Measurment Social and Economic Networks MohammadAmin Fazli Social - - PowerPoint PPT Presentation

Graph Theory and Network Measurment

Social and Economic Networks

MohammadAmin Fazli

Social and Economic Networks 1

ToC

• Network Representation
• Basic Graph Theory Definitions
• (SE) Network Statistics and Characteristics
• Some Graph Theory
• Readings:
• Chapter 2 from the Jackson book
• Chapter 2 from the Kleinberg book

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Network Representation

• N = {1,2,…,n} is the set of nodes (vertices)
• A graph (N,g) is a matrix [gij]n×n where gij represents a link (relation,

edge) between node i and node j

• Weighted network: 𝑕𝑗𝑘 ∈ 𝑆
• Unweighted network: 𝑕𝑗𝑘 ∈ {0,1}
• Undirected network: 𝑕𝑗𝑘 = 𝑕𝑘𝑗

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Network Representation

• Edge list representation: 𝑕 = 12, 23
• Edge addition and deletion: g+ij, g-ij
• Network isomorphism between (N, g) and (N’, g’): ∃𝑔:𝑂→𝑂′𝑕𝑗𝑘 =

𝑕𝑔 𝑗 𝑔(𝑘)

′

• (N’,g’) is a subnetwork of g’ if 𝑂′ ⊆ 𝑂, 𝑕′ ⊆ 𝑕
• Induced (restricted graphs): 𝑕 𝑇 𝑗𝑘 = 𝑕𝑗𝑘 𝑗𝑔 𝑗 ∈ 𝑇, 𝑘 ∈ 𝑇

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Path and Cycles

• A Walk is a sequence of edges connecting a sequence of nodes

𝑋 = 𝑗1𝑗2, 𝑗2𝑗3, 𝑗3𝑗4, … , 𝑗𝑜−1𝑗𝑙 ∀𝑞: 𝑗𝑞𝑗𝑞+1 ∈ 𝑕

• A Path is a walk in which no node repeats
• A Cycle is a path which starts and ends at the same node

𝑗𝑙 = 𝑗1

• The number of walks between two nodes:

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Components & Connectedness

• (N,g) is connected if every two nodes in g are connected by some path.
• A component of a network (N,g) is a non-empty subnetwork (N’,g’) which is
• (N’,g’) is connected
• If 𝑗 ∈ 𝑂′ and 𝑗𝑘 ∈ 𝑕 then 𝑘 ∈ 𝑂′and 𝑗𝑘 ∈ 𝑕′
• Strongly connectivity and strongly connected components for directed

graphs.

• C(N,g) = C(g) = set of g’s connected components
• The link ij is a bridge iff g-ij has more components than g
• Giant component is a component which contains a significant fraction of

nodes.

• There is usually at most one giant component

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Special Kinds of Graphs

• Star:
• Complete Graph:

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Special Kinds of Graphs

• Tree: a connected network with no cycle
• A connected network is a tree iff it has n-1 links
• A tree has at least two leaves
• In a tree, there is a unique path between any pair of nodes
• Forest: a union of trees
• Cycle: a connected graph with n edges in which the degree of every

node is 2.

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Neighborhood

• 𝑂𝑗 𝑕 = 𝑘: 𝑕𝑗𝑘 = 1
• 𝑂𝑗

2 𝑕 = 𝑂𝑗 𝑕 ∪

𝑘∈𝑂𝑗 𝑕 𝑂

𝑘 𝑕

• 𝑂𝑗

𝑙 𝑕 = 𝑂𝑗(𝑕) ∪

𝑘∈𝑂𝑗 𝑕 𝑂

𝑘 𝑙−1 𝑕

• 𝑂

𝑇 𝑙 𝑕 = 𝑗∈𝑇 𝑂𝑗 𝑙

• Degree: 𝑒𝑗 𝑕 = #𝑂𝑗(𝑕)
• For directed graphs out-degree and in-degree is defined

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Degree Distribution

• Degree distribution of a network is a description of relative

frequencies of nodes that have different degrees.

• P(d) is the fraction of nodes that have degree d under the degree

distribution P.

• Most of social and economical networks have scale-free degree

distribution

• A scale-free (power-law) distribution P(d) satisfies:

𝑄 𝑒 = cd−𝛿

• Free of Scale: P(2) / P(1) = P(20)/P(10)

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Degree Distribution

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Degree Distribution

• Scale-free distributions have

fat-tails

• For large degrees the number of

nodes that degree is much more than the random graphs.

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log 𝑄 𝑒 = log 𝑑 − 𝛿log(𝑒)

Diameter & Average Path Length

• The distance between two nodes is the length of the shortest path

between them.

• The diameter of a network is the largest distance between any two

nodes.

• Diameter is not a good measure to path lengths, but it can work as an

upper-bound

• Average path length is a better measure.

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Diameter & Average Path Length

• The tale of Six-degrees of

Separation

• The diameter of SENs is 6!!!
• Based on Milgram’s

Experiment

• The true story:
• The diameter of SENs may be

high

• The average path length is low

[𝑃(log 𝑜 )]

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Diameter & Average Path Length

• The distance

distribution in graph of all active Microsoft Instant Messenger user accounts

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Cliquishness & Clustering

• A clique is a maximal complete subgraph of a given network (𝑇 ⊆ 𝑂 ,

𝑕 𝑇 is a complete network and for any 𝑗 ∈ 𝑂 ∖ 𝑇: 𝑕 𝑇∪ 𝑗 is not complete.

• Removing an edge from a network may destroy the whole clique

structure (e.g. consider removing an edge from a complete graph).

• An approximation: Clustering coefficient,
• This is the overall clustering coefficient

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Cliquishness & Clustering

• Individual Clustering Coefficient for node i:
• Average Clustering Coefficient:
• These values may differ

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Cliquishness & Clustering

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Cliquishness & Clustering

• Average clustering goes to 1
• Overall clustering goes to 0

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Transitivity

• Consider a directed graph g, one can keep track of percentage of

transitive triples:

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Centrality

• Centrality measures show how much central a node is.
• Different measures for centrality have been developed.
• Four general categories:
• Degree: how connected a node is
• Closeness: how easily a node can reach other nodes
• Betweenness: how important a node is in terms of connecting other nodes
• Neighbors’ characteristics: how important, central or influential a node’s

neighbors are

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Degree Centrality

• A simple measure:

𝑒𝑗 𝑕 𝑜 − 1

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Closeness Centrality

• A simple measure:

𝑘≠𝑗 𝑚 𝑗, 𝑘 𝑜 − 1

−1

• Another measure (decay centrality)

𝑘≠𝑗

𝜀𝑚(𝑗,𝑘)

• What does it measure for 𝜀 = 1?

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Betweenness Centrality

• A simple measure:

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Neighbor-Related Measures

• Katz prestige:

𝑄

𝑗 𝐿 𝑕 = 𝑘≠𝑗

𝑕𝑗𝑘 𝑄

𝑘 𝐿(𝑕)

𝑒𝑘 𝑕

• If we define

𝑕𝑗𝑘 =

𝑕𝑗𝑘 𝑒𝑘 𝑕 , we have

𝑄𝐿 𝑕 = 𝑕𝑄𝐿 𝑕

• r

𝐽 − 𝑕 𝑄𝐿 𝑕 = 0

• Calculating Katz prestige reduces to finding the unit eigenvector.

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Eigenvectors & Eigenvalues

• For an 𝑜 × 𝑜 matrix T an eigenvector v is a 𝑜 × 1 vector for which

∃𝜇 𝑈𝑤 = 𝜇𝑤

• Left-hand eigenvector:

𝑤𝑈 = 𝜇𝑤

• Perron-Ferobenius Theorem: if T is a non-negative column stochastic

matrix (the sum of entries in each column is one), then there exists a right-hand eigenvector v and has a corresponding eigenvalue 𝜇 = 1.

• The same is true for right-hand eigenvectors and row stochastic

matrixes.

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Eigenvectors & Eigenvalues

• How to calculate:

𝑈 − 𝜇𝐽 𝑤 = 0

• For this equation to have a non-zero solution v, T − 𝜇𝐽 must be

singular (non-invertible): det 𝑈 − 𝜇𝐽 = 0

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Neighbor-Related Measures

• Computing Katz prestige

for the following

• Katz prestige ≈ degree!
• Not interesting on

undirected networks, but interesting on directed networks.

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Neighbor-Related Measures

• To solve the problem: Eigenvector Centrality: 𝜇𝐷𝑗

𝑓 𝑕 = 𝑘 𝑕𝑗𝑘𝐷 𝑘 𝑓 𝑕

𝜇𝐷𝑓 𝑕 = 𝑕𝐷𝑓(𝑕)

• Katz2: 𝑄𝐿2 𝑕, 𝑏 = 𝑏𝑕𝐽 + 𝑏2𝑕2𝐽 + 𝑏3𝑕3𝐽 + ⋯

𝑄𝐿2 𝑕, 𝑏 = 1 + 𝑏𝑕 + 𝑏2𝑕2 + ⋯ 𝑏𝑕𝐽 = 𝐽 − 𝑏𝑕 −1𝑏𝑕𝐽

• Bonacich:

𝐷𝑓

𝐶 𝑕, 𝑏, 𝑐 = 1 − 𝑐𝑕 −1𝑏𝑕𝐽

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Final Discussion about Centrality Measures

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Matching

• A matching is a subset of edges with no common end-point.
• Finding the maximum matching is an interesting problem specially in

bipartite graphs (recall Matching Markets)

• A bipartite network (N,g) is one for which N can be partitioned into two sets A

and B such that each edge in g resides between A and B.

• A perfect matching infects all vertices.
• Philip-Hall Theorem: For a bipartite graph (N,g), there exists a

matching of a set 𝐷 ⊆ 𝐵, if and only if ∀𝑇⊆𝐷 𝑂𝑇 𝑕 ≥ 𝑇 Proof: see the whiteboard.

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Set Covering and Independent Set

• Independent Set: a subset of nodes 𝐵 ⊆ 𝑊 for which for each 𝑗, 𝑘 ∈ 𝐵, 𝑗𝑘 ∉

𝑕

• Consider two graphs (N,g) and (N,g’) such that 𝑕 ⊂ 𝑕′.
• Any independent set of g’ is an independent set of g.
• If 𝑕 ≠ 𝑕′, there exists an independent sets of g that are not independent set of g’.
• Free-rider game on networks:
• Each player buy the book or he can borrow the book freely from one of the book
• wners in his neighborhood.
• Indirect borrowing is not permitted.
• Each player prefer paying for the book over not having it.
• The equilibrium is where the nodes of a maximal independent set pays for the book.

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Coloring

• Example: We have a network of researchers in which an edge

between node i and j means i or j wants to attends the others

• presentation. How many time slots are needed to schedule all the

presentations?

• In each time slot, we should color the vertices in a way no two

neighboring nodes get the same colors: The Coloring Problem.

• The minimum number of colors needed colors: the chromatic number
• Many number of results, most famous is the 4-color problem: Every

planar graph can be colored with 4 colors.

• A planar graph is a graph which can be drawn in a way that no two edges

cross each other.

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Coloring

• Intuition: The 6-color problem:
• Any planar graph can be colored with 6 colors.
• Proof sktech:
• Euler formula: v+f = e+2
• 𝑓 ≤ 3𝑤 − 6
• 𝜀 ≤ 5
• Recursive coloring
• Four color is needed:

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Eulerian Tours & Hamilton Cycles

• Euler Tour: a closed walk which pass through all edges
• Euler theorem: A connected network g has a closed walk that involves

each link exactly once if and only if the degree of each node is even.

• Proof sketch:
• Induction on the number of edges

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Eulerian Tours & Hamilton Cycles

• Hamilton Cycle: a cycle that passes through all vertices
• Dirac theorem: If a network has 𝑜 ≥ 3 nodes and each node has

degree of at least n/2, then the network has a Hamilton cycle.

• Proof sketch:
• Graph is connected
• Consider the longest path and prove it is in fact a cycle
• Consider a node outside this cycle

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Eulerian Tours and Hamilton Cycles

• Chvatal Theorem: Order the nodes of a network of 𝑜 ≥ 3 nodes in

increasing order of their degrees, so that node 1 has the lowest degree and node n has the highest degree. If the degrees are such that 𝑒𝑗 ≤ 𝑗 for some 𝑗 < 𝑜/2 implies 𝑒𝑜−𝑗 ≥ 𝑜 − 𝑗, then the network has a Hamilton cycle.

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