Evolution-cast: Temporal Evolution in Wireless Social Networks and Its Impact on Capacity Luoyi Fu, Jinbei Zhang, Xinbing Wang Department of Electronic

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

Capacity Capacity

Luoyi Fu, Jinbei Zhang, Xinbing Wang

Department of Electronic Engineering

Shanghai Jiao Tong University

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OutlineOutline IntroductionIntroduction

MotivationsMotivations ObjectivesObjectives

Network Model and DefinitionNetwork Model and Definition

Evolution-cast in Homogeneous TopologyEvolution-cast in Homogeneous Topology

Evolution-cast in Heterogeneous TopologyEvolution-cast in Heterogeneous Topology

DiscussionDiscussion

ConclusionConclusion

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MotivationsMotivations

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

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Motivations (cont’)Motivations (cont’)

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

characteristics of social networks [1][2][3]characteristics of social networks [1][2][3]

[1] E. Dlay and M. Haahr, “Social Network Analysis for Routing in Disconnected Delay-[1] E. Dlay and M. Haahr, “Social Network Analysis for Routing in Disconnected Delay-Tolerant MANETs”, in ACM MobiHoc’07, Montreal,Quebec, Canada, 2007.Tolerant MANETs”, in ACM MobiHoc’07, Montreal,Quebec, Canada, 2007.[2] P. Hui, J. Crowcroft, E. Yoneki, “BUBBLE Rap: Social-based Forwarding in Delay Tolerant [2] P. Hui, J. Crowcroft, E. Yoneki, “BUBBLE Rap: Social-based Forwarding in Delay Tolerant Networks”, in ACM MobiHoc’08, Hong Kong, China, 2008.Networks”, in ACM MobiHoc’08, Hong Kong, China, 2008.[3] W. Gao, Q. Li, B. Zhao and G. Cao, “Multicasting in Delay Tolerant[3] W. Gao, Q. Li, B. Zhao and G. Cao, “Multicasting in Delay TolerantNetworks: A Social Network Perspective”, in Proc. MobiHoc, New Orleans, USA, 2009.Networks: A Social Network Perspective”, in Proc. MobiHoc, New Orleans, USA, 2009.

Capacity receives little investigation under wireless social networks.

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Motivations (cont’)Motivations (cont’)

Several questions arise:Several questions arise: Stringent demand on capacity in wireless social networksStringent demand on capacity in wireless social networks New challenges as well as potentials brought by social New challenges as well as potentials brought by social

networksnetworks Any difference on capacity studied under wireless social Any difference on capacity studied under wireless social

networks?networks? How will capacity be impacted by social network properties, How will capacity be impacted by social network properties,

positively or negatively?positively or negatively?

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ObjectivesObjectives

Capacity in large scale wireless social netowrksCapacity in large scale wireless social netowrks Wireless communication: adjacent interference and transmission Wireless communication: adjacent interference and transmission

rangerange Nodes exhibit social network characteristicsNodes exhibit social network characteristics The network is also evolving (real networks are not fixed objects The network is also evolving (real networks are not fixed objects

[4][5][6][7][8]):[4][5][6][7][8]):

1. New node joins the network over time1. New node joins the network over time

2. New links established between nodes over time2. New links established between nodes over time

[4] M. Starnini, A. Baronchelli, A. Barrat, R. Pastor-Satorras, “Random Walks on Temporal [4] M. Starnini, A. Baronchelli, A. Barrat, R. Pastor-Satorras, “Random Walks on Temporal Networks”, in Phys. Rev. E 85, 056115, 2012.Networks”, in Phys. Rev. E 85, 056115, 2012.

[5] N. Perra, A. Baronchelli, D. Mocanu, B. Goncalves, R. PastorSatorras, A. Vespignani, [5] N. Perra, A. Baronchelli, D. Mocanu, B. Goncalves, R. PastorSatorras, A. Vespignani, “Walking and Searching in Time-varying Networks”, arXiv:1206.2858, 2012.“Walking and Searching in Time-varying Networks”, arXiv:1206.2858, 2012.

[6] L. Rocha, F. Liljeros, P. Holme, “Simulated Epidemics in an Empirical Spatiotemporal [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.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”, [7] L. Rocha, A. Decuyper, V. Blondel, “Epidemics on a Stochastic Model of Temporal Network”, arXiv:1204.5421, 2012.arXiv:1204.5421, 2012.

[8] L. Rocha, V. Blondel, “Temporal Heterogeneities Increase the Prevalence of Epidemics on [8] L. Rocha, V. Blondel, “Temporal Heterogeneities Increase the Prevalence of Epidemics on Evolving Networks”, arXiv:1206.6036, 2012.Evolving Networks”, arXiv:1206.6036, 2012.

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Network Model Network Model

Temporal evolution of networkTemporal evolution of network An algorithm describing the increase of the number of nodes and An algorithm describing the increase of the number of nodes and

that of links established between nodes [5]that of links established between nodes [5]

[9]S. Lattanzi and D. Sivakumar, “Affiliation Networks”, in [9]S. Lattanzi and D. Sivakumar, “Affiliation Networks”, in Proc. ACM STOC’09, Bethesda, Proc. ACM STOC’09, Bethesda, Maryland, USA.Maryland, USA.

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Network Model (cont’)Network Model (cont’)

Geographical Topology:Geographical Topology: Homogeneous distributionHomogeneous distribution Heterogeneous distributionHeterogeneous distribution

Traffic Pattern--evolution-cast:Traffic Pattern--evolution-cast: Evolution unicast: Evolution unicast:

a new arriving node is chosen to be either a source or a a new arriving node is chosen to be either a source or a

destination of a randomly chosen node in existing networkdestination of a randomly chosen node in existing network

message sharing between limited number of individualsmessage sharing between limited number of individuals Evolution multicast: Evolution multicast:

a new arrival randomly chooses k(t) out of n(t)a new arrival randomly chooses k(t) out of n(t)

nodes that already existing before t, acting as a source or nodes that already existing before t, acting as a source or

destinations of these k(t) nodes.destinations of these k(t) nodes.

message broadcast among multiple friends message broadcast among multiple friends Interference Model: widely used protocol modelInterference Model: widely used protocol model

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DefinitionDefinition

Feasible CapacityFeasible Capacity: : We say that a per node capacity We say that a per node capacity λ(t) at time t is λ(t) at time t is said to be feasible if said to be feasible if there exists a spatial and temporal scheduling there exists a spatial and temporal scheduling scheme that yields a per-node capacity of scheme that yields a per-node capacity of λ(t). Consider the caseλ(t). Consider the case

where the network enters stable evolution (the networkwhere the network enters stable evolution (the networkevolves according to a certain rule over time), for an arbitrary evolves according to a certain rule over time), for an arbitrary

duration[(duration[(i−1)T(t), iT (t)], if there are Ψ packets i−1)T(t), iT (t)], if there are Ψ packets transmitted from transmitted from source to destination, then, we say the average per-node capacity issource to destination, then, we say the average per-node capacity is

at time at time t, after t exceeds a specific value tt, after t exceeds a specific value t00. Here t. Here t00 is the is the threshold of threshold of

time after which the network is supposed to enter stable evolution.time after which the network is supposed to enter stable evolution.

Per-node CapacityPer-node Capacity: We say that a per-node capacity at time t in the : We say that a per-node capacity at time t in the network is of order Θ (f(t)) if there is a deterministic constant 0 < c1 < network is of order Θ (f(t)) if there is a deterministic constant 0 < c1 < c2 < +∞ such thatc2 < +∞ such that

=T t

1

2

lim inf Pr is feasible 1

lim inf Pr is feasible 1

n

n

t c f t

t c f t

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Evolution-cast in Homogeneous TopologyEvolution-cast in Homogeneous Topology Evolution UnicastEvolution Unicast Evolution MulticastEvolution Multicast




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Property of Homogeneous TopologyProperty of Homogeneous Topology

Probability distribution of homogeneous topologyProbability distribution of homogeneous topology

Lemma 1: Lemma 1: Consider the geographical distribution of nodes at time Consider the geographical distribution of nodes at time slot slot t, where there are n(t) nodes in the t, where there are n(t) nodes in the network. Then, the network. Then, the positions of nodes follow a uniform distribution over the whole positions of nodes follow a uniform distribution over the whole network when network when t → ∞.t → ∞.

Lemma 2: In homogeneous geographical distribution, Lemma 2: In homogeneous geographical distribution, the the probability that a social path (denoted by probability that a social path (denoted by S = u1 → u2 → u3 → . . . S = u1 → u2 → u3 → . . . → uH = D) composed of a sequence → uH = D) composed of a sequence of consecutive links of consecutive links generated in Algorithm 1 are also reachable within constant hop of generated in Algorithm 1 are also reachable within constant hop of transmission range goes to zero.transmission range goes to zero.

Intuition behind: Intuition behind: Social relations do not affect capacitySocial relations do not affect capacity Only network evolution will affect capacityOnly network evolution will affect capacity

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Routing SchemeRouting Scheme

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

[10]X. Li, “Multicast Capacity of Wireless Ad Hoc Networks”, in [10]X. Li, “Multicast Capacity of Wireless Ad Hoc Networks”, in IEEE/ACM Tracs. Networking, IEEE/ACM Tracs. Networking, Vol. 17, Issue 3 June 2009.Vol. 17, Issue 3 June 2009.

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Evolution UnicastEvolution Unicast

The number of destinations per sourceThe number of destinations per source Lemma 3: In evolution unicast, the average Lemma 3: In evolution unicast, the average number of destinations number of destinations

per source is of order per source is of order Θ(Θ(log log t).t).

The capacity of evolution unicastThe capacity of evolution unicast Theorem 1: With homogeneous geographical Theorem 1: With homogeneous geographical distribution distribution of nodes, the per-node capacity for of nodes, the per-node capacity for evolution unicast traffic isevolution unicast traffic is

when t is sufficiently large.when t is sufficiently large.

1

logt

t t

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Evolution MulticastEvolution Multicast

The number of destinations per sourceThe number of destinations per source Lemma 6: In evolution mutlicast traffic, the average Lemma 6: In evolution mutlicast traffic, the average number of number of

destinations per source is of order , where . destinations per source is of order , where .

The capacity of evolution multicastThe capacity of evolution multicast Theorem 1: With homogeneous geographical Theorem 1: With homogeneous geographical distribution distribution of nodes, the per-node capacity for of nodes, the per-node capacity for evolution multicast traffic isevolution multicast traffic is

when t is sufficiently large.when t is sufficiently large.

1

1 1max ,

logt tt t

t 0 1

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Evolution-cast in Heterogeneous TopologyEvolution-cast in Heterogeneous Topology Evolution UnicastEvolution Unicast


ConclusioConclusionn

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Heterogeneous TopologyHeterogeneous Topology

Generation of heterogeneous topologyGeneration of heterogeneous topology New arrival tends to locate more closer to his friendNew arrival tends to locate more closer to his friend

Probability distribution of heterogeneous topologyProbability distribution of heterogeneous topology

Lemma 9: If the topological generation of the network Lemma 9: If the topological generation of the network evolves evolves according to Mechanism 2, then, when according to Mechanism 2, then, when t is t is sufficiently large, the sufficiently large, the distribution of geographic distance between nodes will yield as distribution of geographic distance between nodes will yield as follows:follows:

The The spatial stationary distribution spatial stationary distribution of a node is assumed to be of a node is assumed to be rotationally invariant rotationally invariant with respect to another node called with respect to another node called support, support, which can be described by a which can be described by a function function ϕ(l) ϕ(l) decaying as a power law decaying as a power law of of exponent σ, exponent σ, i.e., i.e., ϕ(ϕ(l) l∼l) l∼ σσ,, . And here l ranges from . And here l ranges from to to

Θ(1), representing the distance between the Θ(1), representing the distance between the node and the support.node and the support.

12 u

q

c

c

1 t

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Routing SchemeRouting Scheme

Temporal evolution routing scheme:Temporal evolution routing scheme: Message is delivered along a chain of relay nodes whose home Message is delivered along a chain of relay nodes whose home

point is progressively closer to the destination.point is progressively closer to the destination.

①①

②② ③③

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Evolution Unicast CapacityEvolution Unicast Capacity

Theorem 3: For heterogeneous topology distribution,Theorem 3: For heterogeneous topology distribution,under our proposed routing scheme, the achievable per under our proposed routing scheme, the achievable per node capacity of evolution-cast, under uniform trafficnode capacity of evolution-cast, under uniform trafficpattern, ispattern, is

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2

2

1min ,

log

u

q

c

c

t tt

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DiscussionsDiscussions

Impact of evolution-cast on capacityImpact of evolution-cast on capacity Social relations cannot lead to capacity improvement in Social relations cannot lead to capacity improvement in

homogeneous geographical distribution:homogeneous geographical distribution: 1. 1. transmission is only within a certain transmission rangetransmission is only within a certain transmission range 2. the average source-destination distance is2. the average source-destination distance is 3. New arrivals causes more bandwidth allocation3. New arrivals causes more bandwidth allocation

The capacity can be improved in heterogeneous topology:The capacity can be improved in heterogeneous topology:

1. a constant capacity is achievable when 1. a constant capacity is achievable when

1

110

2u

q

c

c

1

Resulting in constant number of highly centralized Resulting in constant number of highly centralized nodes in the networknodes in the network

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Relationship with networks having fixed number of Relationship with networks having fixed number of nodesnodes Network with uniform topologyNetwork with uniform topology

1. Unicast: Fixing t=n, we have 1. Unicast: Fixing t=n, we have

2. Multicast: Fixing t=n, we have2. Multicast: Fixing t=n, we have

1

logn n

Close to the result in [11] Close to the result in [11]

1

10 1

log

1 1

n n

n

Close to the result in [12] Close to the result in [12]

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

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Relationship with networks having fixed number of Relationship with networks having fixed number of nodesnodes Network with heterogeneous topologyNetwork with heterogeneous topology

1. Unicast: Fixing t=n, we have 1. Unicast: Fixing t=n, we have

•Almost constant capacity whenAlmost constant capacity when•Close to the Close to the ΘΘ(1) capacity in [13] (1) capacity in [13]

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

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2

2

1min ,

log

u

q

c

cnn

1

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Conclusion Conclusion

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ConclusionsConclusions

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.

Thank you !Thank you !

Documents

Evolution-cast: Temporal Evolution in Wireless Social Networks and Its Impact on Capacity Luoyi Fu, Jinbei Zhang, Xinbing Wang Department of Electronic