NetEvo is a computational toolkit and collection of end-user tools designed to bring together network topology, dynamics and evolution. With these features integral to many biological systems, these tools can be used to model and analyse a wide range of behaviours; from the dynamics of protein interaction networks to the evolution of genetic regulatory networks. NetEvo is open-source software and is free to use. To find out more check out our website.

Thomas E. Gorochowski, Mario di Bernardo, Claire S. Grierson - University of Bristol, UK

NetEvo Computational Evolution of Dynamical Complex Networks

Introducing NetEvo

Case Study: Network Synchronisation

Evolved Topology

Eigenratio 99.90 99.40↓97.35↓95.83↓

Order Param σ = 0.4Order Param σ = 0.7 (Type 1)

Order Param σ = 0.7 (Type 2)

0.10 0.60↑2.65↑4.17↑

19.71↓ 24.01↓17.35↓

42.36↓

0.20 1.26↑4.68↑8.82↑

0.04↓ 0.49↓0.34↓

1.31↓

Motif Distributions for Evolved Networks

Statistically over ↑ and under ↓ expressed with p-value < 0.01

Parameter Variation

1%5%

10%25%50%

1.17↑1.19↑

1.50↑1.50↑0.94↑

22.84↓24.39↓24.32↓24.32↓0.94↓

2.35↑2.43↑3.14↑3.32↑1.92↑

74.23−72.35−71.84−71.51−

73.72−

0.06↑0.06↑0.13↑0.11↑

0.05↑

Motifs After Parameter Variation

Statistically over ↑ and under ↓ expressed with p-value < 0.01

E!ect of Parameter VariationA bounded synchronisation measure was used to investigate how variation in node dynamics a!ected the topological features we had observed. In support of our conjecture, we found an over expression of triangular feedback motifs with a +ve correlation to parameter variation, up to approximately 25% (see table on right).

The topological measure produced networks with the widest range of full synchronisation [2]. The dynamical measure lead to two types of evolved network. The first, which we called Type 1, displayed increased synchronisation focused around the fixed coupling strength used during evolution. As this coupling strength was increased outside the stable region of the system a topological bifurcation took place and a new form of network, called Type 2, was found. These exhibited only partial synchronisation, but at much higher coupling strengths than that seen before (see figures below).

We used NetEvo to evolve networks with improved synchronisation for purely topological [1,2] and dynamical [3,4] performance measures, assessing the e!ect each had on the evolved topologies. Rossler node dynamics and di!usive coupling with limited stability were chosen and the coupling strength varied. Networks consisted of 100 nodes, 200 undirected edges and evolution was by random rewiring.

Eigenratio Order Parameterσ = 0.1 to 0.7 (Type 1)

Order Parameterσ ! 0.7 (Type 2)

Types of Final Evolved Networks

Evolutionary Trajectories - (A) Type 1 and (B) Type 2 Networks

Order Parameter(Dynamical)

=1

N(N − 1)

N�

i=1

N�

j=1,j �=i

Θ(δ − dij(t))Eigenratio(Topological)

=λN

λ2

Updates Topology

Node/Coupling Dynamicsdxi/dt = f(xi) + coupling

SupervisorComputerised Agent

Network TopologyLaplacian/Adjacency Matrix

QAlters Coupling

Updates Parameters

Performance Measure

NetEvo Network Dynamics, Evolution and Analysis

Use

r Inp

uts

Node Dynamics

Edge Dynamics

Initial Topology

Network Mutations

PerformanceMeasures

Evolutionary Trajectories

Animated Visualisations

Network Topologies

DynamicsTime-Series

Network EvolutionSupervisor based using

simulated annealing

Network DynamicsContinuous ODE SolverDiscrete-Time Simulator

Schematic of a Supervised Network

Analysis of network motifs highlighted a statistically significant over-expression of triangular feedback motifs for all dynamically evolved networks, with large increases for Type 2 networks. We conjecture that these structures aid in stability during evolution.

Network Motifs

http://www.netevo.org

A

B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

Coupling Strength

% S

ynch

roni

sed

EigenratioOP = 0.7 (Type 1)OP = 0.7 (Type 2)

Synchronisation Response - Evolved Networks100

80

60

40

20

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Coupling Strength σ

Type 1

Type 2

Topological Bifurcation

Sync

hron

isatio

n %

DynamicsComplete Synchronisation

Partial Synchronisation

!"#"0.6 !"$"0.8

Transition PeriodEigenratio

Synchronisation is critical to many key processes in biological systems:

Cell cycle

Circadianrhythms

AmazonianFireflies

COMPLEXITYBRISTOL centre for

SCIENCESBCCS

[1] L. Donetti, P. Hurtado, M. Munoz. Entangled networks, synchronization, and optimal network topology. PRL 95, 188701 (2005)[2] L.M. Pecora, T.L. Carroll. Master stability functions for synchronized coupled systems. PRL 80, (1998)[3] T.E. Gorochowski, et al. Evolving enhanced topologies for the synchronization of dynamical complex networks. PRE 81, 056212 (2010)[4] T.E. Gorochowski, et al. A dynamical approach to the evolution of complex networks. Proc. 19th Int. Symp. MTNS (2010)

ConclusionsWe have shown that dynamics and topology cannot be considered in isolation when analysing the evolution of synchronisation.

1. Dynamics → Triangular Feedback Motifs2. Trade-o!s → Topological Bifurcations As we begin to engineer biological systems of our own using synthetic biology, tools such NetEvo provide a way of understanding why certain architectures have been chosen by Nature and to test new ones of our own.