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 [1][2][3] [1] E. Dlay and M. Haahr, “Social Network Analysis for Routing in Disconnected Delay-Tolerant MANETs”, in ACM MobiHoc’07, Montreal,Quebec, Canada, [2] P. Hui, J. Crowcroft, E. Yoneki, “BUBBLE Rap: Social-based Forwarding in Delay Tolerant Networks”, in ACM MobiHoc’08, Hong Kong, China, [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, Capacity receives little investigation under wireless social networks.

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 [4][5][6][7][8]): 1. New node joins the network over time 1. New node joins the network over time 2. New links established between nodes 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, , [5] N. Perra, A. Baronchelli, D. Mocanu, B. Goncalves, R. PastorSatorras, A. Vespignani, “Walking and Searching in Time-varying Networks”, arXiv: , [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): e , [7] L. Rocha, A. Decuyper, V. Blondel, “Epidemics on a Stochastic Model of Temporal Network”, arXiv: , [8] L. Rocha, V. Blondel, “Temporal Heterogeneities Increase the Prevalence of Epidemics on Evolving Networks”, arXiv: , 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] [9]S. Lattanzi and D. Sivakumar, “Affiliation Networks”, in Proc. ACM STOC’09, 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 a new arriving node is chosen to be either a source or a destination of a randomly chosen node in existing network destination of a randomly chosen node in existing network message sharing between limited number of individuals message sharing between limited number of individuals  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 model

10 Definition  Feasible Capacity: We say that a per node capacity λ(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 λ(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 Ψ packets transmitted from source to destination, then, we say the average per-node capacity is at time t, after t exceeds a specific value t 0. Here t 0 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 Θ (f(t)) if there is a deterministic constant 0 < c1 < c2 < +∞ 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 → ∞. 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 Social relations do not affect capacity Only network evolution will affect capacity Only network evolution will affect capacity

13 Routing Scheme 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 time in our work. The only difference lies in that the number of nodes increases over time in our work. [10]X. 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 Θ(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: 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 ϕ (l) decaying as a power law of exponent σ, i.e., ϕ (l) ∼ l σ,. And here l ranges from to 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 ϕ (l) decaying as a power law of exponent σ, i.e., ϕ (l) ∼ l σ,. And here l ranges from to Θ(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. ① ② ③

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 1. transmission is only within a certain transmission range 2. the average source-destination distance is 2. the average source-destination distance is 3. New arrivals causes more bandwidth allocation 3. New arrivals causes more bandwidth allocation  The capacity can be improved in heterogeneous topology: 1. a constant capacity is achievable when 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 t=n, we have 1. Unicast: Fixing t=n, we have 2. Multicast: Fixing t=n, we have 2. Multicast: Fixing t=n, 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 , Mar [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 t=n, we have 1. Unicast: Fixing t=n, we have Almost constant capacity whenAlmost constant capacity when Close to the Θ(1) capacity in [13]Close to the Θ(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

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.

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