CS8803-NS
Network Science
Fall 2013
Instructor: Constantine Dovrolis
constantine@gatech.edu
http://www.cc.gatech.edu/~dovrolis/Courses/NetSci/
Disclaimers
The following slides include only the figures
or videos that we use in class; they do not
include detailed the explanations, derivations
or descriptions covered in class.
Many of the following figures are copied
from open sources at the Web. I do not
claim any intellectual property for the
following material.
Outline
• Basic concepts
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Graphs, paths, adjacency matrix, etc
Centrality metrics
Clustering metrics
Cliques and cores
Assortativity metrics
• Weighted networks
– Application paper: airline networks
• Spatial networks
– Application paper: city networks
• More about betweenness centrality
– Betweenness centrality algorithm by Brandes
– Mini-talk by Oded Green about parallel/streaming BC computation
• Surprise “visitor” will talk about Network Medicine
Undirected, directed, weighted
graphs
Graph adjacency matrix
http://sourcecodemania.com/wp-content/uploads/2012/06/adjacency-matrix-of-graph.jpg
Graph adjacency matrix (cont’)
• Many network properties can be
formulated as properties of the
adjacency matrix
– See “algebraic graph theory”
• For instance:
– A directed network is acyclic if and only if
all eigenvalues of the adjacency matrix are
equal to 0
• Proof?
Planar graphs
http://people.hofstra.edu/geotrans/eng/methods/img/planarnonplanar.png
• Here is an example of a famous graph theory result:
– Kuratowski: Every non-planar graph contains at least one
subgraph that is an expansion of a 5-node clique or of
the “utility graph” K3,3 (shown at the top right)
Node degree, in-out degrees,
degree distribution k’th moment
• Directed graphs: aij=1 if edge from i to j
Degree distribution
http://1.bp.blogspot.com/-QJTJS8wcdtg/T5rio1NHPvI/AAAAAAAAAq8/umNpggK8VAY/s400/p1.png
Average degree, connectance,
sparse and dense graphs
• Undirected graph with n nodes and m edges
• Average node degree: c = 2*m / n
• Connectance: ρ = c / (n-1)
• What happens to ρ as n tends to infinity?
• Sparse graphs: ρ tends to zero
• Dense graphs: ρ tends to positive constant
Paths, Shortest Paths, Diameter,
Characteristic Path Length,
Graph Efficiency
Paths and their length
a
b
d
c
http://www.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture2/Images/A2.gif
• Number of paths of length r from j to i:
Nij(r)=[Ar]ij
Cyclic and Acyclic graphs
http://upload.wikimedia.org/wikipedia/commons/thumb/3/39/Directed_acyclic_graph_3.svg/356pxDirected_acyclic_graph_3.svg.png
• Number of loops of length r anywhere in
network: L(r) = Σi[Ar] ii = Tr[Ar] = Σi[kir]
– ki eigenvalue of adjacency matrix
– Proof? Easier for undirected networks
Eulerian and Hamiltonian graphs
http://www.transtutors.com/Uploadfile/CMS_Images/10131_Hamiltonian%20Graphs.JPG
Weakly Connected Components &
Strongly Connected Components
http://media.tumblr.com/tumblr_m0oeu763ds1qir7tc.png
Min-cut and max-flow
http://scienceblogs.com/goodmath/wp-content/blogs.dir/476/files/2012/04/id36ecaca5a38da5705e2e708e6d84070-max-flow.png
• For a given (source, sink) pair: the max flow
between them is the sum of the weights of
the edges of the min-cut-set that separates
(source, sink)
Network centrality metrics
http://www-personal.umich.edu/~mejn/centrality/labeled.png
http://www.isearchm.com/wp-content/uploads/2012/05/networks.jpg
We will talk later about centrality metrics for directed networks,
such as PageRank or HITS
Cliques, plexes and cores
• Clique of size n: maximal subset of nodes, with
every node connected to every other member
of the subset
• k-plex of size n: maximal subset of nodes, with
every node connected to at least n-k other
members of the subset
– k=1: clique
– k>1: “approximate clique”
• k-core of size n: maximal subset of nodes, with
every node connected to at least k others in
the subset
K-core decomposition
http://www.nature.com/srep/2012/120420/srep00371/images/srep00371-f1.jpg
Transitivity & Clustering coeff
Clustering coefficient
http://www.emeraldinsight.com/fig/202_10_1108_S1479-361X_2012_0000010012.png
Degree correlations
In general, is knn(k) increasing/decreasing with k?
Assortativity – Degree mixing
http://stepsandleaps.files.wordpress.com/2013/08/assortative_disassortative.jpg?w=450&h=235
How would you classify social networks in this axis?
Technological networks such as the Internet?
Core-periphery networks
(“rich club” network)
http://km4meu.files.wordpress.com/2009/11/core-periphery-ross-mayfield1.jpg
Outline
• Basic concepts
–
–
–
–
–
Graphs, paths, adjacency matrix, etc
Centrality metrics
Clustering metrics
Cliques and cores
Assortativity metrics
• Weighted networks
– Application paper: airline networks
• Spatial networks
– Application paper: city networks
• More about betweenness centrality
– Betweenness centrality algorithm by Brandes
– Mini-talk by Oded Green about parallel/streaming BC computation
• Surprise “visitor” will talk about Network Medicine
Node strength
Strength distribution
Relation between strength and degree
Weighted clustering coefficient
Weighted average neighbors degree
Outline
• Basic concepts
–
–
–
–
–
Graphs, paths, adjacency matrix, etc
Centrality metrics
Clustering metrics
Cliques and cores
Assortativity metrics
• Weighted networks
– Application paper: airline networks
• Spatial networks
– Application paper: city networks
• More about betweenness centrality
– Betweenness centrality algorithm by Brandes
– Mini-talk by Oded Green about parallel/streaming BC computation
• Surprise “visitor” will talk about Network Medicine
Spatial networks
• Nodes are embedded in physical space
(2d or 3d)
• Edges have physical length
• Planar graphs constraint
• Spatial embedding affects maximum
degree or maximum edge length
• Spatial networks vs Relational networks
Analyzed one-square-mile maps
from 18 cities
Food for thought
• How do you explain the (major)
difference in the distributions of
betweenness centrality and information
centrality?
– What is a good generative model for selforganized cities?
• How would you cluster similar cities
together based on their spatial network
properties?
Outline
• Basic concepts
–
–
–
–
–
Graphs, paths, adjacency matrix, etc
Centrality metrics
Clustering metrics
Cliques and cores
Assortativity metrics
• Weighted networks
– Application paper: airline networks
• Spatial networks
– Application paper: city networks
• More about betweenness centrality
– Betweenness centrality algorithm by Brandes
– Mini-talk by Oded Green about parallel/streaming BC computation
• Surprise “visitor” will talk about Network Medicine
Counting shortest paths
Accumulation of path-dependencies
Mini-talk by Oded Green about parallel, streaming BC
computation
Outline
• Basic concepts
–
–
–
–
–
Graphs, paths, adjacency matrix, etc
Centrality metrics
Clustering metrics
Cliques and cores
Assortativity metrics
• Weighted networks
– Application paper: airline networks
• Spatial networks
– Application paper: city networks
• More about betweenness centrality
– Betweenness centrality algorithm by Brandes
– Mini-talk by Oded Green about parallel/streaming BC computation
• Surprise “visitor” will talk about Network Medicine
A surprise “visitor” will talk to us
about Network Medicine
http://www.youtube.com/watch?v=10oQMHa
dGos