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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

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From a linear world… Predator Consumer Resource Food chains Predator Consumer Resource Predator Consumer Resource Food tree Consumer

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…to the real world The Biosphere2 project

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

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The email network of a real organization Guimera, Danon, Díaz-Guilera, Giralt, Arenas, PRE (2002)

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The worldwide air transportation network: a real socio-economic network Guimera, Mossa, Turtschi, Amaral, PNAS (2005)

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The protein interactome of yeast: a real biochemical network Jeong, Mason, Barabasi, Oltvai, Nature (2001)

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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?

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

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Lets assume that......individuals live in a square lattice!!

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

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Nodes in real networks (often) have structured neighborhoods

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Real networks are (often) highly inhomogeneous

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Real networks are (often) modular

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What can we learn by studying the interaction network topology?

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Extracting information from complex networks Protein interactions in fruit fly Giot et al., Science (2003)

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We need a cartography of complex networks Modules One divides the system into regions Roles One highlights important players

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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

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The Girvan-Newman algorithm for module detection is remarkably effective

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The community tree of a real organization

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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

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We define a quantitative measure of modularity Low modularity High modularity Newman & Girvan, PRE (2003) Intuitively high modularity = many links within & few links between

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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 )

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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

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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

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Uphill search does not give the best possible partition

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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)

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Simulated Annealing We use simulated annealing to obtain the partition with largest modularity

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The new algorithm for module detection outperforms previous algorithms

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As we already knew, geo-political factors determine the modular structure of the air transportation network Guimera, Mossa Turtschi, Amaral, PNAS (2005)

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Now we need to identify the role of each node

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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.

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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.

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We define the within-module degree Within-module relative degree where: i : number of links of node i inside its own module

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We define the participation coefficient Participation coefficient where: f is : fraction of links of node i in module s

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

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We define seven different roles Hubs Non-hubs Ultra-peripheral Satellite connector Peripheral Provincial hub Global hub

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Our definition of roles enables us to identify important cities

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How does network cartography help us understand the metabolism? Metabolic network of E. coli

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The cartographic representation of the metabolic network of E. coli Guimera & Amaral, Nature (2005) Satellite Global

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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

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Fluxes involving satellite connectors are essential Guimera, Sales-Pardo, Amaral, submitted (2006)

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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?

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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/

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

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To determine the hierarchical modular structure of the network, we sample the whole modularity landscape Sales-Pardo, Guimera, Moreira, Amaral, submitted (2006)

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

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

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