Centrality and Graph Mining

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Centrality and Graph Mining. 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.

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### Centrality and Graph Mining

Introduction
• Many real world systems can be described as networks.
• 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
Notation
• Graph
• G = (V, E)
• Nodes
• Edges
• pair of nodes (u, v)
• Graphs can be undirected, directed, unweighted, or weighted
Scale-free networks, basic properties
• Almost all of real world networks are Scale-free.
• Scale-free networks, basic properties
• Power law degree distribution
• Small World
• Robustness
• Hierarchical Modularity
• Disassortative or Assortative
Structural Property

Random network and scale free network

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
Power Law Network
• 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):

P(K): quantify the probability that a selected node will have K links.

Total number of nodes N(K)/total number of nodes N.

The Barabási-Albert [BA] model

ER Model

WS Model

Actors

Power Grid

www

(a) Random Networks

(b) Power law Networks

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.

Power Law Network

• 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

node failure

Attack Tolerance

• Complex systems maintain their basic functions even under errors and failures (cell  mutations; Internet  router breakdowns)
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 Cluster Size

Path Length

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.
Slide from James Moody

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, C
• Is the average local density.
• A small world graph is any graph with a relatively small L and a relatively large C.

Ci =

Ci directed =

Ci undirected =

• 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 neighbors

maximum number of possible connections between i’s neighbors

# directed connections between i’s neighbors

ni * (ni -1)

# undirected connections between i’s neighbors

ni * (ni -1)/2

ni = 4

max number of connections:

4*3/2 = 6

3 connections present

Ci = 3/6 = 0.5

i

• Average over all n vertices
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

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

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

1.2

140

120

1

100

0.8

Characteristic Path Length

Clustering Coefficient

80

0.6

60

0.4

40

0.2

20

0

0

0

20

40

60

80

100

120

Degree (k)

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

C and L as functions of k

for a Caveman graph of n=1000

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

E. Ravasz et al.,Science, 2002

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.

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

m

1

2

3

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).

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.)

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.

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
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).
Bridging Coefficient

Figure 1. Bridging Coefficient

Bridging Centrality

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

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

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

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

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
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
Experimental Result
• Display of Bridging Nodes

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-20th percentile are highlighted in gray circles.

Experimental Result
• 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

Average Clustering Coefficient Changes between shortest-path based betweenness

Centrality and spanning tree based bridging centrality on the yeast metabolic network

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

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.

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.

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.

Figure 6. Sequential node

removal analysis on the yeast metabolic network

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.
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) showsthe gene expression correlation on the direct neighborsof each percentile. Figure 6(b) shows the average clustering coeffcient of the nodes in each percentile.

Bridge Cut Algorithm
• Iterative Graph Partitioning Algorithm
• Compute Bridging Centrality for each edge
• Cut the highest bridging edge
• 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.
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 Ciand d(Ci ;Cj) is the distance between cluster Ciand 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.

Bridge Cut

Table 1: Comparative analysis. Performance ofbridge cut method on DIP PPI dataset (2339 nodes, 5595 edges) is comparedwith 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 with4 or more components.

Bridge Cut

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

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
Future Works
• Directed network
• Complexity