Networks form the backbones of a wide variety of complex systems,

ranging from food webs, gene regulation and social networks to

transportation networks and the internet. Due to the sheer size and

complexity of many of theses systems, it remains an open challenge to

formulate general descriptions of their large-scale structures.

Although many methods have been proposed to achieve this, many of them

yield diverging descriptions of the same network, making both the

comparison and understanding of their results very

difficult. Furthermore, very few methods attempt to gauge the

statistical significance of the uncovered structures, and hence the

majority cannot reliably separate actual structure from stochastic

fluctuations. In this talk, I will show how these issues can be tackled

in a principled fashion by formulating appropriate generative models of

network structure that can have their parameters inferred from data. I

will also consider the comparison between a variety of generative

models, including different structural features such as degree

correction, where nodes with arbitrary degrees can belong to the same

group, and community overlap, where nodes are allowed to belong to more

than one group. Because such model variants possess an increased number

of parameters, they become prone to overfitting. We demonstrate how

model selection based on the minimum description length criterion and

posterior odds ratios can fully account for the increased degrees of

freedom of the larger models, and selects the most appropriate trade-off

between model complexity and quality of fit based on the statistical

evidence present in the data.

Throughout the talk I will illustrate the application of the methods

with many empirical networks such as the internet at the autonomous

systems level, the global airport network, the network of actors and

films, social networks, citations among websites, co-occurrence of

disease-causing genes and many others.

# Past Industrial and Applied Mathematics Seminar

Noise limits are one of the major constraints when designing

aircraft engines. Acoustic liners are fitted in almost all civilian

turbofan engine intakes, and are being considered for use elsewhere in a

bid to further reduce noise. Despite this, models for acoustic liners

in flow have been rather poor until recently, with discrepancies of 10dB

or more. This talk will show why, and what is being done to model them

better. In the process, as well as mathematical modelling using

asymptotics, we will show that state of the art Computational

AeroAcoustics simulations leave a lot to be desired, particularly when

using optimized finite difference stencils.

Michael Gomez:

Title: The role of ghosts in elastic snap-through

Abstract: Elastic `snap-through' buckling is a striking instability of many elastic systems with natural curvature and bistable states. The conditions under which bistability exists have been reasonably well studied, not least because a number of engineering applications make use of the rapid transitions between states. However, the dynamics of the transition itself remains much less well understood. Several examples have been studied that show slower dynamics than would be expected based on purely elastic timescales of motion, with the natural conclusion drawn that some other effect, such as viscoelasticity, must play a role. I will present analysis (and hopefully experiments) of a purely elastic system that shows similar `anomalous dynamics'; however, we show that here this dynamics is a consequence of the ‘ghost’ of the snap-through bifurcation.

Andrew Krause:

Title: Fluid-Growth Interactions in Bioactive Porous Media

Abstract: Recent models in Tissue Engineering have considered pore blocking by cells in a porous tissue scaffold, as well as fluid shear effects on cell growth. We implement a suite of models to better understand these interactions between cell growth and fluid flow in an active porous medium. We modify some existing models in the literature that are spatially continuous (e.g. Darcy's law with a cell density dependent porosity). However, this type of model is based on assumptions that we argue are not good at describing geometric and topological properties of a heterogeneous pore network, and show how such a network can emerge in this system. Therefore we propose a different modelling paradigm to directly describe the mesoscopic pore networks of a tissue scaffold. We investigate a deterministic network model that can reproduce behaviour of the continuum models found in the literature, but can also exhibit finite-scale effects of the pore network. We also consider simpler stochastic models which compare well with near-critical Percolation behaviour, and show how this kind of behaviour can arise from our deterministic network model.

Abstract:We study an evolving network where the nodes are considered as represent particles with a corresponding state vector. Edges between nodes are created and destroyed as a Poisson process, and new nodes enter the system. We define the concept of a “local state degree distribution” (LSDD) as a degree distribution that is local to a particular point in phase space. We then derive a differential equation that is satisfied approximately by the LSDD under a mean field assumption; this allows us to calculate the degree distribution. We examine the validity of our derived differential equation using numerical simulations, and we find a close match in LSDD when comparing theory and simulation. Using the differential equation derived, we also propose a continuum model for osteocyte network formation within bone. The structure of this network has implications regarding bone quality. Furthermore, osteocyte network structure can be disrupted within cancerous microenvironments. Evidence suggests that cancerous osteocyte networks either have dendritic overgrowth or underdeveloped dendrites. This model allows us to probe the density and degree distribution of the dendritic network. We consider a traveling wave solution of the osteocyte LSDD profile which is of relevance to osteoblastic bone cancer (which induces net bone formation). We then hypothesise that increased rates of differentiation would lead to higher densities of osteocytes but with a lower quantity of dendrites.

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum.

We consider the impact of spatial heterogeneities on the dynamics of

localized patterns in systems of partial differential equations (in one

spatial dimension). We will mostly focus on the most simple possible

heterogeneity: a small jump-like defect that appears in models in which

some parameters change in value as the spatial variable x crosses

through a critical value -- which can be due to natural inhomogeneities,

as is typically the case in ecological models, or can be imposed on the

model for engineering purposes, as in Josephson junctions. Even such a

small, simplified heterogeneity may have a crucial impact on the

dynamics of the PDE. We will especially consider the effect of the

heterogeneity on the existence of defect solutions, which boils down to

finding heteroclinic (or homoclinic) orbits in an n-dimensional

dynamical system in `time' x, for which the vector field for x > 0

differs slightly from that for x < 0 (under the assumption that there is

such an orbit in the homogeneous problem). Both the dimension of the

problem and the nature of the linearized system near the limit points

have a remarkably rich impact on the defect solutions. We complement the

general approach by considering two explicit examples: a heterogeneous

extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous

generalized FitzHugh–Nagumo system (n = 6).

Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.