Levi graphs and concurrence graphs as tools to evaluate designs
R. A. Bailey
The Norman Biggs Lecture, May 2012
1/35 What makes a block design good for experiments?
I have v treatments that I want to compare. I have b blocks. Each block has space for k treatments (not necessarily distinct).
How should I choose a block design?
2/35 Two designs with v = 5, b = 7, k = 3: which is better?
Conventions: columns are blocks; order of treatments within each block is irrelevant; order of blocks is irrelevant.
1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 3 3 4 3 3 4 1 3 3 4 3 3 4 3 4 5 5 4 5 5 2 4 5 5 4 5 5
binary non-binary
A design is binary if no treatment occurs more than once in any block.
3/35 Two designs with v = 15, b = 7, k = 3: which is better?
1 1 2 3 4 5 6 1 1 1 1 1 1 1 2 4 5 6 10 11 12 2 4 6 8 10 12 14 3 7 8 9 13 14 15 3 5 7 9 11 13 15
replications differ by ≤ 1 queen-bee design
The replication of a treatment is its number of occurrences.
A design is a queen-bee design if there is a treatment that occurs in every block.
4/35 Two designs with v = 7, b = 7, k = 3: which is better?
1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 1 4 5 6 7 1 2 3 3 4 5 6 7 1 2
balanced (2-design) non-balanced
A binary design is balanced if every pair of distinct treaments occurs together in the same number of blocks.
5/35 If ω is an experimental unit, put
f (ω) = treatment on ω g(ω) = block containing ω.
For i = 1, . . . , v and j = 1, . . . , b, let
nij = |{ω : f (ω) = i and g(ω) = j}|
= number of experimental units in block j which have treatment i.
The v × b incidence matrix N has entries nij.
Experimental units and incidence matrix
There are bk experimental units.
6/35 For i = 1, . . . , v and j = 1, . . . , b, let
nij = |{ω : f (ω) = i and g(ω) = j}|
= number of experimental units in block j which have treatment i.
The v × b incidence matrix N has entries nij.
Experimental units and incidence matrix
There are bk experimental units. If ω is an experimental unit, put
f (ω) = treatment on ω g(ω) = block containing ω.
6/35 The v × b incidence matrix N has entries nij.
Experimental units and incidence matrix
There are bk experimental units. If ω is an experimental unit, put
f (ω) = treatment on ω g(ω) = block containing ω.
For i = 1, . . . , v and j = 1, . . . , b, let
nij = |{ω : f (ω) = i and g(ω) = j}|
= number of experimental units in block j which have treatment i.
6/35 Experimental units and incidence matrix
There are bk experimental units. If ω is an experimental unit, put
f (ω) = treatment on ω g(ω) = block containing ω.
For i = 1, . . . , v and j = 1, . . . , b, let
nij = |{ω : f (ω) = i and g(ω) = j}|
= number of experimental units in block j which have treatment i.
The v × b incidence matrix N has entries nij.
6/35 I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
Levi graph
The Levi graph G˜ of a block design ∆ has
7/35 I one vertex for each block, I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
Levi graph
The Levi graph G˜ of a block design ∆ has
I one vertex for each treatment,
7/35 I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
Levi graph
The Levi graph G˜ of a block design ∆ has
I one vertex for each treatment, I one vertex for each block,
7/35 It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
Levi graph
The Levi graph G˜ of a block design ∆ has
I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
7/35 It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
Levi graph
The Levi graph G˜ of a block design ∆ has
I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
7/35 Levi graph
The Levi graph G˜ of a block design ∆ has
I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge ω joining vertex f (ω) to vertex g(ω).
It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j.
7/35 ¨¨HH ¨¨ HH ¨¨ HH HH ¨¨ H ¨ HH ¨¨ H¨
4
x 1x2 x
3x
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
8/35 ¨¨HH ¨¨ HH ¨¨ HH HH ¨¨ H ¨ HH ¨¨ H¨
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4
x 1x2 x
3x
8/35 HH HH HH ¨¨ ¨ ¨¨ ¨
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 ¨¨ ¨¨ x ¨¨ HH H1 2 HxH x H 3x
8/35 Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 ¨¨HH ¨¨ x HH ¨¨ HH HH ¨¨ H1 2¨ HxH ¨¨ x H¨ 3x
8/35 Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 ¨¨HH ¨¨ x HH ¨¨ HH HH ¨¨ H1 2¨ HxH ¨¨ x H¨ 3x
8/35 5 8 1 x @ @ x@ x @ 2 x 4 @ @ x@ @
x63 x 7 x
Example 2: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
9/35 Example 2: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 8 1 x @ @ x@ x @ 2 x 4 @ @ x@ @
x63 x 7 x 9/35 I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
10/35 I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment,
10/35 There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
10/35 There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
10/35 If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
There are no loops.
10/35 this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
10/35 Concurrence graph
The concurrence graph G of a block design ∆ has
I one vertex for each treatment, I one edge for each unordered pair α, ω, with α 6= ω, g(α) = g(ω) and f (α) 6= f (ω): this edge joins vertices f (α) and f (ω).
There are no loops. If i 6= j then the number of edges between vertices i and j is
b λij = ∑ nisnjs; s=1
this is called the concurrence of i and j, and is the (i, j)-entry of Λ = NN>.
10/35 can recover design may have more symmetry more vertices more edges if k = 2 more edges if k ≥ 4
4 1 2 ¨H ¨ H @ ¨¨ x HH x@ x ¨¨ HH @ H ¨ HH ¨¨ @ 1Hx¨2 x @ HH¨¨ @ 3x 3x 4x
Levi graph concurrence graph
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
11/35 can recover design may have more symmetry more vertices more edges if k = 2 more edges if k ≥ 4
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 1 2 ¨H ¨ H @ ¨¨ x HH x@ x ¨¨ HH @ H ¨ HH ¨¨ @ 1Hx¨2 x @ HH¨¨ @ 3x 3x 4x
Levi graph concurrence graph
11/35 more vertices more edges if k = 2 more edges if k ≥ 4
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 1 2 ¨H ¨ H @ ¨¨ x HH x@ x ¨¨ HH @ H ¨ HH ¨¨ @ 1Hx¨2 x @ HH¨¨ @ 3x 3x 4x
Levi graph concurrence graph can recover design may have more symmetry
11/35 more edges if k = 2 more edges if k ≥ 4
Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 1 2 ¨H ¨ H @ ¨¨ x HH x@ x ¨¨ HH @ H ¨ HH ¨¨ @ 1Hx¨2 x @ HH¨¨ @ 3x 3x 4x
Levi graph concurrence graph can recover design may have more symmetry more vertices
11/35 Example 1: v = 4, b = k = 3
1 2 1 3 3 2 4 4 2
4 1 2 ¨H ¨ H @ ¨¨ x HH x@ x ¨¨ HH @ H ¨ HH ¨¨ @ 1Hx¨2 x @ HH¨¨ @ 3x 3x 4x
Levi graph concurrence graph can recover design may have more symmetry more vertices more edges if k = 2 more edges if k ≥ 4 11/35 5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xH ¨ x 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
x63 x 7 7x x Levi graph concurrence graph
Example 2: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
12/35 Example 2: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xH ¨ x 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
x63 x 7 7x x Levi graph concurrence graph
12/35 Example 3: v = 15, b = 7, k = 3
1 1 2 3 4 5 6 1 1 1 1 1 1 1 2 4 5 6 10 11 12 2 4 6 8 10 12 14 3 7 8 9 13 14 15 3 5 7 9 11 13 15
8 9 13 7 8 11 7 10 D ¤ bb 4 "" bb 5 "" D ¤ b" b" J x x @ x" b x x" b x 6 xJ D ¤ x@ 11 " b " b QQ J D ¤ x T x x Q J D ¤ x 10 1 T 2 14 5 ````Q 12 x x x x ``Q JD ¤ ` T3 x Q``` x T J Q `` 4 1x 13 xT J Q 12 9 x J QQ x T 6 3 14 x x x @ J T x@ J x 15x 2x 15x
13/35 I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i
I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows:
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows:
14/35 I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
I Lii = valency of i = ∑ λij. j6=i The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows:
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij;
14/35 I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows:
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i
14/35 I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows:
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i
14/35 I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows:
14/35 I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa.
Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows: I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment
14/35 Laplacian matrices
The Laplacian matrix L of the concurrence graph G is a v × v matrix with (i, j)-entry as follows: I if i 6= j then Lij = −(number of edges between i and j) = −λij; I Lii = valency of i = ∑ λij. j6=i The Laplacian matrix L˜ of the Levi graph G˜ is a (v + b) × (v + b) matrix with (i, j)-entry as follows: I L˜ ii = valency of i ( k if i is a block = replication ri of i if i is a treatment I if i 6= j then Lij = −(number of edges between i and j) 0 if i and j are both treatments = 0 if i and j are both blocks −nij if i is a treatment and j is a block, or vice versa. 14/35 Theorem The following are equivalent. 1. 0 is a simple eigenvalue of L; 2. G is a connected graph; 3. G˜ is a connected graph; 4. 0 is a simple eigenvalue of L;˜ 5. the design ∆ is connected in the sense that all differences between treatments can be estimated.
From now on, assume connectivity. Call the remaining eigenvalues non-trivial. They are all non-negative.
Connectivity
All row-sums of L and of L˜ are zero, so both matrices have 0 as eigenvalue on the appropriate all-1 vector.
15/35 From now on, assume connectivity. Call the remaining eigenvalues non-trivial. They are all non-negative.
Connectivity
All row-sums of L and of L˜ are zero, so both matrices have 0 as eigenvalue on the appropriate all-1 vector.
Theorem The following are equivalent. 1. 0 is a simple eigenvalue of L; 2. G is a connected graph; 3. G˜ is a connected graph; 4. 0 is a simple eigenvalue of L;˜ 5. the design ∆ is connected in the sense that all differences between treatments can be estimated.
15/35 Call the remaining eigenvalues non-trivial. They are all non-negative.
Connectivity
All row-sums of L and of L˜ are zero, so both matrices have 0 as eigenvalue on the appropriate all-1 vector.
Theorem The following are equivalent. 1. 0 is a simple eigenvalue of L; 2. G is a connected graph; 3. G˜ is a connected graph; 4. 0 is a simple eigenvalue of L;˜ 5. the design ∆ is connected in the sense that all differences between treatments can be estimated.
From now on, assume connectivity.
15/35 Connectivity
All row-sums of L and of L˜ are zero, so both matrices have 0 as eigenvalue on the appropriate all-1 vector.
Theorem The following are equivalent. 1. 0 is a simple eigenvalue of L; 2. G is a connected graph; 3. G˜ is a connected graph; 4. 0 is a simple eigenvalue of L;˜ 5. the design ∆ is connected in the sense that all differences between treatments can be estimated.
From now on, assume connectivity. Call the remaining eigenvalues non-trivial.
They are all non-negative. 15/35 The Moore–Penrose generalized inverse L˜ − of L˜ is defined similarly.
Generalized inverse
Under the assumption of connectivity, the Moore–Penrose generalized inverse L− of L is defined by
− 1 1 1 L− = L + J − J , v v v v
where Jv is the v × v all-1 matrix. 1 (The matrix J is the orthogonal projector onto the null space v v of L.)
16/35 Generalized inverse
Under the assumption of connectivity, the Moore–Penrose generalized inverse L− of L is defined by
− 1 1 1 L− = L + J − J , v v v v
where Jv is the v × v all-1 matrix. 1 (The matrix J is the orthogonal projector onto the null space v v of L.)
The Moore–Penrose generalized inverse L˜ − of L˜ is defined similarly.
16/35 If experimental unit ω has treatment i and is in block m (f (ω) = i and g(ω) = m), then we assume that
Yω = τi + βm + random noise.
We want to estimate contrasts ∑i xiτi with ∑i xi = 0.
In particular, we want to estimate all the simple differences τi − τj.
Put Vij = variance of the best linear unbiased estimator for τi − τj.
We want all the Vij to be small.
Estimation and variance
We measure the response Yω on each experimenal unit ω.
17/35 We want to estimate contrasts ∑i xiτi with ∑i xi = 0.
In particular, we want to estimate all the simple differences τi − τj.
Put Vij = variance of the best linear unbiased estimator for τi − τj.
We want all the Vij to be small.
Estimation and variance
We measure the response Yω on each experimenal unit ω.
If experimental unit ω has treatment i and is in block m (f (ω) = i and g(ω) = m), then we assume that
Yω = τi + βm + random noise.
17/35 In particular, we want to estimate all the simple differences τi − τj.
Put Vij = variance of the best linear unbiased estimator for τi − τj.
We want all the Vij to be small.
Estimation and variance
We measure the response Yω on each experimenal unit ω.
If experimental unit ω has treatment i and is in block m (f (ω) = i and g(ω) = m), then we assume that
Yω = τi + βm + random noise.
We want to estimate contrasts ∑i xiτi with ∑i xi = 0.
17/35 Estimation and variance
We measure the response Yω on each experimenal unit ω.
If experimental unit ω has treatment i and is in block m (f (ω) = i and g(ω) = m), then we assume that
Yω = τi + βm + random noise.
We want to estimate contrasts ∑i xiτi with ∑i xi = 0.
In particular, we want to estimate all the simple differences τi − τj.
Put Vij = variance of the best linear unbiased estimator for τi − τj.
We want all the Vij to be small.
17/35 How do we calculate variance?
Theorem Assume that all the noise is independent, with variance σ2. If ∑i xi = 0, then the variance of the best linear unbiased estimator of ∑i xiτi is equal to (x>L−x)kσ2. In particular, the variance of the best linear unbiased estimator of the simple difference τi − τj is
− − − 2 Vij = Lii + Ljj − 2Lij kσ .
18/35 . . . Or we can use the Levi graph
Theorem The variance of the best linear unbiased estimator of the simple difference τi − τj is ˜ − ˜ − ˜ − 2 Vij = Lii + Ljj − 2Lij σ .
19/35 Electrical networks
We can consider the concurrence graph G as an electrical network with a 1-ohm resistance in each edge. Connect a 1-volt battery between vertices i and j. Current flows in the network, according to these rules. 1. Ohm’s Law: In every edge, voltage drop = current × resistance = current. 2. Kirchhoff’s Voltage Law: The total voltage drop from one vertex to any other vertex is the same no matter which path we take from one to the other. 3. Kirchhoff’s Current Law: At every vertex which is not connected to the battery, the total current coming in is equal to the total current going out. Find the total current I from i to j, then use Ohm’s Law to define the effective resistance Rij between i and j as 1/I. 20/35 So 2 Vij = Rij × kσ .
Effective resistances are easy to calculate without matrix inversion if the graph is sparse.
Electrical networks: variance
Theorem The effective resistance Rij between vertices i and j in G is
− − − Rij = Lii + Ljj − 2Lij .
21/35 Effective resistances are easy to calculate without matrix inversion if the graph is sparse.
Electrical networks: variance
Theorem The effective resistance Rij between vertices i and j in G is
− − − Rij = Lii + Ljj − 2Lij .
So 2 Vij = Rij × kσ .
21/35 Electrical networks: variance
Theorem The effective resistance Rij between vertices i and j in G is
− − − Rij = Lii + Ljj − 2Lij .
So 2 Vij = Rij × kσ .
Effective resistances are easy to calculate without matrix inversion if the graph is sparse.
21/35 5 ~ 5 >
- [20]- [30] - [47] V = 5 10 17 47
5 } >10 2 } >19
[10] [28] I = 36
5 10 14 7 > } > } 47 5 R = -14 -7 36 [0] [14] ~ 7 > 7
Example calculation: v = 12, b = 6, k = 3
T vT T T T T T T vT vvv T T T T T T T T T vT vT T T T T T T T T T vv T vv T T T T 22/35 v V = 47
I = 36
47 R = 36
Example calculation: v = 12, b = 6, k = 3
T vT 5 ~T 5 > T T - [20] - T[30] - [47] T 5 10 T 17 vT vvv T 5T} > T} >19 T 10 2 T T T [10] T T[28] vT vT 5 T 10 14 T 7 > T} > T} T T 5 T -14 -7 T [0] T [14] vv T vv ~ T > 7 T 7 T T 22/35 v Example calculation: v = 12, b = 6, k = 3
T vT 5 ~T 5 > T T - [20] - T[30] - [47] V = T 5 10 T 17 47 vT vvv T 5T} > T} >19 T 10 2 T T T [10] T T[28] I = 36 vT vT 5 T 10 14 T 7 > T} > T} T T 47 5 R = T -14 -7 T 36 [0] T [14] vv T vv ~ T > 7 T 7 T T 22/35 v . . . Or we can use the Levi graph
If i and j are treatment vertices in the Levi graph G˜ and R˜ ij is the effective resistance between them in G˜ then
2 Vij = R˜ ij × σ .
23/35 23 V = 23 I = 8 R = 8
- 3 R 3 3 ? 63 3 [0] [15] R 5 5 R 5 - 8 [23]
Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xxH ¨ 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 23 V = 23 I = 8 R = 8
3 3 3 3 3 [0] [15] R 5 5 R 5 - 8 [23]
Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 - x ¡A x @ R@ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ H ?x 8H x x ¨ 6 6 H ¨ xxH ¨ 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 23 V = 23 I = 8 R = 8
R 5 5 R 5 - 8 [23]
Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 - x ¡A x @ 3 R@ 1¡ A 2 x@ 3 x 3 @ ¡ A 2 ¨¨ HH ¨ H ?x 8H x x ¨ 6 63 3 H ¨ [ ] xxH ¨ 4 0 [15] A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 23 V = 23 I = 8 R = 8
R 8 [23]
Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 - x ¡A x @ 3 R@ 1¡ A 2 x@ 3 x 3 @ ¡ A 2 ¨¨ HH ¨ H ?x 8H x x ¨ 6 63 3 H ¨ [ ] xxH ¨ 4 0 [15] A ¡ R@ 5 5 @ 4 3 x@ -5 @ xA ¡ x A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 23 V = 23 I = 8 R = 8
Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 - x ¡A x @ 3 R@ 1¡ A 2 x@ 3 x 3 @ ¡ A 2 ¨¨ HH ¨ H ?x 8H x x ¨ 6 63 3 H ¨ [ ] xxH ¨ 4 0 [15] A ¡ R@ 5 5 R@ 4 3 x@ -5 8 @ xA ¡ x [23] A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 Example 2 yet again: v = 8, b = 4, k = 3
1 2 3 4 23 V = 23 I = 8 R = 2 3 4 1 8 5 6 7 8
5 5 8 1 - x ¡A x @ 3 R@ 1¡ A 2 x@ 3 x 3 @ ¡ A 2 ¨¨ HH ¨ H ?x 8H x x ¨ 6 63 3 H ¨ [ ] xxH ¨ 4 0 [15] A ¡ R@ 5 5 R@ 4 3 x@ -5 8 @ xA ¡ x [23] A¡
xx 63 7 7x x Levi graph concurrence graph
24/35 . . . but actually false. There are many counter-examples. It is not even true that the largest resistance corresponds to the largest distance in the graph.
Resistance increases with distance?
This is obviously true . . .
25/35 There are many counter-examples. It is not even true that the largest resistance corresponds to the largest distance in the graph.
Resistance increases with distance?
This is obviously true . . .
. . . but actually false.
25/35 It is not even true that the largest resistance corresponds to the largest distance in the graph.
Resistance increases with distance?
This is obviously true . . .
. . . but actually false. There are many counter-examples.
25/35 Resistance increases with distance?
This is obviously true . . .
. . . but actually false. There are many counter-examples. It is not even true that the largest resistance corresponds to the largest distance in the graph.
25/35 The graph G is distance-regular if A1Ad is a linear combination of Ad−1, Ad and Ad+1 for all d.
Theorem (Biggs) If G is distance-regular then pairwise resistance is an increasing function of distance.
Distance-regular graphs
Suppose that the concurrence graph G is simple (no multiple edges). Let Ad be the v × v matrix whose (i, j)-entry is equal to ( 1 if the distance from i to j in G is d 0 otherwise
26/35 Theorem (Biggs) If G is distance-regular then pairwise resistance is an increasing function of distance.
Distance-regular graphs
Suppose that the concurrence graph G is simple (no multiple edges). Let Ad be the v × v matrix whose (i, j)-entry is equal to ( 1 if the distance from i to j in G is d 0 otherwise
The graph G is distance-regular if A1Ad is a linear combination of Ad−1, Ad and Ad+1 for all d.
26/35 Distance-regular graphs
Suppose that the concurrence graph G is simple (no multiple edges). Let Ad be the v × v matrix whose (i, j)-entry is equal to ( 1 if the distance from i to j in G is d 0 otherwise
The graph G is distance-regular if A1Ad is a linear combination of Ad−1, Ad and Ad+1 for all d.
Theorem (Biggs) If G is distance-regular then pairwise resistance is an increasing function of distance.
26/35 Theorem If the block design is partially balanced with respect to an amorphic association scheme, then pairwise resistance Rij is a monotonic decreasing function of concurrence λij.
Anything else nice?
Theorem If the concurrence graph G is regular (in particular, if the block design is binary and all treatments have the same replication), and the Laplacian matrix L has precisely two non-trivial eigenvalues, then pairwise resistance Rij is a decreasing linear function of concurrence λij.
27/35 Anything else nice?
Theorem If the concurrence graph G is regular (in particular, if the block design is binary and all treatments have the same replication), and the Laplacian matrix L has precisely two non-trivial eigenvalues, then pairwise resistance Rij is a decreasing linear function of concurrence λij.
Theorem If the block design is partially balanced with respect to an amorphic association scheme, then pairwise resistance Rij is a monotonic decreasing function of concurrence λij.
27/35 Put V¯ = average value of the Vij. Then
2kσ2 Tr(L−) 1 V¯ = = 2kσ2 × , v − 1 harmonic mean of θ1,..., θv−1
where θ1,..., θv−1 are the nontrivial eigenvalues of L.
Average pairwise variance
The variance of the best linear unbiased estimator of the simple difference τi − τj is
− − − 2 Vij = Lii + Ljj − 2Lij kσ .
28/35 Average pairwise variance
The variance of the best linear unbiased estimator of the simple difference τi − τj is
− − − 2 Vij = Lii + Ljj − 2Lij kσ .
Put V¯ = average value of the Vij. Then
2kσ2 Tr(L−) 1 V¯ = = 2kσ2 × , v − 1 harmonic mean of θ1,..., θv−1
where θ1,..., θv−1 are the nontrivial eigenvalues of L.
28/35 —equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L;
—equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L:
—equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L; I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv);
> − > I E-optimal if minimizes the largest value of x L x/x x;
Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij;
over all block designs with block size k and the given v and b.
29/35 —equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L:
—equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L;
—equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L; > − > I E-optimal if minimizes the largest value of x L x/x x;
Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij;
I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv);
over all block designs with block size k and the given v and b.
29/35 —equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L;
—equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L;
—equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L:
Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij;
I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv);
> − > I E-optimal if minimizes the largest value of x L x/x x;
over all block designs with block size k and the given v and b.
29/35 —equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L;
—equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L:
Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij; —equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L; I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv);
> − > I E-optimal if minimizes the largest value of x L x/x x;
over all block designs with block size k and the given v and b.
29/35 —equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L:
Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij; —equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L; I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv); —equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L; > − > I E-optimal if minimizes the largest value of x L x/x x;
over all block designs with block size k and the given v and b.
29/35 Optimality
The design is called
I A-optimal if it minimizes the average of the variances Vij; —equivalently, it maximizes the harmonic mean of the non-trivial eigenvalues of the Laplacian matrix L; I D-optimal if it minimizes the volume of the confidence ellipsoid for (τ1,..., τv); —equivalently, it maximizes the geometric mean of the non-trivial eigenvalues of the Laplacian matrix L; > − > I E-optimal if minimizes the largest value of x L x/x x; —equivalently, it maximizes the minimum non-trivial eigenvalue θ1 of the Laplacian matrix L: over all block designs with block size k and the given v and b.
29/35 Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):
product of non-trivial eigenvalues of L = v × number of spanning trees.
So a design is D-optimal if and only if its concurrence graph G has the maximal number of spanning trees.
This is easy to calculate by hand when the graph is sparse.
D-optimality: spanning trees
A spanning tree for the graph is a collection of edges of the graph which form a tree (connected graph with no cycles) and which include every vertex.
30/35 This is easy to calculate by hand when the graph is sparse.
D-optimality: spanning trees
A spanning tree for the graph is a collection of edges of the graph which form a tree (connected graph with no cycles) and which include every vertex.
Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):
product of non-trivial eigenvalues of L = v × number of spanning trees.
So a design is D-optimal if and only if its concurrence graph G has the maximal number of spanning trees.
30/35 D-optimality: spanning trees
A spanning tree for the graph is a collection of edges of the graph which form a tree (connected graph with no cycles) and which include every vertex.
Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):
product of non-trivial eigenvalues of L = v × number of spanning trees.
So a design is D-optimal if and only if its concurrence graph G has the maximal number of spanning trees.
This is easy to calculate by hand when the graph is sparse.
30/35 So a block design is D-optimal if and only if its Levi graph maximizes the number of spanning trees.
If v > b it is easier to count spanning trees in the Levi graph than in the concurrence graph.
What about the Levi graph?
Theorem (Gaffke) Let G and G˜ be the concurrence graph and Levi graph for a connected incomplete-block design for v treatments in b blocks of size k. Then the number of spanning trees for G˜ is equal to kb−v+1 times the number of spanning trees for G.
31/35 If v > b it is easier to count spanning trees in the Levi graph than in the concurrence graph.
What about the Levi graph?
Theorem (Gaffke) Let G and G˜ be the concurrence graph and Levi graph for a connected incomplete-block design for v treatments in b blocks of size k. Then the number of spanning trees for G˜ is equal to kb−v+1 times the number of spanning trees for G.
So a block design is D-optimal if and only if its Levi graph maximizes the number of spanning trees.
31/35 What about the Levi graph?
Theorem (Gaffke) Let G and G˜ be the concurrence graph and Levi graph for a connected incomplete-block design for v treatments in b blocks of size k. Then the number of spanning trees for G˜ is equal to kb−v+1 times the number of spanning trees for G.
So a block design is D-optimal if and only if its Levi graph maximizes the number of spanning trees.
If v > b it is easier to count spanning trees in the Levi graph than in the concurrence graph.
31/35 8 spanning trees 216 spanning trees
Example 2 one last time: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xH ¨ x 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
x63 x 7 7x x Levi graph concurrence graph
32/35 216 spanning trees
Example 2 one last time: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xH ¨ x 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
x63 x 7 7x x Levi graph concurrence graph 8 spanning trees
32/35 Example 2 one last time: v = 8, b = 4, k = 3
1 2 3 4 2 3 4 1 5 6 7 8
5 5 8 1 x ¡A x @ @ 1¡ A 2 x@ x @ ¡ A 2 ¨¨ HH ¨ x x H x 8HH ¨¨ 6 xH ¨ x 4 A ¡ @ @ 4 3 x@ @ xA ¡ x A¡
x63 x 7 7x x Levi graph concurrence graph 8 spanning trees 216 spanning trees
32/35 Lemma Let G have an edge-cutset of size c (set of c edges whose removal disconnects the graph) whose removal separates the graph into components of sizes m and n. Then 1 1 θ ≤ c + . 1 m n
If c is small but m and n are both large, then θ1 is small.
There is a similar result for vertex-cutsets.
E-optimality: the cutset lemma
A design is E-optimal if it maximizes the smallest non-trivial eigenvalue θ1 of the Laplacian L of the concurrence graph G.
33/35 If c is small but m and n are both large, then θ1 is small.
There is a similar result for vertex-cutsets.
E-optimality: the cutset lemma
A design is E-optimal if it maximizes the smallest non-trivial eigenvalue θ1 of the Laplacian L of the concurrence graph G. Lemma Let G have an edge-cutset of size c (set of c edges whose removal disconnects the graph) whose removal separates the graph into components of sizes m and n. Then 1 1 θ ≤ c + . 1 m n
33/35 There is a similar result for vertex-cutsets.
E-optimality: the cutset lemma
A design is E-optimal if it maximizes the smallest non-trivial eigenvalue θ1 of the Laplacian L of the concurrence graph G. Lemma Let G have an edge-cutset of size c (set of c edges whose removal disconnects the graph) whose removal separates the graph into components of sizes m and n. Then 1 1 θ ≤ c + . 1 m n
If c is small but m and n are both large, then θ1 is small.
33/35 E-optimality: the cutset lemma
A design is E-optimal if it maximizes the smallest non-trivial eigenvalue θ1 of the Laplacian L of the concurrence graph G. Lemma Let G have an edge-cutset of size c (set of c edges whose removal disconnects the graph) whose removal separates the graph into components of sizes m and n. Then 1 1 θ ≤ c + . 1 m n
If c is small but m and n are both large, then θ1 is small.
There is a similar result for vertex-cutsets.
33/35 D ¤ bb "" bb "" J xD ¤ x @ b" b" @ x" b x x" b x Q xJ D ¤ x " b " b Q J D ¤ x x x Q x TT `` J D ¤ x x x x ``Q` T Q`JD` ¤ ` x QJ ``` x TT Q ` x xJ Q T Q J Q xx TT x x x @ J T x@ J x x x x 1 1 θ ≤ 2 + θ ≥ 1 1 5 10 1
The only E-optimal designs are the queen-bee designs.
E-optimality when v = 2b + 1 and k = 3
The Levi graph has 3b + 1 vertices and 3b edges, so it is a tree.
34/35 1 1 θ ≤ 2 + θ ≥ 1 1 5 10 1
The only E-optimal designs are the queen-bee designs.
E-optimality when v = 2b + 1 and k = 3
The Levi graph has 3b + 1 vertices and 3b edges, so it is a tree.
D ¤ bb "" bb "" J xD ¤ x @ b" b" @ x" b x x" b x Q xJ D ¤ x " b " b Q J D ¤ x x x Q x TT `` J D ¤ x x x x ```Q T `Q JD ¤`` x QJ ``` x TT Q ` x xJ Q T Q J Q xx TT x x x @ J T x@ J x x x x
34/35 The only E-optimal designs are the queen-bee designs.
E-optimality when v = 2b + 1 and k = 3
The Levi graph has 3b + 1 vertices and 3b edges, so it is a tree.
D ¤ bb "" bb "" J xD ¤ x @ b" b" @ x" b x x" b x Q xJ D ¤ x " b " b Q J D ¤ x x x Q x TT `` J D ¤ x x x x ```Q T `Q JD ¤`` x QJ ``` x TT Q ` x xJ Q T Q J Q xx TT x x x @ J T x@ J x x x x 1 1 θ ≤ 2 + θ ≥ 1 1 5 10 1
34/35 E-optimality when v = 2b + 1 and k = 3
The Levi graph has 3b + 1 vertices and 3b edges, so it is a tree.
D ¤ bb "" bb "" J xD ¤ x @ b" b" @ x" b x x" b x Q xJ D ¤ x " b " b Q J D ¤ x x x Q x TT `` J D ¤ x x x x ```Q T `Q JD ¤`` x QJ ``` x TT Q ` x xJ Q T Q J Q xx TT x x x @ J T x@ J x x x x 1 1 θ ≤ 2 + θ ≥ 1 1 5 10 1
The only E-optimal designs are the queen-bee designs.
34/35 For general block designs, we do not know if we can use the Levi graph to investigate E-optimality.
Can we use the Levi graph to find E-optimal designs?
For binary designs with equal replication, θ1(L) is a monotonic increasing function of θ1(L˜).
35/35 Can we use the Levi graph to find E-optimal designs?
For binary designs with equal replication, θ1(L) is a monotonic increasing function of θ1(L˜).
For general block designs, we do not know if we can use the Levi graph to investigate E-optimality.
35/35