Basic features of a network Degree distribution Clustering coefficients Average shortest path length
Degree of a node (k) Degree of ith node k i = number of nodes linking with it
Degree of a node (k) k in = number of nodes linking in k out = number of nodes linking out
Clustering Coefficient (CC) C i =2E i /k i (k i -1)=2/9 ith node has k i neighbors linking with it E i is the actual number of links between k i neighbors maximal number of links between k i neighbors is k i (k i -1)/2
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 =A ij =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
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 =NP=2 average degree Probability distribution of degree k Erdös & Rényi model Poisson distribution Exponential Network
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 to 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 A.-L.Barabási, R. Albert, Science 286, 509 (1999) Scale Free Network
Are biological networks random, scale free or hierarchical?
Degree distribution of PPI P(k)~k -1.62 Scale free 2617 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 clusters combine 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 Power law degree distribution Power law clustering coefficient distribution
References Albert-László Barabási and Zoltán N. Oltvai, Network Biology: Understanding the Cells's Functional Organization Nature Reviews Genetics 5, 101-113 (2004). O. Mason, and M. Verwoerd, Graph Theory and Networks in Biology, IET Syst. Biol, 1, 89-119, (2007).