Analysis of biological networksAnalysis of biological networks

Part IIIPart III

Shalev ItzkovitzShalev Itzkovitz Uri Alon’s groupUri Alon’s group July 2005July 2005

What is a suitable random ensemble?What is a suitable random ensemble?

Subgraphs which occur many times in the networks, significantly more than in a

suitable random ensemble.

Reminder - Network motifs definitionReminder - Network motifs definition

Types of random ensemblesTypes of random ensembles

Erdos Networks

For a given network with N nodes and E edges define : p=E/N2, the

probability of an edge existing between any one of the N2 possible

directed edges.

Erdos & Renyi, 1960

N

Ek

UMAN ensemble

a canonical version. All networks have the same numbers of Mutual, Antisymetric and Null edges as the real network, Uniformly distributed.

Used in sociology, analytically solvable for subgraph distributions.

Holland & Leinhardt, american journal of sociology 1970

Antisymetric edge

Mutual edge

The configuration model

All networks preserve the same degree sequence of the real network

All networks preserve the same degree sequence of the real network, and multiple edges between two nodes are not allowed

The configuration model+no multiple edges

Bollobas, Random graphs 1985, Molloy & Reed, Random structures and algorithms 1995, Chung et.al. PNAS 1999

Maslov & Sneppen, science 2002, Newman Phys. Rev.Lett. 2002, Milo science 2002

Stubs method for generating random Stubs method for generating random networksnetworks

Problem – multiple edges between nodesSolution – “Go with the winner” algorithm

A

B D

C A

B D

C

Markov chain Monte-Carlo algorithmMarkov chain Monte-Carlo algorithm

Uniform sampling issues : ergodicity, detailed balance, mixing time

Random networks which do not preserve Random networks which do not preserve the degree sequence are not suitablethe degree sequence are not suitable

Network hub

This v-shaped subgraph appears many timeswould be a network motif when comparing with Erdos networks

It is important to filter out subgraphs which appearIt is important to filter out subgraphs which appearin high numbers only due to the degree sequencein high numbers only due to the degree sequence

More stringent ensembles

•Preserve the number of all subgraphs of sizes 3,4..,n-1 when

counting n-node subgraphs [Milo 2002]

•Can be combined with the markov chain algorithm by using

simulated annealing

•Filters out subgraphs which appear many times only because

they contain significant smaller subgraphs

Will appear many times if Is a motif

A

B D

C A

B D

C

Simulated annealing algorithmSimulated annealing algorithm

•Randomize network by making X switches

•Make switches with a metropolis probability exp(-E/T)

•E is the deviation of any characteristic of the real network you

want to preserve (# 3-node subgraphs, clustering sequence etc)

Erdos Networks

UMAN

Degree distribution

Degree sequence

Degreesequence+triads

Subgraphs in Erdos networks: Subgraphs in Erdos networks: exact solutionexact solution

122

N

k

N

kN

N

Ep

NN nodes (8) nodes (8)

EE edges (8) edges (8)

<<kk> > mean degree (1)

Subgraphs in Erdos networks: Subgraphs in Erdos networks: exact solutionexact solution

3

NPossible tripletsPossible triplets

3p Probability of forming a ffl Probability of forming a ffl given specific 3 nodesgiven specific 3 nodes

3333

3

33 ~3

kNkN

kNp

N

# nodes# nodes # edges# edges

Number of ffls does not Number of ffls does not change with network sizechange with network size!!!!!!

• The expectancy of a subgraph with n nodes and g edges is analytically solvable. Scales as N(n-g)

gng N

N

k

n

N ~))((

N

kp

Subgraphs on Erdos NetworksSubgraphs on Erdos Networks

n=3g=3

Select n nodes

place g edges

n=3, g=2, G~O(N3-2)=O(N) n=3, g=3, G~N3-3=O(1)

n=3, g=4, G~N3-4=O(N-1) n=3, g=6, G~N3-6=O(N-3)

Subgraph scaling familiesSubgraph scaling families

P(K)~K-

Natural networks often have scale-freeNatural networks often have scale-free

outdegreeoutdegree

Erdos networkErdos network Scale free networkScale free network

P(k

)P(K)~K

-

2<<3

P(K)~K-

=3

=2

Scale-free networks have hubsScale-free networks have hubs

Edge probability in the configuration modelEdge probability in the configuration model

edgesnetwork #

indegree nodeP(edge)

high edge probabilityhigh edge probability low edge probabilitylow edge probability

Edge probability in the configuration modelEdge probability in the configuration model

edgesnetwork #

indegree) (node2*outdegree) (node1P(edge)

1122

10

2*2P

Networks with E (~N) edges, and arbitrary indegree (Ri ) and outdegree (Ki ) sequences.

Subgraphs in networks that preserve degree Subgraphs in networks that preserve degree sequence: approximate solutionsequence: approximate solution

K1, R1

K2, R2

K3, R3

E

R21KP(edge1) E

R31 )1(K)edge1P(edge2

E

R )1(K,2)edge1P(edge3 32

E

RP(edge)

Subgraph scaling depends on exact Subgraph scaling depends on exact topologytopology

3332211 )1(**)1(KK

)P(subgraphE

RRRK

3

)1()1(~#

K

RRKRKKffl

Subgraph topology effectsSubgraph topology effectsIts expected numbersIts expected numbers

Subgraph scaling depends on exact Subgraph scaling depends on exact topology – as opposed to Erdos networkstopology – as opposed to Erdos networks

Example

O(1)

O(1) O(1)

Erdos Networks

γO( N)

Directed networks with power-law out-degree, compact in-degree :

P(K)~K-

Scale-free Networks ( =2.5)

2K

K

Real networks

O( 1)

O( N)>

Itzkovitz et. al., PRE 2003

Network motifs – a new extensive variableNetwork motifs – a new extensive variable

Milo et. al., science 2002

Global constraints on network structure Global constraints on network structure can create network motifscan create network motifs

• Subgraphs which appear many times in a network (more than random)

• Might stem from evolutionary constraints of selection for some function, or be a result of other global constraints

• Degree sequence is a global constraint with a profound effect on subgraph content

• Are there other global constraints which might result in network motifs?

How do geometrical constraints How do geometrical constraints influence the local structure?influence the local structure?

Examples of geometrically constrained Examples of geometrically constrained systemssystems

• Transportation networks (highways, trains)

• Internet layout

• Neuronal networks, brain layout

• Abstract spaces (www, social, gene-array data)

The neuron network of The neuron network of C. elegansC. elegans

"The abundance of triangular connections in the nervous system of C. elegans may thus simply be a consequence of the high levels of connectivity that are present within neighbourhoods“ (White et. al.)

The geometric modelThe geometric model

• N nodes arranged on d-dimensional lattice

• Connections made only to neighbors within range R

Erdos networks – every node can Erdos networks – every node can connect to every other nodeconnect to every other node

Probability of closing triangles - small

Geometric networks – every node can Geometric networks – every node can connect only to its neighborhoodconnect only to its neighborhood

Probability of closing triangles - large

All subgraphs in geometric networks All subgraphs in geometric networks scale as network sizescale as network size

N/Rd ‘sub-networks’, each one an Erdos network of size Rd

All subgraphs scale as network size

Erdos sub-network

All subgraphs scale as NAll subgraphs scale as N

The Erdos scaling laws determine the The Erdos scaling laws determine the network motifsnetwork motifs

All subgraphs with more edges than All subgraphs with more edges than nodes are motifsnodes are motifs

Motifs – scale as N in geometric networksConstant number in random networks

Not motifs – scale as N in both random and real networks

Feedbacks in neuronal network are much Feedbacks in neuronal network are much more rare than expected from geometrymore rare than expected from geometry

= 1 : 3 = 1 : 3

= 0 : 40 = 0 : 40

geometric model

C elegans neuronal network

Imposing a field changes subgraph ratiosImposing a field changes subgraph ratios

inputsoutputs

Itzkovitz et. al., PRE 2005

A simple model of geometry + directional A simple model of geometry + directional bias is not enoughbias is not enough

abundant in C elegans

Mutual edges rare in geometric networks + directional bias

The mapping of network models and The mapping of network models and resulting network motifs is not a 1-1 resulting network motifs is not a 1-1

mappingmapping

Motif set 1Motif set 1 Motif set 2Motif set 2 Motif set 3Motif set 3

Model 1Model 1 Model 2Model 2 Model 3Model 3 Model 4Model 4

X

conclusionsconclusions

•Biological networks are highly optimized systems aimed at information processing computations.

•These networks contain network motifs – subgraphs that appear significantly more than in suitable random networks.

•The hypothesized functional advantage of each network motif can be tested experimentally.

•Network motifs may be selected modules of information processing, or results of global network constraints.

•The network motif approach can be used to reverse-engineer complex biological networks, and unravel their basic computational building blocks.

AcknowledgmentsAcknowledgments

Ron MiloRon MiloNadav KashtanNadav Kashtan

Uri AlonUri Alon

More information :More information :http://www.weizmann.ac.il/mcb/UriAlon/

PapersPapers

mfinder – network motif detection softwaremfinder – network motif detection software

Collection of complex networksCollection of complex networks