# Day 3: Wednesday 30th Sept.

## Dynamics on Networks, Modularisation and Flows

The study of dynamical processes defined on network topologies starts with the analysis of equilibrium conditions, where the flows into and out of the network equilibrate each other, obeying Kirchhoff's law in the network interior at each node. Such a situation can under standard conditions be associated with mathematical tools based on linearity principles. On this level the network dynamics can also be modularised successfully, relying on the seperability of linear processes. Flows defind on graphs have many applications, and can be successfully used to optimise real world processes.

**10:00am - 10:15am** Welcome Tea & Coffee, Mathematics Common Room.

**10:15am - 12:30am**

Keith Briggs (BT) - * Optimal railway-trip planning*,

**45min**

Markus Kirkilionis (Warwick) - ** Algebraic dissection of dynamical networks - Lessons learned from extremal currents. **45min

Robert MacKay (Warwick) - * Aggregation procedures for Complex Systems*,

**45min**

**12:30am - 1:15pm** Lunch Break, Mathematics Common Room.

**1:15pm - 2:00pm **Ian Stewart (Warwick) - **Symmetry, synchrony, and pattern formation in networks, 45min**

**Bus tour to Stratford! 14:30 - 21:30
**

Abstracts:

**Optimal railway-trip planning**

by Keith Briggs

The problem of optimal trip planning in a timetabled transport network subject to random delays is of clear practical importance, and raises challenging mathematical questions in both graph algorithms and stochastic processes on networks. I will survey these problems and propose some partial solutions. I will demonstrate some software applying these solutions to the UK railway network.

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**Aggregation Procedures for Complex Systems**

by R.S.MacKay

A complex system is something with many interdependent components. An aggregation procedure consists in amalgamating selected groups of components into super-components with derived interactions between the super-components. No information is lost at the higher level and that at the lower level can be derived from the higher level by disaggregation. Aggregation can often be iterated, producing hierarchical aggregation. Aggregation can be useful for computation, by breaking the problem down into pieces and combining them later. It can be useful for planning, since the effects of localised changes can be examined without recomputing the rest of the system. In systems with iterated aggregation, asymptotic trends to higher scales may emerge. Illustrations will be given from statistical physics, Markov processes, selfish traffic flow, synchronisation in oscillator networks, and multi-agent games.

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**SYMMETRY, SYNCHRONY, AND PATTERN FORMATION IN NETWORKS**

by Ian Stewart

In a dynamical system, pattern formation is often associated with symmetry-breaking. The general setting – equivariant bifurcation theory - is fairly well understood and widely used. The analogous theory for networks is less well developed, but it is already clear that the range of typical phenomena is much broader than in generic dynamical systems. The talk will be an informal description of a formal framework for networks of coupled dynamical systems, developed with Marty Golubitsky and others. It will illustrate some of the unexpected behaviour that arises, with particular emphasis on synchrony-breaking bifurcations, both steady-state and time-periodic.

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**Algebraic dissection of dynamical networks - Lessons learned from extremal currents. **

by Markus Kirkilionis

We review the extremal current approach (elementary modes in the bioinformatics literature) developed for mass-action reaction networks. Due to the polynomial structure of the RHS of the associated large system of ODEs describing the dynamics of the network positive equilibria can be expanded in terms of equilibria of sub-networks. This gives a rich algebraic structure where the influence of the sub-networks (associated to extremal currents) can be associated to the stability of the overall network. Moreover also bifurcation behaviours can be re-formulated in terms of