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 Weighted networks Spatial networks 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 Here is an example of a famous graph theory result: 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/356px-Directed_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/i-d36ecaca5a38da5705e2e708e6d84070-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

In general, is knn(k) increasing/decreasing with k? 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 Weighted networks Spatial networks 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 Weighted networks Spatial networks 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 self-organized cities? How would you cluster similar cities together based on their spatial network properties?

Outline Basic concepts Weighted networks Spatial networks 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 Weighted networks Spatial networks 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=10oQMHadGos