Asymptotics of classical spin networks

A spin network is a cubic ribbon graph labeled by representations of $\mathrm{SU}(2)$. Spin networks are important in various areas of Mathematics (3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$-functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) rigorous effective computations of our results for some $6j$-symbols using the Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube $12j$ spin network (including a non-rigorous guess of its Stokes constants), in the appendix.Comment: 24 pages, 32 figure

The geometry of spontaneous spiking in neuronal networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior

An Alternative Approach to the Calculation and Analysis of Connectivity in the World City Network

Empirical research on world cities often draws on Taylor's (2001) notion of an 'interlocking network model', in which office networks of globalized service firms are assumed to shape the spatialities of urban networks. In spite of its many merits, this approach is limited because the resultant adjacency matrices are not really fit for network-analytic calculations. We therefore propose a fresh analytical approach using a primary linkage algorithm that produces a one-mode directed graph based on Taylor's two-mode city/firm network data. The procedure has the advantage of creating less dense networks when compared to the interlocking network model, while nonetheless retaining the network structure apparent in the initial dataset. We randomize the empirical network with a bootstrapping simulation approach, and compare the simulated parameters of this null-model with our empirical network parameter (i.e. betweenness centrality). We find that our approach produces results that are comparable to those of the standard interlocking network model. However, because our approach is based on an actual graph representation and network analysis, we are able to assess cities' position in the network at large. For instance, we find that cities such as Tokyo, Sydney, Melbourne, Almaty and Karachi hold more strategic and valuable positions than suggested in the interlocking networks as they play a bridging role in connecting cities across regions. In general, we argue that our graph representation allows for further and deeper analysis of the original data, further extending world city network research into a theory-based empirical research approach.Comment: 18 pages, 9 figures, 2 table

Modelling Interdependent Cascading Failures in Real World Complex Networks using a Functional Dependency Model

Infrastructure systems are becoming increasingly complex and interdependent. As a result our ability to predict the likelihood of large-scale failure of these systems has significantly diminished and the consequence of this is that we now have a greatly increased risk of devastating impacts to society. Traditionally these systems have been analysed using physically-based models. However, this approach can only provide information for a specific network and is limited by the number of scenarios that can be tested. In an attempt to overcome this shortcoming, many studies have used network graph theory to provide an alternative analysis approach. This approach has tended to consider infrastructure systems in isolation, but has recently considered the analysis of interdependent networks through combination with percolation theory. However, these studies have focused on the analysis of synthetic networks and tend to only consider the topology of the system. In this paper we develop a new analysis approach, based upon network theory, but accounting for the hierarchical structure and functional dependency observed in real world infrastructure networks. We apply this method to two real world networks, to show that it can be used to quantify the impact that failures within an electricity network have upon a dependent water network

Evaluation of Whole-Brain Resting-State Functional Connectivity in Spinal Cord Injury - A Large-Scale Network Analysis Using Network Based Statistic

Large-scale network analysis characterizes the brain as a complex network of nodes and edges to evaluate functional connectivity patterns. The utility of graph-based techniques has been demonstrated in an increasing number of restingstate functional MRI (rs-fMRI) studies in the normal and diseased brain. However, to our knowledge, graph theory has not been used to study the reorganization pattern of resting-state brain networks in patients with traumatic complete spinal cord injury (SCI). In the present analysis, we applied a graph-theoretical approach to explore changes to global brain network architecture as a result of SCI. Fifteen subjects with chronic (\u3e 2 years) complete (American Spinal Injury Association [ASIA] A) cervical SCI and 15 neurologically intact controls were scanned using rs-fMRI. The data were preprocessed followed by parcellation of the brain into 116 regions of interest (ROI) or nodes. The average time series was extracted at each node, and correlation analysis was performed between every pair of nodes. A functional connectivity matrix for each subject was then generated. Subsequently, the matrices were averaged across groups, and network changes were evaluated between groups using the network-based statistic (NBS) method. Our results showed decreased connectivity in a subnetwork of the whole brain in SCI compared with control subjects. Upon further examination, increased connectivity was observed in a subnetwork of the sensorimotor cortex and cerebellum network in SCI. In conclusion, our findings emphasize the applicability of NBS to study functional connectivity architecture in diseased brain states. Further, we show reorganization of large-scale resting-state brain networks in traumatic SCI, with potential prognostic and therapeutic implications

Extracting the Groupwise Core Structural Connectivity Network: Bridging Statistical and Graph-Theoretical Approaches

Finding the common structural brain connectivity network for a given population is an open problem, crucial for current neuro-science. Recent evidence suggests there's a tightly connected network shared between humans. Obtaining this network will, among many advantages , allow us to focus cognitive and clinical analyses on common connections, thus increasing their statistical power. In turn, knowledge about the common network will facilitate novel analyses to understand the structure-function relationship in the brain. In this work, we present a new algorithm for computing the core structural connectivity network of a subject sample combining graph theory and statistics. Our algorithm works in accordance with novel evidence on brain topology. We analyze the problem theoretically and prove its complexity. Using 309 subjects, we show its advantages when used as a feature selection for connectivity analysis on populations, outperforming the current approaches

Physics-inspired Performace Evaluation of a Structured Peer-to-Peer Overlay Network

In the majority of structured peer-to-peer overlay networks a graph with a desirable topology is constructed. In most cases, the graph is maintained by a periodic activity performed by each node in the graph to preserve the desirable structure in face of the continuous change of the set of nodes. The interaction of the autonomous periodic activities of the nodes renders the performance analysis of such systems complex and simulation of scales of interest can be prohibitive. Physicists, however, are accustomed to dealing with scale by characterizing a system using intensive variables, i.e. variables that are size independent. The approach has proved its usefulness when applied to satisfiability theory. This work is the first attempt to apply it in the area of distributed systems. The contribution of this paper is two-fold. First, we describe a methodology to be used for analyzing the performance of large scale distributed systems. Second, we show how we applied the methodology to find an intensive variable that describe the characteristic behavior of the Chord overlay network, namely, the ratio of the magnitude of perturbation of the network (joins/failures) to the magnitude of periodic stabilization of the network