graph theory and network measurment
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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
• Some Graph Theory
• Readings:• Chapter 2 from the Jackson book
• Chapter 2 from the Kleinberg book
Social and Economic Networks 2
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: 𝑔𝑖𝑗 ∈ 𝑅
• Unweighted network: 𝑔𝑖𝑗 ∈ {0,1}
• Undirected network: 𝑔𝑖𝑗 = 𝑔𝑗𝑖
Social and Economic Networks 3
Network Representation
• Edge list representation: 𝑔 = 12, 23
• Edge addition and deletion: g+ij, g-ij
• Network isomorphism between (N, g) and (N’, g’): ∃𝑓:𝑁→𝑁′𝑔𝑖𝑗
= 𝑔𝑓 𝑖 𝑓(𝑗)′
• (N’,g’) is a subnetwork of g’ if 𝑁′ ⊆ 𝑁, 𝑔′⊆ 𝑔
• Induced (restricted graphs): 𝑔 𝑆 𝑖𝑗 = 𝑔𝑖𝑗 𝑖𝑓 𝑖 ∈ 𝑆, 𝑗 ∈ 𝑆
0
Social and Economic Networks 4
Path and Cycles
• A Walk is a sequence of edges connecting a sequence of nodes𝑊 = 𝑖1𝑖2, 𝑖2𝑖3, 𝑖3𝑖4, … , 𝑖𝑛−1𝑖𝑘
∀𝑝: 𝑖𝑝𝑖𝑝+1 ∈ 𝑔
• A Path is a walk in which no node repeats
• A Cycle is a path which starts and ends at the same node𝑖𝑘 = 𝑖1
• The number of walks between two nodes:
Social and Economic Networks 5
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 𝑖 ∈ 𝑁′ and 𝑖𝑗 ∈ 𝑔 then 𝑗 ∈ 𝑁′and 𝑖𝑗 ∈ 𝑔′
• 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
Special Kinds of Graphs
• Star:
• Complete Graph:
Social and Economic Networks 7
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
Neighborhood
• 𝑁𝑖 𝑔 = 𝑗: 𝑔𝑖𝑗 = 1
• 𝑁𝑖2 𝑔 = 𝑁𝑖 𝑔 ∪ 𝑗∈𝑁𝑖 𝑔 𝑁𝑗 𝑔
• 𝑁𝑖𝑘 𝑔 = 𝑁𝑖(𝑔) ∪ 𝑗∈𝑁𝑖 𝑔 𝑁𝑗
𝑘−1 𝑔
• 𝑁𝑆𝑘 𝑔 = 𝑖∈𝑆𝑁𝑖
𝑘
• Degree: 𝑑𝑖 𝑔 = #𝑁𝑖(𝑔)
• For directed graphs out-degree and in-degree is defined
Social and Economic Networks 9
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:𝑃 𝑑 = cd−𝛾
• Free of Scale: P(2) / P(1) = P(20)/P(10)
Social and Economic Networks 10
Degree Distribution
Social and Economic Networks 11
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 𝑃 𝑑 = log 𝑐 − 𝛾log(𝑑)
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
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 [𝑂(log 𝑛 )]
Social and Economic Networks 14
Diameter & Average Path Length
• The distance distribution in graph of all active Microsoft Instant Messenger user accounts
Social and Economic Networks 15
Cliquishness & Clustering
• A clique is a maximal complete subgraph of a given network (𝑆 ⊆ 𝑁, 𝑔 𝑆 is a complete network and for any 𝑖 ∈ 𝑁 ∖ 𝑆: 𝑔 𝑆∪ 𝑖 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
Cliquishness & Clustering
• Individual Clustering Coefficient for node i:
• Average Clustering Coefficient:
• These values may differ
Social and Economic Networks 17
Cliquishness & Clustering
Social and Economic Networks 18
Cliquishness & Clustering
• Average clustering goes to 1
• Overall clustering goes to 0
Social and Economic Networks 19
Transitivity
• Consider a directed graph g, one can keep track of percentage of transitive triples:
Social and Economic Networks 20
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
Degree Centrality
• A simple measure:𝑑𝑖 𝑔
𝑛 − 1
Social and Economic Networks 22
Closeness Centrality
• A simple measure:
𝑗≠𝑖 𝑙 𝑖, 𝑗
𝑛 − 1
−1
• Another measure (decay centrality)
𝑗≠𝑖
𝛿𝑙(𝑖,𝑗)
• What does it measure for 𝛿 = 1?
Social and Economic Networks 23
Betweenness Centrality
• A simple measure:
Social and Economic Networks 24
Neighbor-Related Measures
• Katz prestige:
𝑃𝑖𝐾 𝑔 =
𝑗≠𝑖
𝑔𝑖𝑗𝑃𝑗𝐾(𝑔)
𝑑𝑗 𝑔
• If we define 𝑔𝑖𝑗 =𝑔𝑖𝑗
𝑑𝑗 𝑔, we have
𝑃𝐾 𝑔 = 𝑔𝑃𝐾 𝑔
or
𝐼 − 𝑔 𝑃𝐾 𝑔 = 0
• Calculating Katz prestige reduces to finding the unit eigenvector.
Social and Economic Networks 25
Eigenvectors & Eigenvalues
• For an 𝑛 × 𝑛 matrix T an eigenvector v is a 𝑛 × 1 vector for which
∃𝜆 𝑇𝑣 = 𝜆𝑣
• Left-hand eigenvector:𝑣𝑇 = 𝜆𝑣
• 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 𝜆 = 1.
• The same is true for right-hand eigenvectors and row stochastic matrixes.
Social and Economic Networks 26
Eigenvectors & Eigenvalues
• How to calculate:𝑇 − 𝜆𝐼 𝑣 = 0
• For this equation to have a non-zero solution v, T − 𝜆𝐼 must be singular (non-invertible):
det 𝑇 − 𝜆𝐼 = 0
Social and Economic Networks 27
Neighbor-Related Measures
• Computing Katz prestige for the following
• Katz prestige ≈ degree!
• Not interesting on undirected networks, but interesting on directed networks.
Social and Economic Networks 28
Neighbor-Related Measures
• To solve the problem: Eigenvector Centrality: 𝜆𝐶𝑖𝑒 𝑔 = 𝑗 𝑔𝑖𝑗𝐶𝑗
𝑒 𝑔
𝜆𝐶𝑒 𝑔 = 𝑔𝐶𝑒(𝑔)
• Katz2: 𝑃𝐾2 𝑔, 𝑎 = 𝑎𝑔𝐼 + 𝑎2𝑔2𝐼 + 𝑎3𝑔3𝐼 + ⋯
𝑃𝐾2 𝑔, 𝑎 = 1 + 𝑎𝑔 + 𝑎2𝑔2 +⋯ 𝑎𝑔𝐼 = 𝐼 − 𝑎𝑔 −1𝑎𝑔𝐼
• Bonacich: 𝐶𝑒𝐵 𝑔, 𝑎, 𝑏 = 1 − 𝑏𝑔 −1𝑎𝑔𝐼
Social and Economic Networks 29
Final Discussion about Centrality Measures
Social and Economic Networks 30
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 𝐶 ⊆ 𝐴, if and only if
∀𝑆⊆𝐶 𝑁𝑆 𝑔 ≥ 𝑆
Proof: see the whiteboard.
Social and Economic Networks 31
Set Covering and Independent Set
• Independent Set: a subset of nodes 𝐴 ⊆ 𝑉 for which for each 𝑖, 𝑗 ∈ 𝐴, 𝑖𝑗∉ 𝑔
• Consider two graphs (N,g) and (N,g’) such that 𝑔 ⊂ 𝑔′. • Any independent set of g’ is an independent set of g.
• If 𝑔 ≠ 𝑔′, 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
owners 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 32
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 33
Coloring
• Intuition: The 6-color problem:• Any planar graph can be colored with 6 colors.
• Proof sktech:• Euler formula: v+f = e+2• 𝑒 ≤ 3𝑣 − 6
• 𝛿 ≤ 5
• Recursive coloring
• Four color is needed:
Social and Economic Networks 34
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 involveseach 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 35
Eulerian Tours & Hamilton Cycles
• Hamilton Cycle: a cycle that passes through all vertices
• Dirac theorem: If a network has 𝑛 ≥ 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 36
Eulerian Tours and Hamilton Cycles
• Chvatal Theorem: Order the nodes of a network of 𝑛 ≥ 3 nodes inincreasing 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 𝑑𝑖 ≤ 𝑖 for some 𝑖 < 𝑛/2 implies 𝑑𝑛−𝑖 ≥ 𝑛 − 𝑖, then the network has a Hamilton cycle.
Social and Economic Networks 37
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