Classes of complex networks defined by role-to-role connectivity profiles

1. Classes of complex networks defined by role-to-role connectivity profiles Authors : Roger Guimerà, Marta Sales-pardo, and Luís A. N. Amaral Department of Chemical and Biological Engineering and Northwestern Institute on Complex Systems (NICO), Northwestern University Presented by Tanay Chowdhury

2. Synopsis • Objective • Problem description • Solution formation • Result

3. Objective • To prove the premise that networks are homogenous is wrong. • To classify complex network into two distinct functional classes on the basis of ensemble of properties.

4. Problem description Structure of Complex Network • Characterizing global properties • Average shortest path length between nodes • Clustering coefficient • Assortativity and other degree-degree correlation • Degree distribution

5. Structure of Complex Network • Precondition for using global properties • The network should lack a modular structure. • All modules should be formed according to same mechanism. • The interface between modules should be statistically similar.

6. Using modularity of Complex networks • Networks used • Metabolic networks • Protein interactomes • Global and regional air transportation networks • The Internet at the autonomous system (AS) level. • Significance of modular structure of each network is assessed by comparing it with randomization of the same network. • Result : Global average properties of the network is found to be a not good classifier.

7. Solution formulation : Role based Complex Network • Role of each node is determined using 2 properties • The relative within module degree z • It quantifies how well connected a node is to other nodes in the module. • The participation coefficient p • It quantifies to what extent a node connect to different modules. The participation coefficient Pi of node i is Where kisis the number of links of node i to nodes in module s, and is the total degree of node i.

8. Modular division using z and p • Non-hubs • Nodes that have low within module degree (z<2.5) • Further divided into • Ultra-peripheral nodes (R1) (P ≤ 0.05) • Peripheral nodes(R2) (0.05 < P ≤ 0.62) • Satellite connectors (R3) (0.62 < P ≤ 0.80) • Kinless notes (R4) (P > 0.80) • Hubs • Nodes that have high within module degree(z>2.5) • Further divided into • Provincial hubs (R5) (P ≤ 0.30) • Connector hubs (R6) (0.30 < P ≤ 0.75) • Global hubs (R7) (P > 0.75)

9. Degree-degree correlation • Question 1 • Whether nodes with same degree but different roles have the same or different correlation. • Example : • In 1998 Internet, for example, knn(k = 3) = 43 ± 8 for ultra-peripheral nodes, knn(k = 3) = 196 ± 12 for peripheral nodes and knn(k = 3) = 290 ± 20 for satellite connectors. • Answer • The average degree of the neighborhood of a node strongly depends on the node.

10. Degree-degree correlation • Question 2 • To what extent the observed degree-degree correlations are a by-product of the modular structure of the network. • Observation • The degree distribution of the network is responsible for most of the observed correlations. However, the degree distribution alone does not account for all the observed correlations whereas the modular structure of the network does. • Answer • Modular structure account for most of the degree-degree correlation observed in the network.

11. a–d, Degree normalized by the average neighbours’ degree of all the nodes in the network. e–h, Degree of the neighbours of a node normalized by the average neighbours’ degree in the ensemble of random networks with fixed degree sequence. i–l, Neighbours’ degree of a node normalized by the average neighbours’ degree of the node in the ensemble of random networks with fixed degree sequence and modular structure.

12. Role-to-role connectivity profiles • Connectivity profile plays a role other than degree distribution and the modular structure to classify network. • For each network the number rijof links between nodes belonging to roles i and j, is compared to the number of such links in a properly randomized network. • Figure 2 shows that networks of the same type have highly correlated profiles. • Figure 2c shows that networks of different types have weaker correlations and, at times, even strong anti-correlations.

13. Figure 2 : Role to role connectivity pattern

14. Classification using role-to-role connectivity • Stringy-periphery class • It comprises metabolic and air transportation networks • In networks of this class, ultra-peripheral nodes are more connected to one another than would be expected from chance, which results in long ‘chains’ of ultra-peripheral nodes. • It has hub oligarchy, that are directly reachable from one another.

15. Classification using role-to-role connectivity • Multi-star class • It comprises the protein interactomes and the Internet. • In networks of this links between ultra-peripheral nodes are under-represented, whereas links between ultra-peripheral nodes and provincial hubs are over-represented, giving rise to modules with indirectly connected ‘star-like’ structures. • These networks depend on satellite connectors to bridge connector hubs and modules.

16. Figure 3. Modules and role-to-role connectivity signatures in different network types

17. Result • Global properties that do not take into account the modular organization of the network may sometimes fail to capture potentially important structural features. • Eg 1: In protein interactomes absolute degree alone can’t help distinguish nodes with different roles but the average degree does. • Eg 2: In air transportation network important structural properties may be left unexplained by focusing on degree alone when that is captured properly by the within-module relative degree and participation coefficient. • Same functional needs and growth mechanisms have similar patterns of connections between nodes with different roles.

18. Questions