Evolution-cast:%20Temporal%20Evolution%20in%20Wireless%20Social%20Networks%20and%20Its%20Impact%20on%20%20%20Capacity

1
Evolution-cast Temporal Evolution in Wireless
Social Networks and Its Impact on Capacity

• Luoyi Fu, Jinbei Zhang, Xinbing Wang
• Department of Electronic Engineering
• Shanghai Jiao Tong University

2
Outline

• Introduction
• Motivations
• Objectives
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution-cast in Heterogeneous Topology
• Discussion
• Conclusion

3
Motivations

• Social network has been under intensive study for
  decades.
• Barabasi and Albert Model preferential
  attachment phenomenon
• Watts and Kleinberg small-world phenomen
• Densification shrinking diameter over time

4
Motivations (cont)

• Wireless social network is drawing popularity.
• Cost-effective routing design taking advantage of
  the characteristics of social networks 123

Capacity receives little investigation under
wireless social networks.
1 E. Dlay and M. Haahr, Social Network
Analysis for Routing in Disconnected
Delay-Tolerant MANETs, in ACM MobiHoc07,
Montreal,Quebec, Canada, 2007. 2 P. Hui, J.
Crowcroft, E. Yoneki, BUBBLE Rap Social-based
Forwarding in Delay Tolerant Networks, in ACM
MobiHoc08, Hong Kong, China, 2008. 3 W. Gao,
Q. Li, B. Zhao and G. Cao, Multicasting in Delay
Tolerant Networks A Social Network Perspective,
in Proc. MobiHoc, New Orleans, USA, 2009.
5
Motivations (cont)

• Several questions arise
• Stringent demand on capacity in wireless social
  networks
• New challenges as well as potentials brought by
  social networks
• Any difference on capacity studied under wireless
  social networks?
• How will capacity be impacted by social network
  properties, positively or negatively?

6
Objectives

• Capacity in large scale wireless social netowrks
• Wireless communication adjacent interference and
  transmission range
• Nodes exhibit social network characteristics
• The network is also evolving (real networks are
  not fixed objects 45678)
• 1. New node joins the network over time
• 2. New links established between nodes over
  time

4 M. Starnini, A. Baronchelli, A. Barrat, R.
Pastor-Satorras, Random Walks on Temporal
Networks, in Phys. Rev. E 85, 056115, 2012. 5
N. Perra, A. Baronchelli, D. Mocanu, B.
Goncalves, R. PastorSatorras, A. Vespignani,
Walking and Searching in Time-varying Networks,
arXiv1206.2858, 2012. 6 L. Rocha, F. Liljeros,
P. Holme, Simulated Epidemics in an Empirical
Spatiotemporal Network of 50,185 Sexual
Contacts, in PLoS Comput Biol 7(3) e1001109,
2011. 7 L. Rocha, A. Decuyper, V. Blondel,
Epidemics on a Stochastic Model of Temporal
Network, arXiv1204.5421, 2012. 8 L. Rocha, V.
Blondel, Temporal Heterogeneities Increase the
Prevalence of Epidemics on Evolving Networks,
arXiv1206.6036, 2012.
7
Outline

• Introduction
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution-cast in Heterogeneous Topology
• Discussion
• Conclusion

8
Network Model

• Temporal evolution of network
• An algorithm describing the increase of the
  number of nodes and that of links established
  between nodes 5

9S. Lattanzi and D. Sivakumar, Affiliation
Networks, in Proc. ACM STOC09, Bethesda,
Maryland, USA.
9
Network Model (cont)

• Geographical Topology
• Homogeneous distribution
• Heterogeneous distribution
• Traffic Pattern--evolution-cast
• Evolution unicast
• a new arriving node is chosen to be
  either a source or a
• destination of a randomly chosen node
  in existing network
• message sharing between limited number
  of individuals
• Evolution multicast
• a new arrival randomly chooses k(t) out
  of n(t)
• nodes that already existing before t,
  acting as a source or
• destinations of these k(t) nodes.
• message broadcast among multiple
  friends
• Interference Model widely used protocol model

10
Definition

• Feasible Capacity We say that a per node
  capacity ?(t) at time t is said to be feasible if
  there exists a spatial and temporal scheduling
  scheme that yields a per-node capacity of ?(t).
  Consider the case
• where the network enters stable evolution (the
  network
• evolves according to a certain rule over time),
  for an arbitrary duration(i-1)T(t), iT (t), if
  there are ? packets transmitted from source to
  destination, then, we say the average per-node
  capacity is
• at time t, after t exceeds a specific value t0.
  Here t0 is the threshold of time after which the
  network is supposed to enter stable evolution.
• Per-node Capacity We say that a per-node
  capacity at time t in the network is of order T
  (f(t)) if there is a deterministic constant 0 lt
  c1 lt c2 lt 8 such that

11
Outline

• Introduction
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution Unicast
• Evolution Multicast
• Evolution-cast in Heterogeneous Topology
• Discussion
• Conclusion

12
Property of Homogeneous Topology

• Probability distribution of homogeneous topology

Lemma 1 Consider the geographical distribution
of nodes at time slot t, where there are n(t)
nodes in the network. Then, the positions of
nodes follow a uniform distribution over the
whole network when t ? 8. Lemma 2 In
homogeneous geographical distribution, the
probability that a social path (denoted by S u1
? u2 ? u3 ? . . . ? uH D) composed of a
sequence of consecutive links generated in
Algorithm 1 are also reachable within constant
hop of transmission range goes to zero.
Intuition behind Social relations do not
affect capacity Only network evolution will
affect capacity
13
Routing Scheme

• Evolution-cast Tree (ET)
• The idea is similar to that in 10.
• The only difference lies in that the number
  of nodes increases over time in our
  work.

10X. Li, Multicast Capacity of Wireless Ad Hoc
Networks, in IEEE/ACM Tracs. Networking, Vol.
17, Issue 3 June 2009.
14
Evolution Unicast

• The number of destinations per source

• Lemma 3 In evolution unicast, the average number
  of destinations per source is of order T(log t).
• The capacity of evolution unicast
• Theorem 1 With homogeneous geographical
  distribution of nodes, the per-node capacity for
  evolution unicast traffic is
• when t is sufficiently large.

15
Evolution Multicast

• The number of destinations per source

• Lemma 6 In evolution mutlicast traffic, the
  average number of destinations per source is of
  order , where .
• The capacity of evolution multicast
• Theorem 1 With homogeneous geographical
  distribution of nodes, the per-node capacity for
  evolution multicast traffic is
• when t is sufficiently large.

16
Outline

• Introduction
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution-cast in Heterogeneous Topology
• Evolution Unicast
• Discussion
• Conclusion

17
Heterogeneous Topology

• Generation of heterogeneous topology
• New arrival tends to locate more closer to his
  friend
• Probability distribution of heterogeneous
  topology
• Lemma 9 If the topological generation of the
  network evolves according to Mechanism 2, then,
  when t is sufficiently large, the distribution of
  geographic distance between nodes will yield as
  follows
• The spatial stationary distribution of a
  node is assumed to be rotationally invariant with
  respect to another node called support, which can
  be described by a function ?(l) decaying as a
  power law of exponent s, i.e., ?(l) ls,
  . And here l ranges from
  to
• T(1), representing the distance between the node
  and the support.

18
Routing Scheme

• Temporal evolution routing scheme
• Message is delivered along a chain of relay nodes
  whose home point is progressively closer to the
  destination.

?
?
?
19
Evolution Unicast Capacity

• Theorem 3 For heterogeneous topology
  distribution,
• under our proposed routing scheme, the achievable
  per node capacity of evolution-cast, under
  uniform traffic
• pattern, is

20
Outline

• Introduction
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution-cast in Heterogeneous Topology
• Discussion
• Conclusion

21
Discussions

• Impact of evolution-cast on capacity
• Social relations cannot lead to capacity
  improvement in homogeneous geographical
  distribution
• 1. transmission is only within a
  certain transmission range
• 2. the average source-destination
  distance is
• 3. New arrivals causes more
  bandwidth allocation
• The capacity can be improved in heterogeneous
  topology
• 1. a constant capacity is achievable when
  

Resulting in constant number of highly
centralized nodes in the network
22
Discussions

• Relationship with networks having fixed number of
  nodes
• Network with uniform topology
• 1. Unicast Fixing tn, we have
• 2. Multicast Fixing tn, we have

Close to the result in 11
Close to the result in 12
11 P. Gupta and P. R. Kumar, The Capacity of
Wireless Networks, in IEEE Trans. Inform.
Theory, vol. 46, no. 2, pp. 388-404, Mar.
2000. 12 X. Li, Multicast Capacity of Wireless
Ad Hoc Networks, in IEEE/ACM Tracs. Networking,
Vol. 17, Issue 3 June 2009.
23
Discussions

• Relationship with networks having fixed number of
  nodes
• Network with heterogeneous topology
• 1. Unicast Fixing tn, we have

• Almost constant capacity when
• Close to the T(1) capacity in 13

13 A. Ozgur and O. Leveque, Throughput-Delay
Trade-Off for Hierarchical Cooperation in Ad Hoc
Wireless Networks, in Proc. Int. Conf. Telecom.,
Jun. 2008.
24
Outline

• Introduction
• Network Model and Definition
• Evolution-cast in Homogeneous Topology
• Evolution-cast in Heterogeneous Topology
• Discussion
• Conclusion

25
Conclusions

• We present a mathematically tractable model where
  nodes are associated with each other through
  social relations but employ transmission through
  wireless communications.
• We investigate evolution-cast capacity in terms
  of unicast and multicast in both homogeneous and
  heterogeneous topology.
• This is the first work that studies capacity in a
  both evolving and socially related wireless
  networks. Our result can be flexibly applied to
  more general cases and shed insights into the
  design and analysis of future wireless networks.

26
Thank you !