Network Science: A Brief Overview

1. Network Science: A Short Introductioni3 Workshop Konstantinos Pelechrinis Summer 2014 Figures are taken from: M.E.J. Newman, “Networks: An Introduction”

2. The representation of networks • The network consists of entities connected with each other • The structure of these connections are represented through graphs • A graph is represented by two sets • A vertex set V of the entities participating in the network. In the rest of the slides typically, n will be the number of vertices • Also called node or actor set • An edge set E of the connections between vertices. In the rest of the slides typically, m will be the number of edges • Also called link or tie set

3. Example Edges can have direction, but in this introduction we will only consider undirected edges/networks.

4. Edge attributes • Examples • Weight (e.g., frequency of contacts, bandwidth of the link in a telecommunication network etc.) • Ranking (e.g., primary connection, secondary connection etc.) • Type (e.g., friend edge, family edge, co-worker edge etc.) • …

5. Edge list and the adjacency matrix • If we label the nodes with IDs 1, 2, … n we can denote each edge as a pair (i,j) • This is an edge list specification • Good for storing and processing networks in computers, but not for mathematical development • The adjacency matrix A of a simple graph is a matrix with elements Aij such that:

6. Example Edge list (1,2) (1,5) (2,3) (2,4) (3,4) (3,5) (3,6) Adjacency matrix

7. Adjacency list • Easier to work if the network is • Large • Sparse 1: 2,5 2: 1,3,4 3: 2,4,5,6 4: 2,3 5: 1,3 6: 3

8. Degree • The degree ki of a vertex i in a graph is the number of edges connected to it • For undirected graphs we have: • And the number of edges of a graph is given by: • Mean degree c of a vertex in an undirected graph is:

9. Example Degree of node 2 = 3

10. Density • The maximum number of possible edges in a simple graph is: • Density ρ of a graph is the fraction of these edges that are actually present:

11. Degree sequence and degree distribution • Degree sequence is an (ordered) list of the degree of every node • In our earlier network we have: [4, 3, 2, 2, 2, 1] • Degree distribution is a frequency count of the occurrence of each degree • It is essentially a histogram

12. Paths • A sequence of vertices such that every consecutive pair of vertices in the sequence is connected by an edge in the network • Length of a path is the number of edges traversed along the path • When a path traverses the same edge e two times, e is counted twice • A geodesic path (shortest path) is a path between two vertices such that no shorter path exists • The length of this path is called geodesic (or shortest) distance • If two nodes are not connected with any path their geodesic distance is infinite

13. Connected components • A network for which there exists pairs of vertices that there is no path between them is called disconnected • If there exists a path between any possible pair of vertices in a network the latter is called connected • Component is a maximal subset of vertices of a network such that there exists at least one path from every vertex of the subgroup to any other • Each node within a component can be reached from every other node in the component by following the edges

14. Giant component • If the largest component includes a significant fraction of the network, it is called giant component

15. Transitivity • If A is connected to B and B is connected to C, what is the probability that B is connected to C ? • My friends’ friends are likely to be my friends too C ? A B

16. Local clustering coefficient • The clustering coefficient can be defined for a single vertex i as: 1/(2*1/2)=1 2/(3*2/2)=2/3 3/(4*3/2)=1/2 1/(2*1/2)=1 2/(3*2/2)=2/3

17. Clustering coefficient • Watts and Strogatz have suggested computing the clustering coefficient of a network as the average over all the local clustering coefficients of the vertices: