Probabilistic Analysis of Message Forwarding

1. Probabilistic Analysis of Message Forwarding Louise Moser and Michael Melliar-Smith University of California, Santa Barbara

2. Message Forwarding • Direct multicasting of messages to many nodes is expensive and might be infeasible, if the network is large • Source node transmits its message to a small number of nodes • Each such node retransmits the message to other nodes, chosen at random • Spreads the load across multiple nodes • Introduce a probability of message forwarding • Limit the number of levels of message forwarding • It is easy to calculate an upper bound on the number of nodes reached, when duplicate nodes are ignored ICCCN 2013

3. Forwarding in a Finite Network n0 n1 n2 n3 n4 n11 n12 n13 n14 n21 n22 n23 n24 ICCCN 2013

4. The pdf Algorithm • Input of the algorithm: • n: number of nodes in the network • c: number of nodes to which a node forwards a message • f: probability with which a node forwards a message • l: number of levels of message forwarding • Output of the algorithm: • pdf for the number of nodes reached at level l • expected number of nodes reached at level l • pdf for the number of nodes reached up through level l • expected number of nodes reached up through level l ICCCN 2013

5. Probability of a Duplicate Node Let N be a set of nodes having cardinality n A be a subset of N having cardinality a B be a subset of N having cardinality b where a ≤ b The pdf p(k), 0 ≤ k ≤ a, that A ∩ B has cardinality k, is given by: ICCCN 2013

6. Calculating the pdfs pdf of number of distinct nodes at level l-1 and all prior levels Dl-1 Sl-1 pdf of number of new nodes at level l-1 Clk pdf of number of new nodes sprev[j] scurr[j] Sl Dl ICCCN 2013

7. Upper Bound Analysis • Upper bound on the number of nodes reached at level l • UBs = min(cl,n) • Upper bound on the number of nodes reached up through level l • UBd = min(1 + c + c2 + … + cl , n) = min( (cl+1–1) / (c – 1), n) if c > 1 ICCCN 2013

8. Nodes Reached Upper Bound vs. pdf Analysis c=4 ICCCN 2013

9. pdfs for Total Nodes Reached Varying c l = 10 ICCCN 2013

10. pdfs for Total Nodes Reached Varying l c=4 ICCCN 2013

11. pdfs for Total Nodes Reached Varying f c = 4 l = 10 ICCCN 2013

12. Related Work S. E. Deering and D. R. Cheriton, “Multicast routing in datagram internetworks and extended LANs,” ACM Trans. Computer Systems, vol. 8, no. 2, pp. 85-110, 1990 Z. J. Hass, J. Y. Halpern and L. Li, “Gossip-based ad hoc routing,” IEEE/ACM Trans. Networking, vol. 14, no. 3, pp. 479-491, June 2006 S. M. Hedetniemi, S. T. Hedetniemi and A. L. Liestman, “A survey of gossiping and broadcasting in communications networks,” Networks, vol. 18, pp. 319-349, 1988 D. Shah, “Gossip algorithms,” Foundations and Trends in Networking, vol. 3, no. 1, pp. 1-125, 2008 ICCCN 2013

13. Conclusions • The expected numbers of nodes reached up through a given level, and at a given level, are substantially less than those for the upper bound analysis • The pdfs for the number of nodes reached up through a given level, and at a given level, exhibit a wide range, particularly for smaller values of the • Degree of forwarding • Probability of forwarding • As the forwarding probability decreases, the expected number of nodes reached decreases quite rapidly ICCCN 2013

14. Future Work • Neighborhoods • fully connected within neighborhoods but only partially connected between neighborhoods • Faulty nodes and links • Time to reach a certain number of nodes • Energy and power consumption at nodes • Application to existing iTrust system • decentralized publication, search and retrieval ICCCN 2013

15. Questions? Comments? Louise Moser -moser@ece.ucsb.edu Michael Melliar-Smith - pmms@ece.ucsb.edu This work was supported in part by National Science Foundation Grant NSF CNS 10-16193 ICCCN 2013