Estimating the Information Source under Decaying Diffusion Rates
Abstract
:1. Introduction
- First, we use the MLE to find the source for decaying diffusion and show that the MLE is same as that of the homogeneous diffusion rate of information if the diffusion decays with respect to the distance from the source. This implies that the MLE of our model also has the same graphical centrality property called rumor center in [1]. This enables us to analyze the detection performance for the decaying rate scenario.
- Second, we define two exponential decaying models: Simple exponential decay and Generalized exponential decay. The simple exponential decay is a kind of light tail distribution, but the generalized exponential decay covers light and heavy tail distribution in the sense of the decaying pattern. We then obtain the closed-form of detection probability of the MLE when the underlying graph is a regular tree for both decaying models. Different to the prior result in [1], the detection probability is larger than zero in the line graph and there is a non-neglectable improvement of detection for any degree of a regular tree.
- Third, we consider the case that the decaying model parameter is hidden for two decaying models above. This is a more realistic scenario because knowing the exact parameter of the model is not easy in practice. To do that, we first derive MLE to estimate this parameter and show that it needs exponential computing time. Hence, we design a heuristic estimation algorithm for the true parameter by using the diffusion snapshot information, appropriately.
- Finally, we validate our theoretical result for the regular tree using the MLE and for over popular random graphs (Erdös-Rényi, scale-free and small-world graphs) and real-world networks (US-power grid, Facebook and Wiki vote) using the heuristic Breath-First-Search (BFS) estimator. As a result, we see that the detection probability can be above 80% for the regular tree and it can be above 30% if the diffusion rate decays, whereas it is about 20% without decaying in the Facebook graph.
2. Related Work
3. Model and Estimator
4. Main Results
4.1. Probability of Correct Detection of MLE
4.2. Decaying Parameter Estimation
Algorithm 1 Decaying Parameter Estimation (DPE(K)) |
Input: Diffusion snapshot , sampling cost K, , , increasing step size |
Output: Estimation parameter |
Set the initial decaying parameter ; |
while do |
for each do |
Step1: Compute the rumor centrality by a message passing algorithm [1]; |
Step2: Choose random samples K times and compute its mean by; |
Step3: Set ; |
end for |
Set ←; |
; |
end while |
Compute ; |
Return ; |
5. Proof of Results
5.1. Proof of Proposition 1
5.2. Proof of Theorem 1
5.3. Proof of Theorem 2
5.4. Proof of Theorem 3
6. Numerical and Simulation Results
6.1. Regular Trees
6.2. Random Graphs
6.3. Real World Graphs
7. Discussion
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Proof of Lemmas
Appendix A.1. Proof of Lemma 2
Appendix A.2. Proof of Lemma 3
Appendix A.3. Proof of Lemma 4
Appendix A.4. Proof of Lemma 5
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True p | ER | SF | SW | PG | FB | WiKi |
---|---|---|---|---|---|---|
2 | 1.82 | 2.23 | 2.08 | 1.93 | 1.81 | 1.93 |
3 | 3.26 | 3.72 | 3.35 | 2.83 | 2.84 | 3.11 |
4 | 3.71 | 3.83 | 4.23 | 4.32 | 3.81 | 4.13 |
5 | 4.84 | 4.72 | 4.73 | 5.28 | 4.93 | 4.91 |
6 | 5.84 | 5.82 | 5.88 | 6.28 | 5.83 | 5.91 |
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Woo, J.; Choi, J. Estimating the Information Source under Decaying Diffusion Rates. Electronics 2019, 8, 1384. https://doi.org/10.3390/electronics8121384
Woo J, Choi J. Estimating the Information Source under Decaying Diffusion Rates. Electronics. 2019; 8(12):1384. https://doi.org/10.3390/electronics8121384
Chicago/Turabian StyleWoo, Jiin, and Jaeyoung Choi. 2019. "Estimating the Information Source under Decaying Diffusion Rates" Electronics 8, no. 12: 1384. https://doi.org/10.3390/electronics8121384
APA StyleWoo, J., & Choi, J. (2019). Estimating the Information Source under Decaying Diffusion Rates. Electronics, 8(12), 1384. https://doi.org/10.3390/electronics8121384