vl netzwerke, ws 2007/08 edda klipp 1 max planck institute molecular genetics humboldt university...

VL Netzwerke, WS 2007/08 Edda Klipp 1

Max Planck Institute Molecular Genetics

Humboldt University BerlinTheoretical Biophysics

Networks in Metabolism and Signaling

Edda Klipp Humboldt University Berlin

Lecture 3 / WS 2007/08Random Networks: Scale-free

Networks




Random Networks: Scale-free Networks

Most “natural” networks do nothave a typical degree value,they are free of a characteristic“scale”: they are scale-free.

Their degree distribution followsa Power law: P(k) ~ k-

Heterogeneity: hubs with highconnectivity, many nodes with low connectivity.





The probability that a node is highly connected is statistically more significant than in a random graph, the network’s properties often being determined by a relatively small number of highly connected nodes that are known as hubs.

Barabási–Albert model of a scale-free network: Growth: Start with small number of nodes (m0)At each time point add a new node with m ≤ m0 links to the network

Preferential attachment:Probability for a new edge is

where kI is the degree of node I and J is the index denoting the sum over network nodes.

The network that is generated by this growth process has a power-law degree distribution that is characterized by the degree exponent = 3.

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Evolution of Scale-free Networks

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After t time steps network with N = t + m0 nodes and m t edges.

Continuum theory: ki as continuous real variable, change proportional to (ki)

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Evolution of Scale-free NetworksAfter t time steps network with N = t + m0 nodes and m t edges.

Continuum theory: ki as continuous real variable, change proportional to (ki)

322 kmkPt ~

jj

ii k

kk

Degree distribution independent of time tAnd independent of network size N

But proportional to m2




Evolution of Scale-free NetworksAfter t time steps network with N = t + m0 nodes and m t edges.

Master equation approach: probability p(k,ti,t) that a node i introduced at time ti has degreek at time t.

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Dorogovtsev&Mendes-2001




Scale-free Networks: Degree Distribution




Scale-free Networks: Clustering coefficient

No inherent clustering coefficent C(k)




Scale-free Networks: Average Path Length

lk

1 l

lkN ~

Shortest path length l: distance between two vertices u and v with unit length edges

Fully connected network:

Rough estimation for random network:

Average number of nearest neighbors: <k>

vertices are at distance l or closer

total number of vertices

k

Nl

ln

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Scale-free Networks: Average Path Length

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Is obtained by fitting

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Scale-free Networks: Error and Attack Tolerance

Question: consider arbitrary connected graph of N nodes and assume that a p fraction of edges have been removed. What is the probability of the resulting graph being still connected?Usually: existence of a threshold probability pc.

More severe: removal of nodes




Scale-free Networks: Error and Attack Tolerance

Node RemovalThe relative size S (a),(b) and average path length l (c),(d) of the largest cluster in an initially connected network when a fraction f of the nodes are removed. (a),(c) Erdös-Renyi random network with N=10 000 and <k>=4; (b),(d) scale-free network generated by the Barabasi-Albert model with N=10 000 and <k>=4. , random node removal; º, preferential removal of the most connected nodes.





Difference to classical random networks:

Growth of the network – given number of nodes

Preferential attachment – equal probability for all edges

Examples:

References in www: connections frequently to existing hubs

Metabolism: many molecules are involved in only a few (1,2) reactions, others (like ATP or water) in many

Wagner/Fell: highly connected molecules are evolutionary “old”




Properties of Scale-Free Networks

Small-world – short paths between arbitrary points

Robustness – Topological robustness




Watts & Strogatz ModelProblem: real networks have short average path length and greater clustering coefficients than classical random graphs

Construction: Initially, a regular one dimensional lattice with periodical boundary conditions is present. Each of L vertices has z ≥ 4 nearest neighbors. Then one takes all the edges of the lattice in turn and with probability p rewires to randomly chosen vertices. In such a way, a number of far connections appears. Obviously, when p is small, the situation has to be close to the original regular lattice. For large enough p, the network is similar to the classical random graph.




Watts-Strogatz-Model

By definition of Watts and Strogatz, the smallworld networks are those with “small” average shortest path lengths and “large” clustering coecients.

<k> remains constant During rewiring




Random Networks: Hierarchical Networks

To account for the coexistence of modularity, local clustering and scale-free topology in many real systems it has to be assumed that clusters combine in an iterative manner, generating a hierarchical network. The starting point of this construction is a small cluster of four densely linked nodes.Next, three replicas of this module are generated and the three externalnodes of the replicated clustersconnected to the central node ofthe old cluster, which produces alarge 16-node module. Threereplicas of this 16-node moduleare then generated and the 16peripheral nodes connected tothe central node of the oldmodule, which produces a newmodule of 64 nodes….




Random Networks: Hierarchical Networks

Clustering coefficent scales with the degree of the nodes




Application to Metabolic Networks

Jeong H et al, 2000, Nature




Metabolic Networks: Degree Distributions




Metabolic Networks: Hubs




Metabolic Networks: Pathway Lengths

E.coli 43 different organisms




Metabolic Networks: Robustness

Hubs removed first

Random removal

M=60 – 8% of substrates




Protein-Protein-Interaction NetworksYeast proteomea) Map of protein-protein

interactions Largest cluster: 78% of all

proteinsRed – lethalGreen – non-lethalOrange – slow-growthYellow – unknown

Cut-off: kc=20

b) Connectivity distribution

c) Fraction of essential proteins with

k linksRandom removal – no effectHubs removal – lethal




Protein-Protein-Interaction Networks

Random removal – no effectHubs removal – lethal

93% of proteins have 5 or less links only 21% of them are essential

0.7% of proteins have more than 15 links 62% of them are lethal




Protein-Protein-Interaction Networks

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