Coherent configurations Lecture 3 Weisfeiler-Leman ...karabas/edu/sschool/material/cc03.pdf ·...

Coherent configurationsLecture 3

Weisfeiler-Leman stabilization

Mikhail Klin (Ben-Gurion University)

September 1–5, 2014

M. Klin Coherent configurations September 2014 1 / 53

More about history

Origin 1 Donald G. Higman

Expert in permutation group theory.

Codiscoverer of one of the first new (after

E. Mathieu) sporadic simple groups.

Group was constructed in one day and night

(1967) via SRG(100,22,0,6).

Graph itself was constructed in 1956 by

Dale Mesner (applied statistics).


Higman

It took Higman a couple of years to recognize

(1970) that coherent configurations appear as a

natural generalization of the concept of strongly

regular graph (used and exploited by him in

diverse contexts since 1964).


Weisfeiler and Leman

Origin 2: Boris Weisfeiler and Andrei Leman

Boris Weisfeiler: bright expert in algebraic

groups, student of E. Vinberg.

Due to jewish origin had difficulties in

USSR.

Andrei Leman: brilliant expert in olympiadic

problems; no jewish origin; “axiomatically”

was attributed as a jew.


Adel’son-Vel’skii

Georgii (Gera) Adel’son-Vel’skii was a Ph.D

student (1948) of A. N. Kolmogorov.

Actively attended seminar of I. M. Gel’fand.

Was working in Siberia, returned to

Moscow.

Strongly contributed to development of

initial roots of modern CS and theory of

artificial intelligence.


Graph isomorphism problem

Became interested in the graph

isomorphism problem, due to its links to

chemical informatics (identification of

organic structures).

Involved Weisfeiler and Leman to this

problem and suggested to try to find

polynomial time algorithm for its solution.

Their efforts resulted in the creation of the

concept of a cellular algebra (1968).


Non-Schurian example

At the beginning all involved parties were

sure that each cellular (coherent) algebra is

Schurian: reincarnation of ideas of Schur

(though this language was not used then).

Were extremely surprised to succeed (jointly

with I. Faradzev) to construct (1969) their

own counterexample of SRG(26,10,3,4)

with intransitive automorphism group.

This was done with the aid of a computer.


Main idea of WL algorithm

Main achievement of Weisfeiler-Leman in AGT

(in modern clothes):

Polynomial time algorithm to find coherent

closure of a given set of matrices (graphs in

relational terminology).


Coherent closure - WL-stabilization

Algebraic formulation shows easily -

intersection of coherent algebras is coherent

algebra, and any matrix is in some coherent

algebra (Cn×n is coherent).

We define for matrix A: 〈〈A〉〉 is the

coherent closure of A, the smallest coherent

algebra containing A.

A similar definition is given for a set of

square matrices of the same order n.


Coherent closure - WL-stabilization

Coherent closure of a graph is the coherent

closure of its adjacency matrix.

In other words - it is the smallest coherent

configuration that admits the given graph

as a union of basic graphs.

An efficient polynomial-time algorithm for

calculation — Weisfeiler-Leman

stabilization.


Example 3.1 (WL-stabilization)

Start with graph Γ = •

•

•

•

????

???? 1

2

3

4

.

Create (generalized) adjacency matrix

A = A(Γ) =

(0 1 2 11 0 1 12 1 0 11 1 1 0

)Substitute distinct entries by distinct

(non-commuting) variables

M1 =

(a b c bb a b bc b a bb b b a

)M. Klin Coherent configurations September 2014 11 / 53

Example 3.1 (cont.)

Calculate matrix M21 with polynomial

entries:

M21 =

(a2+2b2+c2 ab+ba+b2+cb ac+2b2+ca ab+ba+b2+cb

ab+ba+b2+bc a2+3b2 ab+ba+b2+bc ab+ba+2b2

ac+2b2+ca ab+ba+b2+cb a2+2b2+c2 ab+ba+b2+cbab+ba+b2+bc ab+ba+2b2 ab+ba+b2+bc a2+3b2

)Substitute distinct polynomials by new

distinct variables, get

M2 =

(x1 x2 x3 x2x4 x5 x4 x6x3 x2 x1 x2x4 x6 x4 x5

)Repeat iteration, that is calculate M2

2 .M. Klin Coherent configurations September 2014 12 / 53

Example 3.1 (cont.)

M22 =

(x2

1 +2x2x4+x23 x1x2+x2x5+x2x6+x3x2 x1x3+2x2x4+x3x1 x1x2+x2x5+x2x6+x3x2

x4x1+x4x3+x5x4+x6x4 2x4x2+x25 +x2

6 x4x1+x4x3+x5x4+x6x4 2x4x2+x5x6+x6x5

x1x3+2x2x4+x3x1 x1x2+x2x5+x2x6+x3x2 x21 +2x2x4+x2

3 x1x2+x2x5+x2x6+x3x2

x4x1+x4x3+x5x4+x6x4 2x4x2+x5x6+x6x5 x4x1+x4x3+x5x4+x6x4 2x4x2+x25 +x2

6

)Obtain

M3 =

(y1 y2 y3 y2y4 y5 y4 y6y3 y2 y1 y2y4 y6 y4 y5

)M2 ≈ M3

End of WL-stabilization.M. Klin Coherent configurations September 2014 13 / 53

Example 3.1 (cont.)

Final analysis: G = Aut(Γ) =

e, (1, 3), (2, 4), (1, 3)(2, 4).There are six 2-orbits of (G , [1, 4]), that is

orbits of the action (G , [1, 4]2).

Here they are as graphs:

••••

••••

__???

??

??? ••••

••••???

??__???

••••

••••


Example 3.1 (cont.)

Summary: In this example we, indeed, obtain

basic graphs of the centralizer algebra

V (G , [1, 4]), using polynomial-time algorithm of

WL-stabilization, avoiding intermediate step of

literal determination of the group (G , [1, 4]).


Algebraic description of algorithm

Start with the adjacency matrix A of an

undirected, directed or colored graph Γ.

Write A in the form A =r−1∑i=0

iAi , where Ai

are (0, 1)-matrices.

At the end of iteration obtain a new set of

(0, 1)-matrices, A′0,A′1, . . . ,A

′r ′−1.



Each time we wish to get a basis of a linear

subspace S , which is closed under

Schur-Hadamard product and transposition

and contains matrices In and Jn.

Initially, S is not closed with respect to

matrix multiplication.

Enough to check this extra property for all

products of pairs of matrices from

considered basis.M. Klin Coherent configurations September 2014 17 / 53


This process is repeated until it is stable.

Criterion of stability: rank of subspace S is

not increasing.

It is convenient to consider matrix

D =r−1∑i=0

tiAi with non-commutative

indeterminates.


Formal description

Input: the adjacency matrix

A = A(Γ) = (auv) of Γ.

Output: a standard basis A0,A1, . . . ,Ar−1of the cellular algebra W (Γ).


Formal description

1 Let [0, s − 1] be the set of different entries of A.For k = 0, 1, . . . , s − 1 do

Define Ak = (a(k)uv ) to be the matrix with a(k)uv = 1if auv = k and a(k)uv = 0 otherwise.

Let r := s.2 Let D =

∑r−1k=0 tkAk, where t0, t1, . . . , tr−1 are distinct

non-commuting indeterminates.3 Compute the matrix product B = (buv ) = D · D. Each entry

buv of B is a sum of products ti tj.4 Determine the set d0, d1, . . . , ds−1 of different

expressions among the entries buv.5 If s > r then

For k = 0, 1, . . . , s − 1 doDefine Ak = (a(k)uv ) to be the matrix with a(k)uv = 1if buv = dk and b(k)uv = 0 otherwise.

r := s. Goto 2.6 STOP.


Example 3.2 (From mathematicalchemistry)

H4

H3

C 1 C 2

H5

H6

?????????

?????????

Here the superscripts denote the numbers

from Ω = 1, 2, 3, 4, 5, 6 associated to

atoms that form the molecule of ethylene.


Example 3.2 (cont)

Let A be the adjacency matrix of the colored

graph Γ associated to the molecular graph.

H – 0

C – 1

single bond – 2

double bond – 3

no bond – 4

A =

0 3 2 2 4 43 0 4 4 2 22 4 1 4 4 42 4 4 1 4 44 2 4 4 1 44 2 4 4 4 1


Example 3.2 (cont)

Then we get that

D =

t0 t3 t2 t2 t4 t4t3 t0 t4 t4 t2 t2t2 t4 t1 t4 t4 t4t2 t4 t4 t1 t4 t4t4 t2 t4 t4 t1 t4t4 t2 t4 t4 t4 t1


Example 3.2 (cont)

B = D · D =

x0 x2 x3 x3 x4 x4x2 x0 x4 x4 x3 x3x5 x6 x1 x7 x8 x8x5 x6 x7 x1 x8 x8x6 x5 x8 x8 x1 x7x6 x5 x8 x8 x7 x1


Example 3.2 (cont)

Wherex0 = t20 + 2t22 + t22 + 2t24x1 = t21 + t22 + 4t24x2 = t0t3 + 2t2t4 + t3t0 + 2t4t2x3 = t0t2 + t2t1 + t2t4 + t3t4 + 2t24x4 = t0t4 + 2t2t4 + t3t2 + t4t1 + t24x5 = t1t2 + t2t0 + t4t2 + t4t3 + 2t24x6 = t1t4 + t2t3 + t4t0 + 2t4t2 + t24x7 = t1t4 + t22 + t4t1 + 3t24x8 = t1t4 + t2t4 + t4t1 + t4t2 + 2t24


Example 3.2 (cont)

Now we proceed with the matrix A′

A′ =

0 2 3 3 4 4

2 0 4 4 3 3

5 6 1 7 8 8

5 6 7 1 8 8

6 5 8 8 1 7

6 5 8 8 7 1


Example 3.2 (cont)

Now we have to start new iteration with the

matrix A′ instead of A and to obtain the

matrix A′′, the result of second iteration.

Check that matrices A′ and A′′ are

equivalent.

Thus the matrix A′ represents the basis of

the coherent closure W (Γ).


Interpretation

There is a natural graph-theoretical

interpretation of WL-stabilization.

The goal is to reach structure constants pkijwith respect to final basis of W (Γ).

That is

AiAj = ppijA0 + p1ijA1 + · · · + pr−1ij Ar−1

for each pair i , j ∈ 0, 1, . . . , r − 1.M. Klin Coherent configurations September 2014 28 / 53

Triangles

In the colored graph ∆, which corresponds

to matrix B , each arc (u, v) of a given color

k is the basis arc of exactly pkij triangles

with first non-basis arc of color i and

second non-basis arc of color j .


Triangles

A triangle consists of three not necessarily

distinct vertices u, v ,w and arcs (u, v),

(u,w) and (w , v).

The arc (u, v) is called the basis arc, the

other arcs are the non-basis arcs of the

triangle.


Iteration in new language

One iteration includes the round along all

arcs of the given graph Γ.

For each arc (u, v) of a fixed color k we

count the number of paths of length 2 such

that the first arc (u,w) is of color i and the

second arc (w , v) is of color j .

These numbers should be equal for all arcs.

If this is true, then these numbers are just

the structure constants pkij .


Re-partitioning

If not, then the arc set Rk of color k has to

be partitioned into subsets

Rk0,Rk1, . . . ,Rkt−1 each consisting of arcs

with the same numbers.

This step is performed for all colors

k ∈ 0, 1, . . . , r − 1.Then the graph Γ is recolored, i.e. we

identify color k0 with the old color k and

introduce the new colors k1, . . . , kt−1.


Stopping condition

The next iteration is performed for the

recolored graph Γ.

If in some iteration no new colors are

introduced, then the process is stable and

we can stop.

In this case, the graph Γ with the final

stable coloring represents the required

coherent algebra W .


Heuristics - symmetry of graph

In the course of practical performance of

WL-stabilization it might be efficient to use

in advance knowledge about symmetry of a

prescribed graph Γ.

Let us consider this claim on a level of

example.


Example 3.3

Γ =1

2

4 6

8

7

53

@@@@@

_____

@@@@@

_____

----------

----------

We wish to describe the coherent closure

〈〈A = A(Γ)〉〉 of the adjacency matrix of Γ.


Example 3.3 (cont)

Consider the four triangles in Γ.

Each vertex from Ω1 = 1, 2, 7, 8 is

involved in precisely two of these triangles

while each vertex of Ω2 = 3, 4, 5, 6 is

involved in only one.

Thus any orbit of G = Aut(Γ) must be a

subset of either Ω1 or Ω2.


Example 3.3 (cont)

Consider now the permutations g1 = (1, 2),

g2 = (7, 8), g3 = (3, 6, 5, 4)(1, 7)(2, 8), and

g4 = (3, 4)(5, 6).

Clearly gi ∈ G for all i = 1, 2, 3, 4 which

proves that Ω1 and Ω2 are indeed the orbits

of G .

In fact, one may check that

G = 〈g1, g2, g3, g4〉, and further that

|G | = |1G | · |3G1| · |7G1,3| = 4 · 2 · 2 = 16.M. Klin Coherent configurations September 2014 37 / 53

Example 3.3 (cont)

We can now verify that the cycle index

polynomial of G in its action on

Ω = Ω1 ∪ Ω2 is given by:

Z (G ,Ω) = 116(x81 + 2x61x2 + 2x41x

22+

2x21x32 + 5x42 + 4x22x4).


Example 3.3 (cont)

By counting fixed points of g ∈ G in the

induced action of G on Ω2 (i.e., ordered

pairs (a, b) ∈ Ω2 for which (a, b)g = (a, b))

we easily obtain the number t2 of 2-orbits of

(G ,Ω):

t2 = 116(82 + 2 · 62 + 2 · 42 + 2 · 22) = 11.


Example 3.3 (cont)

We now start to construct 〈〈A〉〉 using

relational language.

In step 1 of our computation we obtain the

reflexive relations

R1 = (1, 1), (2, 2), (7, 7), (8, 8) and

R2 = (3, 3), (4, 4), (5, 5), (6, 6).


Example 3.3 (cont)

In step 2, we distinguish between ordered

pairs of adjacent vertices by taking into

account to which sets (Ω1 or Ω2) the first

and second members of a pair belong.

Simultaneously, we distinguish between

ordered pairs of non-adjacent vertices in the

same manner.


Example 3.3 (cont)

This gives the following relations:R3 = (1, 2), (2, 1), (7, 8), (8, 7),R4 = (3, 5), (4, 6), (5, 3), (6, 4),R5 = (1, 3), (2, 3), (1, 4), (2, 4), (7, 5), (7, 6), (8, 5), (8, 6) ,R6 = R t

5,R7 = (1, 7), (1, 8), (2, 7), (2, 8), (7, 1), (7, 2), (8, 1), (8, 2),R8 = (3, 4), (3, 6), (4, 3), (4, 5), (5, 4), (5, 6), (6, 3), (6, 5),R9 = (1, 5), (1, 6), (2, 5), (2, 6), (7, 3), (7, 4), (8, 3), (8, 4),R10 = R t

9.


Example 3.3 (cont)

One of our relations must further split into

two parts.

Determining which relation really splits,

constitutes step 3.

Let us look more carefully at relation R8.

We observe that for the pair (3, 4) there

exist two paths of length 2 from 3 to 4 (via

1 and 2).

For the pair (3, 6) no such paths exist.M. Klin Coherent configurations September 2014 43 / 53

Example 3.3 (cont)

This distinction is sufficient to determine

that relation R8 splits into

R ′8 = (3, 4), (4, 3), (5, 6), (6, 5) and

R ′′8 = (3, 6), (4, 5), (5, 4), (6, 3).Thus we have produced 11 relations in all

(viz. R1, . . . ,R7,R′8,R

′′8 ,R9,R10) so that no

further splitting is possible.

This suffices to prove that 〈〈A〉〉 coincides

with the centralizer algebra V (G ,Ω).M. Klin Coherent configurations September 2014 44 / 53

Matrix transpositions

A significant extra warning:

Our starting vector space S should be closed

with respect to transposition of matrices.

Otherwise, at the end of iterations we may

not get a coherent algebra.

This requirement is met if all graphs are

undirected.


Example 3.4 (Regular action of S3)G = e, (1, 2)(3, 5)(4, 6), (1, 3)(2, 4)(5, 6),

(1, 4, 5)(2, 3, 6), (1, 5, 4)(2, 6, 3), (1, 6)(2, 5)(3, 4)

We consider the regular action of S3.

In this action we take three sets:

e, (1, 2)(3, 5)(4, 6),(1, 3)(2, 4)(5, 6), (1, 4, 5)(2, 3, 6) and

(1, 5, 4)(2, 6, 3), (1, 6)(2, 5)(3, 4).


Example 3.4 (cont)

The corresponding (sums of) permutation

matrices are:

A0 =

1 1 0 0 0 01 1 0 0 0 00 0 1 0 1 00 0 0 1 0 10 0 1 0 1 00 0 0 1 0 1

A1 =

0 0 1 1 0 00 0 1 1 0 01 0 0 0 0 10 1 0 0 1 01 0 0 0 0 10 1 0 0 1 0

A2 =

0 0 0 0 1 10 0 0 0 1 10 1 0 1 0 01 0 1 0 0 00 1 0 1 0 01 0 1 0 0 0


Example 3.4 (cont)

The subspace of M6(C), S = 〈A0,A1,A2〉 is

closed under SH-product, since it has a

basis of disjoint (0, 1)-matrices.

S is closed under matrix multiplication

since: A20 = 2A0, A0A1 = 2A1, A0A2 = 2A2,

A1A0 = A1 + A2, A21 = A0 + A2,

A1A2 = A0 + A1, A2A0 = A1 + A2,

A2A1 = A0 + A2 and A2A2 = A0 + A1.


Example 3.4 (cont)

However, S is not a coherent algebra for

two reasons:

I 6∈ S .

S is not closed under transposition: AT1 (as

well as AT2 ) is not an element of S .


Complexity of WL-stabilization

Complexity evaluation of WL-stabilization

requires special attention.

One implementation (Babel, Chuvaeva,

Klin, Pasechnik) has runtime in class O(n7).

Another, more sophisticated (Babel et al) is

in class O(n3 log n).

In any case, it is pretty clear that we have a

polynomial-time algorithm.


Complexity of WL-stabilization

Remark: in fact, the algorithm with higher

complexity for “small” number of vertices is

working much more quickly than the other

algorithm.

Here, “small” means up to a few hundreds

of vertices.


Main references

Babel, L.; Chuvaeva, I. V.; Klin, M.; Pasechnik, D.V. Algebraic combinatorics in mathematicalchemistry. Methods and algorithms. II. Programimplementation of the Weisfeiler-Leman algorithm.http://arxiv.org/pdf/1002.1921.pdf


Thank You!


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