Lecture2

Undirected vs Directed Networks¶

Undirected graph

Links: undirected
Example
- Collaborations
- Friendship on Facebook

Directed

Links: Directed
Example
- Phone calls (If you want to represent the relationship between caller and callee, then you can use undirected graph)
- Following on Twitter

Connectivity of graphs¶

Undirected graph¶

Connected: Any two vertices can be joined by a path
A disconnected graph is made up of two or more connected components.

Bridge edge: If we erase it, the graph becomes disconnected.

Articulation point: If we erase it, the graph becomes disconnected

Isolated node: Node without any connected neighbors.

Directed graph¶

Strongly connected directed graph: Has a path from each node to every other node and vice versa

Weakly connected directed graph: Is connected if we disregard the edge directions

Directed Acyclic Graph: Has no cyclesL if u can reach v, the v cannot reach u.

Strongly connected components: is a set of nodes S so that:

Every pair of nodes in S can reach each other
There is no larger set containing S with this property

Node degree: The number of edges adjacent to node i

Avg. degree: $\hat{k} = \frac{1}{N}$, $K_i = \frac{2E}{N}$

In directed networks we define an in-degree and out-degree. The total degree of a node is the sum of in-and out-degrees.

Complete Graph¶

Maximum number edges in undirected graph on N nodes.

\[ E_{max} = \frac{N(N-1)}{2} \]

A graph with the number of edges $E = E_{max} $ is a complete graph and its average degree is N - 1

Unweighted graph¶

Graph without weight. For example, Friendship and sex.

Weighted graph¶

Graph with weight. For example collaboration with internet roads.

Self-edges (Self-loops)¶

Examples: Proteins, Hyperlinks (A web page link which points to itself)

Multigraph¶

Examples: Communication, collaboration

Examples¶

Example	Type
www	Directed
Facebook friendships	Undirected, unweighted
Citation networks	Unweighted, directed, acyclic
Collaboration networks	Undirected multigraph or weighted graph
Mobile phone calls	directed
Protein interactions	Undirected, unweighted with self-interactions

Bipartite Graph¶

A graph whose nodes can be divided into two disjoint sets U and V such that every link connects a node in U to one in V; that is U and V are independent sets.

Examples¶

Authors-to-papers
Actors-to-movies

Degree distribution P(k):¶

Probability that a randomly chosen node has degree k.

Normalized histogram:¶

$P(k) = N_k / N$

Distance¶

Between a pair of nodes is defined as the number of edges along the shortest path connecting the nodes

In directed graphs paths need to follow the direction of the arrows. So that distance is not symmetric $h_{a,c} \neq h_{c,a}$

Diameter¶

The maximum distance (shortest graph) between any pair of nodes in a graph.

Clustering coefficient¶

$\(C_i = \frac{2e_i}{k_i(k_i - 1)}$\) where $e_i$ is the number of edges between the neighbors of node i.

Web as a graph¶

Nodes = Web pages

Edges = hyperlinks

Side issues¶

Dynamic pages created on the fly
Dark matter - inaccessible database generated pages