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Network Analysis

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Network Analysis

Max Hinne

mhinne@sci.ru.nl

Network Analysis

- 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

- LIGA project

Network Analysis

Network Analysis

- V = vertices, N = |V|
- A = arcs, M = |A|

(x points to y)

Network Analysis

- Neighborhood:
- Degree:
- Path:
Similar concepts for undirected graphs G=(V,E)

Network Analysis

1.

2.

3.

Models for these graphs by:

Erdős-Renyi (1959)

Tsvetovat-Carley (2005)

Barabási-Albert (1999)

Network Analysis

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

Poisson

Power-law (scale-free)

Network Analysis

- 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

- Degree
- Immediate effect

Network Analysis

- Closeness
- ETA of flow to v

cC inverted for visualization

Network Analysis

- Eigenvector
- Influence or risk

Network Analysis

- Betweenness
- Volume of flow/traffic

Network Analysis

- 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

- 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-net: and corresponding edges
- Calculate cB considering only ego(v)
- Let A be the adjacency matrix:

Network Analysis

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

- Very strong positive correlation!

Network Analysis

Network Analysis

- What is a cluster?
- Supervised vs. unsupervised
- Partitional vs. hierarchical

Network Analysis

Cluster adjacency matrix

Cluster adjacency matrix E

Network Analysis

- 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

- 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;

repeat

(i,j) := argmax{∆Q|Ci, Cj ϵ C};

C := C - Cj;

Ci := Ci + Cj;

until |C| = 1

Network Analysis

- 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

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

∆R(C,u) = R(C+u) – R(C)

C := Ø;

v := v0;

repeat

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

Network Analysis

Network Analysis

- For each node v in each global cluster i
- Find the local cluster with the same size
- Average

Network Analysis

- 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

LIGA

Network Analysis

- ‘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

- 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

- 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

- 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

- 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

- 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