Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

Network Modeling of Complex DynamicalSystems

Bosiljka Tadic

Department of Theoretical PhysicsJožef Stefan Institute, Ljubljana , Slovenia

ICCS Workshop “Networks” Krakow June 23-25, 2008

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

THE MAIN IDEA:

⊲In Complex Dynamical Systems:

Local interactions ⇒ Global properties

⊲Representing a Complex Dynamical System by a Network(Graph) then enables use of formal methods:

Graph Theory and Algorithms;

Statistical Physics of Complex Networks;

⊲This approach of quantitative study of complexity is particularlyuseful in biological and social systems, ecology, economy, etc...but also in physics of new functional materials and soft matterphysics, where no exact theories are available!

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

Dynamic State Variables on Networks:

⊲Network function: Dynamic State variables attached to Nodes⊲Nodes activity through interactions ⇒ Noise signals

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Active Nodes Noise SignalsTadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

Mapping CDS to Networks creates Problems...

⊲Mapping CDS ⇒ Network is not unique:

Choise of graph elements (Nodes, Links) and Dynamic.var

Constraints with (given) local, global, mesoscopic structure

⊲Some CDS are Networks, structure dyn.changing, or unknown:

How that structure affects the network function?

⊲ Can we overcome such problems and how?

Depending on the puprose of the resarch & availableresources, comp.power, time, ...

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

NETWORKS FROM EMPIRICAL DATA

⊲Data-Driven Models: New network types & Their properties

New classes of E-Social Networks Bipartitivity...different node types

Gene Networks Characteristic Subgraphs Id.Modularity rel. to Function

Protein Interaction Networks Mesoscopic structure ∼∼

(multigraphs, weighted net.)Enviromental, Eco-systems int. Networks-of-networks

(multinetworks)Communication Networks Adaptivity, wireless, ...

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

FINE STRUCTURE OF NETWORKS

⊲Finding networks structure at local, mesoscopic & global scale ;⊲Modules ⇔Function; Network-of-networks: interacting networks

Pajek

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

METHODS TO EXPLORE NETWORKS

⊲As new networks being discovered (i.e., from CDS), demandsgrow to adjust/develop appropriate algorithms to explore them

New network features require suitablemethods for a quantitative study:

Finding relevant network partition;

Search for dynamical subgraphs(motifs, modules, paths, cycles, trees...)

Visualization related to network type;

Spectra of Aij & other related matrices;

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

DYNAMICS ON/OF NETWORKS

Dynamic var.update ⇔Noise sugnal;

Mirror participation of nodes and otherstructural elements in the dynamics;

Dynamics depends on Structure andcan be used for (noninvasive) structurecharacterization; Need more systematicquantification of Structure ⇔ Dynamics;

Diffusion; Traffic; RW; Spin-Dynamics;Coupled Chaotic Maps;

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Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

APPLICATIONS OF NETWORK THEORY

Nano-science: Networks of GoldNano-Particles as conducting films

Gene interactions: Search forevolutionary robust patterns

E-social interactions: Social networksemerging through communicationnetworks/Agents and Subjects play asimilar role...

...

conduction paths

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Outline

1 Scope & Goals

2 Mapping Complex Dynamical Systems to Networks

3 Current Trends:More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

Tadic Dynamic Phenomena on Networks

Scope & GoalsMapping Complex Dynamical Systems to Networks

Current Trends:

More Networks from DataNetworks’ Fine StructureMore Formal Methods for Network Analysis & VisualizationDynamical Processes on NetworksApplied Networks++New Theoretical Approaches

THEORETICAL MODELS BEYOND SIMPLE SF...

Grown vs. Fluctuating networks Rate Eqs. vs. Stat.Phys.Constraints: given mesoscopic structure; Given degree, CC, ...given spectral properties...Multi—networks == interacting networks diff. nature, or topology—graphs == weighted networks; Links b. any pair—partite networks! vs. Links b.partitionsAdaptive networks: Dynamic variablesinteract ⇒ Network updates optimization principle

Tadic Dynamic Phenomena on Networks