week 03

Node Centralities

Social Network Analysis

Which nodes are important

Motivation

Marriage alliances among leading
Florentine families 15th century.

Determine the most ”important” or ”prominent” actors in the network based on actor location.

Some (obvious) examples

a)

b)

c)

Degree centrality: number of nearest neighbors

Normalized degree centrality

High centrality degree -direct contact with many other actors

Degree Centrality

C_D(i)=k(i)=\sum_j A_{i j}=\sum_j A_{j i}
C_D^*(i)=\frac{1}{n-1} C_D(i)=\frac{k(i)}{n-1}

Closeness centrality: how close an actor to all the other actors in  network

Normalized closeness centrality

High closeness centrality - short communication path to others,  minimal number of steps to reach others

Closeness Centrality

C_C(i)=\frac{1}{\sum_j d(i, j)}
C_C^*(i)=(n-1) C_C(i)=\frac{n-1}{\sum_j d(i, j)}

Betweenness centrality: number of shortest paths going through  the actor σst(i)

Normalized betweenness centrality

Hight betweenness centrality - vertex lies on many shortest paths  Probability that a communication from s to t will go through i

Betweenness Centrality

C_B(i)=\sum_{s \neq t \neq i} \frac{\sigma_{s t}(i)}{\sigma_{s t}}
C_B^*(i)=\frac{2}{(n-1)(n-2)} C_B(i)=\frac{2}{(n-1)(n-2)} \sum_{s \neq t \neq i} \frac{\sigma_{s t}(i)}{\sigma_{s t}}

Bow-tie Structure

Importance of a node depends on the importance of its neighbors  (recursive definition)

Select an eigenvector associated with largest eigenvalue λ = λ1, v = v1

Eigenvector Centrality

\begin{gathered} v_i \leftarrow \sum_j A_{i j} v_j \\ v_i=\frac{1}{\lambda} \sum_j A_{i j} v_j \\ \mathbf{A} \mathbf{v}=\lambda \mathbf{v} \end{gathered}

from www.activenetworks.net

Closeness Centrality

from www.activenetworks.net

Betweenness Centrality

from www.activenetworks.net

Eigenvector Centrality

A) Degree centrality

B) Closeness centrality

D) Eigenvector centrality

C) Betweenness centrality

from Claudio Rocchini

Centrality Examples

Centralization (network measure) - how central the most central  node in the network in relation to all other nodes.

\(C_x \) - one of the centrality measures

\(p_* \)  - node with the largest centrality value

max - is taken over all graphs with the same number of nodes (for degree, closeness and betweenness the most centralizedstructure is the star graph)

Linton Freeman, 1979

Centralization

C_x=\frac{\sum_i^N\left[C_x\left(p_*\right)-C_x\left(p_i\right)\right]}{\max \sum_i^N\left[C_x\left(p_*\right)-C_x\left(p_i\right)\right]}

Directed graph: distinguish between choices made (outgoing  edges) and choices received (incoming edges)

sending - receiving

export - import

cite papers - being cited

Directional Relations

Degree centrality (normalized):

Closeness centrality (normalised):

Betweenness centrality (normalized):

All based on outgoing edges

Centrality Measures

C_D^*(i)=\frac{k^{\text {out }}(i)}{n-1}
C_C^*(i)=\frac{n-1}{\sum_j d(i, j)}
C_B^*(i)=\frac{1}{(n-1)(n-2)} \sum_{s \neq t \neq i} \frac{\sigma_{s t}(i)}{\sigma_{s t}}

Hyperlinks - implicit endorsements

Web graph - graph of endorsements (sometimes reciprocal)

Web as a Graph

”PageRank can be thought of as a model of user behavior. We assume  there is a ”random surfer” who is given a web page at random and keeps  clicking on links, never hitting ”back” but eventually gets bored and starts  on another random page. The probability that the random surfer visits a  page is its PageRank.”

The anatomy of a large-scale hypertextual Web search engine Sergey Brin and Larry Page, 1998 [link]

Pagerank

Random Walk

Random walk on graph

With teleportation

Perron-Frobenius Theorem guarantees existence and uniqueness of  the solution limt→∞ p = π

p_i^{t+1}=\sum_{j \in N(i)} \frac{p_j^t}{d_j^{\text {out }}}=\sum_j \frac{A_{j i}}{d_j^{\text {out }}} p_j
\begin{gathered} \mathbf{P}=\mathbf{D}^{-1} \mathbf{A}, \quad \mathbf{D}_{i i}=\operatorname{diag}\left\{d_i^{\text {out }}\right\} \\ \mathbf{p}^{t+1}=\mathbf{P}^T \mathbf{p}^t \end{gathered}
\mathbf{p}^{t+1}=\alpha \mathbf{P}^T \mathbf{p}^t+(1-\alpha) \frac{\mathbf{e}}{n}
\mathbf{p}=\alpha \mathbf{P}^T \mathbf{p}+(1-\alpha) \frac{\mathbf{e}}{n}

Pagerank

P R\left(p_i\right)=\frac{1-d}{N}+d \sum_{p_j \in M\left(p_i\right)} \frac{P R\left(p_j\right)}{L\left(p_j\right)}
\begin{array}{|c|c|c|c|c|} \hline & \text { it 0 } & \text { it 1 } & \text { it 2 } & \text { PR } \\ \hline \text { A } & 1 / 4 & 1 / 12 & 1.5 / 12 & 1 \\ \hline \text { B } & 1 / 4 & 2.5 / 12 & 2 / 12 & 2 \\ \hline \text { C } & 1 / 4 & 4.5 / 12 & 4.5 / 12 & 4 \\ \hline \text { D } & 1 / 4 & 4 / 12 & 4 / 12 & 3 \\ \hline \end{array}
\begin{array}{lll} \text { 1. GeneRank } & \text { 13. TimedPageRank } & \text { 25. ImageRank } \\ \text { 2. ProteinRank } & \text { 14. SocialPageRank } & \text { 26. VisualRank } \\ \text { 3. FoodRank } & \text { 15. DiffusionRank } & \text { 27. QueryRank } \\ \text { 4. SportsRank } & \text { 16. ImpressionRank } & \text { 28. BookmarkRank } \\ \text { 5. HostRank } & \text { 17. TweetRank } & \text { 29. StoryRank } \\ \text { 6. TrustRank } & \text { 18. TwitterRank } & \text { 30. PerturbationRank } \\ \text { 7. BadRank } & \text { 19. ReversePageRank } & \text { 31. ChemicalRank } \\ \text { 8. ObjectRank } & \text { 20. PageTrust } & \text { 32. RoadRank } \\ \text { 9. ItemRank } & \text { 21. PopRank } & \text { 33. PaperRank } \\ \text { 10. ArticleRank } & \text { 22. CiteRank } & \text { 34. Etc... } \\ \text { 11. BookRank } & \text { 23. FactRank } & \\ \text { 12. FutureRank } & \text { 24. InvestorRank } & \end{array}

Personalized PageRank (PPR) is a measure for node proximity on large graphs. For a pair of nodes s and t, the PPR value πs(t) equals the probability that an αdiscounted random walk from s terminates at t and reflects the importance between s and t in a bidirectional way

Personalized PageRank

Efficient Algorithms for Personalized PageRank Computation: A Survey, 2024, arxiv

\pi_{s} = (1 - \alpha) \mathbf{P} \pi_{s} + \alpha \cdot e_{s}
\pi_{s} = \alpha (\mathbf{I} - (1-\alpha)\mathbf{P})^{-1} e_{s}

is the indicator function. The degree matrix of \(G\) is an \(n \times n\) diagonal matrix \(mathbf{D}\) whose \(i\)-th diagonal entry equals \(d_{\text{out}}(v_i)\). The transition matrix of \(G\) is formalized as \(mathbf{P} = \mathbf{A}^\top \mathbf{D}^{-1}\). To ensure that \(mathbf{P}\) is well-defined, we assume that \(d_{\text{out}}(v) > 0\)

Hubs and Authorities (Hits)

Citation networks. Reviews vs original

research (authoritative) papers

authorities, contain uself information, ai

hubs, contains links to authorities, hi 

Mutual recursion

good authorities referred bu good hubs

good hubs point to good authorities

a_i \leftarrow \sum_j A_{j i} h_j
h_i \leftarrow \sum_j A_{i j} a_j

Hubs and Authorities (Hits)

System of linear equation

Symmetric eigenvalue problem

where eigenvalue λ = (αβ)−1

\begin{aligned} & \mathbf{a}=\alpha \mathbf{A}^T \mathbf{h} \\ & \mathbf{h}=\beta \mathbf{A} \mathbf{a} \end{aligned}
\begin{aligned} & \left(\mathbf{A}^T \mathbf{A}\right) \mathbf{a}=\lambda \mathbf{a} \\ & \left(\mathbf{A A}^T\right) \mathbf{h}=\lambda \mathbf{h} \end{aligned}

Hubs and Authorities (Hits)

Hubs

Authorities

Centralities

Core-Periphery Structure

image from J. Leskovec, K. Lang, 2010

Graph Cores

A k-core is the largest subgraph such that each vertex is connected to at  least k others in subset

Every vertex in k-core has a degree ki ≥ k

(k + 1)-core is always subgraph of k-core

The core number of a vertex is the highest order of a core that contains  this vertex

K-Cores

k-cores: 1:1458, 2:594, 3:142, 4:12, 5:6

k-shells: 1:864-red, 2:452-pale green, 3:130-green, 5:6-blue, 6:6-purple

R:graph.coreness(gcc)

K-Cores

Find 3-core of the given network

Dyads

Text

Dyad is a pair of vertices and possible relational ties between them:

- mutual

- asymmetric

- null (non-existent)

Triads

Triad is a subgraph of three vertices and possible ties between them:

Triad census :16 isomorphism classes

D - down, U - up, T - transitive, C - cyclic.

| mutual diads | assymetric dyads | null dyads |

Network Motifs

Network motifs are recurrent statistically significant subgraphs or patterns  in graphs connected subgraphs that (compare to random network)

Motifs are not induced subgraphs, i.e. they do not contain all the graph  edges between selected vertices.

Network Motifs

Motifs appear in a network more frequently than in a comparable random  network

- calculate the number of occurrences of a sub graph

- evaluate the significance

 

 

For Gt subgraph (motif candidate) of G ,

R - random graph, µ - mean frequency, σ-standard deviatiom

Z_{\text {score }}\left(G^t\right)=\frac{F_G\left(G^t\right)-\mu_R\left(G^t\right)}{\sigma_R\left(G^t\right)}

Network Motifs

Undirected graphs: motifs of size 3 and 4

Network Motifs

Connected triads - motifs of size 3

More complicated motifs:

Ribeiro, 2011, Shen-Orr, 2002

Network Motifs

S. Omidi, 2009

Network Motifs

Ribeiro, 2011, Milo, 2002

Network Motifs

146 nodes, 187 edges

Network Motifs

Network Motifs

Network Motifs

EdMot: An Edge Enhancement Approach for Motif-aware Community Detection, KDD 19, arxiv, git

Network Motifs

References

  1. Centrality in Social Networks. Conceptual Clarification, Linton C. Freeman, Social Networks, 1, 215-239 , 1979​
  2. Power and Centrality: A Family of Measures, Phillip Bonacich, The American Journal of Sociology, Vol. 92, No. 5, 1170-1182, 1987
  3. A new status index derived from sociometric analysis, L. Katz, Psychometrika, 19, 39-43, 1953.
  4. Eigenvector-like measures of centrality for asymmetric relations, Phillip Bonacich, Paulette Lloyd, Social Networks 23, 191-201 , 2001
  5. The PageRank Citation Ranking: Bringing Order to the Web. S. Brin, L. Page, R. Motwani, T. Winograd, Stanford Digital Library Technologies Project, 1998
  6. Authoritative Sources in a Hyperlinked Environment. Jon M. Kleinberg, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms
  7. A Survey of Eigenvector Methods of Web Information Retrieval. Amy N. Langville and Carl D. Meyer, 2004
  8. PageRank beyond the Web. David F. Gleich, arXiv:1407.5107, 2014

Classical

Centralities

Graph Kernels

What if we use Bag of node degrees?

Deg1:  Deg2: Deg3:

       

Both Graphlet Kernel and Weisfeiler-Lehman (WL) Kernel use Bag-of-* representation of graph, where * is more sophisticated than node degrees!

Graph Kernels

Given: A graph С with a set of nodes V.

  • Assign an initial color с) (у) to each node v.
  • Iteratively refine node colors by                                                  \(c^{(k+1)}(v)=HASH(\{ c^{(k)}(v),\{ c^{(k)}(u)\}_{u \in N(v)} \})\)
  • where HASH maps different inputs to different colors.  
  • After К steps of color refinement, \(c^{(K)}(v)\) summarizes the structure of K-hop neighborhood

Graph Kernels

Given: A graph С with a set of nodes V.

  • Assign an initial color с) (у) to each node v.
  • Iteratively refine node colors by                                                  \(c^{(k+1)}(v)=HASH(\{ c^{(k)}(v),\{ c^{(k)}(u)\}_{u \in N(v)} \})\)
  • where HASH maps different inputs to different colors.  
  • After К steps of color refinement, \(c^{(K)}(v)\) summarizes the structure of K-hop neighborhood

Graph Kernels

 Aggregate neighboring colors

Assign initial colors

Graph Kernels

Hash aggregated colors

 Aggregated colors

Graph Kernels

Hash aggregated colors

 Aggregated colors

Graph Kernels

After color refinement, WL kernel counts number of nodes with a given color.

Degree centrality

Key idea: Bag-of-Words (BoW) for a graph

  • BoW simply uses the word counts as features for documents (no ordering considered).
  • Naive extension to a graph: Regard nodes as words.
  • Since both graphs have 4 red nodes, we get the same feature vector for two different graphs
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