Concepts of Graph Theory

Social Networks; Lecture 2

Summary

• Graph representation of social networks

• Matrix representation of social networks

• Node degree; average degree; degree distribution

• Graph density

• Walks, trails and paths

• Cutpoits, cutsets and bridges

What is a Network?

• A set of dyadic ties, all of the same type,among a set of actors

• Actors can be persons, organizations ...

• A tie is an instance of a social relation

Relations Among Persons

• Kinship– Mother of, father of, sibling of

• Role-Based– Boss of, teacher of– Friend Of

• Affective– Likes, trusts

• Interactions– Gives advice to; talks to; sexual interactions

• Affiliations

Content and Coding Matter!

• Each relation yields a different structure and has different effects

• In real data, more then one relation should be studied.

• Coding: – What constitutes an edge?– How to convert interview data into graph data?

Example

Problem Reformulation

Graph Theoretic Concepts

• Consists of a collection of nodes and lines

• Lines also called “ties” or “edges”• Nodes occasionally called “agents” or

“actors”

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Directed and Undirected Ties

• Undirected relations• Attended meeting with...• Communicated with...• Friend of...

• Directed relations• Represent flows or subordination• “Lends money to”, “teacher Of”

• Problem - • Ties that should be symmetric can be measured as non-

symmetric due to measurement error• Friendship relations are not always reciprocal

Tie Strength

• We can attach values to ties, representing quantitative attributes• Strength of relationship• Frequency of communication• Information capacity/bandwidth• Physical distance

• Such graph is called “weighted graph”

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Node Degree

• Degree of a node is a number of lines that connect it to other nodes

• Degree can be interpreted as

• measure of power or importance of a node

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• measure of workload

• In directed graphs:

• indegree: number of incoming edges

• outdegree: number of outgoing edges

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Graph Density

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Components

• Maximal sets of nodes in which every node can reach every other by some path

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Walks, Trails, Paths

• Walk = a sequence of nodes that can be visited by following edges

• Trail = walk with no repeated lines

• Path = walk with no repeated node

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Path Length & Distance

• Length of path = number of links

• Length of shortest path between two nodes = distance or “geodesic”

• Longest geodesic between any two nodes

• = graph diameter

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Cutpoints• Nodes, if deleted, would disconnect the

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• Cutset = set of nodes required to keep a graph connected

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• Local bridge: connects nodes that otherwise would be far removed

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Centralization

• Degree to which network revolves around a single node

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Next Time

• Centrality and Power in Social Networks

• Identification of Key Players