1. Revealing Order in Complex Systems through Graph Representations Dr. Offer Shai Department of Mechanics, Materials and Systems Faculty of Engineering Tel-Aviv University

2. Outline of the talk The methodology will be explained upon the subject of topological rigidity. Applications of topological rigidity in structures, mechanisms, drugs design, geometric constraint systems in CAD, and more. Decomposition of graphs of arbitrary topological structure into atomic sub-structures (Assur groups). Novel method for generating these atomic sub- structures. Canonical form of all the atomic structures – employing only two production rules. Challenges.

3. Checking topological rigidity Framework is said to be rigid if no motions, except for trivial, can be assigned to its vertices. Rigid framework Non-rigid framework

4. Application of topological rigidity for checking structural stability

5. Application of topological rigidity for development of medications

6. Application of topological rigidity for checking validity of geometric constraints

7. Application of topological rigidity for checking the mobility of mechanisms O1O1 O3O3 O 7 A 1 B C D F G H E 0101 0303 2 3 4 5 6 0 8 0 0303 5 8 G A B C D F H E 1 2 3 4 7

8. It was found that there is a canonical form underlying all possible rigid structures. Any rigid structure can be decomposed into basic building blocks called Assur groups

10. Hierarchic Order of Assur Groups

12. Challenges Finding and implementing applications of topological rigidity in additional fields of complex engineering systems. Applying the methodology for revealing additional cases of order in complex system domains.