Cascades on correlated and modular networks

James P. GleesonDepartment of Mathematics and Statistics,University of Limerick, Irelandwww.ul.ie/gleesonj

Collaborators and fundingSergey 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

Some areas of interestNoise effects on oscillatorsApplications: Microelectronic circuit design

Diffusion in microfluidic devicesApplications: Sorting and mixing devices

Complex systemsAgent-based modellingDynamics on complex networksApplications: Pricing financial derivatives

Some areas of interestNoise effects on oscillatorsApplications: Microelectronic circuit design

Diffusion in microfluidic devicesApplications: Sorting and mixing devices

Complex systemsAgent-based modellingDynamics on complex networksApplications: Pricing financial derivatives

OverviewStructure 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).

OverviewStructure 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).

What is a network?A collection of N nodes or vertices which can be labelled iconnected by links or edges, {i,j}.Examples:World wide webInternetSocial networksNetworks of neuronsCoupled dynamical systems

Examples of network structureThe Erds-Rnyi random graph

Consider all possible links,create any link with a given probability p.

Degree distribution is Poissonwith mean z:

The Small World networkStart with a regular ring having links to k nearest neighbours.Then visit every link and rewire it with probability p.Examples of network structure

Scale-free networksMany real-world networks (social, internet, WWW) are found to have scale-free degree distributions.

Scale-free refers to thepower law form:

Examples of network structure

Examples[Newman, SIAM Review 2003]

OverviewStructure 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).

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)

Examples of global cascades:Epidemics, computer virusesSpread of fads and innovationsCascading 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 NetworksStructures and dynamics review see: M.E.J. Newman, SIAM Review 45, 167 (2003). S.N. Dorogovtsev et al., arXiv:0705.0010 (2007)Initially small localized effects can propagate over the whole network, causing a global cascade

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

Threshold dynamicsThe network: aij is the adjacency matrix (N N)un-weightedundirected

The nodes:are labelled i , i from 1 to N;have a state ;and a threshold ri from some distribution.

The fraction of nodes in state vi=1 is r(t):Threshold dynamicsUpdating:

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

Watts` modelCascade condition: Thresholds CDF:

Watts` modelWatts: initially activate single node (of N), determine ifat steady state.

Us: initially activate a fraction of the nodes, anddetermine the steady state value of Conditions for global cascades (and dependence on thesize of the seed fraction) follow

Main resultOur result:withandDerivation: Generalizing zero-temperature random-field Ising modelresults from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna, J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.

Results

Results

Main resultOur result:withand

Cascade conditionqG(q)

Cascade conditionqG(q)

Simple cascade conditionFirst-order cascade condition: usingdemandfor global cascades to be possible. This yields the conditionreproducing Watts percolation result when and slope>1(slope>1)

Simple cascade condition

Extended cascade conditionSecond-order cascade condition: expandto second order and demand no positive zeros of the quadraticfor global cascades to be possible.The extension is, to first order in : above

Extended cascade condition

Gaussian threshold distribution

Gaussian threshold distribution

Bifurcation analysis

Results: Scale-free networks

Results: Scale-free networks

OverviewStructure 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).

Consider undirected unweighted network of N nodes (N is large) defined by degree distribution pkWatts` model of global cascadesUpdating: node i becomes active if the active fraction of its neighbours exceeds its thresholdEach node i has:binary statefixed threshold given by thresholds CDFInitially activate fraction 0

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

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

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

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

Main idea: pick a node A at random and calculate its probability of becoming active. This will give ().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 : 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 childrenDegree distribution of nearest neighbours:

Main idea: pick a node A at random and calculate its probability of becoming active. This will give ().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 : probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.k-1 children

Derivation of resultValid 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

Conclusions Demonstrated an analytical approach to determine the average avalanche size in Wattsmodel 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

OverviewStructure 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).

Extensions Generalized dynamics: SIR-type epidemics Percolation K-core sizes

Degree-degree correlations Modular networks

Asynchronous updating

Non-zero clustering

Derivation of resultOur result for the average fraction of active nodes

Generalization to other dynamical modelsOur result for the average fraction of active nodes

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

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

Adopt approach of M. Newman for percolation problems (PRE 67, 026126 (2003), PRL 89, 208701 (2002)).Degree-degree correlated networksP(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 activen+1..Consider a k-degree node at level n+1:n

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

Pearson correlation rDegree-degree correlated networksInitial damage 0r = 0r = -0.5r = 0.98Correlated networks (105 nodes) generated using Gaussian copula.Theory (curves) vs Numerics (symbols):7-cores in Poisson random graphs with z = 10Case r = 0 considered in S.N. Dorogovtsev et al., PRL 96, 040601 (2006).(zero initial damage)

Predicting K-cores in CAIDA internet router networkInternet router network structure from www.caida.org Degree distributionDegree-degree correlation matrixk

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

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

Modular networks example

SummaryStructure 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).

Cascades on correlated and modular networks

James P. GleesonDepartment of Mathematics and Statistics,University of Limerick, Irelandwww.ul.ie/gleesonj

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

Internet structure using router data from CAIDATransmissibility (bond Occupation probability)

DEPFA Bank collaboration: CDO pricingm1p1mNpNm2p2m3p3m4p4DefinitionsmiNotional of credit i

piDefault probability of credit i, (derived from the CDS quote).

SqFair price for protection against losses in tranche q

ProblemExisting models fail to reproduce the prices (Sq) observed on the market.{m1, m2,,mN}{p1, p2,,pN}{S1, S2,,Ss}Correlation Structure?0 to 5%10% to 15%15% to 25%25% to 35%S1S2S3S4S535% to 100%

An external fieldStochastic Dynamics on Networks

Hysteresis: PRGStochastic Dynamics on Networks

Hysteresis: PRGStochastic Dynamics on Networks

Stochastic dynamicsAim: 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

Noise in oscillators (Theme 1)Prof. M. P. Kennedy, Microelectronic Engineering, UCCNew computational and asymptotic methods for the spectrum of an oscillator subject to white noise Stochastic perturbation methods for effects of coloured noiseCollaboration (Feely/Kennedy):Noise effects in digital phase-locked loops

Microfluidic mixing and sorting (Theme 3)Experimentalists at Tyndall National Institute, CorkAnalysis of MHD micromixing in annular geometriesModelling of micro-sortingmethods

Collaborations: (Lindenberg/Sancho)Noise-induced sorting techniques for microparticles

Cascades on correlated and modular networks

James P. GleesonDepartment of Mathematics and Statistics,University of Limerick, Irelandwww.ul.ie/gleesonj