1 / 40

# Network Analysis - PowerPoint PPT Presentation

Network Analysis. Max Hinne mhinne@sci.ru.nl. 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Network Analysis' - remy

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

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

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 cB and cEB

• Very strong positive correlation!

Network Analysis

Network Analysis

• What is a cluster?

• Supervised vs. unsupervised

• Partitional vs. hierarchical

Network Analysis

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

Local intelligence in global applications

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)

(2 roles)

(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