Patterns And A Generative Model

1. Patterns And A Generative Model Authors: JianweiNiu, Wanjiun Liao, Jing Peng, Chao Tong Published: Performance Computing and Communications Conference (IPCCC), 2012 IEEE 31st International Presenter: Guoming Wang Jan 24, 2014

2. Outline • Introduction • Definitions And Properties • Observations • Model

3.  Temporal evolution of social networks Temporal evolution of social networks has attracted considerable interest among researchers • How do social network evolve through network? • How do the different components of an entire network form and die? • Is the second largest component in a network really very small in size?

4. Method Two types of mergings: 1. mergings between the disconnected components themselves 2. Mergings between the disconnected components and the giant component ? ! Empirical observations Propose model Analyze the mergings Problem Future work Datasets studied

5. G = (V, E) Graph G consists of a set of nodes V and a set of edges E. 1 3 GCC Longevity of DCs The typical directed networks can be stated By a “bow-tie” structure, and usually have a giant connected component(GCC) which involves a significantly large fraction of nodes The length of the period from the birth of a DC to its death. It reflects how long the DCs can live before they merge with other DCs 2 4 DC Final size of the DCs The disconnected components(DC) of a network are defined as the small components that are not connected to any other components in the network The number of nodes in that DC when it dies. It reveals how large DCs can grow before dying out Definitions and interested Properties What properties or patterns can we obtain from the temporal evolution of DCs ?

6. Observation1 Longevity distribution of DCs in each dataset All plots demonstrate a decaying trend, and after the vertical line x = k, the curves begin to oscillate apparently. The units of the longevity values are in snapshots. The longevity of each DC counts the number of snapshots that such a DC can live in the observation period of each dataset.

7. Observation2 Final size distribution of DCs in each dataset. The fitted lines and the slopes are shown. The plots demonstrate power laws

8. Observations • Observation1 • The curve of the overall DC longevity distribution shows a decaying trend, with oscillations at the tail , indicating that the short-lived DCs account for a large fraction of all the DCs, and the longevity of long-lived DCs do not follow a simple fixed pattern. • Observation2(Final Size Power Law(FSPL)) • Let sc denote the final size of a disconnected component, and let nc denote the number of disconnected components whose final size are sc. Then nc and sc follow a power law with exponent B

9. Longevity distribution of DCs that are absorbed by GCC The trend of the curves revealing the similarity between these distributions and the overall longevity distribution shown previously. Which can be regarded as a hint that the majority of DCs are absorbed by the GCC.

10. Final size distribution of DCs that are absorted by GCC The fitted lines and the slopes demonstrate again the FSPL and similarity between these distributions and the overall final size distribution

11. Longevity distribution of DCs that merge with other DCs The curves show a decaying trend, where short-lived DCs are common. There are still small spikes in the decaying part of the curves, and after that, the curves enter a relatively stable state.

12. Final size distribution of DCs that merge with other DCs

13. Different timestamp

14. Mergings of Disconnected Components A DC can either be absorbed by the GCC, or merge with other DCs; The target is to know if there is an obvious difference in number between these two types of mergings Number of DCs is much less compared with the mergings involving the GCC. Long-lived DCs are seldom involved in the DC mergings The count of DCs decreases as the longevity value increases.

15. Observation3 • The mergings among DCs are all small in size, and specially most of the mergings happen between two DCs.

16. Proposed Model • The decaying trend in DC longevity distribution • The FSPL in the final-size distribution of DCs • The small merging sizes

17. Surfer Model – generative steps 1. v chooses a host node u uniformly at random from all the existing nodes in the network, and forms a link to u 2. Generate a random number count which is geometrically distributed with expectation pfrnd/(1-pfrnd).Node v randomly chooses count edges of u expect the edge as it can if there are not enough edges. 3. Let x1, x2…xk denote the nodes on the other ends of these chosen edges in step(2). Node v forms links to x1, x2…xk, and then goes to step(2), recursively visiting all these nodes

18. Surfer Model – Generative steps depiction

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21. Conclusion and Future work Empirical observations and analyse longevity and final sizeof DC Longevity: decaying trend Final-size: power law Detecting anomalies and forcasting the future states…? Surfer model

22. Thank you – Enjoy the rest of your night 21