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Centrality and Graph Mining

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Introduction Many real world systems can be described as networks.

Social relationships: e.g. collaboration relationships in academic, entertainment, business area.

Technological systems: e.g. internet topology, WWW, or mobile networks.

Biological systems: e.g. regulatory, metabolic, or interaction relationships.

Find important nodes Discover network modules (clusters) Inferring important paths

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Notation

Graph G = (V, E)

Nodes Edges

pair of nodes (u, v) Graphs can be undirected, directed,

unweighted, or weighted

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Scale-free networks, basic properties Almost all of real world networks are Scale-free.

Scale-free networks, basic properties

o Power law degree distributiono Small Worldo Robustnesso Hierarchical Modularity

o Disassortative or Assortative

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Structural PropertyRandom network and scale free network

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Degree of nodes

Many nodes on the internet have low degree One or two connections

A few (hubs) have very high degree The number P(k) of nodes with degree k follows a

power law:

Where alpha for the internet is about 2.1

P(k) k

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Power law degree distribution: large events are rare, but small ones quite common.

The probability of finding a highly connected node decreases exponentially with k (degree of node, inversely proportional to k):

Power Law Network

( ) ~P K K

P(K): quantify the probability that a selected node will have K links. Total number of nodes N(K)/total number of nodes N.

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The Barabási-Albert [BA] model

ER Model WS Model Actors Power Grid www

(a) Random Networks (b) Power law Networks

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Small World Property

Shortest path: the path with the smallest number of links between the selected nodes.

Small world networks: the average shortest path length between any two

nodes in the network is relatively small. Any node can be reached within a small number of

edges, 4~5 hops.

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Power-law degree distribution & Small world

phenomena are observed in: communication networks web graphs research citation networks social networks

Classical -Erdos-Renyi type random graphs do not exhibit these properties: Links between pairs of fixed set of nodes picked uniformly: Maximum degree logarithmic with network size No hubs to make short connections between nodes

Power Law Network

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Attack Tolerance

Complex systems maintain their basic functions even under errors and failures (cell mutations; Internet

router breakdowns)

node failure

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Attack Tolerance Robust. For <3, removing

nodes does not break network into islands.

Very resistant to random attacks, but attacks targeting key nodes are more dangerous.

Robustness: Resilient and have strong resistance to failure on random attacks and vulnerable to targeted attacks

Max

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Density of Graph G

n – the number of nodes in G M – the number of edges in G n(n-1)/2 – the number of possible edges in a

complete graph.

)1(/2)( nnmGDen

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Formally, we can characterize a graph through 2 statistics.

1) The characteristic path length, L (the diameter)The average length of the shortest paths connecting any two nodes.

2) The clustering coefficient, CIs the average local density.

A small world graph is any graph with a relatively small L and a relatively large C.

Slide from James Moody

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Local clustering coefficient For a vertex i

The fraction pairs of neighbors of the node that are themselves connected

Let ni be the number of neighbors of vertex i

number of connections between i’s neighborsmaximum number of possible connections between i’s neighbors

# directed connections between i’s neighborsni * (ni -1)

# undirected connections between i’s neighborsni * (ni -1)/2

Slide from Lada Adamic

Ci =

Ci directed =

Ci undirected =

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Local clustering coefficient Average over all n vertices

Slide from Lada Adamic

i

iCn

C 1

i

ni = 4max number of connections:4*3/2 = 63 connections presentCi = 3/6 = 0.5

link absentlink present

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Clustering Coefficient - Example

For an undirected network, vertex i is connected with 4 nodes.

Number of nearest neighbors for vertex i, zi = 4 Number of all possible connections = zi (zi – 1)/2 = 6

Number of actual connections between the nearest neighbors around vertex i, yi = 3

The clustering coefficient of vertex i Ci= 2*3/(4*3) = 3/6 = 0.5

i

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Clustering coefficient

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The most clustered graph is Watt’s “Caveman” graph:

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knLcaveman

Slide from Lada Adamic

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Why does this work? Key is fraction of shortcuts in the network

In a highly clustered, ordered network, a single random connection will create a shortcut that lowers L dramatically

Watts demonstrates that Small world graphs occur in graphs with a small number of shortcuts

Slide from Lada Adamic

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Duncan Watts: Networks, Dynamics and the Small-World Phenomenon

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Clustering Coefficient

N(v) is the set of the direct neighbors of node v and d(v) is the number of the direct neighbors of node v (| N(v)|).

Number of “triangles” that go through v over the total number of triangles that could pass through node v.

The average clustering coefficient of a graph characterizes the overall tendency of nodes to form clusters or groups.

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Hierarchical ModularityA large clustering coefficientHow many of a node’s neighbors are connected to each other

E. Ravasz et al., Science, 2002

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Basic Centralities Several centrality indices have been developed to

measure the components’ importance in a network. Degree centrality: number of neighbors of node v, Betweenness centrality : ratio of the number of shortest

paths passing through a node v out of all shortest paths between all node pairs in a network

σst is the number of shortest paths between node s and t and σst(v) is the number of shortest paths passing on a node v out σst

Closeness centrality: reciprocal of the total distance from a node v to all the other nodes in a network

δ(u,v) is the distance between node u and v.

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Betweenness - Example For an undirected network with 4 nodes

Shortest paths between all pairs: B(1,m)=1,m; B(1,2)=1,m,2;1,3,2; B(1,3)=1,3;

B(2,m)=2,m; B(2,3)=2,3; B(3,m)=3,1,m:3,2,m; Shortest paths pass m:

B(1, m), B(1, m, 2), B(2, m) , B(3, m) Betweenness σ(m) = 4/8

m2

3

1

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Basic Centralities

Eigenvector centrality: a measure of importance of nodes in a network using the adjacency and eigenvector matrices.

A is the adjancy matrix and xi is the eigenvector centrality for node i.

Subgraph centrality: accounts for the participation of a node in all subgraphs of the network.

the number of closed walks of length k starting and ending node v in the network is given by the local spectral moments μk(v).

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New Observation There should be some bridging nodes/edges

between modules in scale-free networks, and we did recognize the bridging nodes/edges by visual inspection of small example networks.

Finding the bridging nodes/edges, which locate between modules, is an interesting and important problem for many applications on many different fields. (Networks’ robustness, paths protection, effective targets finding, etc.)

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Bridging Centrality Bridging node

A bridging node should be located on an important path, e.g., shortest path.

A bridging node should be located between modules.

A bridging node might have a low degree than other central nodes, e.g., hubs.

The neighbor regions of a bridging node should have low range of public domain among them.

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Existing measurements are not good enough for identifying the bridging nodes/edges: those existing indices are dominated by degree of the node of interest.

Betweenness of an edge also has a strong inclination to attach onto high degree nodes.

High tendency of cluttering in the center of the network. So, it is hard to differentiate the bridging nodes/edges from other kinds of nodes/edges.

Our focus in this research is to target vulnerable and central components in a network from a totally different point of view.

Bridging Nodes and Edges

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Betweenness and Bridging Coefficient Betweenness: global importance of a node/edge from shortest paths

viewpoint.

Bridging Coefficient: measuring the extent how well a node or edge is located between well connected regions.

the average probability of leaving the direct neighbor sub-graph of a node v (δ(v): the number of edges leaving the directly-neighboring subgraph of node v).

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Bridging Coefficient

Figure 1. Bridging Coefficient

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Bridging Centrality

)()()( vvBr RRvC

Bridging Centrality is defined as the product of the rank of the betweenness and the rank of the bridging coefficient.

)()()( eeBr RReC

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Bridging Centrality Bridging node

A bridging node should be located on an important path, e.g., shortest path.

A bridging node should be located between modules.

A bridging node might have a low degree than other central nodes, e.g., hubs.

The neighbor regions of a bridging node should have low range of public domain among them.

Use spanning tree centrality from spanning trees in the network.

Use scattering coefficient of the node , modification based on clustering coefficient

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Spanning Tree

a tree composed of all the vertices and some (or perhaps all) of the edges of G

Spanning tree centrality

number of spanning trees : 4

number of spanning trees : 16

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Property 1: The number of spanning trees in a graph is an indicator of density of a graph.

Property 2 : A bridge of the graph must lie on most spanning trees.

Spanning tree centrality

In this case, all spanning trees are passing through e1 and e2

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Spanning tree centrality of an edge e

Spanning tree centrality of an node v

Scattering coefficient of an node v

Scattering coefficient of an edge e

Spanning tree centrality

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Bridging Centrality

Bridging Centrality is defined by utilizing Spanning Tree centrality and the rank of the scattering coefficient through product of their individual rank.

Bridging Centrality

)()()( vCvCBr SCSTCRRvC

)()()( eCeCBr SCSTCRReC

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Experimental Result

The result of spanning tree centrality for the AT &T Web network. The nodes with the highest 0-10th percentile of values for the centrality are highlighted in black circles; the nodes in 11th-20 th percentile are highlighted in gray circles.

Display of Bridging Nodes

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Experimental Result

Average Clustering Coefficient Changes between shortest-path based betweenness Centrality and spanning tree based bridging centrality on the yeast metabolic network

Average Path Length Changes Average Clustering Coefficient Change

Average Path Length Changes between shortest-path based betweenness Centrality and spanning tree based bridging centrality on the yeast metabolic network

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Application on a synthetic network

Figure 2. The network contains 158 nodes and 362 edges was created by adding bridging nodes to three distinct modules. (a) and (b) shows the results of bridging centrality and betweenness centrality

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Application on Web Network Examples

Figure 3. Results for Web Networks: Figure 1A and 1B shows the results for the AT&T Web Network and RPI Web Network, respectively. The nodes with the highest 0-5th percentile of values for the bridging centrality are highlighted in red circles; the nodes with the lowest values of bridging centrality are the 85th-100th percentiles and are highlighted in white circles. The color map for the percentile values is shown in the Figure.

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Application on Social Network Examples

Figure 4. Results for Social Networks: Figure 2A and 2B shows the results for the Les Miserable Character Network and Physics Collaboration Network, respectively. The nodes with the highest 0-5th percentile of values for the bridging centrality are highlighted in red circles; the nodes with the lowest values of bridging centrality are the 85th-100th percentiles and are highlighted in white circles. The nodes corresponding to Valjean (V), Javert (J), Pontmercy (P) and Cosette (C) are labeled in Figure 4A. The nodes corresponding to Rothman (R), Redner (R2), Dodds (D), Krapivsky (K) and Stanley (S) are labeled in Figure 2B. The color map for the percentile values is shown in the Figure.

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Application on Biological Network Examples

Figure 5. Results for Biological Networks: Figure 3A and 3B shows the results the Cardiac Arrest Network and Yeast Metabolic Network, respectively. The nodes corresponding to Src, Shc and Jak2 (J2) are labeled in Figure 3A. The nodes with the highest 0-5th percentile of values for the bridging centrality are highlighted in red circles; the nodes with the lowest values of bridging centrality are the 85th-100th percentiles and are highlighted in white circles. The color map for the percentile values is shown in the Figure.

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Assessing Network Disruption, Structural Integrity and Modularity

Figure 6. Sequential node removal analysis on the yeast metabolic network

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Assessing Ability To Occupy Topological Position

Figure 7A shows the clique affiliation of the nodes detected by three metrics, the bridging centrality (black squares), degree centrality (open circles), betweenness centrality (black circles). Maximal cliques were identified in the Yeast PPI network, and then we measured whether the detected nodes for each metric are in the identified cliques or not. In Figure 7B, random betweenness between detected cliques was measured in the clique graph for each metric, bridging centrality (black squares), degree centrality (open circles), betweenness centrality (black circles). Figure 7C compares the number of singletons that were generated according to sequential node deletion for each metric such as bridging centrality (red line), degree centrality (gray line), betweenness centrality (blue line). The nodes with the highest values for each of these network metrics were sequentially deleted and enumerated the number of singletons that were produced.

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Assessing Ability To Occupy Modulating Position

Figure 8. The biological and the topological characteristics of the direct neighbors of the node ordered by two metrics, the bridging centrality (black bar), betweenness centrality (white bar). Figure 6(a) shows the gene expression correlation on the direct neighbors of each percentile. Figure 6(b) shows the average clustering coeffcient of the nodes in each percentile.

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Bridge Cut AlgorithmIterative Graph Partitioning Algorithm

1. Compute Bridging Centrality for each edge

2. Cut the highest bridging edge

3. Identify an isolated module as a cluster if the density of the isolated module is greater than a threshold.

Density:

n is the number of nodes and e is the number of edges in a sub graph C of a network.

)1(*2)(

nn

eCDensity

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Clustering Validation Precision: |X ∩ F|/|X|, Recall: |X ∩F|/|F| X is the testing cluster, F is ground truth. F-measure

Davies-Bouldin Index

where diam(Ci) is the diameter of cluster Ci and d(Ci ;Cj) is the

distance between cluster Ci and Cj . So, d(Ci ;Cj) is small if cluster i and j are compact and theirs centers are far away from each other. Therefore, DB will have a small values for a good clustering.

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Bridge Cut

Table 1: Comparative analysis. Performance of bridge cut method on DIP PPI dataset (2339 nodes, 5595 edges) is compared with seven graph clustering approaches (Maximal clique, quasi clique, Rives, minimum cut, Markov clustering, Samanta). The fourth column represents the average F-measure of the clusters for MIPS complex modules. The fifth column indicates the Davies-Bouldin cluster quality index. Comparisons are performed on the clusters with 4 or more components.

Methods Clusters Size MIPS complex (F-measure)

DB

Bridge Cut 114 7.6 0.53 4.78 Betweenness Cut 131 6.3 0.49 6.2

Max Cliq 120 4.7 0.49 N/A Quasi Cliq 103 9.2 0.46 N/A

Rives 74 31 0.33 13.5 Mincut 227 8.7 0.35 7.23 MCL 210 8.4 0.47 6.82

Samantha 138 7.2 0.43 6.8

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Bridge Cut

Table 2. Comparative analysis. Performance of bridge cut method on the school friendship dataset (551 nodes, 2066 edges) is compared with seven graph clustering approaches (Maximal clique, quasi clique, Rives, minimum cut, Markov clustering, Samanta). Column descriptions are the same as Table 1

Methods Clusters Size DB Bridge Cut 40 8.6 5.46

Betweenness Cut 48 7.1 5.57 Max Cliq 133 4.4 N/A

Quasi Cliq 109 9.5 N/A Rives 46 10.9 10.4

Mincut 53 9.3 6.29 MCL 50 8.0 5..47

Samantha 40 13.5 7.1

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Discussion The recognition of the bridges should be very valuable

information for many different applications on many different areas. Identifying functional, physical modules, or key components

using the bridging centrality will provide an effective and totally new way of looking at biological systems.

Discovering sub-communities or important components in social network system.

Network robustness improvement, network protection, and paths protection using bridging information.

Drug Target Identification

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Future Works

Directed network Complexity