y

W (xϕ(i) , xϕ(j) ).

− → ij ∈E(H)

The expected value of this then is Z

Y

W (xϕ (i), xϕ (j)) = t(H, W ).

[0,1]|V (H)| − → ij ∈E(H)

This gives probability ϕ is a homomorphism is constant and so proves (i). Using (i) and lemma 3.1.5 we get (ii). For (ii), let H2 be the digraph consisting of two disjoint copies of H, then t(H2 , G) = (t(H, G))2 for any graph G. Similarly, Fubini gives us t(H2 , W ) = (t(H, W ))2 . We apply this to calculate the variance of t(H, Gn,W ). Set R =

|V (H)|2 . n

Then by Lemma 3.1.5,

E[t(H, Gn,W )2 ] = E[t(H2 , Gn,W )] ≤ E[t0 (H2 , Gn,W ) + 2R] = t(H2 , W ) + 2R = t(H, W )2 + 2R.

22

The other inequality is given by

E[t(H, Gn,W )2 ] ≥ E[t0 (H, Gn,W ) − R/2]2 ≥ E[t0 (H, Gn,W )]2 − R = t(H, W )2 − R.

This gives

Var(t(H, Gn,W )) ≤ t(H, W )2 + 2 − (t(H, W )2 − R) = 3R =

3|V (H)|2 , n

completing the proof.

This concentration of t(H, Wn,G ) near t(H, W ) isn’t strong enough for the convergence we want, but it is used to show a stronger concentration result next. Then, we will apply the Borell-Cantelli Lemma and show that for any finite digraph H, t(H, Gn,W ) converges to t(H, W ) with probability 1. For this, we need to show that for each  > 0, the events

En, := {|t(H, Gn,W ) − t(H, W )| > }

have probabilities that have a finite sum together. Lemma 3.1.24 (Borell-Cantelli Lemma). Let En be a sequence of events in a probability space. If ∞ X

P[En ] < ∞

n=1

then the probability that infinitely many of them occur is 0, i.e.,

∞ P [∩∞ n=1 ∪k=n Ek ] = 0.

To get to the point where we can show that the sum of certain events is finite, we will set up a martingale and apply Azuma’s Inequality to find an upper bound on the probabilities

23

of the events we’re will try to sum.

Theorem 3.1.25 (Azuma’s Inequality). Let (Zn )∞ n=1 be a martingale with P [|Xk − Xk−1 | < ck ] = 1. Then for all N ∈ N and  > 0, we have

P [|Xn − X0 | ≥ ] ≤ 2 exp

2

t2 PN

2 k=1 ck

! .

With heading in this direction in mind, we take another Lemma from [5] and tweak it for directed graphs. Lemma 3.1.26. Let H be a digraph with k vertices. Then for every 0 <  < 1,   2 P[|t0 (H, Gn,W ) − t(H, W )| > ] ≤ 2 exp − 2 n , 2k and   2 n . P[|t(H, Gn,W ) − t(H, W )| > ] ≤ 2 exp − 18k 2 Proof. The proof follows the proof of Lemma 2.5 in [5]. The main idea is to form a martingale. The martingale, Zm , is created as follows: Start with Z0 = 1. Generate a uniformly random sequence (Xt )t≥1 ∈ [0, 1]N . Let GW be the infinite graph generated by the vertices Xt and edges with probability W (Xi , Xj ). Fix n ∈ N. For ϕ : V (H) ,→ [n] let Aϕ be the event ϕ is a homomorphism from H to GW . Set Gm to be the graph formed by restricting Gn,W P to the first m vertices in the sequence (Xt ). Define Zm = (n)1 k ϕ P[Aϕ |Gm ]. Then Zm is a martingale. To apply Azuma’s inequality we need to bound |Zm − Zm−1 |. Notice Z0 =

X

P[Aϕ ] = t(H, W )

ϕ

and Zn =

1 Hom0 (H, Gn,W ) = t0 (H, Gn,W ). (n)k

24

Now, P (P[A |G ] − P[A |G ]) ϕ ϕ m ϕ m−1 P ≤ (n)1 k ϕ |P[Aϕ |Gm ] − P[Aϕ |Gm−1 ]| .

|Zm − Zm−1 | =

1 (n)k

(1.1)

Now, this last sum is bounded by the number of ϕ such that m ∈ ϕ([k]). This gives

|Zm − Zm−1 | ≤

k(n − 1)k−1 = k/n. (n)k

This allows us to apply Azuma’s inequality (Theorem 3.1.25):

P[|t0 (H, Gn,W

  2 − t(H, W )| > ] = P[|Bn − B0 | > ] ≤ 2 exp − 2 n . 2k

Then, we get that when |t(H, Gn,W ) − t(H, W )| >  and |t(H, Gn,W ) − t0 (H, Gn,W )| <  2

together |t0 (H, Gn,W ) − t(H, W )| >

 . 3

(Lemma 3.1.5 gives this for  < k 2 /n.) This

immediately gives   2 P[|t(H, Gn,W ) − t(H, W )| > ] ≤ 2 exp − n . 18k 2

With this concentration result of t(H, Gn,W ) near t(H, W ) we have enough background to finish both directions of the main theorem, Theorem 3.0.5. We do so in the next section.

3.2

Convergence and Correspondence

We break the proof of the main theorem into two pieces. The first shows that there is an adjacency function of the limit object in the same sense as that of the undirected case. The other direction states that if you have an adjacency function, there is a sequence of graphs which converge to it.

25 Theorem 3.2.1. Let Gn be a sequence of digraphs such that t(Gn , F ) converges for every finite digraph F . Then, there is a measureable W : [0, 1]2 → [0, 1] such that Z

Y

lim t(F, Gn ) =

n→∞

W (xi ,j ) dx1 dx2 . . . dx|V (F )| .

[0,1]|V (F )| −−→ xi xj ∈E(F )

Proof. Let Gn be sequence of digraphs such that t(F, Gn ) converges for every finite F . We want to construct a W : [0, 1]2 → [0, 1]. But, by Lemma 3.1.20, we have a subsequence, (G0i )i≥1 , that as a subsequence defines the same limits,i.e., limn→∞ t(F, G) = limn→∞ t(F, G0n ), and has associated matrices satisfying (i), (ii). From this sequence of matrices, we can construct, using Lemma 3.1.21, a measureable W : [0, 1]2 → [0, 1] with the properties, (a), (b), and (c). It then is enough to show that limn→∞ t(F, G0n ) = t(F, W ). So, for 1 ≤ j ≤ m, set G∗m,j = G(Pm,j , Qm,j ) (The edge weighted graph with vertices indexed as the partition Pm,j and adjacency matrix Qm,j .) Similarly set G∗∗ m,j = G(Pm,j , Qj ). Then, because Pm,j is a (1/j)-regular partition of G0m , we have that d (G0m , G∗m,j ) ≤

ρm . j

(2.2)

This is the only place the difference of regularity comes into play in the convergence, but since blowing up the matrix by a factor of

1 ρj

and then looking for -regularity actually gives

a tighter bound on d (G0m , G∗m,j ) we needn’t worry. We also have that d (G∗m,j , G∗∗ m,j ) < 1/j

(2.3)

by construction (part (iii) in Lemma 3.1.20.) Setting avoid double subscripts we set Wm,j = WG∗∗ and Wj = WQj . This gives us m,j t(F, Wm,j ) = t(F, G∗∗ m,j ) by the same calculations from Equation 1.1 and the convergence Wm,j → Wj almost everywhere, by construction of Qj in Lemma 3.1.20.

Now, set tF = limn→∞ t(F, Gn ) and let  > 0.

26

We can find M1 ∈ N such that m > M1 implies |t(F, G0m ) − tF | < 

(2.4)

from the definition of tF .  We can also find M2 ≥ max M1 , 2 such that j > M2 implies |t(F, W ) − t(F, G∗∗ m,j )| = |t(F, W ) − t(F, Wj )| < 

(2.5)

from Lemma 3.1.21. Also, we can find M3 ≥ M2 so that for m > M3 we get |t(F, G∗m,j ) − t(F, G∗∗ m,j )| = |t(F, Wm,j ) − t(F, Wj )| ≤ 

(2.6)

from construction of Qj . We can also find M4 ≥ M3 so that for m > M4 we have |t(F, G0m ) − t(F, Wm,j )| = |t(F, G0m ) − t(F, G∗m,j )| ≤ .

(2.7)

Combining these 4 inequalities we get

|TF − t(F, W )| = |TF − t(F, G0m ) + t(F, G0m ) − t(F, G∗m,j ) + t(F, Wm,j ) − t(F, Wj ) + t(F, Wj ) − t(F, W )| < 4.

(2.8)

27 Theorem 3.2.2. Given a measureable W : [0, 1]2 → [0, 1] there exists a sequence of digraphs Gn such that, Z lim t(F, Gn ) =

n→∞

Y

W (xi , xj ) dx1 dx2 . . . dx|V (F )| .

[0,1]|V (F )| −−→ xi xj ∈E(F )

Proof. Let  > 0, and set En, = {|t(H, Gn,W ) − t(H, W )| ≥ } . Then, from Lemma 3.1.26, we have that   2 P[E,n ] ≤ 2 exp − n . 18k 2 This means

P∞

n=1

P[E,n ] converges. Borell-Cantelli, Theorem 3.1.24, then states that

|t(H, Gn,W ) − t(H, W )| < 

almost surely. Letting  → 0, we have convergence of t(H, Gn,W ) to t(H, W ) with probability 1. Since the set of finite directed graphs from which we can choose H is countable, this of convergence holds for every finite digraph H simultaneously with probability 1. As with the undirected case, we now name such functions W . Definition 3.2.3. A (directed) adjacency function of a convergent sequence of digraphs, (Dn ), is a function W : [0, 1]2 → [0, 1] such that t(H, W ) = limn→∞ t(H, Dn ) for every finite digraph H. Again as in the undirected case, an adjacency functions for a convergent sequence is far from being unique.

3.2.1

Relationship Between Directed Graph Limits and Graph 28 Limits

Now that we have a correspondence between limit functions and graph sequence, we show how they relate to the limit functions in the undirected case. Undirecting an edge should be as simple as making the underlying adjacency matrix symmetric by adding its transpose to it. This is infact the case in the limit as well. ~ n )n≥1 a convergent digraph sequence with an adjacency function (W ~ : Lemma 3.2.4. Let (G [0, 1]2 → [0, 1]) = limn→∞ G~n . Then the corresponding sequence of undirected graphs Gn ~ (x, y) + converges and has an adjacency function W : [0, 1]2 → [0, 1] with W (x, y) = W ~ (y, x). W Proof. Let F be a finite graph. We’ll show that t(F, Gn ) → t(F, W ). Let F be the set of all orientations of F . First notice, Z

Y

Z

X

W (xi , xj )dx1 . . . dxe(F ) =

[0,1]e(F ) x x ∈E(F ) i j

Y

~ (xi xj )dx1 . . . dxe(F ) W

[0,1]e(F ) ~ −−x → ~ F ∈F x i j ∈E(F )

=

XZ ~ ∈~F F

Y

~ (xi xj )dx1 . . . dxe(F ) (2.9) W

[0,1]e(F ) −−→ ~) xi xj ∈E(F

Now, lim tinj (F, Gn ) = lim t(F, Gn ), so we show lim tinj (F, Gn ) = t(F, W ). Let ϕ : F → Gn be an injective homomorphism. The orientation of Gn together with ϕ induces an orientation on F . This partitions the set of injective homomorphisms into sets indexed by orienations P ~ ~ of F . That is, tinj (F, Gn ) = ~ ∈~F tinj (F , Gn ). As this sum is finite, this converges to F P ~ ~ ~ ∈~F t(F , W ) which is what we wished to prove. F

The idea of using homomorphisms from subgraphs and supergraphs of H on the same vertex set has other applications as well. To show these, let’s first define some convenient notation.

3.2.2

29 Induced, Injective, and Specified Homomorphism Densities

Definition 3.2.5. We denote the set of supergraphs of F by F¯ = {H : V (H) = V (F ) and E(H) ⊇ E(F )} . In the undirected case Lov´asz and Szegedi in [5] show we can use inclusion and exclusion to calculate injective homomorphism densities from induced densities, and vice-verse. That is,

tinj (F, W ) =

X

tind (H, W )

(2.10)

H∈F¯

This carries over to the directed case as well. Lemma 3.2.6. For all digraphs F we have

tinj (F, W ) =

X

tind (H, W ).

H∈F¯

And conversely, tind (F, W ) =

X

(−1)e(H)−e(F ) tinj (H, W ).

H∈F¯

Proof. By 3.2.2, there is a sequence of digraphs Gn that converge to W . For each F we can partition the injective homomorphisms by looking at the digraphs induced by the image of their vertices. This gives the equality

tinj (F, Gn ) =

X

tind (H, Gn ).

(2.11)

¯ H∈F

A quick application of inclusion and exclusion gives the converse.

tind (F, Gn ) =

X

(−1)e(H)−e(F ) tinj (H, Gn ).

H∈F¯

As these are finite sums, taking the limit finishes the proof.

(2.12)

30 The above lemma allows us to ask what is the density of induced graphs. This means we specify, for each pair of vertices, that there be an edge and its direction or there be a non-edge. It would be beneficial to relax our specifications a bit to say edge, non-edge, and don’t care. The next lemma states that this is in fact possible, but we need some notation to specify mathematically what we mean. Definition 3.2.7. Let F be a directed graph. Define N ⊂ V (F )2 \E(F ) to be a set of → ∈ E(F ) implies non-edges. We say ϕ : V (F ) → V (G) is a specified homomorphism if − xy −−−−−−→ ϕ(x)ϕ(y) ∈ E(G) and xy ∈ N imples ϕ(x)ϕ(y) ∈ / E(G). Let tsp ((F, N ), G) be the proportion of maps from V (F ) → V (G) which are specified homomorphisms. One notation we’ll use here: S c is the complement of a set S. Lemma 3.2.8. Let W : [0, 1]2 → [0, 1] be a graph limit. Given a digraph F and a set of
) non-edges N ⊂ V (F \E(F ), we get 2 tsp ((F, N ), W ) =

X

tind (H, W ).

(2.13)

H∈F¯ N ⊂E(H)c

Proof. By 3.2.2, there exists a sequence of digraphs converging to W , say Gn . Let ϕ be a specified homomorphism from (F, N ) → Gn . The digraph induced by the image of ϕ partitions the set of specified homomorphims as indexed in the sum

tsp ((F, N ), W ) =

X

tind (H, W ).

(2.14)

H∈F¯ N ⊂E(H)c

Since this sum is finite, moving to the limit finishes the proof.

This completes the chapter on Directed graph limits. The next chapter reviews some commonly known results about the well studied class of graphs called threshold graphs and then develops a corresponding class of directed graphs we call directed threshold graphs.

31

Chapter 4 Threshold and Directed Threshold Graphs 4.1

Threshold Graphs

To develop the class of directed threshold graphs, we first need to understand the undirected case. Mahadev and Peled in [6] give a thorough treatment of the class of threshold graphs. Here we give the basic definition and some equivalences. Definition 4.1.1. Let G be a graph. We say G is a threshold graph if there exists a threshold t ∈ R and a vertex weight function W : V (G) → R (for notation, let w¯ = (wi )ni=1 where wi = W (i)) such that e = (x, y) ∈ E if and only if w(x) + w(y) > t. Though this is a fairly simple definition to work with, there are several equivalences that will be worth considering. To understand them we need two definitions: Definition 4.1.2. A graph, G = (V, E), is said to be split if the vertex set V can be partitioned into two classes, K and I such that K induces a clique in G, and I is an independent set in G.

32 Definition 4.1.3. A graph, G = (V, E), on four vertices is an alternating C4 if there is an ordering of the vertices, a, b, c, d such that ab, cd ∈ E and ad, bc ∈ / E. We can now state 4 characterizations of threshold graphs. Theorem 4.1.4. [6] The following are equivalent: (i) G is a threshold graph. (ii) G is a split graph (the vertices of G can be partitioned into a clique K and an independent set I) and the vertex neighborhoods are nested. (iii) G is alternating C4 free. (iv) The graph G can be constructed by starting with a single vertex and sequentially adding either a dominating vertex or an isolated vertex at each step. Let’s look at a quick example showing equivalences (i) and (iv) from Theorem 4.1.4: Using equivalence (iv) of Theorem 4.1.4, we define can define a threshold graph by creating a binary sequence s¯ = (si )ni=1 where si = 1 if the vertex added is dominating, or si = 0 if it is isolated. Given such a sequence we define T (¯ s) to be the threshold graph associated with it. First let’s use equivalence (iv): let s¯ = (1, 1, 0, 1, 0). We get the graph in Figure 4.1. Figure 4.1: The threshold graph corresponding to the sequence (1, 1, 0, 1, ?)

5

2 4

1

3

We obtain the same graph by using equivalence (i) and setting the threshold at t = 5, and the weight function w¯ = (5, 4, 1, 3, 2). This is represented by the weights in the nodes of Figure 4.1.

33 These four equivalences are generalized to the directed case in section 4.2 by Theorem 4.2.6. Let’s look at a quick example of an undirected threshold graph limit before we generalize to the directed case in the next section. W(x,y) 1 Black = 0 White = 1

y

0.75

0.5

0.25

0

0

0.25

0.5 x

0.75

1

Figure 4.2: The limit function W(x,y)

We’ll use the sequential definition to show an example of a graph limit. Let’s look at the sequence of binary sequences s¯(n) is the (2n)-tuple of 0’s and 1’s with 0’s in the even positions and 1’s in the odd positions. Then the graph sequence T (¯ s(n) ) converges to a characteristic function W (x, y) =

   0 if x + y ≤ 1

.

  1 if x + y > 1 This is easily seen with a straightforward rearrangement of the vertices giving the adjacency matrix for T (¯ s(n) ) as Ai,j =

   0 if i + j ≤ n

.

  1 if i + j > n For example, T (¯ s(20) gives the following adjacency matrix, (This is given by the graphical

representation of a matrix from Matlab; White indicates ’1’s; black indicates ’0’s.)

34

20 18 16 14 12 10 8 6 4 2 0

0

5

10

15

20

Figure 4.3: Adjacency matrix of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ?)

In fact the limit of any sequence of threshold graphs generated by sequences of zeros and ones which are formed by uniformly choosing a zero or one for each vertex with probability 1/2 is given by Figure 4.2.

4.2

Directed Threshold Graphs

There are several definitions for a threshold graph in the undirected case, 4.1.4. We’ll begin by developing an analogous vocabulary for directed graphs and then state a theorem presenting several equivalent definitions of a directed threshold graph. Definition 4.2.1. dtgdef A digraph, G = (V, E), is said to be threshold if there exists a → ∈ E if weight function on the vertices w : V → R and a threshold value t ∈ R such that − xy and only if |w(x)| + |w(y)| ≥ t and w(x) > w(y). → Note that we use strict inequality to direct the edge so as to avoid loops and both − xy → in the edge set. and − yx

4.2.1

Background and Directed Threshold Equivalence

35

Now, as in the case of undirected threshold graphs, (see Theorem 4.1.4,) we have several equivalences. To do so, we need to develop a little vocabulary to get corresponding statements for directed graphs as we have for the undirected case. These next definitions will help us generalize the concept of nested neighborhoods. We have in and out neighborhoods to understand. Definition 4.2.2. A directed graph is said to be displit if the vertex set can be partitioned into three classes, V = B ∪ I ∪ T with the properties: i) I is an independent set, ii) the graph induced by B ∪ T is a tournament, iii) all edges between T and B ∪ I are directed from T , and iv) all edges between B and T ∪ I are directed into B. Definition 4.2.3. A vertex, v, in a digraph D is called an out-dominating (in dominated ) if it is adjacent to every other vertex in D and is a source (resp. sink). Definition 4.2.4. Let σ : V → P(V ) be a function from a set to its power set. We say the function σ is top-closed-comparable (TCC), on S ⊂ V if for every x, y ∈ S we have σ(x) ⊆ σ(y) ∪ {y} or σ(y) ⊆ σ(x) ∪ {x}. We denote σ(x) ⊆ σ(y) ∪ {y} by x ≤σ y. We say the function σ is strictly comparable (SC) on S ⊂ V , if for every x, y ∈ S we have either σ(v) ( σ(y) or σ(y) ( σ(x). For x, y ∈ V we denote σ(x) ( σ(y) by x 0 so |w(x)| ≥ |w(y)| therefore |w(x)| + |w(z)| ≥ |w(y)| + |w(z)| ≥ t. So x → z. Case 2) w(y) ≤ 0: Then w(z) < 0 so |w(z)| ≥ |w(y)| therefore |w(x)| + |w(z)| ≥ |w(x)| + |w(y)| ≥ t. So again x → z. This shows that G is transitive. And thus a transitive labelling of a threshold graph.

(b =⇒ c): Because the underlying graph is threshold, we can think of the construction 37 as being from an initial vertex, ?, by consecutively adding dominating vertices 1s, and isolated vertices 0s. The collection of ? and the 0’s form an independent set. And the collection of 1s form a clique. Let I be the collection of ? and the 0s, and K be the clique of 1s. Set T = N− (?) and B = N+ (?). Since every 1 was adjacent to ∗, this partitions K. By the transitivity of the ordering, if t ∈ T and b ∈ B, then t → ? and ? → b so t → b. This gives us the partition T ∪ I ∪ B. We have all edges between T and B are oriented correctly. We need to show yet, that edges between K and I are oriented correctly. That is, we need to show the edges are directed from T to I, and the edges are directed to B from I. To do this we first show that N+ and N− are TCC on I. Since the underlying graph is threshold, the neighborhood function is TCC. So, let i, j ∈ I with i ≤N j. Suppose there is x ∈ N+ (i)\N+ (j). Then x ∈ N− (j) since i ≤N j. But this means i → x → j and transitivity gives i → j but that is impossible as i, j ∈ I. So we must have N+ (i) ⊂ N+ (j). A similar argument gives N− (i) ⊂ N− (j). Using this, and noting that for all i ∈ I we have i ≤N ?, we get that N+ (i) ⊂ N+ (?) = B and N− (i) ⊂ N− (?) = T . That is, the graph is displit. And we have the second condition of properly nested neighborhoods. To show the third condition, let x, y ∈ K with x → y. Then by transitivity N+ (y) ( N+ (x) (the ineequality because y ∈ / N+ (y)) and N− (x) ( N− (y) (inequality because x ∈ / N− (x).) This gives the 3rd condition. (c =⇒ d): Let i ∈ I be minimal in I with respect to total neigbhorhoods. If N (i) = ∅ it is an isolate. If not, then, either its in-neighborhood is non-empty or its out neighborhood is non-empty. Say x ∈ N+ (i). Then, x ∈ N+ (j) for all j ∈ I by TCC neighborhoods on I. Let y be the maximum element in the order given by N− being SC on K. Then since x ∈ N+ (j) for all j ∈ I, we get that j ∈ N− (x) ( N− (y) for all j ∈ I. Also, suppose to get a contradiction, y → z for some z ∈ K. Then y ∈ N− (z) ( N− (y) which is impossible. This means y is an in-dominated vertex. A similar arguement gives that if x ∈ N− (i) then the

maximal vertex with respect to the SC order on N+ is an out-dominating vertex.

38

Now, we simply need to note that removing an independent vertex does not change transitivity or neighborhood conditions of the graph. Similarly, removing an in-dominated or out-dominating vertex does not change the transitivity, and every total neighborhood is decreased by the vertex removed (similarly if it is an out-dominating vertex which is removed, the out-neighborhoods don’t change, but every in-neighborhood is reduced by the removed vertex, a corresponding statement holds for the in-dominated vertex) which doesn’t change the comparability of the neighborhoods. This gives us an inductive deconstruction of the directed threshold graph as required. (d =⇒ a): The assumption gives a sequence of zeros, ones, and negative ones, say (si )ni=1 . If we forget (temporarily) about the sign on the ones, we have a sequence of zeros and ones corresponding to removing the direction on the edges. This underlying graph is constructed by adding isolates or dominating vertices. This means it is an undirected threshold graph. There is an injective weight function and threshold for this underlying graph [6], say (wi )ni=1 and t. It is enough to show then, that the weight function

Wi =

   si wi , ifsi 6= 0   w i ,

ifsi = 0.

gives the correct orientation of the edges. Let’s make an observation about the weights of the vertices. Notice that the weight of a vertex is directly corrolated to the size of it’s neighborhood. This means the later in the sequence a 1 happens, the higher the weight of the vertex associated to it. Conversely, the later in the sequence a 0 happens, the lower the weight of the associated vertex. With this, we see that the above weight function satisfies |Wi | > |Wj | whenever i > j and |si | = 1. This means that the orientation of the graph given by the above weight function is the same as the orientation given by the sequencial construction.

39 An example: Let s¯ = (1, −1, 0, −1, 0). The sequential construction yeilds the following graph. Figure 4.4: Directed graph from the sequence (1, −1, 0, −1, ?)

+5

2 -4

1

-3

The vertex weights shown in the figure with threshold t = 5 also give the same graph. This is a directed version of the graph from Figure 4.1.

4.2.2

Adjacency Matrices of Directed Threshold Graphs

As we will be working with adjacency matrices for the graph limits, we have developed some MATLAB code to take a sequence of zeros, ones, and minus ones to an adjacency matrix and a figure representing the matrix. The MATLAB code is in the appendix. Let’s look at a random directed threshold graph on 30 vertices. Each character 0, +1, or −1 in the sequence was chosen with equal probability. The adjacency matrix has had the columns and rows permuted so that the first indices represent independent vertices, then the clique is represented from least to greatest in the transitive ordering given by the digraph. The lattice points in the figure represent entries in the matrix. The lattice point being the lower left corner of a black square have a 1 in the adjacency matrix.

40 Figure 4.5: The adjacency matrix of the directed threshold graph given by the sequence (+ − 0 − 0 + − + 00 − 000 − − − + − ?) 20 18 16 14 12 10 8 6 4 2 2

4.2.3

4

6

8

10

12

14

16

18

20

Canonical Form of Defining Sequences for Directed Threshold Graphs

A few things before we go further. To draw and think about these directed threshold graphs, the sequential definition is quite a bit more malleable; we’ll work with it. With this in mind, the first vertex drawn is always an independent vertex, so we’ll use the character ? to start all our sequences. Another simplification, instead of +1 and −1 we’ll simply write + and − (resp.) Also, our sequences will start on the right out of convention. Looking more closely at these sequences, things get a little messy. In the undirected case, it is easy to just count the sequences, {0, 1}n−1 (n − 1 as the first vertex drawn does not matter.) Here things are a little more subtle. Notice that the sequence (+ − 0?) gives the same graph as the sequence (− + 0?). The homomorphism being switching the last two (left most two) vertices. This is a characteristic of these sequences. Lemma 4.2.7. Given a sequence s¯ := (si )ni=1 , if there is a k ∈ [n] such that |s(k)| = |s(k−1)| then the sequence s0 = (s1 , s2 , · · · , sk−2 , sk , sk−1 , sk+1 , · · · , sn ) produces a graph isomorphic to the one produced by s.

Proof. Clearly if sk = sk−1 we’re fine. So without loss of generality assume sk = + and sk−1 41 = −−−−−→ −. So there is an edge k(k − 1). Now, just note the neighborhoods N+ (k), N− (k), N+ (k − 1), N− (k − 1) do not change when we swap the order of k and k − 1, as the only edge affected is the one between them, and its order is switched as was needed. Remark 4.2.8. As the ∗ at the beginning of any sequence can be thought of as a +, −, or 0, we can always think of −’s adjacent to it as +’s. Similarly, this gives us a sort of ‘canonical’ representation for any isomorphism class, namely, (pl , ml , zl , pl−1 , ml−1 , zl−1 , · · · , p1 , m1 , z1 , ?) where z1 = 0 implies m1 = 0. This notation simply means pi +’s, mi −’s, and zi 0’s. This gives us an easy way to count the number of isomorphism classes of directed threshold graphs on n vertices. Theorem 4.2.9. The number of isomorphism classes of directed threshold graphs on n vertices is F2n the 2n Fibonacci number (where F0 = 0, F1 = 1.) Proof. We find a recursion relation on the classes by looking at the sequences in canonical form. We can always create a new sequence in cannonical from one in canonical form by augmenting it with a 0 or +, but only sequences that have a 0 or − can be augmented with a − to form a new sequence in canonical form. Let T (n) be the number of sequences in canonical form, and P (n) be the number of sequences in canonical form starting with a +. Because a sequence in canonical form cannot have − before a +, we get that T (n) = T (n − 1) + T (n − 1) + (T (n − 1) − P (n − 1)) where the first two terms come from augmenting a 0 or + to an old sequence, and the last term from augmenting a −. Since we could always have augmented a sequence with a +, we get that P (n) = T (n − 1). This gives us the recurrence T (n) = 3T (n − 1) − T (n − 2). The initial conditions are that T (1) = 1 (being the sequence +) and T (2) = 2 (from the sequences ++ and 0+.) Let’s look at the Fibonacci sequence for a second. Specifically, let’s look at F2n .

42

F2n

= F2n−1 + F2n−2 = 2F2n−2 + F2n−3 = 3F2n−2 − F2n−4 = 3F2(n−1) − F2(n−2)

(2.1)

Also, notice that F0 = 1 and F2 = 2! We have that the number of isomorphism classes of directed threshold graphs is the even Fibonacci numbers. Putting this together with the characterization of directed threshold graphs, 4.2.6, we see that the number of isomorphism classes of orientations of direct threshold graphs which are transitive is F2n . This follows directly from statement b) from 4.2.6 states that every transitive ordering of a threshold graph is directed threshold. This means that the number of transitive orientations come from tertiary sequences and the number of isomorphism classes we’ve just shown is F2n . Similarly, given a threshold graph, let o(T ) be the number of 10 s in the binary sequence. Then the number of transitive orientations of the edges is 2o(T ) .

43

Chapter 5 Limits of Threshold and Directed Threshold Graphs The main idea of graph limits in general was the idea that we can include the set of graphs into the space X = [0, 1]F (see Definition 2.1.3) by t : F → X and look at the limit points of the set t(F). This then idea certainly wants us to delve deeper and ask, what happens if we restrict the function to specific classes of graphs. Say, C ⊂ F is a class of graphs. What can we say about the limit points of t(C)? In [1], Diaconis, Holmes, and Janson do this with the family of threshold graphs and determine that the limit of a sequence of threshold graphs is the characteristic function of an increasing set in the unit square. Here, we’ll develop a theory of directed threshold graphs and in the following section we characterize limit of directed threshold graphs.

5.1

Limits of Threshold Graphs

In [1], Diaconis, Holmes, and Janson characterize the limits of threshold graph limits as characteristic functions of increasing closed sets of the unit square which include the upper two boundaries. They do this by probing the limit’s adjacency functions with simple graphs, stars in fact, to gather information about the limits of the degree distribution. Because these

limits converge, and they can be found by looking at the limits of degree distributions 44 of threshold graphs. The adjacency functions are shown to have a nice form. Theorem 5.1.1. [1] Let Tn be a convergent sequence of threshold graphs. Then there is an adjacency function W : [0, 1]2 → {0, 1} associated to the limit which is the characteristic function of an increasing set in the unit square. With the ideas they used in mind, in the next section we develop some techniques that will help us understand how to probe an adjacency function, W , for information about the types of graphs that arise as W -random graphs.

5.2

Probing for Information in Adjacency Functions

The type of information that might be useful to us, is the limits of degree densities. In the directed case, there are 3 distributions associated to the degree densities of a graph. There is the out-degree density, the in-degree density, and the joint density of the two. We can calculate these for a finite graph easily, but can we calculate them from only having an adjacency function? The answer is yes. ~m,n be the star on m + 1 + n vertices with m edges directed to the Definition 5.2.1. Let S center, and n edges directed from the center. Let the notation ρ+ (x) and ρ− (x) represent the out- and in-degree densities of a vertex at x Lemma 5.2.2. The (m, n)th joint moment of the in and out degree density distributions R ~m,n , W ). µm,n = [0,1]2 xm y n dF (x, y) are equal to t(S

~m,n , W ) : Proof. We calculate t(S

~m,n , W ) = T (S

n Y

Z [0,1]m+1+n

45 ! W (x, yj )

j=1

=

W (zk , x) dy1 . . . dyj dz1 . . . dzk dx

k=1

W (x, y)dy [0,1] Z 1

!

n Z

Z

Z

m Y

[0,1]

m W (z, x)dz

dx

[0,1]

(ρ+ (x))n (ρ− (x))m dx

= 0

= µm,n

(2.1)

A simple corollary of this is that the moments of the individual distributions come from setting n = 0 or m = 0. We will use these in section four to prove the characterization of the limits of directed threshold graphs. Before we do that, we develop the theory behind the directed threshold graphs in the next section. Once we have this information, we need to be able to transfer it back and forth to a sequence of moments from a sequence of directed graphs. This type of problem is known as a moment problem and has been well studied. Solving this moment problem with a sequence of distributions on a compact measure spaces is a well known result from a paper by Hausdorff in 1923. This The second result is from [2], where Hildebrandt and Schoenberg provide the following theorem in regards to moments of joint distributions on a compact space. Definition 5.2.3. Given a sequence (cn )n≥1 , the forward difference operator ∆ acts on the sequence as follows: ∆(cn ) = (∆cn ) = (cn+1 − cn ). The notation ∆k (cn ) will denote k compositions of ∆ applied to the sequence (cn ). In a multi indexed sequence (cn¯ )n¯ ∈Nm we’ll denote the operator on the ith index as ∆i . Note that a quick calculation shows ∆i ∆j = ∆j ∆i .

46 Theorem 5.2.4. Let µn be a sequence of real numbers. Then a necessary and sufficent R1 condistion for there to be a monotonic F (x) such that µn = 0 xn dF (x) is that the system of inequalities ∆j µn ≥ 0 for all j, n ∈ N. Theorem 5.2.5. Let µm,n be a doubly indexed sequence of real numbers. Then a necessary and sufficient condition that there exists a monotonic F (x, y) such that µm,n = R1R1 m n x y dF (x, y) is that the system of inequalities ∆j1 ∆k2 µm,n ≥ 0 for j, k, m, n ∈ N. 0 0 This theorem states then that if we have a sequence of joint distributions on the unit square whose moments converge, then the distributions converge to a distribution. (`)

Theorem 5.2.6. Let f` (x, y), ` = 1, 2, . . . be a sequence of joint distributions. If µm,n = R1R1 m n (`) x y df` (x, y) and lim`→∞ µm,n = µm,n , then there is a joint distribution F (x, y) with 0 0 moments µm,n , unique up to a set of measure 0. Proof. By Theorem 5.2.5, each joint distribution satisfies ∆j1 ∆k2 µ`m,n ≥ 0. This means j k lim ∆j1 ∆k2 µ(`) m,n = ∆1 ∆2 µm,n ≥ 0.

`→∞

So, again by Theorem 5.2.5, there is a monotonic F with joint distributions µm,n . With the information gleaned from looking at homomorphism densities of stars, and being able to understand a bit more about the graphs from using the methods of moments we’re able now to state the characterization of limits of directed threshold graphs.

5.3

Limits of Directed Threshold graphs

What we get here isn’t quite as nice, as the characteristic function of an increasing set, as in [1], but we still get characteristic functions of nicely described sets. The theorem is stated here presently, the proof will follow a couple lemmas.

→ − Theorem 5.3.1. A convergenct sequence of directed threshold graphs, G n has a limit W47: [0, 1] → {0, 1} which is the sum of the characteristic functions of three disjoint subsets of the → − ± unit square. Let ρ± n be the (out/in)-degree distribution functions of G n . Let ρ be the limit of these distributions. The first set,

K = {(x, y) : 1 > x > y > i} , where i = ess inf {ρ+ (x) − x}. This set corresponds to the vertices and edges within the ‘clique’. The second set,

 T = (x, y) : 1 > x > t and 0 < y < ρ+ (x) − (x − i) ,

corresponds to the edges from the ‘clique’ to the ‘independent set’. And the third set,

 B = (x, y) : i < y < i + b and 0 < x < ρ− (i + 1 − y) + (y − 1) ,

corresponds to the edges to the ‘clique’ from the ‘independent set.’ Before we begin the proof, let’s look at an example. Let sn ∈ {0, +, −}n be uniformly chosen from all the such sequences. That is, each coordinate has equal probability of being 0, + or −. The adjacency matrix in Figure 4.5 is from one such randomly generated sequence on 20 vertices. Let’s look at a sequence of figures generated on 30, 300, and 3000 vertices. These figures are formed by ordering the vertices left to right (top to bottom) as follows: vertices associated to zeros in the threshold graph sequence that appear closer to the ? come first. Next we order the negative ones from furthest from ? first. Lastly, the ones closest to ? first. That is, the independent set is first ordered by increasing time of appearance followed by the clique from least to greatest ordered by the transitivity of the graph. Now, with this ordering, it is easy to see that the triangle in the upper right corner corresponds to the edges in the clique. The shaded area on the left correspond to edges from the independent set

48

30

25

20

15

10

5

0

0

5

10

15

20

25

30

Figure 5.1: Uniformly randomly chosen sequence s30 ∈ {0, +, −}30

Figure 5.2: Uniformly randomly chosen sequence s300 ∈ {0, +, −}300

Figure 5.3: Uniformly randomly chosen sequence s3000 ∈ {0, +, −}3000

to the clique. The shaded area on the bottom corresponds to edges from the clique to the independent set. By chosing whether a vertex is 0, +, − with uniform probability, we have made it so that the sizes of T, I, and B converge to 1/3 of the number of vertices. Similarly, the uniformity again makes it so that the cdf of degrees between these sets converges to a uniform cdf. That is, the limit function W (x, y) is the characteristic function of the union

49

of the sets K = {(x, y) : 1/3 < x < 1 and 1/3 < y < x} , T = {(x, y) : 2/3 < x < 1 and 0 < y < x − 2/3} , and B = {(x, y) : 1/3 < y < 2/3 and 0 < x < 2/3 − y} . The following figure displays this limit. 1

y

2/3 1/3 (0,0)

1/3 2/3

1

x

Figure 5.4: The limit object W (x, y).

This, as we stated in 5.3.1, is characteristic of all directed threshold graph limits. We need to define a bit more precisely what we mean by convergence of the distribution functions, and we need a technical measure theoretic limit as well. We give these next. Lemma 5.3.2. Let Gn be convergent sequence of digraphs. Let v be a uniformly chosen vertex of Gn . Set  ±  ρ± n (z) = P deg (v)/|V (Gn )| ≤ z , and   ρjn (s, t) = P deg+ (v)/|V (Gn )| ≤ s and deg− (v)/|V (Gn )| ≤ t . ± Then, the two sequences of distributions ρ± n converge to distributions ρ and the sequence of

joint distributions ρjn converge to a joint distribution ρj . m rm l,m Proof. Let νnm be the mth moment of ρ+ moment of ρ− be the n , µn be the m n and ηn

50 (l, m)th moment of ρjn . Then, by 5.2.4 we have ∆k νnm ≥ 0, ∆k µm n ≥ 0, and by 5.2.5 we get l,m ∆p ∆q ηnl,m ≥ 0. We also have that νnm , µm are all given by homomorphism densities by n , ηn

5.2.2. By the convergence of Gn we get that these moments converge say, to ν m , µm η l,m . These by continuity of finite addition then satisfy ∆k ν m ≥ 0, ∆k µm ≥ 0, and ∆p ∆q η l,m ≥ 0. Applying the other directions of 5.2.4 and 5.2.5 we get distributions ρ± and ρj . Lemma 5.3.3. Let X be measure space. Let fn : X → R be a sequence of essentially bounded functions which converge pointwise almost everywhere to a function f . Then,

ess inf {f } ≥ lim sup ess inf {fn } . n→∞

Moreover, if µ(X) < ∞ then the two are actually equal. Proof. We have pointwise convergence almost everywhere; therefore, we have convergence in measure. That means, m {f −1 ((−∞, α))} = limn→∞ m {fn−1 ((−∞, α))}. Let

E = {α : m {x : f (x) < α} = 0}

and En = {α : m {x : f (x) < α} = 0} . We get that Eˆ :=

∞ [ ∞ \

Em

n=1 m=n

is the set of α such that α ∈ En eventually always. If we include the point at −∞ this set is necessarily a non-empty interval and α∗ := sup Eˆ = lim supn→∞ ess inf {fn }. Suppose α∗ − ess inf {f } = δ > 0. Then, m {f −1 ((−∞, α∗ + δ/2))} > 0 but m {fn−1 ((−∞, α∗ + δ/2)} = 0 contradicting convergence in measure. This gives ess inf {f } ≥ α∗ . By convergence almost everywhere on X and Egorov’s theorem 3.1.18, we have almost uniform convergence. That means, given  > 0, there exists A with µ(A ) <  and N ∈ N

51 such that for n > N we have m {x ∈ X\A : |fn (x) − f (x)| ≥ } < . Looking at the sets Dn = {x : fn (x) > f (x) + } we see that for n > N we have Dn ⊂ A . But this shows that m {Dn } → 0 as  → 0 and n → ∞. This gives ess inf {f } ≤ α∗ completing the proof. In this case, we could have gotten the other direction in the same way. Say, by setting Cn = {x : f (x) > fn (x) + } . We again have Cn ⊂ A and we have µ(Cn ) → 0 as  → 0 and n → ∞. And concluding ess inf {f } ≥ α∗ .

This gives us enough background now to prove 5.3.1. ± ± ± Proof of Theorem 5.3.1. By 5.3.2, we have ρ± n converging to ρ . Setting fn (x) = ρn (x)−x =:

f (x) we get fn± (x) converging to ρ± (x) − x. Set i = ess inf {f } . Then by 5.3.3 we get that i = lim sup ess inf {fn } . Let in = ess inf {fn } . Define the step functions sn (x) =

d|V (Gn )|xe −in . |V (Gn )|

This gives us sn (x) converging to x − i. Now, set Kn = {(x, y) : x > in and sn (x) + in > y > in } .

This set corresponds to the clique of Gn . By 4.2.6, the out-neighborhoods of Gn are strictly nested on Kn . We also have that for every v ∈ In the out neighborhoods N + (v) ⊂ N + (w) for every w ∈ Tn . This means that if we define tn = 1 − |Tn |/|V (Gn )|, then the degree distribution from Tn to In can be modeled by the set  Tn = (x, y) : x > tn and 0 < y < ρ+ n (x) − sn (x) . Similarly, set bn = 1 − |Bn |/|V (Gn )|. Then the degree distribution from Bn to In can be modeled by the set

 Bn = (x, y) : in < y < in + bn and 0 < x < ρ− n (in + 1 − y) + (y − 1) .

52 These three sets in union are given by measure preserving transformations of Wn , the function formed by the scaled adjacency matrix of Gn . Since these are measure preserving, we have convergence in k · k of χn = {(x, y) : (x, y) ∈ Kn ∪ Tn ∪ Bn } . To finish the theorem, we simply note we have the following three pointwise limits. First,

lim Kn = {(x, y) : 1 > x > y > i} ,

n→∞

where i = ess inf {ρ+ (x) − x}. This set corresponds to the vertices and edges within the ‘clique’. The second set,

 lim Tn = (x, y) : 1 > x > t and 0 < y < ρ+ (x) − (x − i) ,

n→∞

corresponds to the edges from the ‘clique’ to the ‘independent set’. And the third set,

 lim Bn = (x, y) : i < y < i + b and 0 < x < ρ− (i + 1 − y) + (y − 1) ,

n→∞

corresponds to the edges to the ‘clique’ from the ‘independent set.’ As pointwise convergence is stronger than convergence in k · k (3.1.19), we’re done with the proof.

53

Chapter 6 Population Projection Matrices 6.1

Mathematical Biology Background

The next sections diverge from graph theory and will need some additional notation. We’ll introduce these sections with the new notation here. The symbols , ≤, ≥ when used in vector or matrix expressions denote entrywise inequality; ei denotes the standard basis vector with a 1 in the ith entry; xT denotes the transpose of x; I is the identity matrix; the spectrum of A is denoted by

 σ(A) = z ∈ C det(zI − A) = 0 ;

the spectral radius of A is denoted by

 ρ(A) = max |z| z ∈ σ(A) .

We consider populations which are modeled using population projection matrices (PPM). If A is an n by n projection matrix for a population, and the population vector during year

k is denoted xk , then (xk )∞ k=0 satisfies the discrete time equation

xk+1 = Axk ,

54

(1.1)

The total population of a population with population vector x is

kxk := x1 + x2 + . . . + xn ,

the 1-norm of x. The long term growth rate is determined by the eigenvalue of A of maximum modulus (also called the leading eigenvalue). Most PPM are primitive matrices, see Definition 6.2.1 below. From the Perron-Frobenious theorem (see Theorem 6.2.2 below), the eigenvalue of largest modulus is guaranteed to be real and positive. For any matrix M, we denote the eigenvalue of maximum modulus by λ(M). The following is well-known: If λ(A) > 1 the total population increases geometrically, i.e. there exists m > 0, ρ > 1 such that kxn k ≥ mρn , n = 0, 1, . . . . If λ(A) < 1, the total population decreases geometrically, i.e. there exists M ≥ 1, ρ < 1 such that kxn k ≤ M ρn , n = 0, 1, . . . . If λ(A) = 1, the total population is asymptotically constant, i.e.

lim xn = cx,

n→∞

where x is the eigenvector of A associated with an eigenvalue 1 and c is some positive constant. Determining when an eigenvalue in the spectrum of a matrix is the leading eigenvalue is important in studying the growth of a population. For a fixed matrix, it is easy to

55 use standard software such as MATLAB or Maple to determine which eigenvalue is the leading eigenvalue. However, when dealing with parameterized matrices, this can be quite difficult. In this paper we first describe such a problem which occurs naturally in the analysis of the effect of data uncertainty on the asymptotic growth rate. We then give a precise mathematical definition of a PPM, which includes most (but not all) actual population projection matrices. We then show that for a mathematically defined PPM A, if some eigenvalue of A is greater than or equal to 1, then this eigenvalue is actually the leading eigenvalue. We then apply this to the analysis of uncertainty in a model for a population of thistles. We will denote most matrices by bold capital letters, such as A. We will denote vectors and n by 1 matrices (i.e. column vectors) by bold lower case letters, such as t. We will denote 1 by n matrices (i.e. column vectors) by bold lower case letter with a T superscript, to denote transpose, such as pT . The matrix A has n2 entries. The uncertainties are typically structured, and can be described by m parameters (p1 , p2 , . . . , pm ), where m ≤ n2 . We can denote the explicit dependence of A and λ on (p1 , p2 , . . . , pm ) by writing

A = A(p1 , p2 , . . . , pm ),

λ = λ(p1 , p2 , . . . , pm ).

We say that (p1 , p2 , . . . , pm ) is admissible if A(p1 , p2 , . . . , pm ) is an “acceptable” projection matrix (“acceptable” is made precise in Definition 6.2.6), and we let S be the set of admissible (p1 , p2 , . . . , pm ). Now consider the subset of S given by

C := {(p1 , p2 , . . . , pm ) ∈ S | λ(p1 , p2 , . . . , pm ) = 1}.

(1.2)

This is the set of (p1 , p2 , . . . , pm ) which lead to a leading eigenvalue of 1. Mathematically, this set is a variety. If we are considering two uncertain parameters, then m = 2 and C is a curve. If we are considering three uncertain parameters, then m = 3 and C is an ordinary

56 surface (that is, a two dimensional object in three dimensions). If we are concerned with maintaining a particular growth rate, say 3%, then we would replace C by

C1.03 := {(p1 , p2 , . . . , pm ) ∈ S | λ(p1 , p2 , . . . , pm ) = 1.03}.

Furthermore, it should be pointed out that for some applications one will be interested in maintaining population decay, in which case the good perturbations will be on the side of C which guarantees that λ(p1 , p2 , . . . , pm ) < 1 (we’ll show that in most cases the variety having ‘sides’ makes sense.) It still remains to find an equation for C. We first find the variety on which some eigenvalue of A is 1: Letting I denote the n × n identity matrix, this variety is

Γ := {(p1 , p2 , . . . , pm ) ∈ S | det(I − A(p1 , p2 , . . . , pm )) = 0}.

(1.3)

Since it is easy to find Γ, it would be useful to have conditions under which C is the same as Γ. Here, we develop mathematical results which show that Γ is typically the same as C.

6.2

The Leading Eigenvalue of a PPM

We first need to give a precise mathematical definition of a PPM. This definition should encompass a large class of population projection matrices found in population ecology. We say that a vector or matrix is positive if all of it’s entries are positive, and non-negative if all of it’s entries are nonnegative. Definition 6.2.1. A primitive matrix is a non-negative matrix A for which there exists n0 ∈ N such that An0 is positive. The vast majority of population projection matrices are primitive. The following wellknown theorem about primitive matrices makes it easier to identify the leading eigenvalue.

57 Theorem 6.2.2. (The Perron Frobenius Theorem, [9]) Suppose that A is primitive. Then the following hold: 1. λ(A) is real and positive; 2. The left and right eigenvectors associated with λ(A) are positive, and these are the only eigenvectors with this property; and 3. λ(A) has geometric and algebraic multiplicity 1. We will also be working with stochastic matrices. Definition 6.2.3. A column (row) stochastic matrix is a non-negative matrix and whose column (row) entries sum to 1. The following results are well known. Theorem 6.2.4. Let S be a stochastic matrix. The eigenvalue of largest modulus of S is 1. Let v be the right (left) eigenvector of S associated with λ = 1, which is positive and unique. Furthermore, there exists v such that

k





lim S = v . . . v .

k→∞

(2.4)

v is called the limiting probability vector, and satisfies kvk = 1. This is so that each entry can be interpreted as a probability. We now give a mathematical definition of a Population Projection Matrix. Definition 6.2.5. A Population Projection Matrix (PPM) is a primitive matrix which can be written as S + tpT where S is a primitive column stochastic matrix whose first row has strictly positive entries, t is a non-zero, non-negative column vector, and pT is a row vector. Definition 6.2.6. An admissible perturbation of a PPM A is a matrix P such that A + P is still a P P M .

58

Examples:

Example 6.2.7. We show how a population projection matrix for a thistle we will be analyzing satisfies the conditions in Definition 6.2.5. Let 



6.75 30.67   0   . A :=  .121 .110 0     .0157 .267 .171

(2.5)

To find the correct stochastic matrix, take the last two rows, and choose the top row so that the matrix is stochastic, i.e. so that the columns add up to 1: 



 .8633 .623 .829    . S :=  .121 .110 0     .0157 .267 .171 Then A − S is guaranteed to have all but the top row as zero: 



 −.8633 6.127 29.841    , tpT = A − S =  .0 0 0     0 0 0 Thus we can choose 



 1     t=  0    0

T

and p =



 −.8633 6.127 29.841

.

This is typical of a situation where the top row consists of fecundities or seed bank rates. Example 6.2.8. The PPM above is the bottom-right 3 by 3 matrix of the general 4 by 4

59

matrix





0 c1 f 3 c1 f 4  0   s21 0 c2 f3 c2 f4    A=  0 s s33 0    32   0 s42 s43 s44

(2.6)

We will show that any matrix of this form is a PPM; this includes the nominal matrix and any perturbation which maintains this structure. Letting

k1 =

c2 , c1 + c2

k2 =

c1 , c1 + c2

we can write A = S + tpT = 

 k1 [1 − (s32 + s42 )] k1 [1 − (s33 + s43 )] k1 (1 − s44 )  k1 (1 − s11 )   s11 + k2 (1 − s11 ) k2 [1 − (s32 + s42 )] k2 [1 − (s33 + s43 )] k2 (1 − s44 )    +   0 s32 s33 0     0 s42 s43 s44 



 1     c2 /c1       −k1 (1 − s11 ) −k1 [1 − (s32 + s42 )] c1 f3 − k1 [1 − (s33 + s43 )] c1 f4 − k1 (1 − s44 ) .  0      0 We now present a few simple results which we will need to prove our main results. We first discuss how the spectrum of a population projection matrix is affected by the addition of a positive matrix. Unless otherwise stated, assume that A and P are n × n matrices. The following simple result shows that if A is primitive, then the addition of a nonnegative matrix does not change this. Lemma 6.2.9. If A is primitive and P ≥ 0 then A + P is primitive.

60 Proof. Since A is primitive, there exits k for which Ak has only positive entries. The same K then works for (A + P). We now show that if A is primitive, then the addition of a nonnegative matrix does not decrease the largest eigenvalue. Theorem 6.2.10. Let A be primitive and P ≥ 0 Then λ(A) ≤ λ(A + P). Proof. First note that by Lemma 6.2.9, A+P is also primitive. Let xT be the left eigenvector of A associated with λ(A), and let y denote the right eigenvector of A + P associated with λ(A + P). Since A and A + P are primitive, the Peron-Frobenius theorem tells us that x > 0 and y > 0, so xT Py ≥ 0 and λ(A)xT y = xT Ay ≤ xT Ay + xT Py = xT (A + P)y = λ(A + P)xT y. Since xT y > 0, we see that λ(A) ≤ λ(A + P). We now state our main theorem. We show that if 1 is an eigenvalue of a PPM A, then it must be the largest eigenvalue. This of course is not true for arbitrary matrices, and in fact is not true for arbitrary primitive matrices. Let σ(A) denote the set of eigenvalues of A. Theorem 6.2.11. Let A be PPM. Then 1 ∈ σ(A) implies 1 = λ(A). Proof. Since A is a PPM in the sense of Definition 6.2.5, we can write it as

A = S + tpT .

(2.7)

Let K = lim Sk . k→∞

KS = lim Sk+1 = lim Sk = K. k→∞

k→∞

(2.8)

61

Using this with (2.4), K = v[1 1 . . . 1].

(2.9)

Let x be the right eigenvector of A associated with the eigenvalue of 1. Then

Ax = Sx + tpT x = x,

so Sx − x = −tpT x. Applying K to both sides KSx − Kx = −KtpT x. Using (2.8) with this, we see that KtpT x = 0.

(2.10)

Kt = v([1 1 . . . 1]t).

(2.11)

Note that, using (2.9),

Since t is a positive column vector, [1 1 . . . 1]t is a positive scalar. So, because we have v > 0, we get that Kt > 0 and so (2.10) implies that

pT x = 0,

so using (2.7), we get that x = Ax = Sx. Hence 1 is an eigenvalue of S with associated eigenvector x. Since A is a primitive matrix, by Theorem 6.2.2 x is real and positive. Since x is also an eigenvector of S, which is also primitive, Theorem 6.2.2 tells us x is the unique positive eigenvector associated with λ(S). Therefore λ(S) = 1.

Corollary 6.2.12. Suppose that A is a PPM, λ ∈ σ(A), and A/λ is a PPM. Then ρ(A) 62 = λ. Proof. Let B = (A/λ), so λ ∈ σ(A) if and only if 1 ∈ σ(B). By hypothesis, B is primitive, so by Theorem 6.2.11 λ(B) = 1. This is true if and only if λ(A) = λ. Corollary 6.2.13. Suppose that A is a PPM, and λ ≥ 1 is such that λ ∈ σ(A). Then λ(A) = λ. Proof. Suppose the hypotheses of the corollary hold. From Corollary 6.2.12, we just need to prove that A/λ is a PPM. Note that

A/λ = S/λ + tpT /λ.

(2.12)

Let 1 α = (1 − )/ktk, λ

qT = [α α, . . . α]

and S1 = S/λ + tqT .

(2.13)

We need to show that S1 is primitive and column stochastic. To see that S/λ is primitive, first note that it is nonnegative, and then note that since S is primitive, there exists n0 ∈ N such that Sn0 > 0, so (S/λ)n0 > 0. Since λ ≥ 1, α is nonnegative, and so is the matrix tqT . Hence by Lemma 6.2.9, S1 is also primitive. We now need to show that S1 is column stochastic. Since S is stochastic, each column of S/λ has sum 1/λ. Furthermore, each column of tqT is αt, which has sum αktk = 1 − (1/λ). Hence each colum of S1 = S/λ + tqT has sum 1. Using (2.12) and (2.13), we can write

A/λ = S1 + t(

pT − qT ). λ

Hence, A/λ is a PPM, so using Corollary 6.2.12, λ(A) = λ.

63

We now return to the problem addressed at the beginning of this chapter. We can be now more precise about the set S. Let

S = {(p1 , p2 , . . . pm ) ∈ Rm | A(p1 , p2 , . . . pm ) is a P P M }.

We consider the varieties

Cµ := {(p1 , p2 , . . . , pm ) ∈ S | λ(p1 , p2 , . . . , pm ) = µ}

(2.14)

and Γµ := {(p1 , p2 , . . . , pm ) ∈ S | det(I − A(p1 , p2 , . . . , pm )) = µ}. As explained in the introduction, in the analysis of the effect of uncertainty on long-term population growth, we want to know Cµ , especially for µ = 1, while Γµ is what is generally easily computable and (in low dimensions) graphable. The following follows from Corollary 6.2.13. Corollary 6.2.14. For µ ≥ 1, Cµ = Γµ . Proof. Γµ is the set of all (p1 , p2 , . . . , pm ) ∈ S such that some eigenvalue is µ. From Corollary 6.2.13, if some eigenvalue of A(p1 , p2 , . . . , pm ) is µ, then the leading eigenvalue of A(p1 , p2 , . . . , pm ) is µ, so Γµ must be the same as the set of all (p1 , p2 , . . . , pm ) ∈ S such that the leading eigenvalue is µ. Hence Γµ = Cµ . We give an of examples of this. This example of the salmon population in the Columbia river basin will be explained in greater detail later in the paper to demonstrate the applications of Corollary 6.2.12.

64 Example 6.2.15. Chinook Salmon in the Columbia river basin ( [4]) have a population projection matrix 

 0 .3262 5.0157 39.6647     .0131 0  0 0 0      A= 0 .8 0 0 0  .     0 .7896 0 0   0   0 0 0 .6728 0 0

This PPM has a leading eigenvalue of .7602. We look at the effects of a percent change in the fecundities and in the survivorship of the first age class. These changes are represented by perturbations of the form  0   0   P 1 = p1  0   0  0

  0 .3262 5.0157 39.6647 0     .0131 0 0 0 0       0 , P = p 0 0 0 0  2 2       0 0 0 0   0   0 0 0 0 0

 0 0 0 0   0 0 0 0   0 0 0 0 .   0 0 0 0  0 0 0 0

Here, p1 , p2 represent the percentage change of the fecundities and survivorship respectively. We use techniques developed by Hodgeson and Townley ( [3]) to find the perturbations required to yield an eigenvalue of .95, 1, 1.05 respectively. Later we prove that these are indeed the maximum eigenvalues. Perturbations on the λmax = .95 line will lead to continued, but slower, decline in the salmon population, whereas perturbations on the line λmax = 1 cause the population will remain stationary. Finally, perturbations on the line λmax = 1.05 yield growth of 5%. This examples show the importance of determining the leading eigenvalue of a matrix. Certainly it is easy to determine the maximum eigenvalue of a nominal matrix using an assortment of techniques already developed. When the matrix is variable the same techniques used on each variation is a very time consuming. While Townley and Hodgeson [3] give

65

2 Variable Perturbation of Salmon PPM 400 λmax=1 λmax=1.05

350

% Change in Fecundities (p1)

λmax=.95 300 250 200 150 100 50 0

0

15

30

45 60 75 90 105 120 % Change in 2nd Stage Survivability (p2)

135

150

Figure 6.1: Perturbations leading to specified maximum eigenvalues.

techniques to find curves in the parameter space that have a specific eigenvalue, λ, in the spectrum of the variable matrix, there is no guarantee these techniques ensure that the perturbations in fact give us λ is the leading eigenvalue. Our theorem tells us when this is true. This is an important thing to know for controlling invasive species. It is easy to determine if 1 is an eigenvalue of a population projection matrix. This does not guarantee however that there is not larger eigenvalue. For controlling an invasive species, it is essencial to make sure no eigenvalues are larger than 1. This theorem tells us that for PPMs defined in the manor of 6.2.5 we can guarantee if there is an eigenvalue of 1 it is the leading eigenvalue. So, if we can perturb the ppm of an invasive species to have an eigenvalue of 1, we have perturbed the matrix to have no eigenvalues greater than 1. Why is this a reasonable definition? While our definition for a PPM is a precise mathematical definition, ecologists use the term much more loosely. Age-based PPM’s for single species are however a subset of the PPM we defined.

For a given stage-based population vector, stage 1 is typically the youngest members 66 of the population, such as newborns or newly fallen seeds. The time step is typically chosen so that every new member of the population must stop at this stage before moving to the other stages of its life. For each class of the population, k that has a probability to move to a stage which can reproduce, we have a row in the matrix. If the class cannot effectively reproduce in some finite number of time steps, tracking its progress is not reasonable. For each stage, i, for which we have a row in the matrix, let pi be the number of elements of the population produced by the ith age/stage class that will survive to be in the stage 1 class in the next time step, i.e., the reproduction/fecundity value of stage i. Let di be the percentage of the population in the ith stage that does not move on to another stage represented in the matrix. Let Pi = pi −di . To create S, let s1j = di +{probability of moving f rom stage j to stage 1}. For j > 1, let sij be the probability of the elements of stage/age j move to stage i. Now, for all j, X j

sij =

{probability of i not moving to any stage j} P + j {probability to move f rom i to j} .

(2.15)

Certainly this sum is 1 since it is the sum of all movement probabilities for an element of stage i. Clearly then this makes S column stochastic. As for primitivity, we refer the reader to Chapter 4 of Ross [7] and note that from stage 1 we can get to every other stage, and from every other stage, we can get to stage 1. This means that stage 1 and stage j communicate with each other. Then since communication is an equivalence relation, stage i, j communicate for all i, j. The other property we must check, is that the period of the matrix is 1. Certainly though, s11 > 0 since there is always the probability of an element of the probability dying before it can move to the next stage. So, since each entry of S is non-negative, we know that sk11 > 0 for all k. Therefore the period is 1. Since the period is identical for all stages that communicate, the matrix has

period 1. Then, because a matrix with a single communication class and a period of 1 67 is primitive, S is primitive. Alternatively, the argument given below that A is primitive is valid for S as well. Now, we look at A = S + e1 P . Since a1i = pi and pi is the fecundity of the ith stage class, we have that the first row of A is suitable for the PPM modeling our population. Similarly, aij is the probability of moving from stage j to stage i. Again, this makes it suitable to model the population. Then, A is the P P M for our population and is written as S + e1 P . The final thing we must show is that A is primitive. Since for every stage, i, j there exists a finite number of time steps from j to 1 through a reproduction stage. There also exist a finite number of steps from 1 to i. Let kij be the total number of time steps to get from j (k)

to i. Let k = lcm(kij ). Then aij > 0 for all i, j. Now, since A > 0 and there exists at least one aij > 0 for each j, Ak A > 0, inductively then, A is primitive. This, then, shows that every age based single species ecological population modeling matrix, can be written as a mathematically defined PPM. Now, certain ecological models often are not written this way because increasing the time step makes the number of stage classes smaller and thus reducing the number of computations needed to analyze the model. Any model of this sort, can however be reformed with a proper time step to meet the requirements of our definition of a PPM. Example 6.2.16. The population projection matrix for Helmeted Honeyeaters is as given:  0 .48 .48 .48     .7 0 0 0   A=    0 .72 0 0     0 0 .75 .72 

can be decomposed into

68

  .3 .28 .25 .28       7 0 0 0   S+P =  + e1 .3 −.2 −.23 −.2 .  0 .72 0  0     0 0 .75 .72 Now, S is primitive and column stochastic and P is of the form e1 P¯ , so by Theorem 6.2.11 and Corollary 6.2.12, A can only have a single eigenvalue greater than or equal to 1. Since we can easily show, that the leading eigenvalue of A is approximately 1.047, it the only eigenvalue greater than or equal to 1. For, if 1 ≤ λ < 1.047 and λ ∈ σ(A), then Corollary 6.2.12 tells us that λ = ρ(A) a contradiction to 1.047 = ρ(A). We now return to the claims made in the salmon example. Example 6.2.17. Recall that we claimed that Figure 6.1 gives perturbations leading to ρ(A) = .95, 1, 1.05 for the respective loci. To prove this claim, we note that all of these perturbations are an increase in matrix values so that primitivity is retained in any case. The only question is if the a12 value increases enough so that A cannot be decomposed into S + e1 P . The only way this would happen is if a12 ≥ 1. The nominal value of a12 is .0131. It would take an p2 value of over 7500 to achieve this. Certainly the graph does not achieve p2 values of this magnitude. Now, since these perturbations retain that A + P1 + P2 is still a PPM, and 1, 1.05 ≥ 1, corollary 6.2.12 and remark 6.2.13 tell us that the perturbed matrices that lead to 1 and 1.05 in their spectrum, in fact have a spectral radius of 1 and 1.05 respectively. As for .95, we look again at A0 = A/.95 + P1 /.95 + P2 /.95. It is clear that A/.95 is a PPM. Since P2 only effects the fecundity values, division by .95 does not effect whether A0 can be written as S + e1 P . We look then at P1 /.95. This only effects the 2, 1 entry. The 2, 1 entry of A/.95 is .0138. Now, P2 /.95 = .0138e2 et1 . Since again it would take p2 value of

over 72 to bring this entry to 1. This means A0 is a PPM and so .95 in the spectrum of 69 A0 implies that ρ(A0 ) = .95. As we claimed then, the perturbations shown in Figure 6.1 give λmax = .95, 1, 1.05

70

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