Graphs and social networks Social networks Active area of research - - PowerPoint PPT Presentation

csci 210: Data Structures

Graphs and social networks

Social networks

• Active area of research
• motivated in part by growing public fascination withe the complex “connectedness” of modern society
• Social networks have grown in complexity
• technological advances, global travel, global communication
• Networks come up in many other contexts
• information networks
• economic networks
• trade networks
• biological networks
• Necessary to study and understand the structure and behavior or complex networks

Networks

• How should we think about networks?
• Structurally, networks are graphs
• we can find structural properties of the underlying graph
• But structure is not enough. We also need a framework for reasoning about behavior

and interaction in complex networks.

• Traditionally networks have been studied across many disciplines
• sociology, economics,...
• now coming together with computer science, math, physics.
• From computer science point of view, we are primarily interested in
• the structural properties of the network
• modeling networks as classes of graph and studying their properties

Graphs and social networks

• Paths, cycles, connectivity
• Are two people connected?
• Are any two people connected?
• Are there cycles of friends?
• What is the size of the largest group of friends?
• Distances
• What is the distance of person A to person B?
• What is the average distance between two people in a network?
• What is the diameter of the network?
• Centrality
• What is the center of the network?
• Clustering
• What are the underlying clusters in the network?

Graphs and social networks

• Paths, cycles, connectivity
• Traverse the graph (BFS or DFS)
• Time: O(V+E)

Graphs and social networks

• Distances
• Compute the distance between two vertices in a graph
• BFS (from one vertex until reaching the other)
• Time: O(V+E)
• Compute the average distances between two vertices in a graph
• BFS from all vertices in G and record all distances
• Time: O(V(V+E)) = O(V2 + VE)
• Compute the diameter of a network
• diameter = the length of the longest shortest path between two vertices.
• Meaning: gives an idea of the time it takes to spread something over the network.
• To compute:
• need to compute shortest paths between all pairs of vertices
• run BFS from every vertex as source and record pairwise distances
• this takes O(V(V+E)) time and O(V2) space.

• Centrality
• Degree centrality
• degree(v) = the number of edges of a node
• the center is the node with largest degree
• time to compute: ?
• Closeness centrality
• closeness(v) = the average path length to all vertices that are reachable from v
• the center is the node with lowest closeness
• time to compute: ?
• Betweenness centrality
• idea: vertices that occur on many shortest path have higher betweenness than those that do not.
• time to compute: ?
• Various other measures of centrality, depending on specific application

Graphs and social networks

Clustering (graph partitioning)

• Goal: Identify the underlying clusters in a graph

from http://projects.si.umich.edu/netlearn/GUESS/betweennessclust.html

Clustering (graph partitioning)

• Goal: Identify the underlying clusters in a graph
• A wide range of methods
• in social networks: clustering using betweenness
• Betweenness of an edge e:
• the total number of times the edges appears on a shortest path between vertices in the graph
• How to compute betweenness values for all edges?
• Time?

Clustering (graph partitioning)

• Goal: Identify the underlying clusters in a graph
• A wide range of methods
• in social networks: clustering using betweenness
• Betweenness of an edge e:
• the total number of times the edges appears on a shortest path between vertices in the graph
• How to compute betweenness values for all edges?
• compute shortest path between all pairs of vertices, keep track of the edges on the shortest paths, and

update the frequency of each edge

• Time:
• O(V2 + VE)

Clustering (graph partitioning)

• Goal: Identify the underlying clusters in a graph
• A wide range of methods
• in social networks: clustering using betweenness
• Betweenness of an edge e:
• the total number of times the edges appears on a shortest path between vertices in the graph
• Clustering using betweenness:
• repeatedly remove the edge of highest betweenness.
• [demo]: http://projects.si.umich.edu/netlearn/GUESS/betweennessclust.html