Network Theory Introduction

Transcription

1 The 13 th Taiwan Nuclear Physics Summer School 2009 June 29 ~ July 4 National Chiao-Tung University Network Theory Introduction HC Lee (Notes prepared with Dr. Rolf Sing-Guan Kong) Computational Biology Laboratory Graduate Institute of Systems Biology and Bioinformatics National Central University

2 Outline Introduction Network history Types of network Nomenclature Properties of network Networks in the real world Model

3 Introduction Network theory is an area of applied mathematics and part of graph theory. Network theory concerns itself with the study of graphs as a representation of either symmetric relations or, more generally, of asymmetric relations between discrete objects. A network is a set of items, which we ll call vertices (nodes), which connections between them, called edges.

4 Network history

5 Königsberg Bridge Problem The city of Königberg was built on the banks of the Pregel River in what was the Prussia ( ), and on two islands that lie in midstream. Seven bridges connected the land masses. Does there exist any single path that crosses all seven bridges exactly once each? 1736, Leonard Euler used graph theory to prove that it s impossibility of its existence. In the past three centuries, graph theory has become the principal mathematical language for describing the properties of networks.

6 Graph theory By abstracting away the details of a problem, graph theory is capable of describing the important topological features with a clarity that would be impossible were all the details retained.

7 Applications of graph theory 1950s, in response to a growing interest in quantitative methods in sociology and anthropology, the mathematical language of graph theory was coopted by social scientists to help understand data from ethnographic studies. Much of the terminology of social network analysis was borrowed directly from graph theory, to address questions of status, influence, cohesiveness, social roles, and identities in social networks. 1950s, mathematicians began to think of graphs as the medium through which various modes of influence information and disease in particular could propagate.

8 Types of network

9 Illustration A set of vertices joined by edges is only the simplest type of network; there are many ways in which networks may be more complex than this. For instance, there may be more than one different type of vertex in a network, or more than one different type of edge. And vertices or edges may have a variety of properties, numerical or otherwise, associated with them. Edges can carry weights, or can be also directed. Directed graphs can be either cyclic or acyclic. Graphs may also evolve over time, with vertices or edges appearing or disappearing, or values defined on those vertices and edge changing.

10 Diagram

11 Nomenclature

12 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Vertex: The fundamental unit of a network, also called a site (physics), a node (computer science), or an actor (sociology). Edge: The line connecting two vertices. Also called a bond (physics), a link (computer science), or a tie (sociology).

13 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Directed/undirected: An edge is directed if it runs in only direction (such as a oneway road between two points), and undirected if it runs in both directions.

14 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Degree: The number of edges connected to a vertex. A directed graph has both an in-degree and an out-degree for each vertex, which are the numbers of incoming and out-going edges respectively.

15 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Component: The component to which a vertex belongs is that set of vertices that can be reached from it by paths running along edges of the graph. In a directed graph a vertex has both an incomponent and outcomponent, which are the sets of vertices from which the vertex can be reached and which can be reached from it.

16 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Protein-protein interaction network A network consisting of proteins known to be involved in infertility (yellow nodes) and interaction partners (grey nodes). Adopted:

17 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Geodesic path: A geodesic path is the shortest path through the network from one vertex to another.

18 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Diameter: The diameter of a network is the length (in number of edges) of the longest geodesic path between any two vertices. A few authors have also used this term to mean the average geodesic distance in a graph, although strictly the two quantities are quite distinct.

19 Nomenclature Vertex Edge Directed/ undirected Degree Component Geodesic path Diameter Adopted:

20 Properties of network

21 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Stanley Milgram (Aug.15,1933 Dec.20,1984) was a social psychologist at Yale University, Harvard University and the City University of New York. While at Harvard, he conducted the small-world experiment (the source of the six degrees of separation concept).

22 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Consider an undirected network, and let us define l to be the mean geodesic distance between vertex pairs in a network: where d ij is the geodesic distance from vertex i to vertex j. It s problematic in networks that have more than one component.

23 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es In multi-component networks, there exist vertex pairs that have no connecting path. Conventionally one assigns infinite geodesic distance to such pairs, but then the value of l also becomes infinite. Infinite values of d ij then contribute nothing to sum.

24 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es In many networks it is found that if vertex A is connected to vertex B and vertex B to vertex C, then there is a heightened probability that vertex A will be also be connected to vertex C. In terms of network topology, transitivity means the presence of a heightened number of triangles in the network sets of three vertices each of which is connected to each of the others.

25 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es It can be quantified by defining a clustering coefficient C (0 to1) thus: where a connected triple means a single vertex with edges running to an unordered pair of others.

26 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es C is the mean probability that two vertices that are network neighbors of the same other vertex will themselves be neighbors. An alternative definition of the clustering coefficient: For vertices with degree 0 or 1, for which both numerator and denominator are zero, we put C i = 0.

27 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Then the clustering coefficient for the whole network is the average Definition [3] is easier to calculate analytically, but [4] is easily calculated on a computer and has found wide use in numerical studies and data analysis. The clustering coefficient measures the density of triangles in a network.

28 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Degree of a vertex in a network is the number connected to that vertex. P(k) is the fraction of vertices in the network that have degree k. A plot of P(k) for any given network can be formed by making a histogram of the degrees of vertices. This histogram is the degree distribution for the network.

29 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es In a random graph, each edge is present or absent with equal probability. Real-world networks are mostly found to be very unlike the random graph in their degree distribution, and their distribution has a long right tail of values.

30 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es

31 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Related to degree distributions is the property of resilience of networks to the removal of their vertices. If vertices are removed from a network, the typical length of these paths will increase, and ultimately vertex pairs will become disconnected and communication between them through the network will become impossible.

32 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es There are also a variety of different ways in which vertices can be removed and different networks show varying degrees of resilience to these also. For example, one could remove vertices at random from a network, or once could target some specific class of vertices, such as those the highest degrees.

33 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es

34 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Authors measured average vertexvertex distances as a function of number of vertices removed, both for random removal and for progressive removal of the vertices with the highest degrees.

35 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es The distance was almost entirely unaffected by random vertex removal. On the other hand, when removal is targeted at the highest degree vertices, it is found to have devastating effect. The Internet is highly resilient against the random failure of vertices in the network, but highly vulnerable to deliberate attack on its highest-degree vertices.

36 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es In most kinds of networks there are at least a few different types of vertices, and the probabilities of connection between vertices often depends on types. In social networks this kind of selective linking is called assortative mixing or homophily.

37 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es A study of 1958 couples in the city of San Francisco, California. The study recorded the race of study participants in each couple. Participants appear to draw their partners preferentially from those of their own race.

38 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Assortative mixing can be quantified by an assortativity coefficient. Let E ij be the number of edges in a network that connect vertices of types i and j, with i, j = 1 N, and let E be the matrix with elements E ij, then [6] where x means the sum of all the elements of the matrix x. The elements e ij measure the fraction of edges that fall between vertices of types i and j.

39 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es One can ask about the conditional probability P(j i) that my network neighbor is of type j given that I am of type i, which is given by P(j i) = e ij / j e ij. Another definition of assortative mixing is It s 1 for a perfectly assortative network (every edge falls between vertices of the same type), and 0 for randomly mixed networks.

40 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Two shortcomings for Q: (1) Q has two different values, depending on the horizontal axis, and it s unclear which is correct one for the network; (2) the measure weights each vertex type equally, regardless of how many vertices there are of each type, which can give rise to misleading figures for Q in cases where community size is heterogeneous, as it often is.

41 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es An alternative assortatively coefficient that remedies these problems is: This quantity is also 0 in a randomly mixed network and 1 in a perfectly assortative one. We find that r = in this example.

42 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Do the high-degree vertices in a network associate preferentially with other high-degree vertices? Or do they prefer to attach to low-degree ones? Calculation of the mean degree of the network neighbors of a vertex as a function of the degree k of that vertex. That gives a one-parameter curve which increases with k if the network is assortatively mixed. And we call disassortativity if it s decrease with k.

43 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Newman reduced the measurement to a single number by calculating the Pearson correlation coefficient of the degrees at either ends of an edge. This gives a single number that should be positive for assortatively mixed networks and negative for disassortatively ones. All social networks measured appear to be assortative, but other types of networks appear to be disassortative.

44 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Community structure means groups of vertices that have a high density of edges within them, with a lower density of edges between groups. A visualization of the friendship network of children in a US school taken from a study by Moody. The network appears to have strong enough community structure.

45 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Moody colors the vertices according to the races, it becomes clear that one of the principal divisions in the network is by individual s race, and this is presumably what is driving the formation of communities in this case.

46 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es The traditional method for extracting community structure from a network is cluster analysis, sometimes also called hierarchical clustering. In this method, one assigns a connection strength to vertex pairs in the network of interest.

47 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Clustering is possible according to many different definitions of the connection strength. Reasonable choices include various weighted vertex-vertex distance measures, the sizes of cut-sets, and weighted path counts between vertices.

48 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Tree of life Adopted: item=2757

49 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Stanley Milgram s famous smallworld experiment, in which letters were passed from person to person in an attempt to get them to a desired target individual, showed that there exist short paths through social networks between apparently distant individuals. The participants had no special knowledge of the network connecting them to the target person. Nonetheless it proved possible to get a message to a distant target in only a small number of steps.

50 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es This is indicates that there is something quite special about the structure of the network. If it were possible to construct artificial networks that were easy to navigate in the same way that social networks appear to be, it has suggested they could be used to build efficient database structures or better peer-to-peer computer networks.

51 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es In some networks the size of the largest component is an important quantity, i.e., in a communication network the size of the largest component represents the largest fraction of the network within which communication is possible and hence is a measure of the effectiveness of network. The size of the largest component is often equated with the graph theoretical concept of the giant component.

52 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Adopted: networkx.lanl.gov/.../giant_component.html

53 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es The betweenness centrality of a vertex i is the number of geodesic paths between other vertices that run through i. And betweenness appears to follow a power law for many networks and propose a classification of networks into two kinds based on the exponent of this power law. Betweenness centrality can also be viewed as a measure of network resilience it tell us how many geodesic paths will get longer when a vertex is removed from the network.

54

55 Proper&es of network A. The small world effect B. Transi7vity or clustering C. Degree distribu7on D. Network resilience E. Mixing paeerns F. Degree correla7ons G. Community structure H. Network naviga7on I. Other network proper7es Milo et al. Network motifs: Simple building block of complex networks. Science 298, (2002).

56 Networks in the real world 1. Social networks 2. Information networks 3. Technological networks 4. Biological networks

57 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds A social network is a set of people or groups of people with some pattern of contacts or interactions between them. Traditional social network studies often suffer from problems of inaccuracy, subjectivity, and small sample size. Data collection is usually carried out be querying participants directly using questionnaires or interviews. Such methods are labor-intensive and therefore limit the size of the network that can be observed.

58 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds Watts and Strogatz, Collective dynamics of small-world networks, Nature, 393, (1998). Small-world networks: Networks that are partly disordered but not random, are highly clustered but have small characteristic path lengths, like random graphs.

59 Watts and Strogatz, Nature, 393, (1998)

60 Watts and Strogatz, Nature, 393, (1998) Make random reconnections

61 Watts and Strogatz, Nature, 393, (1998) Smallworld Ordered Random L(0) > L(p) ~ L(1) ; C(0) >~ C(p) >> C(1)

62 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds Newman, M. E. J., Scientific collaboration networks: I. Network construction and fundamental results, Phys. Rev. E 64, (2001); Newman, The structure of scientific collaboration networks. PNAS, 98, (2001). Newman, M. E. J., Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality, Phys. Rev. E 64, (2001).

63 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds Power law for number of papers versus authors (Newman (I), Phys. Rev. E 64, (2001)) Connection is co-authorship γ ~ , C ~

64 Inter-author distance is small-world 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds (Newman (II), Phys. Rev. E 64, (2001)) <l> autho r ~ <l> random grath

65 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds Adamic L. and Adar E., How to search a social network, Social Networks 27, (2005). How to find short paths in a social network using only local information? Simulate network experiments with network of contacts and online student social networking websites. Finds local information strategy works well when underlying organizational hierarchy is known, but not so well otherwise.

66 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds Adamic and Adar. Social Networks 27, (2005).

67 Adamic and Adar, Social Networks 27, (2005). 1. Social networks A. The paeerns of friendships B. Business rela7onships C. Intermarriages between families D. PaEerns of sexual contacts E. Small world network F. Collabora7on network G. Communica7on records of certain kinds communication (grey lines) clings to organizational chart (black lines) Local information search strategy works well when underlying organizational hierarchy is known, but not so well otherwise.

68 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Network of word classes in thesaurus F. Preference network Information networks also sometimes called knowledge networks.

69 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Network of word classes in thesaurus F. Preference network Redner, S., How popular is your paper? An empirical study of the citation distribution, Eur. Phys. J. B 4, (1998). Citation distribution from 783,000 papers N(x) ~ x γ, γ ~ 3

70 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Network of word classes in thesaurus F. Preference network A. Albert and A.-L. Barabasi, Statistical mechanics of complex network, Rev. Mod. Phys. 74, (2002). Connections in WWW are hyperlinks; in Internet are physical wires.

71 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Network of word classes in thesaurus F. Preference network Albert & Barabasi Rev. Mod. Phys. 74, (2002). Degree distribution of 326,000 (square) and 200 million (circle) samples. P(k) = k γ, γ out ~ , γ in ~

72 General characteristics of some real networks (Albert & Barabasi Rev. Mod. Phys. 74, (2002)) l ~ l rand, C >> C rand

73 General characteristics of some real scale-free networks (Albert & Barabasi Rev. Mod. Phys. 74, (2002)) γ ~ 2-3, l ~ l rand,

74 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Word network F. Preference network Cancho and Sole, The small world of human language, The Royal Society B268, (2001).

75 2. Informa&on networks A. Cita7ons between academic papers B. World Wide Web C. Cita7ons between US patents D. Peer to peer networks E. Word network F. Preference network Cancho and Sole, C>>C random, d~d random, γ~ -2.7 (small world)

76 3. Technological networks A. Distribu7on of some commodity/resource B. Airline routes C. Roads D. Railway E. Pedestrian traffic F. River G. Telephone H. Delivery networks I. Electronic circuits J. Internet Third class of networks is technological networks, manmade networks designed typically for distribution of some commodity or resource.

77 3. Technological networks A. Distribu7on of some commodity/resource B. Air transporta7on network C. Roads D. Railway E. Pedestrian traffic F. River G. Telephone H. Delivery networks I. Electronic circuits J. Internet Guimera, The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles, PNAS 31, (2005). Betweenness of node i: number of shortest paths that passes through i.

78 3. Technological networks A. Distribu7on of some commodity/resource B. Air transporta7on network C. Roads D. Railway E. Pedestrian traffic F. River G. Telephone H. Delivery networks I. Electronic circuits J. Internet Guimera, PNAS 31, (2005) High centrality does not necessarily imply large betweenness bogus

79 3. Technological networks A. Distribu7on of some commodity/resource B. Air transporta7on network C. Roads D. Railway E. Pedestrian traffic F. River G. Telephone H. Delivery networks I. Electronic circuits J. Internet Balthrop, et al. Technological networks and the spread of computer viruses, Science 304, (2004). (Right) IP and Admin. networks are from a large corporation. (Left) Address books and traffic data are from a large university. Throttling is the best strategy for slowing virus and worm spread.

80 4. Biological networks A. Metabolic pathways B. Sta7s7cal proper7es of metabolic networks C. Protein interac7on D. Gene7c regulatory networks E. Disease network F. Cell cycle G. Blood vessels and the equivalent vascular network in plants Giot, L., et al., A protein interaction map of Drosophila melanogaster, Science 302, (2003). Top 3000 interactions among 3522 proteins in fruit fly

81 Transcriptional regulation of gene (cartoon) Biological functions controlled by transcription factors (TFs, a type of proteins) acting as switches DNA upstream of a gene aaaaaaaa Coding of gene Machine (ribosome) that transcribes gene aaaaaaaa Activator: activating TF Suppressor: suppressing TF TF binding sites

82 4. Biological networks A. Metabolic pathways B. Sta7s7cal proper7es of metabolic networks C. Protein interac7on D. Gene7c regulatory networks E. Disease network F. Cell cycle G. Blood vessels and the equivalent vascular network in plants Tong Ihn Lee, et al., Transcription regulatory networks in Saccharomyces cerevisiae, Science 298, (2002). Yeast. Much larger number of promoter region shared by many regulators Random system Eukaryo7c cellular func7ons are highly connected through networks transcrip7onal regulators that regulate other transcrip7onal regulators.

83 4. Biological networks A. Metabolic pathways B. Sta7s7cal proper7es of metabolic networks C. Protein interac7on D. Gene7c regulatory networks E. Disease network F. Cell cycle G. Blood vessels and the equivalent vascular network in plants Tong Ihn Lee, et al., Science 298, (2002). aaaaaaaaaaaaaaaaaaaaaaa

84 4. Biological networks A. Metabolic pathways B. Sta7s7cal proper7es of metabolic networks C. Protein interac7on D. Gene7c regulatory networks E. Disease network F. Cell cycle G. Blood vessels and the equivalent vascular network in plants K. Goh et al., The human disease network, PNAS 104, (2007). Conclusion Proteins/genes contributing to common disease tend to interact with each other Study offers network-based explanation for complex disorders: a phenotype correlating with malfunction of a particular functional module

85 Use map of diseasome to construct networks

86 Nodes: diseases, connection: shared gene

87 Nodes: genes; connection: shared disease

88 4. Biological networks A. Metabolic pathways B. Sta7s7cal proper7es of metabolic networks C. Protein interac7on D. Gene7c regulatory networks E. Disease network F. Cell cycle G. Blood vessels and the equivalent vascular network in plants Class Exercise Li FT, et al., The yeast cell-cycle network is robustly designed, PNAS 101, (2003). Simplified regulatory network of yeast cycle (down from> 400 genes)

89 Li FT, et al., PNAS 101, (2003). Network of cycle paths Each (green) node is a configuration: {S i i= 1 to 11}. Genes have two states: S i = 1, active S i = 0, inactive

90 Li FT, et al., PNAS 101, (2003). Fixed-points of cell-cycle network Thirteen configurations of temporal evolution of a pathway to the largest fixed-point

91 Models 1. Help us to understanding the construction of networks 2. Predict the behaviors of networks.

92 Three classical models

93 The small-world model

94 References PDF copies of the following papers can be obtained at the website hep://sansan.phy.ncu.edu.tw/~hclee/ref/nucl_school09/papers.html A. Albert and A.-L. Barabasi, Statistical mechanics of complex network, Rev. Mod. Phys. 74, (2002) - Watts and Strogatz, Collective dynamics of "small-world" networks, Nature, 393, (1998). - Newman, M. E. J., Scientific collaboration networks: I & II. Network construction and fundamental results, Phys. Rev. E 64, (2001). - Cancho and Sole, The small world of human language, The Royal Society B268, (2001). - Guimera et al., The worldwide air transportation network: Anomalous centrality, community structure, and cities' global roles, PNAS 31, (2005). - Balthrop, et al. Technological networks and the spread of computer viruses, Science 304, (2004). - Tong Ihn Lee, et al., Transcription regulatory networks in Saccharomyces cerevisiae, Science 298, (2002). - K. Goh et al., The human disease network, PNAS 104, (2007). - Sporns, et al., Organization, development and function of complex brain networks, TRENDS in Cognitive Sciences 8(9), (2004). - Li FT, et al., The yeast cell-cycle network is robustly designed, PNAS 101, (2003). - Giot, L., et al., A protein interaction map of Drosophila melanogaster, Science 302, (2003).

95 Thank for your attention