Global topological properties of biological

networks

Node: protein

Edge: protein-protein interaction

Protein-Protein Interaction Network

Saccharomyces cerevisiae

E. coli metabolic network

Basic features of a network

• Degree distribution

• Clustering coefficients

• Average shortest path length

Degree of a node (k)Degree of ith node ki= number of nodes

linking with it

Degree of a node (k)kin= number of nodes linking in

kout= number of nodes linking out

Clustering Coefficient (CC)

Ci=2Ei/ki(ki-1)=2/9

ith node has ki neighbors linking with it

Ei is the actual number of links between ki neighbors

maximal number of links between ki neighbors is ki(ki-1)/2

Average shortest path length

jil

lNN

l

ij

jiij

and nodebetween length path shortest theis

)1(

2

Shortest path length

)pathshortest (2l

All pair shortest path Algorithm• Floyd Algorithm: d(k)

ij: shortest path between i,j with intermediate node’s label not higher than k

j

k

i

d(k)ij=min(d(k-1)

ij,d(k-1)ik+d(k-1)

kj)

d(k-1)ik d(k-1)

kj

d(k-1)ij

Pseudocode

• D(0)ij=Aij=adjacency matrix

• For k=1 -> N• for i=1 -> N• for j=1 -> N• D(k)

ij=min(D(k-1)ij,D(k-1)

ik+D(k-1)kj)

• Return D

Small world network

Three ways to generate networks

Random networks

• Paul Erdös & Alfréd Rényi model : Hugarian mathematicians in 1959

Paul Erdös Alfréd Rényi

1913~1996 1921~1970

Randomly connect two nodes with probability P=1/5 linking probability

N=10 number of nodes

<K>=NP=2 average degreeProbability distribution of degree k

Erdös & Rényi model

Poisson distribution

Exponential

NetworkNPkk

ke

ppCkkPk

kN

kNkNki

!

)1()( 11

Scale free network Albert-László Barabási

“Statistical mechanics of complex networks” Review of Modern Physics 74, 47-97 (2002)

Scale free Network• A new node is added and deleted randomly t

o and from the network, i.e. N is not fixed

• The new node preferably connects with other node with higher connections with m edges, i.e

P(k)~k-γ

ii

ii k

kkPP )(

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

Scale Free Network

Scale free network

Mean Field Theory

γ = 3

t

k

k

kAkP

t

k i

j j

ii

i

2)(

ii t

tmtk )(

, with initial condition mtk ii )(

)(1)(1)())((

02

2

2

2

2

2

tmk

tm

k

tmtP

k

tmtPktkP ititi

33

2

~12))((

)(

kktm

tm

k

ktkPkP

o

i

A.-L.Barabási, R. Albert and H. Jeong, Physica A 272, 173 (1999)

degree distribution

Hierarchical Networks

kkP ~)(

kkC ~)(

!~)(

k

ekP

k

Are biological networks random, scale free or

hierarchical?

Degree distribution of PPI

P(k)~k-

1.62Scale free

32617 proteins

11855 interactions

Data from HMS-PCI, Yeast two hybrid, and TAP

data

Degree distribution of metabolic network

a: Archaeoglobus fulgidus

b: E.coli

c: C. elegans

d: Averaged over 43 organisms

Scale free !!!

Hierarchy in biological networks

Metabolic networks Protein networks

What does it mean?

Real Networks Have a Hierarchical Topology

Many highly connected small clusterscombine into

few larger but less connected clusters combine into

even larger and even less connected clusters

The degree of clustering follows:

Biological networks are hierarchical

kkP ~)(

kkC ~)(

Power law degree distribution

Power law clustering coefficient distribution

References

• Albert-László Barabási and Zoltán N. Oltvai,Network Biology: Understanding the Cells's Functional OrganizationNature Reviews Genetics 5, 101-113 (2004).

• O. Mason, and M. Verwoerd, Graph Theory and Networks in Biology, IET Syst. Biol, 1, 89-119, (2007).