Modeling Nature February 2009 1 Modeling Nature LECTURE
Modeling Nature February 2009 1
Modeling Nature LECTURE 3: Network models 2
Overview • • • Some definitions Basic characteristics of networks Special network topologies Examples from nature and sociology Relation to the tasks 3
Definition of a Network A network is a system of N similar entities called nodes (a. k. a. edges), where each node interacts with (i. e. ‘has a relation to’) certain other nodes in the system. This interaction is visualized through a connection (a. k. a. vertex). 4
Some examples Undirected network Directed network Selfconnection and multiple edges 5
A more complex example 6
A large network 7
A gene regulatory network 8
Characteristics of Networks • Characteristic Path Length (L) – The average number of associative links between a pair of concepts • Clustering Coefficient (C) – The fraction of associated neighbors of a concept that are also connected 9
Q A possible path between nodes P and Q P 10
Picture pathlengths and clustercoefficients in these networks 11
Characteristics of Networks • The coordination number z is the average number of links per vertices, i. e. there a total of Nz/2 connections in an undirected network. • The network diameter D is the maximum degree of separation between all pairs of vertices. For a network with N vertices and coordination number z we thus have a diameter D of: z. D ≈ N 12
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Characteristics of Networks • Branching Factor a. k. a. Degree (k) – The number of other nodes connected to this node i. e. the number of vertices of a node • Degree distribution – The relation between the degree and number of nodes in the network that have exactly this degree. 14
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Connection probability p The probability that a given edge occurs is called the connection probability p. 16
Characteristics of Networks Clique A clique is a set of vertices for which: (i) every node of the same clique is connected by an edge to every other member of the clique and (ii) no node outside the clique is connected to all members of the clique. 17
Cliques 18
Special Network Topologies In many situations networks can have a special structure (topology) or properties. We will consider the following cases. 19
1. Regular network A regular network is a network where each node has an identical connection scheme. ? YES ? NO 20
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2. Fully connected network A fully connected network is a network where each node is connected to all other nodes. 22
3. A sparse network is a network that exhibits a (very) small amount of connections. (opposite: dense) 23
dense network sparse network 24
4. Random network A random network is a network that is generated by some random process. 25
Random network 26
Small-World (SW) network A SW network is a property of the network rather than a specific topology – though the SW-property has implications for the network architecture. 27
Small-world networks In such a network there are numerous clusters of richly interconnected elements and a small number of connections between the clusters. This type of network has been offered as a model of the "six-degrees-of-separation" concept. A small world network falls between a regular and random network in its properties, as depicted in this figure: 28
Small-world networks p is the probability that a randomly chosed connection will be randomly redirected elsewhere (i. e. , p=0 means nothing is changed, leaving the network regular; p=1 means every connection is changed and randomly reconnected, yielding complete randomness). 29
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Small-world networks 32
Four network types a fully connected c regular b d random “small world 33
Network Evaluation Type of network k C L Fully-connected N-1 Large Small Random <<N Small Regular <<N Large Small-world <<N Large Small 34
Varying the rewiring probability p: from regular to random networks 1 C(p)/C(0) L(p)/L(0) 0 0. 00001 0. 1 1. 0 p 35
First an example … the accumulation of knowledge and the growth of the ‘semantic network’ in children 36
Example : Semantic Network lemon gravitation pear apple orange Newton Einstein 37
Growth of knowledge semantic networks lemon gravitation pear apple orange Newton Einstein • Average separation should be small • Local clustering should be large 38
Strongest links of/with APPLE PIE PEAR ORANGE TREE CORE FRUIT (20) (17) (13) ( 8) ( 7) ( 4) NEWTON APPLE ISAAC LAW ABBOT PHYSICS SCIENCE (22) (15) ( 8) ( 6) ( 4) ( 3) 39
Semantic net at age 3 40
Semantic net at age 4 41
Semantic net at age 5 42
The growth of semantic networks obeys a logistic law 43
L as a function of age (× 100) = semantic network = random network 44
C as a function of age (× 100) = semantic network = random network 45
Small-worldliness Walsh (1999) • Measure of how well small path length is combined with large clustering • Small-worldliness = (C/L)/(Crand/Lrand) 46
Small-worldliness as a function of age adult 47
Small-Worldliness Some comparisons 5 4 3. 5 3 2. 5 2 1. 5 1 0. 5 0 Semantic Network Cerebral Cortex Caenorhabditis Elegans 48
What causes the smallworldliness in the semantic net? Optimal efficient organization 49
Strongest links in semantic net of adult males [Shields, 2001] • TOP 40 of concepts • Ranked according to their k-value (number of associations with other concepts) 50
Semantic top 40 51
Scale-Free (SF) networks A SF network is a network where the degree distribution has a very specific structure 52
Scale-Free Networks (Barabasi et al, 1998) • In the late 1990 s: Analysis of large data sets became possible • Finding: the degree distribution often follows a power law: many lowly connected nodes, very few highly connected nodes: • Examples – Biological networks: metabolic, protein-protein interaction – Technological networks: Internet, WWW – Social networks: citation, actor collaboration – Other: earthquakes, human language 53
Scale-Free Networks Some small-world networks are also what are called scale-free. In a scall-free networks the characteristic clustering is maintained even as the networks themselves grow arbitrarily large. The mathematical properties and methods of analysis of such scale-free networks allow broad types of analysis, modeling and simulation. 54
Scale-Free Networks * Here is a picture of a part of such a scale-free network : 55
Scale-Free Networks (Barabasi et al, 1998) 56
Scale-Free Networks In any real network some nodes are more highly connected than others. P(k) is the proportion of nodes that have k-links. For large, random graphs only a few nodes have a very small k and only a few have a very large k, leading to a bell-shaped distribution, such as this one: 57
Scale-Free Networks Scale-free networks fall off more slowly than random ones. Such networks are governed by a power law of the form Because of this power law relationship, a log-log plot of P(k) versus k gives a straight line. 58
Scale-Free Networks (Barabasi et al, 1998) 59
Random versus scale-free 60
Random (□) and scale free (○) Linear axes Logarithmic axes 61
In scale-free networks, some nodes act as "highly connected hubs" (high degree), although most nodes are of low degree. Scale-free networks' structure and dynamics are independent of the system's size N, the number of nodes the system has. In other words, a network that is scale-free will have the same properties no matter what the number of its nodes is. 62
Scale-free networks can grow by the process of preferential attachment : new links are made preferably to hubs: the probability of a new link is proportional to the links of a node. 63
Some examples… 64
Nodes: email-addresses, links: emails 65
Nodes: people, links: # of sexual partners 66
Protein network C. elegans 67
The WWWeb is scale free Web pages : Inlinks and outlinks (red and blue) Network nodes (green) 68
100 000 Internet routers and the physical connections between them 69
Mycoplasma genitalium Metabolic Network Degree distribution Horizontally log of degree (= number of connections), vertically log of number of genes with this degree Mycoplasma genitalium 500 nm 580 Kbp 477 genes 74% coding DNA Obligatory parasitic endosymbiont Metabolic Network Nodes are genes, edges are gene co-expressions 70
Degree distributions in human gene coexpression network. Coexpressed genes are linked for different values of the correlation r, King et al, Molecular Biology 71 and Evolution, 2004
Social Networks A social network is a social structure made of nodes (which are generally individuals or organizations) that are tied by one or more specific types of interdependency, such as values, visions, ideas, financial exchange, friendship, kinship, dislike, conflict or trade. The resulting graph-based structures are often very complex. 72
Many more examples… 73
Relation to the Tasks Task 3 a. Global Social Networks Task 3 b. The neural network structure of consciousness. 74
END of LECTURE 3 75
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