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Statistical Analysis of Network Data with RNetwork Cohesion & Graph Partitioning

Kim Seonghyeon

April 14, 2017

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 1 / 27

Network Cohesion

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 2 / 27

subgraph & censuses

cliqueclique: Complete subgraphmaximal clique: A clique that is not a subset of a larger clique

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 3 / 27

subgraph & censuses

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Figure 1: karateKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 4 / 27

subgraph & censuses

Table 1: number of clique

1 2 3 4 5count 34 78 45 11 2

Table 2: number of maximal clique

2 3 4 5count 11 21 2 2

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subgraph & censuses

core & corenessk-core: weakened notion of cliqueA subgraph of G for which all vertex degrees are at least k.No other subgraph obeying the same condition contains it. (i.e., it ismaximal in this property)coreness: coreness(v) = max{k|H is k-core, v ∈ VH}

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subgraph & censuses

Figure 2: visualization with corenessKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 7 / 27

subgraph & censuses

Censuses (directed graph)mutual:Cmut = {{u, v} ⊂ VG |(u, v), (v , u) ∈ EG}asymmetric:Casym = {{u, v} ⊂ VG |(u, v) ∈ EG} \ Cmutnull:Cnull = {{u, v}|{u, v} ⊂ VG} \ (Cmut ∪ Casym)

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 8 / 27

subgraph & censuses## aidsblog

## $v## [1] 146#### $e## [1] 183#### $mut## [1] 3#### $asym## [1] 177#### $null## [1] 10405

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 9 / 27

Density and Related Notions of Relative Frequency

DensityDensity:

den(H) = |EH ||VH |(|VH | − 1)/2

*In the case that G is a directed graph, the denominator is replaced by|VH|(|VH| − 1).

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 10 / 27

Density and Related Notions of Relative Frequency

clustering coefficientglobal clustering coefficient:

clT (G) = 3τ∆(G)τ3(G)

local clustering coefficient:

cl(v) = τ∆(v)τ3(v)

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 11 / 27

Density and Related Notions of Relative Frequency

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Figure 3: transitivityKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 12 / 27

Density and Related Notions of Relative Frequency

reciprocity (directed graph)type 1:

rec1(G) = |Cmut ||Cmut ∪ Casym|

type 2:

rec2(G) = 2|Cmut ||EG |

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 13 / 27

Density and Related Notions of Relative Frequency

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Figure 4: reciprocityKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 14 / 27

Connectivity, Cuts, and Flows

ConnectivityA graph G is said to be connected if every vertex is reachable fromevery other.Connected component of a graph is a maximally connected subgraph.diameter: length of the longest path.

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Connectivity, Cuts, and Flows

k-vertex-connectedA graph G is called k-vertex-connected if(i) the number of vertices Nv > k(ii) the removal of any subset of vertices X ⊂ V of cardinality |X | < kleaves a subgraph that is connected.connectivity: connectivity(G) = max{k|G is k-vertex-connected }

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Graph Partitioning

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Graph Partition

Graph Partitionpartition: C = {C1, ...,CK}, partition of the vertex set VGE (Ck ,Ck′): edges connecting vertices in Ck to vertices in Ck

′

We want to seek partition C where E (Ck ,Ck′) is relatively small in sizecompared to the set E (Ck ,Ck)

Kim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 18 / 27

Hierarchical Clustering

modularity

eij = |E (Ci ,Cj)|2|E | , ai =

K∑j=1

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mod(C) =K∑

i=1(eii − ai

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mod(C) = 0 if eij = aiajmod(C) is large if ∑K

i=1 eii = 1

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Hierarchical Clustering

## fraction1

## 1 2 3 sum## 1 0.04 0.04 0.12 0.2## 2 0.04 0.04 0.12 0.2## 3 0.12 0.12 0.36 0.6## sum 0.20 0.20 0.60 1.0

## fraction2

## 1 2 3 sum## 1 0.2 0.0 0.0 0.2## 2 0.0 0.2 0.0 0.2## 3 0.0 0.0 0.6 0.6## sum 0.2 0.2 0.6 1.0

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Hierarchical Clustering

Hierarchical methodsagglomerative: begin with partition {{v1}, ..., {vNv}}divisive: begin with partition {V }

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Hierarchical Clustering

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Figure 5: Agglomerative ClusteringKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 22 / 27

Spectral Partitioning

graph Laplaciangraph Laplacian: L = D − A, where A is adjacency matrix andD = diag [(dv )]λ1 ≤ ... ≤ λNv are the eigenvalues of L.graph G will consist of K connected components if and only ifλ1(L) = ··· = λK (L) = 0 and 0 < λK+1.

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Spectral Partitioning

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Figure 6: Agglomerative ClusteringKim Seonghyeon Statistical Analysis of Network Data with R April 14, 2017 24 / 27

Spectral Partitioning

## [1] 0 0 0 1 2 2 2 2 3 3 4 5

## A B C D E F G H I J K L## 1 0.00 0.00 0.00 0.00 0.00 0.5 0.5 0.5 0.5 0.00 0.00 0.00## 2 0.00 0.00 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.58 0.58 0.58## 3 0.45 0.45 0.45 0.45 0.45 0.0 0.0 0.0 0.0 0.00 0.00 0.00

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Spectral Partitioning

spectral bisectionIf λ2(L) is close to zero, we might expect that there is good candidatefor bisection.partition vertices by separating them according to the sign of theirentries in the corresponding eigenvector x2S = {v ∈ V : x2(v) ≥ 0}, S̄ = {v ∈ V : x2(v) < 0}

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Spectral Partitioning

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Figure 7: Agglomerative Clustering

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