Network analysis
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Network Analysis. Max Hinne [email protected] Social Networks. Networks & Digital Security. Interdisciplinary Combination formal & ‘soft’ interpretation Security in the sense of a detective. Overview. Primer on graph theory Centrality Who is important? Clustering

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Network Analysis

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Network analysis

Network Analysis

Max Hinne

[email protected]

Social networks

Social Networks

Network Analysis

Networks digital security

Networks & Digital Security

  • Interdisciplinary

  • Combination formal & ‘soft’ interpretation

  • Security in the sense of a detective

Network Analysis



  • Primer on graph theory

  • Centrality

    • Who is important?

  • Clustering

    • Who belong together?

  • Detecting & predicting changes

    • LIGA project

      Central theme: global vs. local approaches

Network Analysis

Graph primer

Graph primer

Network Analysis

Graph primer basics

Graph primer - basics

  • V = vertices, N = |V|

  • A = arcs, M = |A|

(x points to y)

Network Analysis

Graph primer concepts

Graph primer - concepts

  • Neighborhood:

  • Degree:

  • Path:

    Similar concepts for undirected graphs G=(V,E)

Network Analysis

Graph primer graph types

Graph primer – graph types




Models for these graphs by:

Erdős-Renyi (1959)

Tsvetovat-Carley (2005)

Barabási-Albert (1999)

Network Analysis

Graph primer degree distributions

Graph primer – degree distributions

Degree distributions: what is the chance a node has degree k?

  • Erdős-Renyi: number of vertices N, each edge occurs with probability p

  • Barabási-Albert: start with a small set of vertices and add new ones. Each new vertex is connected to others with a probability based on their degree


Power-law (scale-free)

Network Analysis

Graph primer small world effect

Graph primer – small world effect

  • Famous experiment by Milgram (1967)

  • Everyone on the world is connected to everyone else in at most 6 steps

  • Social graphs exhibit the ‘small world effect’: the diameter of a social graph scales logarithmically with N

Network Analysis



Network Analysis



  • Importance, control of flow

  • Ranking of most important (control) to least important (control)

Network Analysis

Node centrality measures 1 4

Node centrality measures 1/4

  • Degree

    • Immediate effect

Network Analysis

Node centrality measures 2 4

Node centrality measures 2/4

  • Closeness

    • ETA of flow to v

cC inverted for visualization

Network Analysis

Node centrality measures 3 4

Node centrality measures 3/4

  • Eigenvector

    • Influence or risk

Network Analysis

Node centrality measures 4 4

Node centrality measures 4/4

  • Betweenness

    • Volume of flow/traffic

Network Analysis

Obtaining c b

Obtaining cB

  • Fastest current algorithm by Brandes in O(nm)

  • Solves all shortest paths in one pass

    • For each vertex, consider all d=1 nearest neighbors, then d=2 and so on

    • For each shortest path, store which vertices are on it

    • Derive cB

Network Analysis

Local approach

Local approach

  • No known algorithms calculate cB(v) faster than cB(v) for all v!

  • We only want to rank nodes of interest, not all

  • Local approach

    • Find cB for some specific nodes

    • If we can estimate cB, we can rank relevant nodes

Network Analysis

Ego betweenness

Ego betweenness

  • Ego-net: and corresponding edges

  • Calculate cB considering only ego(v)

  • Let A be the adjacency matrix:

Network Analysis

No direct link between c b and c eb

No direct link between cB and cEB

Red circles + ego form a n+1 node star

Green triangles form an p node complete graph Kp

Red circles + ego form a p+1 node star

Green triangles + ego form an n node complete graph Kn

Network Analysis

Correlation c b and c eb

Correlation cB and cEB

  • Very strong positive correlation!

Network Analysis

Graph clustering

Graph Clustering

Network Analysis

Types of clustering

Types of clustering

  • What is a cluster?

  • Supervised vs. unsupervised

  • Partitional vs. hierarchical

Network Analysis

Clustering quality modularity

Clustering quality – modularity

Cluster adjacency matrix

Cluster adjacency matrix E

Network Analysis

Newman girvan clustering algorithm

Newman & Girvan clustering algorithm

  • Edges that are the most ‘between’ connect large parts of the graph

    • Calculate edge betweenness Aij in n x n matrix A

    • Remove edge with highest score

    • Recalculate edge betweenness for affected edges

    • Goto 2 until no edges remain

  • O(m2n), may be smaller on graphs with strong clustering

Network Analysis

Greedy clustering algorithm

Greedy clustering algorithm

  • Maximize Q to find clustering

  • Greedy approach:

  • Creates a bottom-up dendogram

  • Cut corresponding to maximum Q is optimal clustering

  • Still a costly process, O(n2)

C := V;


(i,j) := argmax{∆Q|Ci, Cj ϵ C};

C := C - Cj;

Ci := Ci + Cj;

until |C| = 1

Network Analysis

Practical applications of social clusters

Practical applications of social clusters

  • Find people related to someone

  • Find out if people belong to the same cluster

  • This does not require a partitioning of the entire network!

Network Analysis

Local modularity

Local modularity

C: cluster

U: universe

B: boundary

C= collection nodes v ∈ V with known link structure

U(C) = all nodes outside C to which nodes from C point: U(C) = {u ∈ V-C|A(C,u) ≠ ∅}

B(C) = all nodes in C with at least one neighbor outside C: B(C) = {b ∈ C|A(b,U) ≠ ∅}

Network Analysis

Local cluster algorithm

Local cluster algorithm

∆R(C,u) = R(C+u) – R(C)

C := Ø;

v := v0;


C := C+v;

v := argmax{R(C+u)|u∈U(C)}

until |C| = k or R ≥ d

Arcs removed from arcs(B(C),V)

Arcs newly added to arcs(B(C),V)

Arcs removed from arcs(B(C),C)

Arcs newly added to arcs(B(C),C)

∆R(C+v4) = 1/3 – 1/4 = 1/12

Network Analysis

Example 1 on zachary s karate club d 0 65

Example 1 on Zachary’s Karate Club (d=0.65)

Network Analysis

Example 2 on zachary s karate club d 0 65

Example 2 on Zachary’s Karate Club (d=0.65)

Network Analysis

Local cluster quality vs global clusters

Local cluster quality vs. global clusters

  • For each node v in each global cluster i

    • Find the local cluster with the same size

    • Average

Network Analysis

Preliminary results on real graphs

Preliminary results on real graphs

  • Experiment too small for real conclusions, but

    • edge vertices ruin the fun,

    • edge betweenness?

  • Usefulness of local approach depends on the seed node

Network Analysis

Local intelligence in global applications


Local intelligence in global applications

Network Analysis

Web graph

Web graph

  • ‘Social’ network of blogs and news sites

  • Most graph models are static, but the Web is highly dynamic

  • Stored copy is infeasible, continuous crawling intractable

  • Change in relevance -> change in link structure

Network Analysis

Node roles

Node roles

  • Frequently recurring sub graphs: motifs

  • Nodes share a role iff there is a permutation of nodes and edges that preserves motif structure

  • On the Web:

Feedback with two mutual dyads

(2 roles)

Uplinked mutual dyad

(2 roles)

Fully connected triad

(1 role)

Network Analysis

Dynamic graphs

Dynamic graphs

  • Changes in relevance cause changes in link structure

  • Changes in specific roles imply changes in other node roles

    • Fanbase links to itself and their authorities

    • Learning relevant links through affiliated sites

    • etc.

  • Relevance decays (half-life λ)

Network Analysis

Liga research questions

LIGA research questions

  • How to model (Web) node relevance ?

  • How does acquired or lost relevance change linkage?

  • How can we predict consequential changes?

  • How can such prediction models be approximated by local incremental algorithms?

  • A. m. o. ...

Network Analysis

Putting it together

Putting it together

  • Networks can be analyzed using an array of tools

  • Network analysis is useful in various disciplines:

    • Information Retrieval

    • Security

  • But also in:

    • Sociology

    • (Statistical) physics

    • Bioinformatics

    • AI

Network Analysis

Most cited literature

Most cited literature

  • Centrality:

    • Borgatti S. P.: Centrality and Network Flow. Social Networks 27 (2005) 55-71

    • Brandes U.: A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2) (2001) 163-177

    • Freeman L. C.: A Set of Measures of Centrality Based on Betweennes. Sociometry 40 (1977) 35-41

  • Clustering:

    • Clauset A.: Finding local community structure in networks. Physics Review E 72 (2005) 026132

    • Girvan M., Newman M. E. J.: Community structure in social and biological networks. PNAS 99(12) (2002) 7821-7826

    • Newman M. E. J.: Fast algorithm for detecting community structure in networks. Physics Review E 69 (2004) 066133

Network Analysis

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