1. Gossiping with IOIMCs Pepijn Crouzen Saarland University

2. Gossiping models: the basics Networks consist of simple nodes. Broadcasts are forwarded to a (small) number of neighbors. A node does not have to know the entire network. A node does not have to know who has received which messages.

3. What did we model? Constant, but arbitrary, number of nodes, Constant, but arbitrary, interconnections, Multiple messages from multiple sources, Individual message reception, Delayed, probabilistic message forwarding, Resulting model: labeled CTMC, Scalable model generation with CADP. Goal: stochastic validation on message reception times. Focus: Scalable model generation + Information spread

4. What did we leave out? Dynamics: – New nodes appearing, – Nodes dying, – Interconnections changing. Message buffers, Message content.

5. How did we model gossiping? Using Input/Output Interactive Markov Chains: Each node is modeled by an I/O-IMC, Messages are sent through output signals and received through input signals. New messages are received through system-inputs and message reception is signaled using system- outputs. Network model is constructed through composition of the node models.

6. Simple node model A B C M(B)? M(C)? REC(A)! M(A)! p.λ (1-p).λ Waiting rate = λ, Sending probability = p, Messages are identified by sending node, While waiting to send, incoming messages are ignored Node also waits when not sending! START(A)?

7. Scalability: Adding links M(B)? ADD(A)? REC(A)! M(A)! p.λ (1-p).λ ADD(A)! M(C)? |[ADD(A)]| hide ADD(A) in ADD2(A)? rename ADD2(A) -> ADD(A) in

8. Scalability: Adding links, result M(B)? M(C)? REC(A)! M(A)! p.λ (1-p).λ ADD(A)? Now we can generate any gossiping network using: – Composition – Abstraction – Minimization – Renaming On: – Node model (0 links) – Add-neighbor model

9. Case study 15 node network, Each node has 3 neighbors, Convert each node to an I/O-IMC, Compute the total network model using compositional aggregation, Compose the network model with a message generation model and a message reception models, Compute probability that an incoming message reaches all nodes after some time period using resulting labeled CTMC.

10. Message generation and reception START(NODE1)!REC(NODE2)? Node 2 has received the message Network START signals REC signals Hide the START and REC signals, Weak bisimulation minimization Labeled CTMC x15

11. Case study: results Generation time: +/- 2 hours Largest appearing model: 223743 states, 1241054 transitions CTMC size (anonymous reception): 233 states Analysis time: <1 second And now the probability that all nodes receive a message with send-rate 0.01 and send-probability 70%

12. Conclusion Scalable complete state space generation for gossiping networks is possible using very simple base models, but: We run into the state space explosion fairly early, Advanced maximal progress cutting is needed to make it feasible, No dynamics!