Cascades on correlated and modular networks

1. Cascades on correlated and modular networks James P. Gleeson Department of Mathematics and Statistics, University of Limerick, Ireland www.ul.ie/gleesonj

2. Collaborators and funding • Sergey Melnik, UL • Diarmuid Cahalane, UCC (now Cornell) • Rich Braun, University of Delaware • Donal Gallagher, DEPFA Bank • SFI Investigator Award • MACSI (SFI Maths Initiative) • IRCSET Embark studentship

3. Some areas of interest • Noise effects on oscillators • Applications: Microelectronic circuit design • Diffusion in microfluidic devices • Applications: Sorting and mixing devices • Complex systems • Agent-based modelling • Dynamics on complex networks • Applications: Pricing financial derivatives

4. Some areas of interest • Noise effects on oscillators • Applications: Microelectronic circuit design • Diffusion in microfluidic devices • Applications: Sorting and mixing devices • Complex systems • Agent-based modelling • Dynamics on complex networks • Applications: Pricing financial derivatives

5. Overview • Structure of complex networks • Dynamics on complex networks • Derivation of main result • Extensions and applications • J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007). • J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).

6. Overview • Structure of complex networks • Dynamics on complex networks • Derivation of main result • Extensions and applications • J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007). • J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).

7. What is a network? A collection of N “nodes” or “vertices” which can be labelled i… …connected by links or “edges”, {i,j}. • Examples: • World wide web • Internet • Social networks • Networks of neurons • Coupled dynamical systems

8. Examples of network structure The Erdós-Rényi random graph Consider all possible links, create any link with a given probability p. Degree distribution is Poisson with mean z:

9. [Watts & Strogatz, 1998] Examples of network structure The Small World network Start with a regular ring having links to k nearest neighbours. Then visit every link and rewire it with probability p.

10. Examples of network structure Scale-free networks Many real-world networks (social, internet, WWW) are found to have scale-free degree distributions. “Scale-free” refers to the power law form:

11. Examples [Newman, SIAM Review 2003]

12. Overview • Structure of complex networks • Dynamics on complex networks • Derivation of main result • Extensions and applications • J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007). • J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).

13. Dynamics on networks • Binary-valued nodes: • Epidemic models (SIS, SIR) • Threshold dynamics (Ising model, Watts) • ODEs at nodes: • Coupled dynamical systems • Coupled phase oscillators (Kuramoto model)

14. Examples of global cascades: Epidemics, computer viruses Spread of fads and innovations Cascading failures in infrastructure (e.g. power grid) networks Similarity: initial failures increase the likelihood of subsequent failures Cascade dynamics depends strongly on: Network topology (degree distribution, degree-degree correlations, community structure, clustering) Resilience of individual nodes (node response function) Global Cascades and Complex Networks Initially small localized effects can propagate over the whole network, causing a global cascade • Structures and dynamics review see: • M.E.J. Newman, SIAM Review 45, 167 (2003). • S.N. Dorogovtsev et al., arXiv:0705.0010 (2007)

15. Watts` model D.J. Watts, Proc. Nat. Acad. Sci. 99, 5766 (2002).

16. Threshold dynamics • The network: • aij is the adjacency matrix (N ×N) • un-weighted • undirected • The nodes: • are labelled i , i from 1 to N; • have a state ; • and a thresholdri from some distribution.

17. Node i has state and threshold Neighbourhood average: Threshold dynamics Updating: The fraction of nodes in state vi=1 is r(t):

18. Watts` model D.J. Watts, Proc. Nat. Acad. Sci. 99, 5766 (2002).

19. Watts` model Cascade condition: Thresholds CDF:

20. Watts` model Watts: initially activate single node (of N), determine if at steady state. Us: initially activate a fraction of the nodes, and determine the steady state value of Conditions for global cascades (and dependence on the size of the seed fraction) follow…

21. Main result Our result: with and Derivation: Generalizing zero-temperature random-field Ising model results from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna, J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.

22. Results

23. Results

24. Main result Our result: with and

25. Cascade condition G(q) q

26. Cascade condition G(q) q

27. slope=1 Simple cascade condition First-order cascade condition: using slope>1 demand (slope>1) for global cascades to be possible. This yields the condition reproducing Watts’ percolation result when and

28. Simple cascade condition

29. slope=1 Extended cascade condition above Second-order cascade condition: expand to second order and demand no positive zeros of the quadratic for global cascades to be possible. The extension is, to first order in :

30. Extended cascade condition

31. Gaussian threshold distribution

32. Gaussian threshold distribution

33. Bifurcation analysis

34. Results: Scale-free networks

35. Results: Scale-free networks

36. Overview • Structure of complex networks • Dynamics on complex networks • Derivation of main result • Extensions and applications • J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007). • J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).

37. Watts` model of global cascades Consider undirected unweighted network of N nodes (Nis large) defined by degree distribution pk • Each node i has: • binarystate • fixed threshold • given by thresholds CDF (probability that a node has threshold < r) Initially activate fraction ρ0<<1 of N nodes. Updating: node i becomes active if the active fraction of its neighbours exceeds its threshold The average fraction of active nodes

38. Derivation of result Derivation: Generalizing zero-temperature random-field Ising model results from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna, J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.

39. A Derivation of result Main idea: pick a node A at random and calculate its probability of becoming active. This will give ρ(∞).

40. ∞ A … … ……………… … … n+2 n+1 … n ………………….. Derivation of result Main idea: pick a node A at random and calculate its probability of becoming active. This will give ρ(∞). Re-arrange the network in the form of a tree with A being the root. : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

41. ∞ A … … ……………… … … n+2 n+1 … n ………………….. Derivation of result Main idea: pick a node A at random and calculate its probability of becoming active. This will give ρ(∞). Re-arrange the network in the form of a tree with A being the root. (initially active) (initially inactive) : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

42. ∞ A … … ……………… … … n+2 n+1 … n ………………….. Derivation of result Main idea: pick a node A at random and calculate its probability of becoming active. This will give ρ(∞). Re-arrange the network in the form of a tree with A being the root. (initially active) (initially inactive) (has degree k; k-1 children) (m out of k-1 children active) k-1 children : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive. Degree distribution of nearest neighbours:

43. ∞ A … … ……………… … … n+2 n+1 … n ………………….. Derivation of result Main idea: pick a node A at random and calculate its probability of becoming active. This will give ρ(∞). Re-arrange the network in the form of a tree with A being the root. (initially active) (initially inactive) (has degree k; k-1 children) (m out of k-1 children active) (activated by m active neighbours) k-1 children : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

44. Derivation of result Our result for the average fraction of active nodes Valid when: (i) Network structure is locally tree-like (vanishing clustering coefficient). (ii) The state of each node is altered at most once.

45. Conclusions • Demonstrated an analytical approach to determine the average avalanche size in Watts’ • model of threshold dynamics. • Derived extended condition for global cascades to occur; noted strong dependence on seed size. • Results apply for arbitrary degree distribution, but zero clustering important. • Further work…

46. Overview • Structure of complex networks • Dynamics on complex networks • Derivation of main result • Extensions and applications • J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007). • J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).

47. Extensions • Generalized dynamics: • SIR-type epidemics • Percolation • K-core sizes • Degree-degree correlations • Modular networks • Asynchronous updating • Non-zero clustering

48. Derivation of result Our result for the average fraction of active nodes

49. Fraction of active neighbours (Watts): Absolute number of active neighbours: Bond percolation: Site percolation: Generalization to other dynamical models Our result for the average fraction of active nodes

50. Generalization to other dynamical models Our result for the average fraction of active nodes K-core: the largest subgraph of a network whose nodes have degree at least K Initially activate (damage) fraction ρ0 of nodes. A node becomes active if it has fewer than Kinactive neighbours: Final inactive fraction (1- ρ) of the total network gives the size of K-core