Measuring Preferential Attachment in Evolving Networks

1
Measuring Preferential Attachment in Evolving
Networks

• H.JEONG1,2, Z.NEDA1, A.L.BARABASI1
• 1-Department of Physics,
  University of Nore Dame,USA
• 2- Department of Physics,
  Korea Advanced Institute of Science and
  Technology
• accepted in
• 4 December 2002
• (last revision July 9,2004)

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Abstract

• A key ingredient of many current models proposed
  to capture the topological evolution of complex
  networks is the hypothesis that highly connected
  nodes increase their connectivity faster than
  their less connected peers, a phenomenon called
  preferential attachment.
• Measurements on four networks
• Science citation network
• Internet
• Actor collaboration
• Science coauthorship network
• indicate that the rate at which nodes
  acquire links depends on the nodes degree,
  offering direct quantitative support for the
  presence of preferential attachment.

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Introduction

• Many networks as social, biological and
  communication systems were seen as random
  networks. However recent studies show that they
  have scale-free property.
• The studies resulted that these networks are
  evolving dynamical systems rather than static
  graphs.
• Evolving network models are based on two
  ingredients
• Growth
• Preferential attachment
• Growth Networks continuously expand through the
  addition of new nodes and new links between nodes
• Preferential attachment rate ?(k) with which a
  node with k links acquire new links is a
  monotonically increasing function of k.

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Preferential attachment

• Most results show that ?(k) is linear in k,
  however recently several authors proposed it
  could follow a power law.
• Time evolution of the degree ki of node i can be
  obtained from
• where m is constant and ?(k) has the form
• with a gt 0 an unknown scaling exponent. For a1
  these models reduce to the scale-free model , for
  which the degree distribution P(k), giving the
  probability that a node has k links, follows P(k)
  ? kexp(-?) with ? 3.

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Preferential attachment (cont.)

• for a lt1 the degree distribution follows a
  stretched exponential, while for a gt1 a
  gelation-like phenomenon is expected.

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Questions about Preferential Attachment

• There are fundamental questions that are not yet
  supported by experimental data.
• Is preferential attachment indeed present in real
  networks?
• If ?(k) does indeed depend on k, what is its
  functional form? Is it linear or does it follow a
  power law?
• Could ?(k) follow some unknown and yet
  unexplored functional form?
• In the paper, a numerical method is
  proposed that allows us to extract functional
  form of ?(k) and its characteristic (power law).
  This paper also shows that for Internet and
  citation networks, the value of a is 1 while for
  science collaboration and actor network alt1

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Methods

• To measure ?(k) we need to monitor to which old
  node, the new nodes link, as a function of degree
  of the old node.
• However there is an important problem with this
  approach, normalization constant, C(t), depends
  on the time at which a given node joins the
  system. C(t) creates unwanted biases in
  measurement.
• Solution is to collect data in very tiny time
  intervals such that nodes in the network in time
  T0 will be T0 nodes. Call T1 nodes as the
  nodes added between T1,T1?T where ?TltltT1 and
  T1gtT0.
• When a T1 node joins, we record the degree of T0
  node to which the new node links.
• The histogram providing the number of links
  acquired by the T0 nodes with exactly k degree,
  after normalization, gives the ?(k,T0,T1)

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Methods(cont.)

• If the growing network develops a stationery
  state then ?(k,T0,T1) independent of T0 and T1.
• Large networks with hundreds of thousands of
  nodes, ?(k) has significant fluctations for large
  k.
• To reduce the noise level, instead of ?(k) we
  study the cumulative function
• If ?(k) follows the previous definition,we
  expect

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Measurements

• 4 networks are analyzed
• In the coauthorship network of neuro-science (NS)
  the nodes are scientists, two nodes being linked
  if they coauthored a paper. The database consist
  of journals published between 1991-98. Papers
  published between 1991-9x are used to reveal the
  network topology so that papers published in
  199x1 are used to measure ?(k).
• In the citation network the nodes are papers
  published in 1988 in Physical Review Letters, and
  links represent the citations these articles
  received. T0 chosen as the year 1989.
• In the actor network nodes are actors which are
  linked if they acted together in a movie. The
  ?(k) values for actors that debuted between 1920
  and 1940 are chosen. T01940. And evolution of
  new links between 1942 and 1993 is followed.
• For the Internet data the investigated nodes
  represent Autonomous Systems (AS) and links are
  direct connections between them. The network
  structure data contains nodes from 1997 to
  present. The ?(k) was determined for nodes
  existing in 2000.

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Results

• The ?(k) functions are obtained for the discussed
  databases. If preferential attachment is absent
  i.e. ?(k) is independent of k, we expect ?(k)? k
• However in the figures below, the increase of
  ?(k) is faster than linear, offering direct
  evidence that preferential attachment is present
  in each system. For internet, measurement was
  performed for only one year, while for citation
  network the ?(k) values are observed for eight
  different years.

Citation Network
Internet
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Results(cont.)
Collaboration
Actor network

• Furthermore, curves follow a straight line on a
  log-log plot, indicating that with a good
  approximation power law hypothesis at the
  beginning is valid.
• In Citation network and Internet, we obtain
  ?1.05 and ?0.95 0.1. For these two networks,
  linear preferential attachment hypothesis offers
  a good approximation.
• For scientific collaboration and actor networks,
  we find ?lt1, on the average 0.81 0.1 and 0.79
  0.1 respectively.
• The observed sub-linear behavior in the
  scientific collaboration network predicts that
  P(k) should follow a stretched exponential.
  However the measured P(k) indicates that a power
  law offers a better fit. This is a contradiction.

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Internal and external links

• The links in these networks dont occur by
  addition of new nodes, existing nodes can link to
  each other. For this reason, the measurement
  contained both external and internal links for
  the latter 2 network.
• For science collaboration and actor networks,
  there exist internal links. When determining ?(k)
  the measurement is limited first only to
  external links, and then only to internal links.
• Note that, for the citation network, the internal
  links are not allowed and data resolution for the
  Internet does not allow us to perform the same
  experiment.

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Internal and external links

• A new measurement is performed for external links
  and internal links separately in the actor
  network.
• Probability that a new internal link appears
  between two nodes with degree k1 and k2 scales
  with k1k2 product.
• The results shows that, internal links are also
  governed by preferential attachment, which scales
  linearly with k.
• The P(k) distribution in the network is believed
  to be majorly driven by the characteristic of the
  internal attachment.

Preferential attachment of new internal nodes
Preferential attachment of new nodes
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Initial attractiveness

• Dorogovtsev, Mendes and Samukhin have suggested
  that nodes with no links can acquire links so
  ?(k) should have an additive term, k0 , called
  initial attactiveness, so that ?(k) ? k0kexp(a)
• According to available statistics, k0 has a small
  value in the 10exp(-6).
• Thus it has no effect on the scaling of K(k) at
  large k.

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Results (cont.)

• Summary of the investigated databases

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Conclusion

• Measurements shows preferential attachment exists
  in real evolving networks.
• ?(k) follows a power law distribution, however ?
  is system dependent. For scale-free network ?1.
• For Internet and citation network a linear ?(k)
  offers a reasonable fit, for actor and
  collaboration network attachment rate is
  sublinear.

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Further Work

• What is the microscopic origin of the
  preferential attachment?
• What determines the exponent ? in general?

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• Thanks for listening
• QUESTIONS are WELCOME