nonlocality, nonlinearity and complexity

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NONLOCALITY, NONLINEARITY AND COMPLEXITY: ON THE MATHEMATICS OF

MODELLING NCW AND EBO

22nd International Symposium on

Military Operational Research

29 August – 2 September 2005

Michael F. Ling

Defence Systems Analysis Division

Defence Science and Technology Organisation, Australia

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1. Introduction

2. Nonlocal Interactions in Complex Systems

and their implications

3. Network Modelling and Analysis – and some

of its dark secrets

4. NCW Modelling and Simulation

5. Summary

Overview

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¶ Network-Centric Warfare (NCW) is fundamentally about using cooperative efforts to bring together spatially distributed warfighting capabilities and projecting the desired effect to the right place at the right time.

¶ In a similar fashion, the concept of Effects-Based Operations (EBO) involves the use of capabilities and propagation of effects across space, time and different domains of the DIME (Diplomatic, Information, Military, Economic) construct.

Introduction

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¶ That is, the concepts of NCW and EBO both involve the use of interactions of different length scales (in the temporal, spatial or graph-theoretic sense) that may coexist, and may even compete with one and other, in the complex systems called networked forces (e.g. Agile Mission Groups).

¶ These interactions may lead to outcomes that are nonlinear and could not be understood or even approximated without explicitly taking all the essential interactions into account.

Introduction (2)

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¶ Therefore, in modelling and analysing these complex systems, it is essential that the mathematical models are commensurate with the nature of the interactions:

¶ Failing to do that would make it difficult if not impossible to develop a proper and thorough understanding of the problems.

¶ Worse still, it could lead to misinterpretations and wrong conclusions.

Introduction (3)

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As in many areas of contemporary scientific research, ideas and theoretical models developed in diverse disciplines may be employed in solving seemingly unrelated military operational research problems, which can be cast into similar conceptual frameworks and mathematical formulations.

Introduction (4)

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In this talk, we will see a common theme amongst examples taken from a number of topics (physics, biology and networks) in which nonlocal interactions play a crucial role in the understanding of the complex systems, and the underlying symmetries in these systems are intimately linked to the nonlocal interactions.

Introduction (5)

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¶ One challenging problem in itinerant magnetism

has been the discovery of long-range diffused and

polarised neutron diffraction patterns in a number

of itinerant magnetic metal alloys:

¶ The origin of these patterns was difficult to

understand as both the atomic and magnetic

ordering forces were known to be short-range in

nature with a high degree of symmetry.

¶ Fitting of the neutron data to short-range models

led only to confusion and misinterpretation.

Physical Systems

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¶ First-principles all-electron theory that captures

the nonlocal interactions in these systems reveals

that the nonlocal patterns are a direct result of the

competing interactions between the magnetic and

atomic ordering forces:

¶ The long-range neutron scattering patterns have

emerged out of the interactions of atomic and

magnetic short-range ordering forces.

¶ This class of alloys exhibit true emergent behaviour

in the complexity-theoretic sense.

Physical Systems (2)

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¶ Maybe of particular relevance to this Symposium is the fact that in some of the aforementioned alloys, the atomic ordering tendency can be altered by changing the state of magnetisation:

¶ This possibility to achieve a particular goal (a specific atomic ordering) by a different means (altering the magnetisation state) via some subtle mechanism (the underlying electronic structure) is perhaps one of Nature’s best examples of effects-based operations.


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¶ Countless other examples in physics can be found where the long-range global behaviours arise out of short-range local interactions, and yet the emergent global patterns cannot be understood without explicitly including the long-range interactions in the mathematics.

¶ Other well-known examples include

¶ Turbulence in fluids and the annual soliton waves (tidal bore) in the River Severn.

¶ Photon interference and diffraction patterns.


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¶ In all the examples above, the length scale of the interaction is intimately related to the underlying symmetries in the systems:¶ In the magnetic alloys, the symmetries are broken

by the presence of magnetic fields.

¶ Turbulence in fluids and the solitons waves emerge with lower symmetries from the high-symmetry state of molecular interaction.

¶ Interference patterns arising out of localised photon interactions have originated from nonlocal interactions with the interference slits.


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¶ The melting and folding of DNA protein chains have long been described by pair-wise interaction models, for example, variations of the Ising model.

¶ However, the reality is that the folding and melting processes in many DNA chains can involve tens, hundred or even thousands of base pairs – not very localised at all.

¶ Also, we have to account for the vastly different folding speeds and highly complicated but precise actions in 3-D space to reach stable topologies.

Biological Systems

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¶ As in the examples of physical systems, both long and short-range interactions have been found to play important roles in the DNA folding and melting processes.

¶ Proper and thorough understanding of these processes becomes possible only after both long and short-range interactions, along with the corresponding symmetries, have been

incorporated into the mathematical modelling.

Biological Systems (2)

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¶ Much effort has been devoted on analysing large networks by focusing on the number of connections to each node and the average graph-theoretic length of links between the nodes.

¶ Analytical models such as small-world, exponential or scale-free have provided much insight into network topology and behaviours, including the susceptibility to failures and/or attacks.

¶ An effective-medium approximation has implicitly been made here that all the nodes are assumed to be the same and all the links are identical.

Network Systems

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¶ Unlike most networks one encounters in the literature, a military network (a networked force) is a highly structured organisation with a number of specific objectives shared by all its members, though each member has their individual role to play and own goal to achieve.

¶ That is, every member in a military network may be assigned a specific value to the organisation, though the values of the members may vary with changing mission demands and circumstances.

Network Systems (2)

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¶ Consequently, a networked force, especially a relatively small one like the Australian Defence Force, can be very heterogeneous in the capability and functionality of its members.

¶ Also, the networking requirements of a force on a mission will have to vary in response to changing mission requirements and circumstances.

¶ Therefore, realistic mathematical expressions for military networks must account for the differences and the changing values of the nodes.

Network Systems (3)

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¶ As an example, the key elements of the previous slide are captured in the definition of Network Reach or normalised Connectivity Measure :

(see Symposium paper for further details)

¶ In an effective medium approximation, CM reduces

to the connectivity index

Network Systems (4)

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1 11

)()()(

1)(

TT N NN

RM d

tFtLtK

CtC

)1(

TTi NN

kC

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¶ In defining the normalised Connectivity Measure, we have introduced the concept of a reference network in which all nodes are the same with a value of unity, and likewise for all the links and the flow coefficients which are also all bi-directional.

¶ A real network with the same number of nodes and same topology can then be considered as a network with broken symmetries.

¶ The absolute value of a network connectivity has little meaning, but its value relative to that of the reference network has much more information content.

Network Systems (5)

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Network Systems (6)

¶ Furthermore, useful insight into the robustness of a military network can be gained by examining the

rate of change of CM(t) as the nodes and links are

randomly or systematically damaged, severed or removed:¶ This is a direct measure of a network’s robustness

against (susceptibility to) degradation under attack as a function of different network topologies and symmetries.

¶ Symmetry concepts may prove to be a powerful tool for network analysis in general.

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¶ Due to the complexity of dynamic networks and

the interactions therein, simulation is often the

only means for analysing their behaviours.

¶ The challenge here is to ensure that the

mathematics underpinning the modelling and

simulation is indeed commensurate with the

nature of the interactions in these systems.

¶ One possible solution is to use intelligent agents

to manage the interactions and in effect let them

do the mathematics themselves.

NCW Modelling and Simulation

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¶ The Dynamic Agents Representation of Networks of Systems (DARNOS) is a tool being developed in Defence Systems Analysis Division, DSTO, for studying the complex interactions in networked operations in the C2, information, decision-making and physical domains.

¶ In DARNOS, there are no constraints on the interaction length scales (spatial or graph-theoretic), the organisational structure, or network topology and symmetry.

NCW Modelling and Simulation (2)

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¶ DARNOS employs an organisation-oriented intelligent agent technology to model the dynamics of C2, networking (who-needs-to-talk-to-who), and decision-making in networked operations.

¶ The simulation infrastructure and representation of all the physical activities in the battle-space are currently provided by a DSTO product called the BattleModel.

¶ DARNOS + BattleModel can be used for close-loop simulations or human-in-the-loop CD&E.


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The DARNOS + BattleModel combination is at present probably the only M&S package capable of supporting modelling and analysis of the operational effectiveness of a networked force to bring together its spatially separated warfighting capabilities and to project the desired effect to the right place at the right time.


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¶ Operational effectiveness of a networked force with different degrees of networking, and/or different networking and C2 structures.

¶ Force mix studies.

¶ Organisational interoperability and multi-agency (e.g. security and counter-terrorism) operations.

¶ (Joint) force synchronisation and dynamic re-tasking problems (e.g. time-critical-targeting).

¶ NCW experimentation.

¶ The human dimension in NCW – under development.

DARNOS Applications

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¶ Nonlocal interactions and low degrees of symmetry are common occurrence in many complex dynamic systems, and probably no more so than in networked operations.

¶ These problems can present great challenges to our ability to understand and manage them, both at the conceptual level and in the mathematical modelling.

¶ Symmetry concepts and agent-enabled simulations with the requisite mathematical models may prove to be powerful tools for tackling and unravelling the highly complex interactions in these systems.

SUMMARY

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Questions?