Persistence on Disordered Networks

1. Persistence on Disordered Networks Omar Malik

2. Outline • Diffusion • Percolation • Persistence • Other Spreading Processes

3. Diffusion • The net statistical movement of particles. • The spread of the smell of perfume throughout a room. • The spread of news on social media. • The spread of a highly contagious disease through a population.

4. Diffusion • Described by a differential equation. The diffusive field. Diffusion constant

5. Diffusion • Must be discretized to describe the process on networks • F bundles the rate of diffusion and the length of the discrete time step. • If it is too large the equation does not converge properly. • For networks the maximum number that F can take is related to the largest degree of the network Field at i-th location at (n+1) time step Diffusion constant

6. Percolation • With probability (1-p) we remove a connection between nodes. • The classical example on this lattice is that of a liquid pouring through the lattice. • Can the liquid move from the top to the bottom? [3] Stauffer, D. et al. Introduction to Percolation Theory. 2003

7. Percolation • There is a critical threshold of p above which an infinite cluster spanning the length of the lattice appears. • This value is called the percolation threshold, and it varies with network topology. • What is the percolation threshold for a 1D chain? [3] Stauffer, D. et al. Introduction to Percolation Theory. 2003

8. Percolation on a 2D Lattice • Percolation on a 2D lattice creates a fractal lattice [1]. • The percolation threshold marks the point at which a cluster that spans the space of the lattice first appears. • For a 2D lattice the fractal dimension above the percolation threshold is 2, while at the percolation it is 1.896 [2]. • The percolation threshold therefore marks a transition in the network topology. [1] Arkadiusz Kosior and Krzysztof Sacha. Localization in random fractal lattices. Phys. Rev. B, 95:104206, Mar 2017. [2] M. B. Isichenko. Percolation, statistical topography, and transport in random media. Rev. Mod. Phys., 64:961-1043, Oct 1992.

9. Giant Component and Percolation • In the context of network theory, the percolation threshold marks the point at which a giant component first appears in the network. • Different networks can have drastically different resilience to edge removal.

10. Diffusion on Percolating Lattice Distance Traversed in Random Walk • Diffusion on a lattice (such as the ant in a labyrinth problem) is associated with power law behavior [3]. • In disordered media there are different power law exponents at the critical percolation threshold and for the ordered site. • R ∝ t1/2 for 𝜙=1. • R ∝ t1/3 for 𝜙=𝜙c=0.59. • For intermediate values of 𝜙 the slope is between the 𝜙=𝜙cand 𝜙=1 cases. 1-𝜙 is the fraction of removed nodes. 𝜙=1 𝜙=0.7 𝜙c=0.6 𝜙=0.4 [3] Stauffer, D. et al. Introduction to Percolation Theory. 2003

11. Introduction to Persistence • The local persistence probability is defined as the probability that the field at a node does not change sign up to a certain time [4]. We define persistence, P(t), as the fraction of nodes that have not changed sign up to a certain time. • Persistence is associated with power-law decay on d-dimensional regular lattices, with the exponent, 𝜃, depending on d [5]. • For a 2-dimensional diffusive process this exponent is known to be 𝜃 ≅ 0.1875. [4] Satya N. Majumdar, Clement Sire, Alan J. Bray, and Stephen J. Cornell. Nontrivial exponent for simple diffusion. Phys. Rev. Lett., 77:2867-2870, Sep 1996. [5] H. Hinrichsen, H.M. Koduvely. Numerical study of local and global persistence in directed percolation. Eur. Phys. J. B5, 257-264 (1998)

12. Diffusive Persistence on Disordered Networks • We found that persistence on a 2D network is characterized by a power law with 𝜃 = 0.187 ± 0.02, which agrees with the literature value [6]. • As the network becomes disordered, characterized by 1-𝜙, the fraction of removed edges, the exponent maintains its value until it gets close to the percolation threshold. • At the critical bond percolation threshold of 𝜙 = 0.5, 𝜃 is about 0.10. • These exponents remain unaffected by network size. [6] T. J. Newman and Z. Toroczkai. Diffusive persistence and the “sign-time" distribution. Phys. Rev. E, 58:R2685-R2688, Sep 1998.

13. Diffusive Persistence on Disordered Networks • To ensure that our exponent is not being affected by our selection of data we pick different starting points, ti (normalized to the total run time), to fit the data. • We see that as we get closer to the tail of the curves (i.e. when ti is close to 1) the values of 𝜃 away from the percolation threshold converge towards 𝜃 = 0.187. • However, at the percolation threshold the exponent seems to converge to a different value entirely.

14. Diffusive Persistence on Disordered Random Networks • In contrast to the clear power-law scaling of the 2D network, random networks, such as Erdős-Rényi and k-regular random networks, show no clear scaling behavior. • 1-𝜙 is the fraction of removed edges.

15. SIS Model • The SIS model is another model of spread. • Originally this model was proposed for population level characterization of epidemic spread. • However, it can be extended to networks. [3] Newman, M. Networks: An Introduction. 2010

16. SIS Model • We pick a node, and if it is susceptible we select one of its neighbors randomly. • If the neighbor is infected the node becomes infected with probability 𝛽. • If the node was infected it becomes susceptible with probability 𝛾. [3] Newman, M. Networks: An Introduction. 2010

17. Critical Point for the SIS Model • For any given value of 𝛽 there is a critical value of 𝛾 above which the infection is highly likely to die out. • This critical value changes depending on the value of 𝜙.

18. Persistence of the SIS Model • Like the diffusive case, we see that the behavior changes drastically below the percolation threshold.

19. Threshold Model • Yet another model of spread is the threshold model. • Typically used to model opinion formation in social situations. • If a fraction 𝛽 of a node’s neighbors are infected, the node becomes infected. • With probability 𝛾 an infected node can become susceptible. [7] Granovetter, M. Threshold Models of Critical Behavior. 1978

20. Critical Point for the Threshold Model • Like the SIS model, there is an interplay between the infection and recovery parameters. • Fixing the infection parameter and changing the recovery parameter we eventually reach a point where the infection dies out. • This point is also dependent on 𝝓. • Why is the 𝝓=1 case the same for both graphs?

21. Persistence on the Threshold Model • As before, we see a marked difference in the persistence behavior above and below the percolation threshold.

22. Conclusion • Network topology determines the persistence behavior • Networks with regular structure show power-law scaling. • This scaling is absent in random graphs. • For diffusive persistence the exponent of the power-law does not change above the percolation threshold. • For persistence on the SIS and threshold model we also see the persistence behavior changes drastically at the percolation threshold.