Measuring and Extracting Proximity in Networks

1
Measuring and Extracting Proximity in Networks

• By - Yehuda Koren, Stephen C.North and Chris
  Volinsky
• 
  
• 
  - Rahul Sehgal

2
Introduction

• Network
• Information hidden in a network
• What is Proximity?
• Why do we need proximity in a network?
• Methods for measuring and extracting proximities
  in a network.
• CFEC ( Cycle free effective conductance) and how
  to compute cycle-free escape probability
• Extracting proximity through proximity graphs
  obtained by CFEC
• Working with large networks
• Experiments
• Conclusion
• Questions

3
Network

• Collection of edges or links
• Collection of nodes
• These edges and links help in deciding the
  proximity in a given network.

4
Hidden information in a network
C
D
E
B
A
F
5
Hidden information in a network
C

• If two people speak on the phone to many common
  friends, the probability is high that they will
  talk to each other in the future, or perhaps that
  they already communicate through some other
  medium as email.

D
E
B
A
F
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Hidden information in a network
C

• If two nodes are connected to a common node then
  it might be possible that they have a strong
  relationship or they dont have any relationship
  at all.
• for example in a telephone network where
  many people call to service center but it might
  be possible that two people who are calling they
  dont know each other at all.

D
E
B
A
F
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What is proximity?

• Proximity is a method of measuring distance or
  closeness between different objects.
• It is a method of finding hidden relationship
  between the objects.
• It is a method of finding similarities and helps
  in clustering objects or nodes.

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Need of proximity

• It measures potential information exchange
  between two non-linked objects through
  intermediaries
• It can measure the extent to which two nodes
  belong to each other.
• It helps in knowing the likelihood that a link
  will exist in future.
• In social network settings proximity helps to
  predict or track the propagation of an idea,
  product or disease .
• It helps in discovering unexpected communities in
  a network.

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Various Methods

• Graph-theoretic distance
• It is length of shortest path connecting two
  nodes measured either by number of hops or sum of
  weight of edges
• Limitations
• Proximity decays as nodes become farther apart.
• Information may be lost due to friction or noise
  at a particular node.
• This method doesnt assume that proximity can
  exist via multiple paths.

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Various Methods contd

• Network Flow (maximal network flow)
• Limited capacity is assigned to each edge,
  depending upon the weight of that edge and then
  compute the maximal number of units that can be
  simultaneously delivered from node s to node t.
• It prefers high weight edges and captures the
  premise that an increasing number of alternative
  paths increases the proximity.
• For example we consider the adjacent figure.

a1
B
A
b1
A
B
a1
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Various Methods contd

• Network Flow (maximal network flow)
• Limitations
• It disregards the length of the path.
• It also follow that the maximal s-t flow (that is
  graph flow from node s to node t) in a graph
  equals the minimal s-t cut that is the minimal
  edge capacity we need to remove to disconnect s
  from t, therefore we cannot implement it in a
  robust system.

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Various Methods contd

• Effective Conductance (EC)
• Modeling of networks as an electric circuit by
  treating the edges as resistors whose conductance
  is the given edge weight.
• In this method we keep the starting node (s) as 1
  and the end node (t) as 0. and then we solve the
  linear equations for getting voltage and current
  on each edge.
• It accounts for both path length and number of
  alternative paths .
• It avoids dependence on single shortest path and
  bottlenecks
• It has a monotonicity property which states that
  in an electrical resistor network, increasing the
  conductance of any resistor or increasing the
  number of resistors increase conductance between
  any two nodes.

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Various Methods contd

• Limitations of Effective Conductance-
• Monotonicity has its limitations consider
  following example

s1
a1
t1
same EC
a1
t
s
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Various Methods contd

• Sink augmented effective conductance
• Each node in a network is connected to a
  universal sink which is at voltage zero.
• This universal sink competes with the node t.
• Universal sink tax each node that absorbs a
  portion of out going current. Consequently it
  forces all the node to have degree greater than
  1. So, our restriction for degree one node
  doesnt exist any more.

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Various Methods contd

• Limitations of sink augmented effective
  conductance
• Its required to know how much current will flow
  through each node . Understanding how such
  parameter will influence the proximity is a
  difficult task.
• It destroys the concept of monotonicity . It
  means whenever a new node is added to the network
  it has a direct link to the sink but not to t. It
  strengthens sink and it compete with node t and
  thus the proximity between s and t is lost.
• So, we look for a solution that can overcome the
  limitations of above methods..

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Random Walk

• Definitions
• Random Walk is a transition from one state to
  another without depending on the previous state.
  The transition could be to the same state also.

t
a3
a4
s
a2
a1
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Random Walk

• Random walk in network proximity is the infinite
  number of attempts that is made to reach from
  starting node s to end node t. It might be
  possible that when we traverse this path we might
  return back to the starting node s.

s
t
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Random Walk

• Explanation
• In network it might be possible that during
  random walk we might go back to s without going
  to t.
• As we can see in the diagram it might be possible
  that we return back to s via path sa1-a2-a3-s
  without going to t.
• We are having a cyclic path which is leading us
  back to starting point s.
• continued gt

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Random Walk

• Diagram

a3
a2
s
a1
a5
t
a4
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What is our goal?

• We have to improve the effective conductance
  measure and avoid any cyclic path..

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CFEC( Cycle Free Effective Conductance)

• It considers random walk interpretation
• DEFINITION The cycle-free escape probability
  from s to t is the probability that a random walk
  originating at s will reach t without visiting
  any node more than once.
• In random walk, a probability of transition from
  node i to node j is
• probability of transition
  from node i to node j
• weight of edge from i
  to j
• degree of node i.

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CFEC( Cycle Free Effective Conductance)

• The probability that a random walk starting at v1
  for path P v1-v2--vr, will follow this path is
  given by

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Features of CFEC

• We have following equalities
• Multiplying by the degree
• R set of simple paths from s to t, simple path
  are those that never visit the same node twice
• CFEC discourages long paths as the probability of
  following the path decays exponentially with its
  length. It is given by
• It supports proximity measure for multiple paths.
• Degree-1 nodes dilute the significance of path
  from s to t. So we cut the main graph into
  sub-graph. And we preserve the original degree of
  the nodes.

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Explanation of CFEC features

• CFEC discourages long paths

ck
s
t
c1
c2
Proximity decreases
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Computing Cycle Free Escape Probability

• Restrict the sum in following equation to K most
  probable simple path.
• Edge weights are transformed into edge lengths,
  establishing 1-1 correspondence between path
  probability and path length.
• Path probability is given as

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Extracting Proximity Through Proximity Graphs

• Extract a subgraph that maximizes the
  ratio
• subgraph
• constant,
• Find subset of R, which required for above
  problem.
• Set of simple paths R is sorted in ascending
  order of weights of paths.

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Extracting Proximity Through Proximity Graphs

• Optimizing the given formula using branch and
  bound path merging algorithm

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Branch and Bound path merging algorithm
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Working with large networks

• Growing candidate graphs via s, t neighborhoods
• Producing the candidate graph is to find a sub
  graph containing the highest weight paths
  originating at either s or t.
• Now, the problem becomes , find a sub graph
  containing shortest paths originating at either s
  or t. Our objective is to expand the
  neighborhoods of s and t. This is done by
  Dijkstras algorithm for computing shortest path
  on graphs with non-negative edge lengths.

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Working with large networks contd.

• Determining neighborhood size
• we determine paths which are not longer than L
  log(). Path lengths greater than is are not
  useful. Here L is the length of shortest s-t
  path.
• In practice, (L log())/2 neighborhoods might
  be too large.

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Working with large networks contd.

• Pruning the neighborhood
• We prune the neighbor in such a way that
  dist(s,i)dist(t,i) gt ß , where s is our starting
  node and t is terminating node.
• Then, we exclude from the neighborhood any i for
  which dist(s,i)dist(i,t) gt L log().

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Experiments

• Online movie database IMDB.
• Co-authorship graph.
• Telecommunication graph.

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Experiments
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Telecommunication Graph

• Extract the candidate graph by growing
  neighborhoods from the two nodes of interest.
• 2000 random pairs of telephone numbers to
  calculate the CFEC value.
• For 1808(90) of these pairs paths between them
  were found. Others do not have any known path
  between them, this could be due to, these number
  were not in frequent use.
• We have values for as 1,5,10 and 50.

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Telecommunication Graph (contd)
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Conclusion

• CFEC proximity allows us to readily compute
  proximity graphs, which are small portions of the
  network that are aimed at capturing a related
  proximity value.
• An analyst studying proximity in a graph has to
  focus on the most relevant part of the graph.
• It is extension of connection graph which is
  capable of presenting compact relationship
  between objects of a network.
• We can deduce relationship between more than two
  endpoints, the flexibility to handle edge
  direction, and the fact that they are obtained by
  solving an intuitively tunable optimization
  problem.

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• QUESTIONS??????

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References
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References