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

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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

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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

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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

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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).

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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).

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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

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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 :

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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. [Watts & Strogatz, 1998] Examples of network structure

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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: Examples of network structure

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Examples [Newman, SIAM Review 2003]

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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).

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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)

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Global Cascades and Complex Networks Structures and dynamics review see: M.E.J. Newman, SIAM Review 45, 167 (2003). S.N. Dorogovtsev et al., arXiv:0705.0010 (2007) 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) Initially small localized effects can propagate over the whole network, causing a global cascade

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

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

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

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

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Watts` model Cascade condition: Thresholds CDF:

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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…

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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.

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Results

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Main result Our result: with and

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Cascade condition q G(q)

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Cascade condition q G(q)

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

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Simple cascade condition

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slope=1 Extended cascade condition 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 : above

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Extended cascade condition

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Gaussian threshold distribution

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Bifurcation analysis

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Results: Scale-free networks

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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).

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Consider undirected unweighted network of N nodes (N is large) defined by degree distribution p k Watts` model of global cascades Updating: node i becomes active if the active fraction of its neighbours exceeds its threshold Each node i has: binary state fixed threshold given by thresholds CDF Initially activate fraction ρ 0 <<1 of N nodes. The average fraction of active nodes (probability that a node has threshold < r)

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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. Derivation of result

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

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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. Derivation of result ………………….. ∞ n+2 n+1 n … ……………… … … … A … : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

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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) Derivation of result ………………….. ∞ n+2 n+1 n … ……………… … … … A … : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

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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) Derivation of result ………………….. ∞ n+2 n+1 n … ……………… … … … A … : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive. (m out of k-1 children active) k-1 children Degree distribution of nearest neighbours:

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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) Derivation of result ………………….. ∞ n+2 n+1 n … ……………… … … … A … : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive. k-1 children

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

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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…

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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).

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

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

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

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K-core: the largest subgraph of a network whose nodes have degree at least K Initially activate (damage) fraction ρ 0 of nodes. A node becomes active if it has fewer than K inactive neighbours: Final inactive fraction (1- ρ) of the total network gives the size of K-core Generalization to other dynamical models Our result for the average fraction of active nodes

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K-core sizes on degree-degree correlated networks Initial damage ρ 0 r = 0 r = -0.5 r = 0.98 Theory vs Numerics: 7-cores in Poisson random graphs with z = 10 Case r = 0 considered in S.N. Dorogovtsev et al., PRL 96, 040601 (2006).

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Adopt approach of M. Newman for percolation problems (PRE 67, 026126 (2003), PRL 89, 208701 (2002)). Degree-degree correlated networks P(k,k’) – joint PDF that an edge connects vertices with degrees k, k’ – probability that a k-degree node is active (conditioned on its parent being inactive) – probability that a child of an inactive k-degree node is active n+1 ………………….. ……………… … … … Consider a k-degree node at level n+1: n

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Degree-degree correlated networks (Also obtain a cascade condition in matrix form).

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Pearson correlation r Degree-degree correlated networks Initial damage ρ 0 r = 0 r = -0.5 r = 0.98 Correlated networks (10 5 nodes) generated using Gaussian copula. Theory (curves) vs Numerics (symbols): 7-cores in Poisson random graphs with z = 10 Case r = 0 considered in S.N. Dorogovtsev et al., PRL 96, 040601 (2006). (zero initial damage)

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Predicting K-cores in CAIDA internet router network Internet router network structure from www.caida.orgwww.caida.org Degree distribution Degree-degree correlation matrix k

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Predicting K-cores in CAIDA internet router network Predicted from analysis of degree distribution only (see S.N. Dorogovtsev et al., PRL 96, 040601 (2006)). Actual size Us: Predicted from analysis of degree distribution and degree-degree correlation. Internet router network structure from www.caida.orgwww.caida.org

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Similar idea, but instead of P(k,k’) use the mixing matrix e, which quantifies connections between different communities. Modular networks; asynchronous updating Asynchronous updating gives continuous time evolution:

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Modular networks example

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Summary 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).

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

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What is best “random” model for the Internet? Jellyfish model: Siganos et al., J. Comm. Networks ‘06 Medusa model: Carmi et al., Proc. Nat. Acad. Sci. ‘07

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Internet structure using router data from CAIDA Transmissibility (bond Occupation probability)

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DEPFA Bank collaboration: CDO pricing m1m1 p1p1 mNmN pNpN m2m2 p2p2 m3m3 p3p3 m4m4 p4p4 Definitions m i Notional of credit i p i Default probability of credit i, (derived from the CDS quote). S q Fair price for protection against losses in tranche q Problem Existing models fail to reproduce the prices (S q ) observed on the market. {m 1, m 2,…,m N } {p 1, p 2,…,p N } {S 1, S 2,…,S s } Correlation Structure ? 0 to 5% 10% to 15% 15% to 25% 25% to 35% S1S1 S2S2 S3S3 S4S4 S5S5 35% to 100%

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An external field Stochastic Dynamics on Networks

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Hysteresis: PRG Stochastic Dynamics on Networks

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Hysteresis: PRG Stochastic Dynamics on Networks

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Stochastic dynamics Aim: Fundamental understanding of the interactions between nonlinear dynamical systems and random fluctuations. External noise sources e.g. transistor noise, thermal noise. Heterogeneity within system e.g. agent-based models, large-scale networks. Tools: Numerical simulations …guiding fundamental understanding via… Asymptotic methods Perturbation techniques Exact solutions

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Noise in oscillators (Theme 1) Prof. M. P. Kennedy, Microelectronic Engineering, UCC New computational and asymptotic methods for the spectrum of an oscillator subject to white noise Stochastic perturbation methods for effects of coloured noise Collaboration (Feely/Kennedy): Noise effects in digital phase- locked loops Papers: SIAM J. Appl. Math. IEEE TCAS

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Microfluidic mixing and sorting (Theme 3) Experimentalists at Tyndall National Institute, Cork Analysis of MHD micromixing in annular geometries Modelling of micro-sorting methods Collaborations: (Lindenberg/Sancho) Noise-induced sorting techniques for microparticles Papers: SIAM J. Appl. Math. Phys. Rev E Phys. Fluids

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

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