An Introduction of Complex networks

Zhao Jing2006.11.22

Zjane_cn@sjtu.edu.cn

Outline

I. Network metrics and topological features

II. Modularity and network decomposition

III. Topological diversity of networks with a given degree sequence

I. Network metrics and topological features

Zhao J, Yu H, Luo J, Cao Z, Li Y: Complex networks theory for analyzing metabolic networks. Chinese Science Bulletin 2006, 51(13):1529-1537.

1.1 Degree distribution vs. scale-free networks

Degree distribution p(k) :

the occurrence frequency of nodes with degree k, (k=1,2,…).

Barabasi, A.L., Albert, R., Emergence of scaling in random networks, Science, 1999, 286:509-512

BA model for network evolution:(1) Growth: the continuous addition of new nodes.

(2) Preferential attachment: “the rich get richer” principle. The high-degree nodes should appear in the earlier stag

e of network formation.

Thirteen hub metabolites in E.coli metabolic network

Wagner, A., Fell, D.A., The small world inside large metabolic networks, Proc R Soc Lond B, 2001, 268:1803-1810.

Performance of scale-free networks:error tolerance: high resistance to random perturbations

attack vulnerability : the removal of a few hub nodes will destroy the whole network.

Albert, R., Jeong, H., Barabasi, A.-L., Error and attack tolerance of complex networks, Nature, 2000, 406:378-382.

Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai, Z.N., Lethality and centrality in protein networks, Nature, 2001, 411:41-42.

=>The most highly connected proteins in the cell are the most important for its survival.

Notice: Computation of the exponent

cumulative distribution :

ki

ipkxP )()(

kkp ~)( )1(~)( kkxP

Log-log plot of the degree distribution (A) and cumulative degree distribution (B) for a network of 20000 nodes constructed by Barabasi-Albert preferential attachment model.

A.-.Barabási R. Albert

Norte Dame Univ.Barabási is the 2006 recipient of the John von Neumann Medal. The award has been presented since 1976 to a maximum of three

individuals who have gained distinction in the dissemination of computer culture. Previous recipients of the award include Microsoft founder Bill Gates, former IBM chairman Louis Gerstner and Intel Corporation board chair Andrew Grove

1.2 Clustering coefficient vs. Hierarchical modular networks

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)(2)(

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Ravasz E, Somera A L, Mongru D A, Oltvai Z N, Barabasi A L, Hierarchical organization of modularity in metabolic networks, Science,2002,297: 1551-1556

Life’s complex Pyramid: from the particular to the universal

Oltvai, Z.N., Barabási, A.-L., Life’s Complexity Pyramid, SCIENCE, 2002, 298:763-764.

Mayr E., “How biology differs from the physical sciences”, Evolution as a crossroad: the new biology and the new philosophy of science, MIT press, Cambridge,1985.

Davis Paul, The cosmic blueprint, Simon and Schuster,1988.

Complex systems usually have a hierarchical structure, the entities of one level being compounded into new entities at the next higher lever, as cells into tissues, tissues into organs, and organs into functional systems.

The whole is greater than the sum of its parts!

At each new level of complexity in biology new and unexpected qualities appear, qualities which apparently cannot be reduced to the properties of the component parts.

1.3 Mean path length vs. small-world networks

Watts, D.J., Strogatz, S.H., Collective dynamics of `small-world' networks, Nature, 1998, 393:440-442.

Small-world cell networks=>the cell may react quickly to changes of the surroundings

Small-world network: small mean path length; high clustering coefficient

1.4 Assortativity coefficient vs. degree-degree correlation

Newman , M.E.J., Assortative mixing in networks, Phys Rev Lett, 2002, 89:208701.

The average connectivity <knn> of the nearest neighbors of a node depending on its connectivity k for the 1998 snapshot of the Internet, the generalized BA model and the fitness model.

Romualdo Pastor-Satorras, Alexei Vázquez, and Alessandro Vespignani, Dynamical and Correlation Properties of the Internet, PHYSI CAL REV IEW LETTERS, VOLUME 87, NUMBER 25(2002)

Correlation profiles of protein interaction network in yeast. Z-scores for connectivity correlations :Z(K0,K1) = (P(K0,K1) − Pr(K0,K1))/r(K0,K1) where r(K0,K1) is the standard deviation of Pr(K0,K1) in 1000 realizations of a randomized network.

Maslov, S., Sneppen, K., Specificity and Stability in Topology of Protein Networks, Science, 2002, 296:910-913.

1.5 Rich-club coefficient and rich-club phenomenon

Colizza V, Flammini A, Serrano MA, Vespignani A: Detecting rich-club ordering in complex networks. Nat Phys 2006, 2(2):110-115.

Notice: Rich-club Assortative mixing

1.6 k-core

1, 2 and 3-core. Two basic properties of cores: first, cores may be disconnected subgraphs; second, cores are nested: for i>j, an i-core is a subgraph of a j-core of the same graph.

=> The probability of nodes both being essential and evolutionary conserved successively increases toward the innermost cores.

Wuchty, S., Almaas, E., Peeling the yeast protein network, Proteomics, 2005, 5:444-449.

Zhao J, Tao L, Yu H, Luo J-H, Cao ZW, Li Y: Bow-tie topological features of metabolic networks and the functional significance. eprint q-bioMN/0611013 2006.

3-core of E.coli metabolic network

1.7 Betweenness centrality Betweenness centrality is based on the assumption t

hat information is transmitted along shortest paths.

Node betweenness : the number of shortest paths between pairs of nodes that run along this node.

Edge betweenness: the number of shortest paths between pairs of nodes that run along this edge.

=> Nodes and edges of high betweenness centrality could be bottlenecks of the network, thus could be important enzymes or metabolites.

Rahman, S.A., Schomburg, D., Observing local and global properties of metabolic pathways: 'load points' and 'choke points' in the metabolic networks, Bioinformatics, 2006, 22:1767-1774.

1.8 Null Model and Z-score

r

r

P

PPZ

Maslov S, Sneppen K, Zaliznyak A: Detection of topological patterns in complex networks: correlation profile of the internet. Physica A: Statistical and Theoretical Physics 2004, 333:529-540.

Maslov, S., Sneppen, K., Specificity and Stability in Topology of Protein Networks, Science, 2002, 296:910-913.

II. Modularity and network decomposition

Zhao J, Yu H, Luo J, Cao Z, Li Y: Complex networks theory for analyzing metabolic networks. Chinese Science Bulletin 2006, 51(13):1529-1537.

2.1 Modularity:

From functional view:

Modularity: the system can be decomposed in parts (modules), such that the function of each part is more complex than a basic combination of the input to a new output.

From topological view:

Assumption:

A densely connected subnetwork "part with complex function."

Modularity: network could be divided into groups of vertices that have a high density of edges within them, with a lower density of edges between groups.

Hartwell LH, Hopfield JJ, Leibler S, Murray AW: From molecular to modular cell biology. Nature 1999, 402:C47-C52.

Papin JA, Reed JL, Palsson BO: Hierarchical thinking in network biology: the unbiased modularization of biochemical networks ,Trends in Biochemical Sciences 2004, 29:641-647.

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The modularity metric of a network is defined as the largest modularity metric of all possible partitions of the network.

The modularity of networks must always be compared to the null case of a random graph.

For a given decomposition of a network, the modularity metric is defined as:

Newman M: Detecting community structure in networks EurPhysJB 2004, 38:321-330.

Guimera R, Sales-Pardo M, Amaral LAN: Modularity from fluctuations in random graphs and complex networks. Physical Review E 2004, 70:025101.

2.2 Simulated annealing method:

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Guimera R, Nunes Amaral LA: Functional cartography of complex metabolic networks. Nature 2005, 433(7028):895-900.

2.3 Hierarchical clustering method:

Similarity index(or dissimilarity index):

to signify the extent to which two nodes would like in the same cluster.

Agglomerative method:

to start off with each node being its own cluster. At each step, it combines the two most similar clusters to form a new larger cluster until all nodes have been combined into one cluster.

Divisive method:

to begin with one cluster including all the nodes, and attempts to find the splitting point at which two clusters are as dissimilar as possible.

Topological overlap algorithm: Substrate graph

),min(

),(),(

ji

nT kk

jiJjiO

Jn(i,j) denotes the number of nodes to which both i and j are linked ( pl

us 1 if there is a direct link between i and j ); ki, kj is the degree of i and

j, respectively.

Agglomerative method.

Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabasi AL: Hierarchical Organization of Modularity in Metabolic Networks. Science 2002, 297(5586):1551-1555

Shortest path algorithm: enzyme graph

)),(),,(min(),( ijdjidjiitydissimilar

d(i, j) is the number of arcs in the shortest directed path from i to j .

Agglomerative method.

Ma H-W, Zhao X-M, Yuan Y-J, Zeng A-P: Decomposition of metabolic network into functional modules based on the global connectivity structure of reaction graph. Bioinformatics 2004, 20(12):1870-1876.

Betweenness method: substrate-enzyme bipartite graph

ts

r

inB ts

ts

rkrC

),(

),(

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is the number of shortest paths between s and t that passes through r, is the total number of shortest paths between s and t, is the in-degree of node r.

Divisive method.

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Holme P, Huss M, Jeong H: Subnetwork hierarchies of biochemical pathways. Bioinformatics 2003, 19(4):532-538.

Corrected Euclidean-like dissimilarity algorithm: substrate graph

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jikk

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d(i, j) is the number of arcs in the shortest directed path from i to j .

Agglomerative method.

Zhao J, Yu H, Luo J, Cao Z, Li Y: Hierarchical modularity of nested bow-ties in metabolic networks. BMC Bioinformatics 2006:7:386.

2.4 Relationship between topological modules and functional modules

Case 1: some modules are dominated by one major category of metabolisms

Zhao J, Yu H, Luo J, Cao Z, Li Y: Hierarchical modularity of nested bow-ties in metabolic networks. BMC Bioinformatics 2006:7:386.

Case 2 : A standard textbook pathway can break into several modules.

Case 3 : Some modules are mixtures of pieces of several conventional biochemical pathways.

III. Topological diversity of networks with a given degree sequence

--Degree sequence tells us few things

Graphs with the same degree sequence have significantly topological diversity.

Zhao J, Tao L, Yu H, Luo J-H, Cao Z-W, Li Y-X: The spectrum of degree correlations: topological diversity of networks with a given degree sequence. e-print physics/0611078 2006.

Holme P, Zhao J: Exploring the assortativity-clustering space of a network's degree sequence. eprint q-bioOT/0611020 2006.

Thanks!