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Network Dynamics and Optimization: Balancing Global Goals and Local Interactions

This overview explores the intricate dynamics within networks, addressing the balance between global and local goals in optimizing network performance. We'll discuss flow in complex networks and how ideas, innovations, and computer viruses propagate through interconnected systems. Key concepts include evolutionary models, optimization mechanisms like simulated annealing, and the impact of decision-making at the node level. Additionally, we analyze the SIS model of virus spreading and the epidemiological implications in various network structures. Understanding these dynamics is crucial for effective design and resilience against potential disruptions.

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Network Dynamics and Optimization: Balancing Global Goals and Local Interactions

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  1. 5. DYNAMICS IN THE NETWORKS Something going on

  2. Network dynamics: • global goal • local goal • Flow in complex networks: • ideas • innovations • computer viruses • problems

  3. Network dynamics • The time scale governing the dynamics of the network is comparable to that characterizing the network connectivity • Evolutionary models with optimization mechanisms: • Parameterization • Simulated annealing

  4. Global vs local optimization • Design: the goal is to optimize global quantity (distance, clustering, density, ...) • Evolution: decision taken at node level

  5. Evolution • Bornholdt & Rohlf: Global criticality from local dynamics • Network of interconnected binary elements • The dynamics reaches an attractor • Change the connectivity of a node according to its behavior during the attractor • Evolution towards critical value of connectivity • Phase transition at the critical value: frozen state- dynamical state

  6. Optimization • Global goal: • Distance: related to minimal cost in transportation • Number of connections: costly connections • A combination of parameters • Initial configuration: random graph • Change connections • Accept if there is an improvement • Stars vs trees

  7. Flow in complex networks • Viruses • Information

  8. Virus spreading • SIS (susceptible-infected-susceptible) model • Each healthy (susceptible) individual is infected with rate  when it has at least one infected neighbor • Infected nodes are cured (become susceptible) with rate  (=1 without lost of generality)

  9. Known results • Regular lattices • Random graphs •  Non zero epidemic threshold • >= c: spreads and become persistent • < c: the infection dies out exponentially • Equivalent to a nonequilibrium phase transition

  10. Scale free networks • Absence of an epidemic threshold • Due to the unboundedness of the connectivity fluctuations (<k2>with a power law distribution) • The same fact that make scale-free networks to be robust against random failures makes it very sensitive to the spread of infections

  11. Virus prevalence • Density of infected nodes in surviving infections

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