Cartography of complex networks: From organizations to the metabolism Cartography of complex networks: From organizations to the metabolism Roger Guimerà Department 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 Food tree Consumer

…to the real world The Biosphere2 project

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

The 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?

Lets assume that......proteins/people interact at random with other proteins/people

Lets 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 fly Giot 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(N 3 ) 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) f s : fraction of links within module s F s : expected fraction of links within module s, for a random partition of the nodes Modularity of a partition: M = (f s – F s )

We define a quantitative measure of modularity Modularity of a partition: Where: l s is the number of links within module s d s is the sum of the degrees of the nodes in module s L is the total number of links in the network

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: f is : 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-peripheral Satellite connector Peripheral 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) Satellite Global

Satellite connectors are more conserved across species than provincial hubs Comparison between 12 organisms: 4 archea 4 bacteria 4 eukaryotes Ultra-peripheralPeripheralSatellite connectorsProvincial hubsGlobal 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:

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

We are able to identify the modules at each of the hierarchical levels Sales-Pardo, Guimera, Moreira, Amaral, submitted (2006)