Ronald Westra

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• Ronald Westra
• Department of Mathematics
• Faculty of Humanities Sciences
• Universiteit Maastricht

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• Lecture 3
• Network Models

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• 3.1. Properties of Networks

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Network properties
Networks consist of nodes connections
(directed or undirected) update rules for the
nodes
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Network properties
x
Node
Connection (directed arrow)
Update rule xt1 some_function_of (xt)
f(xt)
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The meaning of networks
Until now we have encountered a number of
interesting models 1 entity that interacts only
with itself
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The meaning of networks
Examples population growth, exponential growth,
Verhulst equation xt1 some_function_of
(xt) f(xt)
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The meaning of networks
2 entities that interact
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The meaning of networks
Example predator-prey-relations as the
Lotka-Volterra equation xt1 f( xt, yt)
yt1 g( xt, yt)
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The meaning of networks
Multiple entities that interact in a network
structure
This is a general model for multi-agent
interaction
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• 3.2. Small-World Networks

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Growth of knowledgesemantic networks

• Average separation should be small
• Local clustering should be large

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Semantic net at age 3
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Semantic net at age 4
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Semantic net at age 5
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The growth of semantic networks obeys a logistic
law
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• We now anticipate on task 3
• Given the enormous size of our semantic networks,
  how do we associate two arbitrary concepts?

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Clustering coefficient and Characteristic Path
Length

• Clustering Coefficient (C)
• The fraction of associated neighbors of a concept
  
• Characteristic Path Length (L)
• The average number of associative links between a
  pair of concepts
• Branching Factor (k)
• The average number of associative links for one
  concept

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Example
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Four network types
a
b
fully connected
random
c
d
regular
small world
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Network Evaluation
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Varying the rewiring probability p from regular
to random networks



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Data set two examples

• APPLE
• PIE (20)
• PEAR (17)
• ORANGE (13)
• TREE ( 8)
• CORE ( 7)
• FRUIT ( 4)

NEWTON APPLE (22) ISAAC (15) LAW ( 8) ABBOT (
6) PHYSICS ( 4) SCIENCE ( 3)
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L as a function of age ( 100)
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C as a function of age ( 100)
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Small-worldlinessWalsh (1999)

• Measure of how well small path length is combined
  with large clustering
• Small-wordliness (C/L)/(Crand/Lrand)

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Small-worldliness as a function of age
adult
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Some comparisons
Small-Worldliness
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What causes the small-worldliness in the semantic
net?

• TOP 40 of concepts
• Ranked according to their k-value (number of
  associations with other concepts)

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Semantic top 40
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• 3.3. Special Networks

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Special Networks

• Small-world networks
• Scale-free networks

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Network properties branching factor k
Consider a set of nodes
x1
x2
x3
x4
x5
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Network properties branching factor k
Now make random connections
x1
x2
x3
x4
x5
This is a random network
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Network properties branching factor k
This approach results in an average branching
number kav If we plot a histogram of the number
of connections we find
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Network properties branching factor k
Now consider an structured network
x1
x2
x3
x4
x5
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Network properties branching factor k
This approach results in an equal branching
number kav for all nodes If we plot a histogram
of the number of connections we find
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Network properties branching factor k
The same for a fully connected network this
results in an equal branching number kav n - 1
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Network properties branching factor k
Now what for a small-world network?
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Scale-Free Networks (Barabasi et al, 1998)
History Using a Web crawler, physicist
Albert-László Barabási at the University of Notre
Dame mapped the connectedness of the Web in 1999
(Barabási and Albert, 1999). To their surprise,
the Web did not have an even distribution of
connectivity (so-called "random connectivity").
Instead, some network nodes had many more
connections than the average seeking a simple
categorical label, Barabási and his collaborators
called such highly connected nodes "hubs".
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Scale-Free Networks (Barabasi et al, 1998)
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Scale-Free Networks (Barabasi et al, 1998)
History (Ctd) In physics, such right-skewed or
heavy-tailed distributions often have the form of
a power law, i.e., the probability P(k) that a
node in the network connects with k other nodes
was roughly proportional to k-?, and this
function gave a roughly good fit to their
observed data.
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Scale-Free Networks (Barabasi et al, 1998)
History (Ctd) After finding that a few other
networks, including some social and biological
networks, also had heavy-tailed degree
distributions, Barabási and collaborators coined
the term "scale-free network" to describe the
class of networks that exhibit a power-law degree
distribution. Soon after, Amaral et al. showed
that most of the real-world networks can be
classified into two large categories according to
the decay of P(k) for large k.
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Scale-Free Networks (Barabasi et al, 1998)
A scale-free network is a noteworthy kind of
complex network because many "real-world
networks" fall into this category. Real-world"
refers to any of various observable phenomena
that exhibit network theoretic characteristics
(see e.g., social network, computer network,
neural network, epidemiology).
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Scale-Free Networks (Barabasi et al, 1998)
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.
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Scale-Free Networks (Barabasi et al, 1998)
The defining characteristic of scale-free
networks is that their degree distribution
follows the Yule-Simon distribution a power law
relationship defined by where the probability
P(k) that a node in the network connects with k
other nodes was roughly proportional to k-?, and
this function gave a roughly good fit to their
observed data. The coefficient ? may vary
approximately from 2 to 3 for most real networks.
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Scale-Free Networks (Barabasi et al, 1998)

• In the late 1990s 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

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Random versus scale-free
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Random (?) and scale free (?)
Linear axes
Logarithmic axes
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Nodes people, links of sexual partners
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Nodes email-addresses, links emails
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Protein network C.elegans
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100 000 Internet routers and the physical
connections between them
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• c. Web pages
• Inlinks and outlinks (red and blue)
• d. Network nodes (green)

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Many more examples
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END LECTURE 3