Once upon a time …

1. 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?

2. 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;

3. 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.

4. Graphs theoryUseful elements • A graph G • G (V, E) • V = {v1, v2, …, vn} that is nVertices • E = {e1, e2, …, em } that is mEdges Size of the graph • A graph G (V, E) • V = {1, 2, 3, 4, 5, 6, 7, 8, 9, } • E = {(1, 2), (1, 3), (1, 4), …, (8, 9)}

5. Graphs theoryUseful 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

6. Graphs theoryUseful 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 (1, 2), (2, 3) & (3, 1) = a cycle

7. Graphs theoryUseful 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 ?

8. Useful elements for SRHow 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.

9. Useful elements for SRHow 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

10. Let us return to our example …

11. constraints the means to have relations social integration 1 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 Arête (7, 6) = (6,7) 5 The degree of similitude between two elements can be associated with the graph

12. 1 Example 1 (inspired from Abric, 2003)Stage 4 – From similitude matrix to the structure of the relations between elements of a representation

13. 1 Example 1 (inspired from Abric, 2003)Stage 4 – From similitude matrix to the structure of the relations between elements of a representation

14. Important ! (1) Connected graph & without cycle

15. Important ! (2)  

16. Example 1 (inspiré de Abric, 2003)Two populations = 2 graphs Workers Young students