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 July-August 2013ICCCN 20132

3. Forwarding in a Finite Network n 11 n 12 n 13 n 14 n1n1 n2n2 n3n3 n4n4 n 21 n 22 n 23 n 24 n0n0 July-August 2013ICCCN 20133

4. 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 July-August 2013ICCCN 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: July-August 2013ICCCN 20135

6. Calculating the pdfs July-August 2013ICCCN 20136 D l -1 S l -1 C l k SlSl DlDl sprev[j] scurr[j] pdf of number of distinct nodes at level l -1 and all prior levels pdf of number of new nodes at level l -1 pdf of number of new nodes

7. Upper Bound Analysis  Upper bound on the number of nodes reached at level l  UB s = min(c l, n)  Upper bound on the number of nodes reached up through level l  UB d = min(1 + c + c 2 + … + c l, n) = min( (c l+1 – 1) / (c – 1), n) if c > 1 July-August 2013ICCCN 20137

8. Nodes Reached Upper Bound vs. pdf Analysis July-August 2013ICCCN 20138 c=4

9. pdfs for Total Nodes Reached Varying c July-August 2013ICCCN 20139 l = 10

10. pdfs for Total Nodes Reached Varying l July-August 2013ICCCN 201310 c=4

11. pdfs for Total Nodes Reached Varying f July-August 2013ICCCN 201311 c = 4 l = 10

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 July-August 2013ICCCN 201312

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 July-August 2013ICCCN 201313

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 July-August 2013ICCCN 201314

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 July-August 2013ICCCN 201315