Once upon a time …
Euler’s problem (1936)
Königsberg’s city (nowadays Kaliningrad) is crossed by Pregel river, which runs
around the island of Kneiphof on both sides, and has seven bridges
During a walk, is-it possible to pass on all the bridges of the city once and only
once?
Some references …
• König, D. (1936).Theorie der endlichen und
unendlichen Graphen.
König, D. (1990).Theory of finite and infinite graphs. Berlin:
Birkhauser
• Berge, C. (1958). Théorie des graphes et ses
applications. Paris: Dunod.
English edition, Wiley 1961; Methuen & Co, New York
1962; Dover, New York 2001.
Russian, Moscow 1961;
Spanish, Mexico 1962;
Roumanian, Bucharest 1969;
Chinese, Shanghai 1963;
Some references …
3 fundamental articles to use similitude analysis in the
social representations’ domain
Flament (1962). L’analyse de similitude. Cahiers du
Centre de Recherche Opérationnelle, 4, 63-97
Degenne, A. & Vergès, P. (1973). Introduction à
l’analyse de similitude. Revue française de
Sociologie, 14, 471-512
Flament, C., Degenne, A. & Vergès, P. (1971). Similarity
Analysis. Paris: Maison des Sciences de l’Homme.
Graphs theory
Useful elements
• A graph G
Size of the graph
 G (V, E)
 V = {v1, v2, …, vn} that is n Vertices
 E = {e1, e2, …, em } that is m Edges
• A graph G (V, E)
 V = {1, 2, 3, 4, 5, 6, 7, 8, 9, }
 E = {(1, 2), (1, 3), (1, 4), …, (8, 9)}
m
9   9  1
2
72

 36
2
m
n   n  1
2
Graphs theory
Useful elements
G(V,E)
–
–
V = {1, 2, 3, 4, 5}
E = {(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5),
(3,4), (3,5), (4,5)}
G(V,E)
–
–
V’ = {1, 2, 3}
E’ = {(1,2), (1,3), (2,3)}
Some vertices = subgraph of G
G(V,E)
–
V’ = {1, 2, 3, 4, 5}
–
E’ = {(1,2), (3,4), (4,5)}
All the vertices, some edges = Spanning Subgraph of G
Graphs theory
Useful elements
G(V,E)
–
–
V = {1, 2, 3, 4, 5}
E = {(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5),
(3,4), (3,5), (4,5)}
Symmetrical relations : (5, 4) = (4, 5)
(1, 3) , (3, 5), (5, 4) = a chain of G
G(V,E)
–
–
V’ = {1, 2, 3, 4, 5}
E’ = {(1,2), (1,3), (2,3), (3,4), (4,5)}
(1, 2), (2, 3), (3, 4), (4, 5), (5, 6) = a chain
Graphs theory
Useful elements
A complete graph
– A cycle
– A chain
A TREE
A connected tree without cycle
– A chain which connects all the
verticies
– A chain without cycle
How to pass from a complete graph to a tree ?
Useful elements for SR
How to pass from a complete graph to a tree ?
Searching for the structure of the relations
=
Searching for the skeleton of the representation
=
Searching for a tree
Each edge has a weight
=
Similitude analysis ￫ weight = similitude index
=
Co-occurrence, symmetrical co-occurrence, Phi square measure,
Correlation, squared index of similitude (Guimelli), etc.
Useful elements for SR
How to pass from a complete graph to a tree ?
Searching for the structure of the relations
=
Searching for a maximum tree
Searching for a connected graph without cycle
+
Searching for the heaviest tree
=
Searching for a tree which retains the most similarity
Let us return to our example …
Example 1 (inspired from Abric, 2003)
Stage 4 – From similitude matrix to the structure of the relations
between elements of a representation
1
Arête (7, 6) = (6,7)
5
constraints
the means to
have relations
The degree of similitude between
two elements can be associated
with the graph
social
integration
an
obligation
self-confidence
finance its
leisure activities
personal blooming
the means to
earn the keep
Example 1 (inspired from Abric, 2003)
Stage 4 – From similitude matrix to the structure of the relations
between elements of a representation
1
Example 1 (inspired from Abric, 2003)
Stage 4 – From similitude matrix to the structure of the relations
between elements of a representation
1
Edges S1
Edges S1
Edges
S1
Edges
S1
(2, 6)
6
(1, 7)
2
(1, 4)
1
(3, 6)
1
(2, 8)
6
(2, 5)
2
(1, 5)
1
(3, 8)
1
(5, 6)
5
(3, 4)
2
(1, 6)
1
(4, 5)
1
(6, 7)
5
(3, 5)
2
(1, 8)
1
(4, 7)
1
(1, 2)
3
(3, 7)
2
(2, 3)
1
(4, 8)
1
(1, 3)
3
(5, 7)
2
(2, 4)
1
(5, 8)
1
(4, 6)
3
(7, 8)
2
(2, 7)
1
(6, 8)
1
Important ! (1)
Connected graph & without cycle
Edges
S
Edges
1
S
Edges
1
S
Edges
1
S
1
(2, 6)
6
(4, 6)
3
(1, 4)
1
(3, 6)
1
(2, 8)
6
(2, 5)
2
(1, 5)
1
(3, 8)
1
(5, 6)
5
(3, 4)
2
(1, 6)
1
(4, 5)
1
(6, 7)
5
(3, 5)
2
(1, 8)
1
(4, 7)
1
(1, 3)
3
(3, 7)
2
(2, 3)
1
(4, 8)
1
(1, 2)
3
(5, 7)
2
(2, 4)
1
(5, 8)
1
(1, 7)
3
(7, 8)
2
(2, 7)
1
(6, 8)
1
Important ! (2)
Edges
S1
Edges
S1
Edges
S1
Edges
S1
(2, 6)
6
(4, 6)
3
(1, 4)
1
(3, 6)
1
(2, 8)
6
(1, 7)
2
(1, 5)
1
(3, 8)
1
(5, 6)
5
(2, 5)
2
(1, 6)
1
(4, 5)
1
(6, 7)
5
(3, 4)
2
(2, 1 )
1
(4, 7)
1
(7, 8)
5
(3, 5)
2
(2, 3)
1
(4, 8)
1
(1, 8)
3
(3, 7)
2
(2, 4)
1
(5, 8)
1
(1, 3)
3
(5, 7)
2
(2, 7)
1
(6, 8)
1


Example 1 (inspiré de Abric, 2003)
Two populations = 2 graphs
Workers
Young students