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Graph Theory and Network Measurment Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1 ToC Network Representation Basic Graph Theory Definitions (SE) Network Statistics and Characteristics

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Graph Theory and Network Measurment

Social and Economic Networks

Jafar Habibi MohammadAmin Fazli

Social and Economic Networks 1

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SLIDE 2

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

Social and Economic Networks 2

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Network Representation

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

edge) between node i and node j

  • Weighted network: π‘•π‘—π‘˜ ∈ 𝑆
  • Unweighted network: π‘•π‘—π‘˜ ∈ {0,1}
  • Undirected network: π‘•π‘—π‘˜ = π‘•π‘˜π‘—

Social and Economic Networks 3

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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’): βˆƒπ‘”:π‘‚β†’π‘‚β€²π‘•π‘—π‘˜

= 𝑕𝑔 𝑗 𝑔(π‘˜)

β€²

  • (N’,g’) is a subnetwork of g’ if 𝑂′ βŠ† 𝑂, 𝑕′ βŠ† 𝑕
  • Induced (restricted graphs): 𝑕 𝑇 π‘—π‘˜ = π‘•π‘—π‘˜ 𝑗𝑔 𝑗 ∈ 𝑇, π‘˜ ∈ 𝑇

Social and Economic Networks 4

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Path and Cycles

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

𝑋 = 𝑗1𝑗2, 𝑗2𝑗3, 𝑗3𝑗4, … , π‘—π‘œβˆ’1𝑗𝑙 βˆ€π‘ž: π‘—π‘žπ‘—π‘ž+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:

Social and Economic Networks 5

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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 π‘—π‘˜ ∈ 𝑕 then π‘˜ ∈ 𝑂′and π‘—π‘˜ ∈ 𝑕′
  • 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

Social and Economic Networks 6

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Special Kinds of Graphs

  • Star:
  • Complete Graph:

Social and Economic Networks 7

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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.

Social and Economic Networks 8

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Neighborhood

  • 𝑂𝑗 𝑕 = π‘˜: π‘•π‘—π‘˜ = 1
  • 𝑂𝑗

2 𝑕 = 𝑂𝑗 𝑕 βˆͺ

π‘˜βˆˆπ‘‚π‘— 𝑕 𝑂

π‘˜ 𝑕

  • 𝑂𝑗

𝑙 𝑕 = 𝑂𝑗(𝑕) βˆͺ

π‘˜βˆˆπ‘‚π‘— 𝑕 𝑂

π‘˜ π‘™βˆ’1 𝑕

  • 𝑂𝑇

𝑙 𝑕 = π‘—βˆˆπ‘‡ 𝑂𝑗 𝑙

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

Social and Economic Networks 9

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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 = cdβˆ’π›Ώ

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

Social and Economic Networks 10

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Degree Distribution

Social and Economic Networks 11

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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.

Social and Economic Networks 12

log 𝑄 𝑒 = log 𝑑 βˆ’ 𝛿log(𝑒)

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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.

Social and Economic Networks 13

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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 π‘œ )]

Social and Economic Networks 14

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Diameter & Average Path Length

  • The distance

distribution in graph of all active Microsoft Instant Messenger user accounts

Social and Economic Networks 15

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Cliquishness & Clustering

  • A clique is a maximal complete subgraph of a given network (𝑇 βŠ† 𝑂

, 𝑕 𝑇 is a complete network and for any 𝑗 ∈ 𝑂 βˆ– 𝑇: 𝑕 𝑇βˆͺ 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

Social and Economic Networks 16

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Cliquishness & Clustering

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

Social and Economic Networks 17

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Cliquishness & Clustering

Social and Economic Networks 18

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Cliquishness & Clustering

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

Social and Economic Networks 19

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Transitivity

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

transitive triples:

Social and Economic Networks 20

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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

Social and Economic Networks 21

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Degree Centrality

  • A simple measure:

𝑒𝑗 𝑕 π‘œ βˆ’ 1

Social and Economic Networks 22

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Closeness Centrality

  • A simple measure:

π‘˜β‰ π‘— π‘š 𝑗, π‘˜ π‘œ βˆ’ 1

βˆ’1

  • Another measure (decay centrality)

π‘˜β‰ π‘—

πœ€π‘š(𝑗,π‘˜)

  • What does it measure for πœ€ = 1?

Social and Economic Networks 23

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Betweenness Centrality

  • A simple measure:

Social and Economic Networks 24

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Neighbor-Related Measures

  • Katz prestige:

𝑄

𝑗 𝐿 𝑕 = π‘˜β‰ π‘—

π‘•π‘—π‘˜ 𝑄

π‘˜ 𝐿(𝑕)

π‘’π‘˜ 𝑕

  • If we define

π‘•π‘—π‘˜ =

π‘•π‘—π‘˜ π‘’π‘˜ 𝑕 , we have

𝑄𝐿 𝑕 = 𝑕𝑄𝐿 𝑕

  • r

𝐽 βˆ’ 𝑕 𝑄𝐿 𝑕 = 0

  • Calculating Katz prestige reduces to finding the unit eigenvector.

Social and Economic Networks 25

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Eigenvectors & Eigenvalues

  • For an π‘œ Γ— π‘œ matrix T an eigenvector v is a π‘œ Γ— 1 vector for which

βˆƒπœ‡ π‘ˆπ‘€ = πœ‡π‘€

  • Left-hand eigenvector:

π‘€π‘ˆ = πœ‡π‘€

  • 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 πœ‡ = 1.

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

matrixes.

Social and Economic Networks 26

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Eigenvectors & Eigenvalues

  • How to calculate:

π‘ˆ βˆ’ πœ‡π½ 𝑀 = 0

  • For this equation to have a non-zero solution v, T βˆ’ πœ‡π½ must be

singular (non-invertible): det π‘ˆ βˆ’ πœ‡π½ = 0

Social and Economic Networks 27

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Neighbor-Related Measures

  • Computing Katz prestige

for the following

  • Katz prestige β‰ˆ degree!
  • Not interesting on

undirected networks, but interesting on directed networks.

Social and Economic Networks 28

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Neighbor-Related Measures

  • To solve the problem: Eigenvector Centrality: πœ‡π·π‘—

𝑓 𝑕 = π‘˜ π‘•π‘—π‘˜π· π‘˜ 𝑓 𝑕

πœ‡π·π‘“ 𝑕 = 𝑕𝐷𝑓(𝑕)

  • Katz2: 𝑄𝐿2 𝑕, 𝑏 = 𝑏𝑕𝐽 + 𝑏2𝑕2𝐽 + 𝑏3𝑕3𝐽 + β‹―

𝑄𝐿2 𝑕, 𝑏 = 1 + 𝑏𝑕 + 𝑏2𝑕2 + β‹― 𝑏𝑕𝐽 = 𝐽 βˆ’ 𝑏𝑕 βˆ’1𝑏𝑕𝐽

  • Bonacich:

𝐷𝑓

𝐢 𝑕, 𝑏, 𝑐 = 1 βˆ’ 𝑐𝑕 βˆ’1𝑏𝑕𝐽

Social and Economic Networks 29

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Final Discussion about Centrality Measures

Social and Economic Networks 30

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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 βŠ† 𝐡, if and only if βˆ€π‘‡βŠ†π· 𝑂

𝑇 𝑕

β‰₯ 𝑇

Social and Economic Networks 31

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Matching

  • Proof sketch:

Social and Economic Networks 32

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Set Covering and Independent Set

  • Independent Set: a subset of nodes 𝐡 βŠ† π‘Š for which for each 𝑗, π‘˜ ∈ 𝐡, π‘—π‘˜

βˆ‰ 𝑕

  • Consider two graphs (N,g) and (N,g’) such that 𝑕 βŠ‚ 𝑕′.
  • Any independent set of g’ is an independent set of g.
  • If 𝑕 β‰  𝑕′, 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.

Social and Economic Networks 33

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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.

Social and Economic Networks 34

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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 ≀ 3𝑀 βˆ’ 6
  • πœ€ ≀ 5
  • Recursive coloring
  • Four color is needed:

Social and Economic Networks 35

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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

Social and Economic Networks 36

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Eulerian Tours & Hamilton Cycles

  • Hamilton Cycle: a cycle that passes through all vertices
  • Dirac theorem: If a network has π‘œ β‰₯ 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

Social and Economic Networks 37

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Eulerian Tours and Hamilton Cycles

  • Chvatal Theorem: Order the nodes of a network of π‘œ β‰₯ 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 < π‘œ/2 implies π‘’π‘œβˆ’π‘— β‰₯ π‘œ βˆ’ 𝑗, then the network has a Hamilton cycle.

Social and Economic Networks 38