Reachability estimation in temporal networks

How to efficiently estimate reachability from all nodes at all times in a temporal network.

Transcript

1. Arash Badie-Modiri, Márton Karsai, Mikko Kivelä Efficient (limited time) reachability estimation in temporal networks Norman Rockwell (1894-1978), “The Gossips,” 1948. Open access on PRE: https://doi.org/10.1103/PhysRevE.101.052303
2. Reachability • How much can an effect/epidemic/information spread? • Works great on static networks and teaches us a lot. How to “export” this to temporal networks? • One can do an O(|E|) Connection Scan/BFS from a representative sample of nodes and starting times? Is there a way to calculate reachability for all times and nodes faster than this? Holme, P. (2005). Network reachability of real-world contact sequences. Physical Review E, 71(4), 046119.
3. Is there a path from point A at time t 1 to point B at time t 2 ? versus How many different destinations can be reached? How many events can participate in the process? How long will the epidemic last? Reachability (cont.)
4. Temporal Network
5. Temporal Network One possible spreading path
6. It’s going to take some time... How long it takes to calculate reachability from every seed?
7. Adjacency & Causal relationship Alice hears some quality gossip during a call with Bob at time t 2 . Can this be logically attributed to Bob’s call with Carol at time t 1 ? • Did event e 1 cause event e 2 ? • Can the effect of e 2 be plausibly attributed to e 1 ?
8. Adjacency & Causal relationship Alice hears some quality gossip during a call with Bob at time t 2 . Can this be logically attributed to Bob’s call with Carol at time t 1 ? • Did event e 1 cause event e 2 ? • Can the effect of e 2 be plausibly attributed to e 1 ?
9. Event graphs Second order graph where events are the nodes and adjacent events are connected with a directed link. The link can have the weight equal to time difference between the events. Cycles are not logically possible, therefore a weighted temporal event graphs is a directed acyclic graph. Kivelä, M. et al. (2018). Mapping temporal-network percolation to weighted, static event graphs. Scientific reports, 8(1), 12357.
10. Event graphs
11. Adjacency & Causal relationship (cont.) You can also think of causal chains: a chain of events where each two consecutive events are adjacent to each other. Each directed path in the event graph is a causal chain of adjacent events. Also known as a directed time-respecting path.
12. Weighted temporal event graphs
13. Weighted temporal event graphs
14. Adjacency & Causal relationship (cont.) “Logically attributable” is a low bar. We can narrow it down by putting a maximum waiting time cutoff between events to be considered adjacent; i.e. t(e 1 , e 2 ) ≤ Δt
15. Limited waiting-time reachability
16. In- and out-components on the event graph Out-component represents the maximum possible set of “affected” events (or nodes). In event graph terms it is the set of events “downstream” of an event. Example: If Alice shares some killer gossip with Bob at time t: • What is the maximum possible times it would come up in conversations? • What about maximum number of people hearing it? • How long it can possibly circulate before dying out?
17. In- and out-components on the event graph Out-component represents the maximum possible set of “affected” events (or nodes). In event graph terms it is the set of events “downstream” of an event. Example: If Alice shares some killer gossip with Bob at time t: • What is the maximum possible times it would come up in conversations? • What about maximum number of people hearing it? • How long it can possibly circulate before dying out?
18. In-component Out-component Saramäki, J. (2019). Temporal Networks: Past, present, future. Slides for keynote at Complex Networks 2019. Available from: https://jarisaramaki.com.
19. Measuring in- and out-component sizes (cont.)
20. Probabilistic cardinality estimators: HyperLogLog++ The idea is to store a sketch of elements instead of every single element or hashes of all elements. In this case: maximum observed number of leading zeros in the hashes of inputs. Relative error of 1.04/√m with a space of 6m bits, e.g. 3.25% error with 640 bytes. Flajolet et al. (2007). Hyperloglog: the analysis of a near-optimal cardinality estimation algorithm. In Discrete Mathematics and Theoretical Computer Science (pp. 137-156). Heule et al. (2013). HyperLogLog in practice: algorithmic engineering of a state of the art cardinality estimation algorithm. In Proceedings of the 16th International Conference on Extending Database Technology (pp. 683-692). ACM.
21. Upshot Estimate reachability from all starting nodes and times with O(|E|log|E|) time complexity compared to current O(|E|2) algorithms for exact results.
22. Slides available at https://speakerdeck.com/arashbm Implementation on GitHub at https://github.com/CxAalto/event-graph Publication: Badie-Modiri, A., Karsai, M., & Kivelä, M. (2020). Efficient limited-time reachability estimation in temporal networks. Physical Review E, 101(5), 052303.