Cartography of complex networks:Cartography of complex networks:From organizationsFrom organizations

to the metabolismto the metabolism

Roger GuimeràDepartment of Chemical and Biological Engineering

Northwestern University

Oxford, June 19, 2006

From a linear world…

Predator

Consumer

Resource

Food chains

Predator

Consumer

Resource

Predator

Consumer

Resource Resource Resource

Food “tree”

Consumer

…to the real world

The Biosphere2 project

Trophic interactions in the North Atlantic fishery: a real food web

The email network of a real organization

Guimera, Danon, Díaz-Guilera, Giralt, Arenas, PRE (2002)

The worldwide air transportation network: a real socio-economic network

Guimera, Mossa, Turtschi, Amaral, PNAS (2005)

The protein interactome of yeast: a real biochemical network

Jeong, Mason, Barabasi, Oltvai, Nature (2001)

Summary

What is (was) missing in the analysis of complex systems?

Cartography of complex networks: Modules in complex networks Roles in complex networks

Can we discover new therapeutic drugs by analyzing complex networks?

Let’s assume that...

...proteins/people interact at random with other proteins/people

Let’s assume that...

...individuals live in a square lattice!!

Nodes in real networks are (often) “close” to each other

Nodes in real networks (often) have structured neighborhoods

Real networks are (often) highly inhomogeneous

Real networks are (often) modular

What can we learn by studying the interaction network topology?

Extracting information from complex networks

Protein interactions in fruit flyGiot et al., Science (2003)

We need a “cartography” of complex networks

Modules One divides the system into “regions”

Roles One highlights important players

Heuristic methods to identify modules in complex networks: Girvan-Newman algorithm

Girvan & Newman, PNAS (2002)

Identify the most central edge in the network

Remove the most central edge in the network

Iterate the process

A

B

C

D

E

F

H I

G

The Girvan-Newman algorithm for module detection is remarkably effective

The community tree of a real organization

Shortcomings of the GN algorithm

It is very slow: O(N3)

One needs to decide where to stop the process

It does not work that well when the modular structure becomes fuzzy

We define a quantitative measure of modularity

Low modularity

High modularity

Newman & Girvan, PRE (2003)

Intuitively

high modularity = many links within & few links between

We define a quantitative measure of modularity

Newman & Girvan, PRE (2003); Guimera, Sales-Pardo, Amaral, PRE (2004)

fs: fraction of links within module s

Fs: expected fraction of links within module s, for a random partition

of the nodes

Modularity of a partition: M = (fs – Fs)

But now that we have modularity, we can try optimization-based approaches

Brute force: Find all possible partitions of the network, calculate their modularity, and keep the partition with the highest modularity.

Uphill search:1. Start from a random partition of the network.

2. Try to randomly move a node from one module to another. Does the modularity increase?– Yes: Accept the movement.– No: Reject the movement.

3. Repeat from 2

Uphill search does not give the best possible partition

We use simulated annealing to obtain the partition with largest modularity

Simulated annealing:1. Start from a random partition of the network.

2. Define a “computational temperature” T. Set T to a high value.

3. Try to randomly move a node from one module to another. Does the modularity increase?– Yes: Accept the movement.– No: Is the decrease in modularity much larger

than T?– Yes: Reject the movement.– No: Sometimes accept the movement.

4. Decrease T and repeat from 3.

Guimera & Amaral, Nature (2005)

Simulated Annealing

We use simulated annealing to obtain the partition with largest modularity

The new algorithm for module detection outperforms previous algorithms

As we already knew, geo-political factors determine the modular structure of the air transportation network

Guimera, Mossa Turtschi, Amaral, PNAS (2005)

Now we need to identify the role of each node

Previous approaches to role identification:Structural equivalence

Definition Two nodes are structurally equivalent if, for all actors, k=1, 2, …, g (k=i, j), and all relations r =1, 2, …, R, actor i has a tie to k, if and only if j also has a tie to k, and

i has a tie from k if and only if j also has a tie from k. (Wasserman & Faust)

‘Translation’ Two nodes are structurally equivalent if they have the exact same connections.

Previous approaches to role identification:Regular equivalence

Definition If actors i and j are regularly equivalent, and actor i has a tie to/from some actor, k, then actor j must have the same kind of tie to/from some actor, m, and k and m

must be regularly equivalent. (Wasserman & Faust)

‘Translation’ Two nodes are regularly equivalent if they have identical connections to equivalent nodes.

We define the within-module degree

Within-module relative degree

where: i: number of links of node i inside its own module

We define the participation coefficient

Participation coefficient

where: fis: fraction of links of node i in module s

The within-module degree and the participation coefficient define the role of each node

We define seven different roles

Hubs

Non-hubs

Ultra-peripheralSatellite connectorPeripheral

Provincial hub

Global hub

Our definition of roles enables us to identify important cities

How does network cartography help us understand the metabolism?

Metabolic network of E. coli

The cartographic representation of the metabolic network of E. coli

Guimera & Amaral, Nature (2005)

SatelliteGlobal

Satellite connectors are more conserved across species than provincial hubs

Comparison between 12 organisms: 4 archea 4 bacteria 4 eukaryotes

Ultra-p

eriphera

l

Periphera

l

Satellite co

nnectors

Provin

cial h

ubs

Global hubs

Fluxes involving satellite connectors are essential

Guimera, Sales-Pardo, Amaral, submitted (2006)

Questions for us to think

Can we design better organizations / transportation systems / … by using these new tools?

What can we learn from organizations / … that could help us design better drugs?

How are topology, dynamics, and function related?

Acknowledgements

Luís A. N. Amaral, Marta Sales-Pardo Fulbright Commission and Spanish Ministry of

Education, Culture, and Sports.

More information:

http://amaral.northwestern.edu/

http://amaral.northwestern.edu/roger/

What happens if the modular structure of the network is hierarchically organized?

To determine the hierarchical modular structure of the network, we sample the whole modularity landscape

Sales-Pardo, Guimera, Moreira, Amaral, submitted (2006)

We are able to identify the modules at each of the hierarchical levels

Sales-Pardo, Guimera, Moreira, Amaral, submitted (2006)

Nodes

No

des

We are able to identify the modules at each of the hierarchical levels

Sales-Pardo, Guimera, Moreira, Amaral, submitted (2006)