Many Parts, One Behaviour

Patterns, Networks and Avalanches in 10 chapters — from “Emergence and self-organisation: collective order from local rules” to “Self-organised criticality and avalanches”.

Chapters

  1. Emergence and self-organisation: collective order from local rules
    • A traffic jam with no cause
    • Emergent behaviour and levels of description
    • The Nagel-Schreckenberg traffic model
    • Order parameters and thresholds
    • Self-organisation away from equilibrium
    • Active matter
    • Run-and-tumble motion as a random walk
    • The Vicsek model of flocking
    • What emergence has meant here
    • Problems
  2. Pattern formation: instabilities that choose a wavelength
    • Fish stripes and convection rolls
    • Growth rate of a Fourier mode
    • The finite-wavenumber instability
    • The Swift-Hohenberg equation
    • Pattern amplitude saturation
    • The reaction-diffusion system
    • The Turing instability
    • The Brusselator model worked through
    • Patterns in the laboratory and in living tissue
    • Settled and handed on
    • Problems
  3. Networks as graphs: degree, distance and clustering
    • Six handshakes
    • Nodes, edges and the adjacency matrix
    • Node degree and the handshake sum
    • The degree distribution
    • Walks and the shortest path length
    • The small-world effect
    • Triangles and the clustering coefficient
    • Eigenvector centrality
    • The friendship paradox
    • What a network description contains
    • Problems
  4. Heavy tails: power laws and how to measure them
    • Earthquakes, words and cities
    • Heavy-tailed distributions
    • The power-law distribution and its normalisation
    • Scale invariance of the power law
    • Divergence of moments
    • The cumulative tail plot
    • The maximum-likelihood exponent estimate
    • Multiplicative random processes and the lognormal
    • The first-return time distribution of a walk
    • What a power law tells you
    • Problems
  5. Random graphs, the giant component and percolation on networks
    • Random wiring and the probability generating function
    • The Erdos-Renyi random graph
    • Exploring a component as a branching process
    • The giant component appears at mean degree one
    • Small components and the size distribution at threshold
    • The Watts-Strogatz small-world model
    • The configuration model
    • Excess degree and the Molloy-Reed criterion
    • Percolation on a network by deleting nodes
    • What random wiring explains
    • Problems
  6. Growing networks: preferential attachment and scale-free degree distributions
    • Citations and the mean-field rate equation
    • Growing networks
    • The rate equation for uniform random attachment
    • Preferential attachment
    • The Barabasi-Albert model and its rate equation
    • Scale-free networks and the exponent three
    • The Price model of citations
    • The first-mover advantage
    • Nonlinear preferential attachment
    • Node fitness and vertex copying
    • The measured attachment kernel
    • What growth explains
    • Problems
  7. Communities: finding the groups in a network
    • A club that split in two
    • Community structure against a null model
    • Graph bisection
    • The graph Laplacian and the cut size
    • Spectral bisection
    • Modularity counts edges beyond chance
    • The modularity matrix and its leading eigenvector
    • Greedy optimisation and the Louvain algorithm
    • The resolution limit
    • The stochastic block model
    • The detectability threshold
    • What community detection can and cannot do
    • Problems
  8. Epidemics on networks
    • Why some outbreaks fizzle and others do not
    • The compartmental epidemic model
    • The basic reproduction number
    • The herd immunity threshold
    • The final-size relation
    • Transmissibility and bond percolation
    • Hubs and the vanishing epidemic threshold
    • Heterogeneous mean-field theory for SIS
    • Targeted immunisation
    • Settled and handed on
    • Problems
  9. Synchronisation, cascades and the resilience of networks
    • Fireflies, blackouts and the phase oscillator
    • Phase locking of two oscillators
    • The Kuramoto model and its order parameter
    • The synchronisation threshold
    • Synchronisation on a network
    • Random failure and targeted attack on hubs
    • Cascading failure
    • What makes a network resilient
    • Settled and handed on
    • Problems
  10. Self-organised criticality and avalanches
    • Avalanches in rice piles and brain slices
    • Slowly driven threshold systems
    • The Bak-Tang-Wiesenfeld sandpile
    • The abelian property of the sandpile
    • The self-organised critical state
    • Avalanches as a critical branching process
    • Avalanche data collapse
    • Crackling noise in nature
    • Tuned versus self-organised criticality
    • Thresholds and branching across the book
    • Problems