recap descriptive analysis of network graph connectivity · 2014. 10. 9. · recap introduction...

Graph descriptives

Sacha Epskamp

Recap

Introduction

ConnectivityShortest Path Length

Diameter and Density

ClusteringLocal Clustering

Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness

Eigenvector centrality

Weighted andDirected networksShortest Path length

Centrality

References

Descriptive Analysis of Network GraphCharacteristics

Network Analysis: Lecture 3

Sacha Epskamp

University of AmsterdamDepartment of Psychological Methods

16-09-2014

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

A common method for constructing networks is by takingsome measure of similarity or association, sim(u, v)between each pair of nodes u, v ∈ V and either using thatsimilarity as edge weights for a weighted graph:

wuv = sim(u, v)

or only connecting two nodes in an unweighted graph iftheir similarity is not zero:

auv =

{1 if sim(u, v) 6= 00 if sim(u, v) = 0

In the lecture we used the correlation coefficient asassociation measure.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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NeuroticismExtraversionOpennessAgreeablenessConscientiousness

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Cutoff: 0.4Minimum: 0.25 Maximum: 0.77

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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Cutoff: 0.4Minimum: 0.25 Maximum: 0.77

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

To interpret qgraph networks, three values need to beknown:

Minimum Edges with absolute weights under thisvalue are omitted

Cut If specified, splits scaling of width and colorMaximum If set, edge width and color scale such that

an edge with this value would be the widestand most colorful

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

If the one-factor model is true, then the covariancebetween two variables equals the product of factorloadings!

I Correlations are non-zero if and only if factorloadings are non-zero

I The correlation-network of data generated by asingle factors portrays a fully connected cluster ofnodes:

y1

y2

y3

y4

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

If variable i has a stronger covariance than variable j onsome third variable and the one-factor model is true, i hasa stronger covariance than j to all other variables.

y1

y2

y3

y4

In line with one−factor model

y1

y2

y3

y4

Violates one−factor model

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Descriptive Analysis of Network GraphCharacteristics

Network Analysis: Lecture 3

Sacha Epskamp

University of AmsterdamDepartment of Psychological Methods

16-09-2014

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Prevent the outbreak:http://vax.herokuapp.com/

http://vax.herokuapp.com/

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

I How fast will the disease spread?I Connectivity

I What nodes should be vaccinated?I Centrality

I Which parts of the network should be quarantined?I Clustering

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

We will first analyze only the unweighted, undirectedsimple graph G:

G = (V ,E)

With |V | nodes and |E | edges, encoded using |V | × |V |adjacency matrix A:

auv = avu =

{1 if {u, v} ∈ E0 Otherwise

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

The shortest path length between nodes v and u,dist(v ,u), is defined in an unweighted graph as theminimum number of steps you need to take from node vto reach node u:

dist(v ,u) = min (avx + . . .+ ayu)

I Can be computed using Dijkstra’s algorithm (Dijkstra,1959) with weights fixed to 1.

I Commonly referred to as the shortest path lengthor geodesic distance

The mean shortest path length is called the averageshortest path length (APL) and is an important measurefor how well connected a graph is:

APL(G) =

∑v ,u dist(v ,u)

|V | (|V | − 1)/2

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

shortest.paths(G)

## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]## [1,] 0 1 1 1 1 1 2 2## [2,] 1 0 2 2 2 2 3 3## [3,] 1 2 0 2 2 2 3 3## [4,] 1 2 2 0 2 2 3 3## [5,] 1 2 2 2 0 2 1 1## [6,] 1 2 2 2 2 0 1 1## [7,] 2 3 3 3 1 1 0 2## [8,] 2 3 3 3 1 1 2 0## [9,] 2 3 3 3 1 1 2 2## [10,] 2 3 3 3 1 1 2 2## [11,] 2 3 3 3 1 1 2 2## [,9] [,10] [,11]## [1,] 2 2 2## [2,] 3 3 3## [3,] 3 3 3## [4,] 3 3 3## [5,] 1 1 1## [6,] 1 1 1## [7,] 2 2 2## [8,] 2 2 2## [9,] 0 2 2## [10,] 2 0 2## [11,] 2 2 0

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

average.path.length(G)

## [1] 2

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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Disorders usually first diagnosed in infancy, childhood or adolescenceDelirium, dementia, and amnesia and other cognitive disordersMental disorders due to a general medical conditionSubstance−related disordersSchizophrenia and other psychotic disordersMood disordersAnxiety disordersSomatoform disordersFactitious disordersDissociative disordersSexual and gender identity disordersEating disordersSleep disordersImpulse control disorders not elsewhere classifiedAdjustment disordersPersonality disordersSymptom is featured equally in multiple chapters

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

3.5

3

2.5

2

1.5

1

MDE x

DYS

MDE x

AGPH

MDE x

SOP

MDE x

SIP

MDE x

PD

MDE x

APD

DYS X A

GPH

DYS X S

OP

DYS X S

IP

DYS X P

D

DYS X A

PD

AGPH

X S

OP

AGPH

X S

IP

AGPH

X P

D

AGPH

X A

PD

SOP X

SIP

SOP X

PD

SOP X

APD

SIP X

PD

SIP X

APD

PD X

APD

Correlation Average Shortest Path Length

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

The diameter of a graph is its longest shortest pathlength:

diameter(G) = maxu,v

[dist(u, v)]

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

diameter(G)

## [1] 3

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

The density of a graph is the proportion of the presentnumber of edges to the total possible amount of edges:

den(G) =|E |

|V | (|V | − 1)/2

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

1

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

graph.density(G)

## [1] 0.2727

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Are two connected nodes also connected to each other?Or more general, does a graph exhibit cliques?

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

The local clustering coefficient, cl(v), gives for node vthe proportion of neighbors of v that are also connectedto each other.This corresponds for dividing the amount of “triangles” ofwhich node v is part, τ∆(v) to the amount of possibletriangles of which v could be part: τ3(v):

cl(v) =τ∆(v)τ3(v)

τ3(v) also corresponds to the number of triplets in whichnode v is the middle node.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Triangle Triplet

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

1

1

1

1

0.43

1

1

1

1

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

I The local clustering coefficient can also be seen as ameasure for redundancy

I A node with clustering of 1 has connected neighbors.Deleting this node will not hugely change thestructure of a network

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Traditionally, a global clustering coefficient for thewhole graph can be obtained by averaging all the localclustering coefficients:

cl(G) =1|V |

|V |∑i=1

cl(i)

This is an average of averages, and should be properlyweighted to obtain a more informative coefficient:

clT (G) =

∑|V |i=1 τ∆(i)cl(i)τ3(i)

=3τ∆(G)

τ3(G)

This is the more modern clustering coefficient, alsotermed transitivity

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

1

1

1

1

0.43

1

1

1

1

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

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A <- matrix(0,9,9)A[1:5,1:5] <- 1A[5:9,5:9] <- 1

library("igraph")

G4 <- graph.adjacency(A, mode = "undirected", diag = FALSE)

transitivity(G4,"local")

## [1] 1.0000 1.0000 1.0000 1.0000 0.4286 1.0000## [7] 1.0000 1.0000 1.0000

transitivity(G4,"global")

## [1] 0.7895

transitivity(G4,"average")

## [1] 0.9365

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

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References

The qgraph function clustcoef_auto() will alsoreturn the local clustering coefficient

clustcoef_auto(G4)

## clustWS## 1 1.0000## 2 1.0000## 3 1.0000## 4 1.0000## 5 0.4286## 6 1.0000## 7 1.0000## 8 1.0000## 9 1.0000

Also works for any valid input to qgraph and can returnweighted generalizations of the clustering coefficient (notdiscussed in this course).

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Closeness

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References

The famous paper of Watts and Strogatz (1998)—alreadycited 23623 times—describes the “small world” principlethat frequently occurs in natural graphs.

I “Six degrees of separation”I High clustering and low average path length

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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mouseover for friend details

Hbased on data from 190 of 203 friendsL

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Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

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Centrality

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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A graph exhibits a small world structure if it has a muchhigher clustering than a random graph of the samedimensions while still having a low APL.

Graph descriptives

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Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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# Simulate a graph:set.seed(1)G5 <- watts.strogatz.game(1, 100, 5, 0.05)plot(G5)

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Function to compute average path length of comparablerandom graph:

APLr <- function(x){if ("qgraph"%in%class(x)) x <- as.igraph(x)if ("igraph"%in%class(x)) x <- get.adjacency(x)

N=nrow(x)p=sum(x/2)/sum(lower.tri(x))

eulers_constant <- .57721566490153l = (log(N)-eulers_constant)/log(p*(N-1)) +.5l

}

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Function to compute clustering of comparable randomgraph:

Cr <- function(x){if ("qgraph"%in%class(x)) x <- as.igraph(x)if ("igraph"%in%class(x)) x <- get.adjacency(x)

N=nrow(x)p=sum(x/2)/sum(lower.tri(x))

t=(p*(N-1)/N)t

}

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Is there a small world?

# Clustering in graph:transitivity(G5)

## [1] 0.5402

# Clustering in random graph:Cr(G5)

## [1] 0.1

# Average path length in graph:average.path.length(G5)

## [1] 2.867

# Average path length in random graph:APLr(G5)

## [1] 2.249

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Small world index:

(transitivity(G5) / Cr(G5)) /(average.path.length(G5) / APLr(G5))

## [1] 4.237

Higher than 3? There is a Small world!

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Recap

Introduction




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Alternatively, the qgraph function smallworldness canbe used:

smallworldness(G5)

## smallworldness trans_target## 4.97977 0.54016## averagelength_target trans_rnd_M## 2.86747 0.08413## trans_rnd_lo trans_rnd_up## 0.06950 0.10128## averagelength_rnd_M averagelength_rnd_lo## 2.22391 2.20969## averagelength_rnd_up## 2.23838

Also works for any valid input to qgraph()

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

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References

Centrality measures assign numeric values to theimportance of nodes in the graph and answer thequestion “what is the most central node?”.

I DegreeI ClosenessI BetweennessI Eigenvector centrality

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Recap

Introduction




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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

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Centrality

References

The degree of node v , CD(v) is simply the number ofedges connected to node v , which we can compute byeither summing over row v or column v of A:

CD(v) =|V |∑i=1

aiv =

|V |∑j=1

avj

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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degree(G)

## [1] 5 1 1 1 6 6 2 2 2 2 2

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The degree distribution, fd , gives the probability that anode in G has degree d :

fd = P (CD(v) = d)

For a given graph the observed degree distribution cansimply be computed by dividing the number of nodes thathave degree d with the total number of nodes:

fd =# of nodes with degree d

|V |

and can easily be represented with an histogram.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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table(degree(G)) / vcount(G)

#### 1 2 5 6## 0.27273 0.45455 0.09091 0.18182

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Sacha Epskamp

Recap

Introduction




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library("ggplot2")qplot(degree(G), geom = "histogram")

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Sacha Epskamp

Recap

Introduction




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A random graph is a graph in which each edge ispresent with probability p. The degree distribution of arandom graph follows a binomial distribution:

fd =

(|V | − 1

d

)pd(1− p)|V |−1−d

Or Poisson for large graphs:

fd =ρde−ρ

d !

where ρ = |V |p

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

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References

Random graphG2 <- erdos.renyi.game(100, 0.3)plot(G2)

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Random graph

qplot(degree(G2), geom = "histogram")

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Sacha Epskamp

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Introduction




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High School dating

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Scale-free networks

I Many natural networks do not have a binomial orpoisson degree distribution

I These graphs usually have many nodes with a verylow degree and few nodes with a very high degree

I These are termed scale-free networks, and forthese networks a power-law holds approximately trueat least in part.

fd ∝∼ d−α

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

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Preferential Attachment

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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G3 <- barabasi.game(50, 1.2, directed=FALSE)plot(G3)

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qplot(degree(G3), geom = "histogram")

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Global Clustering

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Closeness CC(v) defines that a node is central if it is‘close’ to other nodes. This can be computed by takingthe inverse of the sum of all path lengths going from nodev to all other nodes:

CC(v) =1∑|V |

i=1 dist(v , i)

This is only an interesting measure for fully connectedgraphs or components.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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CentralityDegree

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closeness(G)

## [1] 0.06667 0.04167 0.04167 0.04167 0.07143## [6] 0.07143 0.04762 0.04762 0.04762 0.04762## [11] 0.04762

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Betweenness of node v is defined as the sum ofproportions of the number of shortest paths between allpairs of nodes that go through node v :

CB(i) =n∑

i 6=j 6=k∈V

σ(i , j | v)σ(i , j)

Where σ(i , j) is the total number of shortest pathsbetween any two nodes and σ(i , j | v) the amount ofthose paths that go through v .

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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betweenness(G)

## [1] 24.1667 0.0000 0.0000 0.0000 15.0000## [6] 15.0000 0.1667 0.1667 0.1667 0.1667## [11] 0.1667

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Eigenvector centrality states that a node if central if itsneighbors are central, and is recursive:

CEV (v) = α∑{u,v}∈E

CEV (u)

Since A contains only zeroes and ones we can write thisas:

CEV (v) = α

|V |∑i=1

aiv CEV (i)

.We can write this in matrix form:

cEV = αAcEV

, in which:

cEV =

CEV (1)CEV (2)

...CEV (|V |)

Graph descriptives

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Recap

Introduction




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An eigenvector x of matrix A is defined as follows:

Ax = λx

, in which λ is some scalar and called an eigenvalue.Rearranging the expression on the previous slide leads tothis eigenvalue problem with λ = α−1 and cE I = x

AcEV =1α

cEV

This shows CEV (v) to be the v th element of aneigenvector. Since centrality measures are positive,Perron-Frobenius theorem dictates that this should be theeigenvector corresponding to the largest eigenvalue.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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evcent(G)

## $vector## [1] 0.7384 0.2078 0.2078 0.2078 1.0000 1.0000## [7] 0.5629 0.5629 0.5629 0.5629 0.5629#### $value## [1] 3.553#### $options## $options$bmat## [1] "I"#### $options$n## [1] 11#### $options$which## [1] "LA"#### $options$nev## [1] 1#### $options$tol## [1] 0#### $options$ncv## [1] 0#### $options$ldv## [1] 0#### $options$ishift## [1] 1#### $options$maxiter## [1] 3000#### $options$nb## [1] 1#### $options$mode## [1] 1#### $options$start## [1] 1#### $options$sigma## [1] 0#### $options$sigmai## [1] 0#### $options$info## [1] 0#### $options$iter## [1] 1#### $options$nconv## [1] 1#### $options$numop## [1] 5#### $options$numopb## [1] 0#### $options$numreo## [1] 5

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Centrality recap

Degree How well connected is a node?Closeness How easy is it to reach all other nodes from

a node?Betweenness How well does a node connect other

nodes?Eigenvector Centrality How important are the neighbors

of a node?

Graph descriptives

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Recap

Introduction




Global Clustering

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Degree

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207208

insomnia / difficu...psychomotor agitat...

psychomotor retard...

depressed mood

tachycardia / acce...

often easily distr...

is often touchy or...

anxietynausua

sweating / perspir...Weight loss

difficulty concent...

transient visual, ...

fatigue / fatigue ...

increased appetite

Clinically signifi...

delusions

coma

extreme negativism

wei

ght g

ain

Hypersomnia

feelings of worthl...

Loss of appetite

excessive motor ac...peculiarities of v...

mar

kedl

y di

min

ishe

...

echolalia

echopraxia

recurrent thoughts...

disorganized speech

flight of ideas or...

disorganized behav...

inflated self−este...decreased need for...

more talkative tha...

increase in goal−d...

excessive involvem...

A d

istin

ct p

erio

d ...

motoric immobilitypupillary dilation

mutism

derealization

depersonalization

palpitations

vomiting

fear of one or mor...nystagmus / vertic...

pounding heart

sensations of shor...

abdominal distressThe situations are...

chills or hot flus...

There is evidence ...

fear of losing con...

stupor

elevated blood pre... trembling or shaking

ches

t pai

n or

dis

c...

fear of dying

paresthesias

persistent concern...

worry about the im...

a significant chan...

incoordination

memory impairment ...

The person recogni...

Marked and persist...

rambling flow of t...

slur

red

spee

ch

blurring of vision

tremors

Anxiety about bein...

A distinct period ...

often lies to obta...

often bullies, thr...

often initiates ph...

has used a weapon ... has been physicall...

has been physicall...

has stolen while c...

has forced someone...

has deliberately e...

has deliberately d...

has broken into so...

has stolen items o...

often stays out at...

has run away from ...is often truant fr...

The development of...

often fails to giv...

often has difficul...

often does not see...

ofte

n do

es n

ot fo

l...

often has diff icu...

often avoids, disl...

often loses t hing...is often forgetful...

often fidgets with...

often leaves seat ...

often runs about o...


is often "on the g...

often talks excess...

often blurts out a...


often interrupts o...

Excessive anxiety ...

Disturbance of con...

unsteady gait

bradycardia

chills

confusion, seizure...

flat or inappropri...

muscular weakness,...

lowered blood pres...

generalized muscle...

the person experie...

dizziness

lethargy

depressed reflexes

euphoria

Presence of two or...

lacks close friend...

cardiac arrhythmia

piloerection

nervousness

excitement

flushed face

diuresis

gastroint estinal ...

mus

cle

twitc

hing

periods of inexhau...

muscle aches

lacr

imat

ion

or r

hi...

diarrhea

yawning

fever

seiz

ures

Focal neurological...

aphasia

apraxia

agnosia disturbance in exe...

numbness or dimini...

ataxia

dysa

rthr

ia

mus

cle

rigid

ity

hyperacusis

Confusion about pe...

The

per

son

finds

i...

mind going blank

mus

cle

tens

ion

odd beliefs or mag...unusual perceptual...

decreased heart rate

vivid, unpleasant ...

suspiciousness or ...

The traumatic even...

transient, stress ...

Marked avoidance o...

ideas of reference

A change in cognit...

development of a p...

Recent ingestion o...

autonomic hyperact...

increased hand tre...

grand mal seizures

feelings of hopele...

dissociative amnesia

behavior or appear...

frantic efforts to...

a pa

ttern

of u

nsta

...

impulsivity in at ...

recurrent suicidal...

affective instabil...

chronic fee lings ...

failure to conform...deceitfulness

Per

sist

ent a

void

an...

often loses temperoften argues with ...

ofte

n de

liber

atel

y...

often blames other...

is often angry and...

is often spiteful ...

hypervigilance

exaggerated startl...

impulsivity or fai...

reckless disregard...

consistent irrespo...

lack of remorse

often actively def... Dysphoric mood

drowsiness

neither desires no...

has little, if any...

takes pleasure in ...

Pupillary constric...

catalepsy

almost always choo...

appears indifferen...

conjunctival injec...

dry

mou

th

Relative unrespons...

No detailed dream ...

Rec

urre

nt e

piso

des.

..

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Closeness

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201

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203

204

205

206

207208

insomnia / difficu...psychomotor agitat...

increased appetitedepressed mood


nausua




Weight loss

weight gain

sweating / perspir...is often touchy or...

anxiety

tach

ycar

dia

/ acc

e...

vomiting


pupillary dilation

delusions

disorganized speech

diso

rgan

ized

beh

av...

extreme negativism

Hypersomnia

feel

ings

of w

orth

l...

echolalia

echopraxia

mutism

com

a

palpitations

incoordination

mus

cle

ache

s

diarrheayawning

fever chills

mus

cula

r w

eakn

ess,

...

tremors

excitement

flush

ed fa

ce

stupor

leth

argy

ataxia

unexpected travel ...

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Betweenness

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110111

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116117

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132133

134135

136137

138139

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142143

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151152

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154155

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177178

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194195

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201

202

203

204

205

206

207208

is often touchy or...often easily distr...

depressed mood

insomnia / difficu...

anxietypsychomotor agitat...



naus

ua

sweating / perspir...flat or inappropri...

increased appetitefear of one or mor...



rambling flow of t...

pupillary dilation



Weight loss

coma

disorganized speech

disorganized behav...

extreme negativism


derealization

depersonalization

delusions


vom

iting

weight gainvivid, unpleasant ...

Con

fusi

on a

bout

pe.

..



Excessive anxiety ...

palpitations

elevated blood pre...


stupor

nystagmus / vertic...

incoordination

blurring of vision

tremors

odd beliefs or mag...

unusual perceptual...

excessive motor ac...

peculiarities of v... echolalia

echopraxia

Hypersomnia

slurred speech


aphasia

apraxia

agnosia disturbance in exe...


Loss of appetite

unsteady gait

markedly diminishe...recurrent thoughts...often lies to obta...



has used a weapon ...

has been physicall... has been physicall...

has stolen while c... has forced someone...

has deliberately e...has deliberately d...




has run away from ...

is often truant fr...

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Eigenvector Centrality

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2930

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61

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656667

68

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76

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78

79

8081

82

83

8485

86

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88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103104

105

106

107

108

109

110111

112

113

114

115

116117

118

119

120

121

122

123

124

125

126

127

128

129130

131

132133

134135

136137

138139

140

141

142143

144

145

146

147

148

149

150

151152

153

154155

156

157

158

159

160

161

162

163

164

165

166

167

168

169170

171

172

173174

175

176

177178

179

180

181

182

183

184 185

186

187

188

189

190

191

192

193

194195

196

197

198

199

200

201

202

203

204

205

206

207208

psychomotor agitat...insomnia / difficu...

psychomotor retard...Weight loss

depressed moodtransient visual, ...



increased appetite


wei

ght g

ain


Loss of appetite

Hypersomnia

mar

kedl

y di

min

ishe

...

recurrent thoughts...

decreased need for...

excessive involvem...

more talkative tha...

delusions

echopraxia

echolalia

mot

oric

imm

obili

tymutismA distinct period ...

Presence of two or...


anxietynausua

sweating / perspir...

is often touchy or...


pupillary dilation

vomitingcoma

feel

ings

of h

opel

e...

stupor

decreased heart rate

elev

ated

blo

od p

re...

vivid, unpleasant ...

bradycardia

chills

confusion, seizure...

muscular weakness,...lowered blood pres...

Dysphoric mood

Exc

essi

ve a

nxie

ty ..

.

mind going blank

The person finds i...

muscle tension

incoordination

blurring of vision

tremors

catalepsy

palpitations

grand mal seizures

increased hand tre...

auto

nom

ic h

yper

act..

.


nystagmus / vertic...

ram

blin

g flo

w o

f t...


slurred speech

exci

tem

ent

flushed face

card

iac

arrh

ythm

ia

nervousness

diuresis

mus

cle

twitc

hing

gastroint estinal ...

periods of inexhau...

depersonalization

derealizationyawning

piloerection

fever

lacrimation or rhi...

diarrhea

muscle aches

unsteady gait

hypervigilancePersistent avoidan...

exaggerated startl...

euphoria

leth

argy

generalized muscle...

depressed reflexes

dizziness

fear of one or mor...

paresthesias

chest pain or disc...

fear

of d

ying

The situations are...

sensations of shor...

abdominal distress

fear of losing con...

persistent concern...

pounding heart

a significant chan...

chills or hot flus...

worry about the im...

trembling or shaking

The

per

son

reco

gni..

.


has been physicall... often lies to obta...

has used a weapon ...

has forced someone...

has deliberately e...

has been physicall...

has deliberately d...




has stolen while c...

has run away from ...


is often truant fr...

The

dev

elop

men

t of..

.

Marked and persist...

Anxiety about bein...

aphasia

apraxia

agnosia

disturbance in exe...


flat or inappropri...

often avoids, disl... is often forgetful...


often does not see...

often fails to giv...often has diff icu...

often runs about o...

often loses t hing...

often fidgets with...

often leaves seat ...

often interrupts o...

often does not fol...

is often "on the g...


ofte

n ha

s di

fficu

l...

often talks excess...

often blurts out a...

conjunctival injec...

dry mouth


numbness or dimini...muscle rigidity

seizures

hype

racu

sis

dysarthria ataxia


Recent ingestion o...

unusual perceptual...

odd beliefs or mag...

drowsiness

Pup

illar

y co

nstr

ic...

Marked avoidance o...

The traumatic even...

dissociative amnesia

A change in cognit...

development of a p...

impulsivity or fai...

reckless disregard...

deceitfulness

failure to conform...

lack of remorse

consistent irrespo...

Confusion about pe...

frantic efforts to...

recurrent suicidal...

impulsivity in at ...

affective instabil...

transient, stress ...

a pattern of unsta...

chronic fee lings ...

often argues with ...

often deliberately...

often blames other...

often loses temper

is often angry and...

often actively def...

is often spiteful ...

No detailed dream ...

Recurrent episodes...

Relative unrespons...


behavior or appear...

ideas of reference

suspiciousness or ...

On awakening from ...

almost always choo...has little, if any...

neither desires no...

appears indifferen...

takes pleasure in ...

unexpected travel ...

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

How to win VAX

I Delete nodes that haveI High centralityI Low clustering

I Reduce small-worldnessI Increase the average shortest path length

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

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What is the most central node?

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A weighted graph is a network in which the strength ofconnections can vary. For a graph of n nodes the networkstructure is defined by the |V | × |V | adjacency matrix Asuch that:

aij =

{1 if there is an edge from node i to node j0 otherwise

.Its weights are defined by the |V | × |V | weights matrix Wsuch that:

wij

{= 0 if aij = 0∈ R otherwise

.The diagonals of both A and W are set to zero.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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A graph is undirected only if A and W are symmetric:

A = A>

W = W>

A graph is unweighted only if A is equal to W multipliedby some scalar c:

A = cW

Note that both these cases do not imply that a graph isundirected or unweighted.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

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References

Finally, we define the length of an edge from node i tonode j as the inverse of the absolute weight:

lij =

{1|wij | if i 6= j

0 if i = j

Because the denumerator is always positive, we will takethe limit in the case of a weight of 0:

limwij→0

1|wij |

=∞

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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I In unweighted graphs network descriptives are welldefined

I In weighted graphs this is less the case. There is alot of debate on whether the structure (adjacencymatrix) or the weights (weights matrix) are the mostimportant

I Classically only the weights matrix is analyzedI Opsahl, Agneessens, and Skvoretz (2010) proposes

a set of centrality measures with a tuning parameter,α, to manually assign importance to both matrices

I α of 0 only regards the structure, and α of 1 onlyregards the weights

Graph descriptives

Sacha Epskamp

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Introduction




Global Clustering

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A =

0 1 10 0 10 0 0

W =

0 1 20 0 30 0 0

L =

0 1 1/2∞ 0 1/3∞ ∞ 0

1

2

3

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Sacha Epskamp

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The shortest path length between nodes i and j , d(i , j) isdefined in an unweighted graph as the minimum numberof steps you need to take from node i to node j :

d(i , j) = min(aih + . . .+ ahj

)which can be obtained through Dijkstra’s algorithm(Dijkstra, 1959) with weights fixed to 1.Similarly, for weighted graphs the shortest path length isdefined as the minimum number of distance needed tocross on the graph to reach node i from node j :

d(i , j) = min(

1|wih|

+ . . .+1|whj |

)= min

(lih + . . .+ lhj

)

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Opsahl et al. (2010) propose the following combination:

d(i , j) = min(

lαih + . . .+ lαhj

)Which generalizes to the unweighted form near α = 0and the weighted form if α = 1.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

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Of node i the in-degree k (in)i is the number of incoming

edges and the out-degree k (out)i the number of outgoing

edges:

k (in)i =

n∑j=1

aji

k (out)i =

n∑j=1

aij

In weighted graphs often the node strengths s(in)i and

s(out)i are used:

s(in)i =

n∑j=1

wji

s(out)i =

n∑j=1

wij

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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References

Opsahl et al. (2010) propose the following combination:

C(in)D (i) = k (in)

i ×

(s(in)

i

k (in)i

)α

C(out)D (i) = k (out)

i ×

(s(out)

i

k (out)i

)α

Note that for both:

CD(i) =

{ki if α = 0si if α = 1

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

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Closeness is defined as the inverse of the total shortestdistance from node i to all other nodes in the graph,which is only defined for connected clusters:

CC(i) =

n∑j=1

d(i , j)

−1

Here the α parameter is already used in computing theshortest path lengths.

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

Degree distribution

Closeness

Betweenness



Centrality

References

Betweenness of node i is defined as the sum ofproportions of the number of shortest paths between allpairs of nodes that go through node i :

CB(i) =n∑

i 6=j 6=k

gjk (i)gjk

Where gjk is the total number of shortest paths betweenany two nodes and gjk (i) the amount of those paths thatgo through i .

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Dout

Din

Din

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Global Clustering

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Sacha Epskamp

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Global Clustering

Small-worldness

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Dout

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Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Centrality

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Sacha Epskamp

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Introduction




Global Clustering

Small-worldness

CentralityDegree

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Centrality

References


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Sacha Epskamp

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Global Clustering

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Literature on Blackboard:

I Node centrality in weighted networks: Generalizingdegree and shortest paths

I Collective dynamics of ‘small-world’ networksI The Small World of PsychopathologyI State of the aRt personality research: A tutorial on

network analysis of personality data in R

Graph descriptives

Sacha Epskamp

Recap

Introduction




Global Clustering

Small-worldness

CentralityDegree

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Centrality

References

References I

Dijkstra, E. (1959). A note on two problems in connexionwith graphs. Numerische mathematik, 1(1),269–271.

Opsahl, T., Agneessens, F., & Skvoretz, J. (2010). Nodecentrality in weighted networks: Generalizingdegree and shortest paths. Social Networks,32(3), 245–251.

Watts, D., & Strogatz, S. (1998). Collective dynamics ofsmall-world networks. Nature, 393(6684),440–442.

recap descriptive analysis of network graph connectivity · 2014. 10. 9. · recap introduction...

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