1. Class 12: Communities Network Science: Communities Dr. Baruch Barzel

2. The Modular Structure of Networks

3. Is a Network Modular Clustering implies modularity Small Worldness tends to wipe out modularity Functionality requires modularity

4. Is a Network Modular Clustering implies modularity Small Worldness tends to wipe out modularity Functionality requires modularity Hubs tends to wipe out modularity

5. Is a Network Modular Clustering at the periphery only Low degree nodes typically belong to a single module Hubs bridge between different modules

6. Is a Network Modular Clustering at the periphery only Low degree nodes typically belong to a single module Hubs bridge between different modules But how do we unveil the modules

7. The Modular Structure of Networks Functional modularity Natural partition lines

8. Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks

9. Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks, while minimizing the transmission between tasks

10. Network Partitioning Optimally dividing the network into a predefined number of partitions Dividing a task into sub-tasks, while minimizing the transmission between tasks

11. Network Partitioning Minimizing the Cut: The index vector: The Laplacian Matrix:

12. The Laplacian Matrix Minimizing the Cut: Consider the Eigenvector: Choose the Eigenvector with the minimal Eigenvalue

13. The Laplacian Matrix Minimizing the Cut: Consider the Eigenvector: Choose the Eigenvector with the minimal Eigenvalue

14. The Laplacian Matrix The matrix: The trivial partitioning – put the entire network together: or

15. The Laplacian Matrix The matrix: The case of isolated components The number of Eigenvectors with λ = 0 equals the number of connected components

16. The Laplacian Matrix The matrix: The case of almost isolated components The Eigenvectors with λ close to zero capture the partitioning

17. From Partitioning to Communities The number of communities and their size should be given by the network itself.

18. Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

19. Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

20. Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

21. Hierarchical Clustering Edges 4 Sides Stable Equal 1. Square + + + + 2. Rectangle + + + -- 3. Circle -- -- -- -- 4. Triangle + -- + +

22. Dendograms

24. Topologically Induced Weights

25. Betweeness Edge Betweeness – the number of paths through an edge

26. Football and Karate Networks Zacharys Karate Club College Football

27. Football and Karate Networks Zacharys Karate Club College Football

28. Ising and Potts Models

29. Groups of nodes with high link density will tend to have the same polarization Sparseness of connections between groups will allow different communities to have unrelated spins

30. Ising and Potts Models Groups of nodes with high link density will tend to have the same polarization Sparseness of connections between groups will allow different communities to have unrelated spins Potts Model

31. Ising and Potts Models

36. Link Communities Community - A group of densely connected nodes A group of topologically similar links Project Presentations (5 min.) 1.Define your network (nodes, links) 2.How will you get the data 3.Estimated size of network 4.Why