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Computational Systems Biology TUM WS 2010/11 Lecture 9: Hierarchical Networks and Network Motifs 2011-01-13 Dr. Arthur Dong Emergence of Networks Many real-world complex networks Have relatively many hubs scale-free Are locally

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Computational Systems Biology

TUM WS 2010/11

Lecture 9: Hierarchical Networks and Network Motifs

2011-01-13

Dr. Arthur Dong

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Emergence of Networks Many real-world complex networks

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Have relatively many hubs → scale-free

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Are locally clustered (high CC)

CC mostly is independent of system size (number of nodes N) → modular
CC mostly scales with 1/k (hubs are less clustered) → hierarchical

Scale-free and clustering coexist in real networks. Need a model to reconcile those 2 features! Evolution of Models

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ER (random graph) is the baseline

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WS (small-world) focuses on short L (CC just inherited)

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BA (scale-free) addresses relative abundance of hubs Now we need a model to incorporate CC while keeping those other features.

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Network Model Construction L CC Hubs Real Evolution short high; N-indep; CC ~ 1/k abundant Random ER Fixed N, uniform p short low; CC ~ 1/N; k- indep rare Small-world WS A little rewiring from regular graphs short high (inheritance); N-indep; k-indep rare Scale-free BA Growth and preferential attachment short high; CC ~ 1/N**0.75; k-indep abundant Hierarchical Fractal Deterministic short high; N-indep; CC ~ 1/k abundant

Putting it all together

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Deterministic Model (Pseudo-fractal Construction) Hierarchy

Dense intra-module connections, sparse inter-module connections
Modules get less and less cohesive as level goes up

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Deterministic (Fractal) vs probabilistic (BA) scale-free models Like BA, Fractal produces power-law degree distribution. Unlike under BA, under Fractal CC is independent of N and scales with 1/k. The analytics is nontrivial, but you should be able to run numerical simulations!

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Explaining the figures

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What are the points and lines?

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What are the appropriate controls? Like the Fractal model, metabolic networks also exhibit high clustering

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That is independent of N (modular)

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That scales with 1/k (hierarchical)

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Network Motifs

We have looked at some global features of real complex networks:

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Short distance between nodes (small-world)

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High local clustering (modular)

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Abundance of hubs (scale-free) Much of the advance was propelled by the desire to

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Explain key features

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Reconcile conflicting features Now we look at “patterns” in complex networks Network motif = small subgraphs that are significantly over-represented Example of a 3-node motif: Do you expect this motif to be over-represented? First focus on directed networks and look at 3- and 4-node motifs

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What is a 2-node motif?

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How many 3-node motifs are there?

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How many 4-node motifs are there? Beware of overcounting due to isomorphisms!

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Enumeration of directed 3-node motifs Again, interpretations (what those formalisms actually mean)! Does X ↔ Y make sense in the food web context? Exercise: How many undirected 3-node motifs are there? (8)

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Example: Feed-forward loop

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Count how many times it appears in the real network

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Count how many times it appears in “comparable” random networks (through edge- swapping)

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Compute empirical p-value or z-score.

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Different classes of networks prefer different network motifs

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Wolf Wolf Human Tiger sheep sheep sheep wolf veggie veggie veggie sheep Wolf sheep veggie rabbit Tiger wolf sheep 3-chain Feed-forward loop Bi-parallel Feed-back loop

Exercise your common sense

How to generalize 3-chain? Do you think it's over- or under-represented? Carnivore → Herbivore → Flora How about feed-forward loop? Incompatibility, omnivore, competition

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