Session Y42: Spatio-temporal dynamics of complex networks: From mean field to large deviations

Noise-Induced Rare Events in Population Dynamics: The Effect of Spatial Heterogeneity

We study the influence of the network topology on the statistics of interest including rare events of interacting particle systems on complex graphs. As an example, we investigate the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system’s metastable state and leads to anomalous fixation compared to the non-spatial well-mixed case. In particular, we analytically and numerically show that the probability and mean time to fixation are characterized by stretched-exponential behaviors with exponents depending on the network’s degree distribution.   

Probabilistic stability measures and the dynamics of power grids.

I present results on the impact of the topology of the underlying network on the dynamics of oscillator networks, in particular, dynamical models of power grids. In order to study such high-dimensional networked systems I will introduce probabilistic stability measures that allow for sampling based estimation. By designing these measures with explicit reference to the network structure, we can uncover novel relationships between dynamical and structural properties.   

Spatio-temporal propagation of cascading failures in complex networks

When the functionality of a node in a complex network requires the simultaneous functioning of a node in another network, even a single failed node can lead to a cascade of failures throughout the system[1,2,3]. In spatial networks, this cascade has distinctive spatio-temporal properties, including a new kind of nucleation transition and a new metastable phase[4,5]. In this talk we will review recent advances on interdependent spatial network cascades including spatially localized attacks [6,7] and the impact of tunable embedding strength [8]. We will also compare these results with the spatio-temporal patterns of overload cascades [9]. The ubiquity of spatial networks and the distinctiveness of the cascading patterns on them, make it essential that we fully understand the fundamental cascade mechanisms underlying the observed behavior.


[1] Buldyrev et al. Nature 464, 08932 (2010)
[2] Danziger et al., Chapter 5 in Interconnected Networks (ed. A. Garas), Springer (2016)
[3] Danziger et al., arXiv:1705.00241
[4] Li et al. PRL 108, 228702 (2012)
[5] Bashan et al Nature Physics 9, 667 (2013)
[6] Berezin et al. Scientific Reports 5, 8934 (2015)
[7] Vaknin et al. New Journal of Physics 19, 073037 (2017)
[8] Danziger et al. EPL 115, 36002 (2016)
[9] Zhao et al. Nature Communications 7, 10094 (2016)   

Making rare events happen: prediction and control of network extinction, switching, and other extreme processes

Many complex networks must operate in uncertain and dynamic environments. Over long time scales, combinations of random internal interactions and dynamical perturbations can organize to drive a network from one collective state to another. Such noise-induced large fluctuations may be associated with desirable outcomes, such as epidemic extinctions, or undesirable ones, such as switching in collective order, or loss of network synchrony. In this talk I will discuss a general formalism for predicting rare events in networks with internal and external noise, the role of topology in facilitating the most extreme network events, techniques for optimal network control that leverage uncertainty, and numerical solutions for the aforementioned when explicit formulas are unknown. Along the way, I will consider many examples: from epidemic dynamics to opinion formation and synchronization of coupled oscillators   

Densification, Emergent Modularity, and Related Mysteries of Complex Networks

We discuss unusual features of networks that grow by: (a) node copying and (b) link redirection. In copying, a new node attaches to a randomly selected target node and to each of its neighbors with probability p. The resulting network is sparse for p<1/2 and dense (average degree increasing with number of nodes N) for p>1/2. In the dense regime, there is an infinite sequence of transitions at p = 2/3, 3/4, 4/5, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, etc., change. When second-neighbor linking to the target occurs, the resulting graph is effectively complete as N → ∞. Redirection leads to highly modular networks for redirection probability equal to 1. Individual realizations consist of almost entirely of leaves (nodes of degree 1), with a vanishingly small "nucleus" (nodes of degree greater than 1), and multiple hubs.