Regions of Reduced Dynamics in Dynamic Networks

We consider complex networks where the dynamics of each interacting agent is given by a nonlinear vector field and the connections between the agents are defined according to the topology of undirected simple graphs. The aim of the work is to explore whether the asymptotic dynamic behavior of the entire network can be fully determined from the knowledge of the dynamic properties of the underlying constituent agents. While the complexity that arises by connecting many nonlinear systems hinders us to analytically determine general solutions, we show that there are conditions under which the dynamical properties of the constituent agents are equivalent to the dynamical properties of the entire network. This feature, which depends on the nature and structure of both the agents and connections, leads us to define the concept of regions of reduced dynamics, which are subsets of the parameter space where the asymptotic solutions of a network behave equivalently to the limit sets of the constituent agents. On one hand, we discuss the existence of regions of reduced dynamics, which can be proven in the case of diffusive networks of identical agents with all-to-all topologies and conjectured for other topologies. On the other hand, using three examples, we show how to locate regions of reduced dynamics in parameter space. In simple cases, this can be done analytically through bifurcation analysis and in other cases we exploit numerical continuation methods.

General polytropic hydrodynamic cylinder under self-gravity

For filamentary clouds on various scales obeying general polytropic (GP) equation of state, their hydrodynamic collapses, expansions and shocks are investigated. Our cylindrical model is axisymmetric, infinitely long with axial uniformity and involves Newtonian gravity. For such GP cylinders, we explore various analytical and numerical similarity solutions. Based on a singular hydrostatic solution, we derive a quasi-static asymptotic dynamic solution approaching the axis. There, we also derive the asymptotic cylindrical free-fall solution for polytropic index γ ≤ 1 and show the absence of such solutions for γ > 1. We find new asymptotic solutions for expanding cylindrical central voids with no matter inside, and examine the asymptotic expansion solutions to higher orders far from the axis. We classify the sonic critical curve (SCC) into three (or five) types and analyse their properties. The asymptotic behaviors of the SCC towards the axis and infinity are examined. Examples are shown for solutions crossing the SCC twice with the global features of cylindrical envelope expansion or contraction with core collapses. We numerically construct new types of global similarity solutions with or without outgoing shocks. For γ > 1, a shock is necessary to connect the inner and outer parts. The collapse and fragmentation of massive filaments or strings may give clues and implications to the formations of chains of stellar objects, chains of black holes, chains of galaxies or even chains of galaxy clusters in proper astrophysical and cosmological contexts.

Bounded-Size Rules: The Barely Subcritical Regime

Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [26] proved that for such rules, there exists a (rule-dependent) critical time tc such that when t < tc the size of the largest component is of order log n, while for t > tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n2γ log4n, on the size of the largest component, at time instants tn = tc−n−γ, where γ ∈ (0,1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behaviour of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space $\mathbb{R}_+\times \mathcal{D}([0,\infty):\mathbb{N}_0)$, where $\mathcal{D}([0,\infty):\mathbb{N}_0)$ is the Skorokhod D-space of functions that are right continuous and have left limits, with values in the space of non-negative integers $\mathbb{N}_0$, equipped with the usual Skorokhod topology. The coupling construction also gives an alternative characterization (from the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces.