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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).
M. Klin Coherent configurations September 2014 2 / 53
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).
M. Klin Coherent configurations September 2014 3 / 53
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.
M. Klin Coherent configurations September 2014 4 / 53
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.
M. Klin Coherent configurations September 2014 5 / 53
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).
M. Klin Coherent configurations September 2014 6 / 53
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.
M. Klin Coherent configurations September 2014 7 / 53
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).
M. Klin Coherent configurations September 2014 8 / 53
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.
M. Klin Coherent configurations September 2014 9 / 53
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.
M. Klin Coherent configurations September 2014 10 / 53
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:
••••
••••
__???
??
??? ••••
••••???
??__???
••••
••••
M. Klin Coherent configurations September 2014 14 / 53
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]).
M. Klin Coherent configurations September 2014 15 / 53
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.
M. Klin Coherent configurations September 2014 16 / 53
Algebraic description of algorithm
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
Algebraic description of algorithm
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.
M. Klin Coherent configurations September 2014 18 / 53
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).
M. Klin Coherent configurations September 2014 19 / 53
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.
M. Klin Coherent configurations September 2014 20 / 53
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.
M. Klin Coherent configurations September 2014 21 / 53
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
M. Klin Coherent configurations September 2014 22 / 53
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
M. Klin Coherent configurations September 2014 23 / 53
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
M. Klin Coherent configurations September 2014 24 / 53
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
M. Klin Coherent configurations September 2014 25 / 53
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
M. Klin Coherent configurations September 2014 26 / 53
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).
M. Klin Coherent configurations September 2014 27 / 53
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 .
M. Klin Coherent configurations September 2014 29 / 53
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.
M. Klin Coherent configurations September 2014 30 / 53
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 .
M. Klin Coherent configurations September 2014 31 / 53
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.
M. Klin Coherent configurations September 2014 32 / 53
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 .
M. Klin Coherent configurations September 2014 33 / 53
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.
M. Klin Coherent configurations September 2014 34 / 53
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.
M. Klin Coherent configurations September 2014 35 / 53
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.
M. Klin Coherent configurations September 2014 36 / 53
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).
M. Klin Coherent configurations September 2014 38 / 53
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.
M. Klin Coherent configurations September 2014 39 / 53
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).
M. Klin Coherent configurations September 2014 40 / 53
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.
M. Klin Coherent configurations September 2014 41 / 53
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.
M. Klin Coherent configurations September 2014 42 / 53
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.
M. Klin Coherent configurations September 2014 45 / 53
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).
M. Klin Coherent configurations September 2014 46 / 53
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
M. Klin Coherent configurations September 2014 47 / 53
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.
M. Klin Coherent configurations September 2014 48 / 53
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 .
M. Klin Coherent configurations September 2014 49 / 53
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.
M. Klin Coherent configurations September 2014 50 / 53
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.
M. Klin Coherent configurations September 2014 51 / 53
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
M. Klin Coherent configurations September 2014 52 / 53
Thank You!
M. Klin Coherent configurations September 2014 53 / 53