Download - 2 Graph Theory
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Concepts of Graph Theory
Social Networks; Lecture 2
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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
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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
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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
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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?
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Example
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Problem Reformulation
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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
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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”
4
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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
• or
• measure of workload
• In directed graphs:
• indegree: number of incoming edges
• outdegree: number of outgoing edges
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Degree Distribution
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Graph Density
• Defined as ratio of number of edges in the graph to the total POSSIBLE number of edges:
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Density and Network Survival: Help with rice harvest
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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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Seven Bridges of Königsberg
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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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Bridges• An edge, if removed, would disconnect the
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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