Scalefree and Hierarchical Structures in Complex Networks

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Scale-free and Hierarchical Structures in Complex
Networks

• L. Barabasi, Z. Dezso, E. Ravasz, S.H. Yook and
  Z. Oltvai
• Presented by Arzucan Özgür

2
Outline

• Network Models
• Random Networks
• Scale-Free Networks
• Scale-Free Model
• Hierarchical Organization in Complex Networks
• Hierarchical Network Model
• Hierarchical Organization in Real Networks
• Halting Viruses in Scale-Free Networks
• Outlook

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Introduction

• Behavior of natural and social systems depends on
  the web through which the systems constituents
  interact with each other.
• Cells metabolizm is maintained by a cellular
  network
• Nodes?substrates
• Links?chemical reactions
• Complex networks describe human societies
• Nodes? individuals
• Links? social interactions
• WWW
• Nodes? Web documents
• Links? URL
• Scientific literature
• Nodes? publications
• Links? citations
• Language
• Nodes? words
• Links? syntaxical or grammatical relationships
• Networks describing these real life systems
  constantly evolve by the addition and removal of
  new nodes and links.
• Due to the diversity and large number of nodes
  and interactions? until recently topology of
  these complex evolving networks was largely
  unknown and unexpected.
• Aim is ? review some advances in the area in
  order to convey the potential for understanding
  complex systems through the evolution of the
  networks behind them.

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Network Models

• Random Networks
• Scale-Free Networks
• Hierarchical Network Model

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Random Networks

• Random graphs? since the 1950s described as
  large networks with no apparent design principles
• Erdos-Renyi (ER) model of random graphs
• start with N nodes and connect every pair of
  nodes with probability p
• A graph is created with approximately pN(N-1)/2
  edges distributed randomly.

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Scale-Free Networks

• P(k)? probability that a randomly selected node
  has exactly k edges.
• In random graphs edges are placed at random? the
  majority of nodes have approximately the same
  degree close to the average degree ltkgt of the
  network.
• Degrees in random graph follow a Poisson
  Distribution with a peak at ltkgt.
• It has been shown that most complex networks such
  as the WWW, Internet, protein networks, language
  or sexual networks have Power Law degree
  distribution.? scale-free networks.
• In random networks, the exponential decay of P(k)
  guarantees the absance of nodes with
  significantly more links than ltkgt.
• In scale-free networks, power low distribution
  implies that nodes with only a few links are
  numerous, but a few nodes have a very large
  number of links.

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Some Scale-Free Networks
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Scale-Free Model

• Two mechanisms, not present in classical random
  network models played role in the development of
  scale-free network model that leads to a network
  with power-law degree distribution
• Growth ? start with a small number of nodes (m0),
  at every timestep we add a new node with m edges
  (mltm0) that link the new node to m different
  nodes already present in the network.
• Preferential attachment ? When choosing the nodes
  to which the new node connects,we assume that the
  probability ? that a new node will be connected
  to node i depends on the degree ki of node i,
  such that

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Scale-Free Model

• Simulations show that this network evolves into a
  scale-invariant state with the probability that a
  node has k edges follows a power-law with an
  exponent ?3
• Scaling exponent is independent of m, the only
  parameter in the model.
• Degree distribution of the scale-free model, with
  N m0t 300,000 and m0 m1 (circles), m0 m
  3 (squares), m0 m 5 (diamonds) and m0 m 7
  (triangles). The slope of the dashed line is ?
  2.9, providing the best fit to the data. The
  inset shows the rescaled distribution P(k)/2m2
  for the same values of m, the slope of the dashed
  line being ? 3. (b) P(k) for m0 m 5 and
  system sizes N 100,000 (circles), N 150,000
  (squares) and N 200,000 (diamonds). The inset
  shows the time-evolution for the degree of two
  vertices, added to the system at t 1 5 and t2
  95. Here m0 m 5, and the dashed line has
  slope 0.5

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Continuum Theory

• The dynamical properties of the scale-free model
  can be addressed using analytical approaches.
• Continuum theory is such an approached focusing
  on the dynamics of node degrees.
• Continuum approach calculates the time dependence
  of the degree ki of a given node i.
• This degree will increase every time a new node
  enters the system and links to node i.
• The probability of this process is ?(ki).

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Continuum Theory

• ki is a continuous real variable
• The rate at which ki changes is proportional to
  ?(ki).
• So, ki satisfies the dynamical equation

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Continuum Theory
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Hierarchical Organization in Complex Networks

• In addition of being scale-free, measurements
  indicate that most networks show a high degree of
  clustering.
• Clustering coefficient for node i with ki links
  is displayed below. Here ni is the number of
  links between the ki neighbours of i.

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Hierarchical Organization in Complex Networks

• Empirical results show that Ci, averaged over all
  nodes is significantly higher for most real
  networks that for a random network of similar
  size.
• Clustering coefficient of real networks is to a
  high degree independent of the number of nodes in
  the network.
• In order to combine modularity, high degree of
  clustering and scale free topology ? it is
  assumed that modules combine into each other in a
  hierarchical manner ? generating hierarchical
  network.
• Scaling-Law

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Hierarchical Network Model
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Scaling Properties of Hierarchical Model

• (N 5 7). (a) The numerically determined degree
  distribution. The assymptotic scaling, with slope
  ?1ln5/ln4, is shown as a dashedline. (b) The
  C(k) curve for the model. The open circles show
  C(k) for a scale-free model of the same size,
  illustrating that it does not have a hierarchical
  architecture. (c) The dependence of the
  clustering coefficient, C, on the size of the
  network N. While for the hierarchical model C is
  independent of N (diamond), for the scale-free
  model C(N) decreases rapidly (circle).

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Hierarchical Organization in Real Networks

• The scaling of C(k) with k for six large
  networks (a) Actor network, two actors being
  connected if they acted in the same movie
  according to the www.IMDB.com database.
• (b) The semantic web, connecting two English
  words if they are listed as synonyms in the
  MerriamWebster dictionary.
• (c) TheWorldWideWeb.
• (d) Internet at the Autonomous System level, each
  node representing a domain, connected if there is
  a communication link between them.
• (e) The metabolic networks of 43 organisms with
  their averaged C(k) curves.
• (f) The protein-protein physical interaction
  networks using four different databases.
• The dashed line in each figure has slope -1.

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Summary

• Measurements indicate that some real networks
  lack a hierarchical architecture, and do not obey
  the scaling law.
• In particular, the power grid and the router
  level Internet topology have a k independent
  C(k).
• In summary, it is shown that for several large
  networks C(k) is well approximated by C(k) 1/k,
  in contrast to the k-independent C(k) predicted
  by both the scale-free and random networks.
• This indicates that these networks have an
  inherently
• hierarchical organization.
• In contrast, hierarchy is absent in networks with
  strong geographical contraints, possibly because
  limitation on the link length strongly
  constraints the network topology.

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Halting Viruses in Scale-Free Networks

• Classical epidemiological models predict that
  infectious diseases with transmission probability
  under an epidemic threshold will inevitably die
  out.
• Thus, lowering transmission probability by
  universally available cure seems an effective
  action agains virus spreading.
• However,
• It has been shown that in scale-free networks the
  epidemic threshold is zero. ? even extremely
  weakly infectious viruses spread and prevail.
• Network of human sexual contacts has a scale-free
  topology.
• So, infected hubs increase the transmission
  probability of the epidemics (HIV) by reaching an
  unusually high percentage of other nodes.
• Given the high cost of cure and immunization
  there are two approaches that can be taken
• Random immunization ? not very effective as the
  scale-free nature of the network is not altered.
• Immunizing hubs with higher degree of
  connectivity ? the optimum approach.

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Thank You.