An economical approach to control congestion in scalefree networks

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An economical approach to control congestion in
scale-free networks

• Zonghua Liu
• Institute of theoretical physics, East China
  Normal University

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Motivation

• Internet has become a very useful tool in daily
  life.
• Internet is not perfect. The intermittent
  congestion in internet has been observed
• A key problem of communication in network is how
  to control the intermittent congestion.

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Communication in Internet
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Previous works Cayley tree
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2-dimensional lattice
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Our model

• Attachment probability

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Properties

• Connectivity distribution

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Algorithm of routing

• (1)At each time step, a node has a possibility
  to create packets. The newly created packets will
  be put at the end of the queue if the node
  already has several packets waiting for
  forwarding to their destinations, which may be
  created at the previous steps or received from
  other nodes.
• (2)At each time step, a node has an ability to
  forward packets one step toward their
  destinations, where the fractal part of is
  implemented by possibility. The packet will be
  removed once it reaches its destination.

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(3)Once a packet is created, a destination node,
different from the original one, is chosen at
random in the network. The router will find a
shortest path between the newly created packet
and its destination and the packet will be
forwarded along this path during the following
time steps. If there are several shortest paths
for one packet, we choose the one whose next
station (selected node) has the shortest queuing
length. (4) At each time step, the first
packets (at the head of its queue) of the node
will be forwarded one step toward their
destinations and be put at the end of the queues
of the selected nodes, respectively, if the
number of packets in its queue is greater than
otherwise, all the packets in the queue
will be forwarded one step. This procedure
works for every node at the same time.
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Little law

• Total number of created packets at each time
  step
• Total number of packets of forwarding one hop
  when each node has enough packets
• When the created packets equal the arrived
  packets at each time step, the network runs in
  the range of Little law and there will be no
  congestion.

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Economical approach

• We choose a fraction of nodes with the heaviest
  links, , and increase their communication
  capacities ( ) and let the other nodes stay at
  the status of .
• The total number of packets of forwarding one hop
  at each time step equals to
• if every node has enough packets.
• Correspondingly, we call the the case with the
  same to all the nodes as Normal approach.

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Numerical simulations
average number of packets on nodes with
link (a) (b)
circles squares stars
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• Evolution of the average number of packets on
  each node (a) and on the nodes with heavy
  links (b) where the three lines from top to
  bottom in both (a) and (b) denote the cases of
  0.05, 0.059, and 0.07, respectively.

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Theoretical explanation

• There is a critical point, , for a given . At
  the critical point,there will be packets
  at the node with the heaviest link.
• The packets on a node comes from two aspects the
  creation and pass by.
• The packets at the critical point at node is
• is a nonlinear decreasing function of
  which reflects the nonlinear contribution of
  the respect of pass by and when
  .

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• For the case of , under the restriction of
  only packets at the nodes with the
  heaviest links, the average packets at those
  nodes with small and middle links will be less
  than one, implying economical approach normal
  approach.
• There are new packets created at each time
  step.
• Suppose the diameter of the network is ,then
  the total number of hops for forwarding the new
  packets to their destinations is

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• According to Little's law we have
• Let
• We have

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Comparing with numerical simulation

• For ,we take and have
• and the slope of is 7.34
• For ,we take and have
• and the slope of is 5.58
• The largest for is
• which gives 0.0016 for and 0.0072 for .
• These results are consistent with the numerical
  simulations.

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Conclusions

• we have constructed a model to describe the
  communication on general networks and given an
  economical approach to increase the capacity of
  communication in scale-free network.
• Theoretic formula is given to determine the
  critical point for a given and has been
  confirmed by numerical simulations.
• The equivalent effect between the economical and
  normal approaches in scale-free network suggests
  us that we can significantly reduce the
  congestion in the internet by increasing only the
  capacity of nodes with heavy links, which
  will save a lot of money of our society.