The study of complex systems questions the bottom up approach of traditional science, the reductionist approach. The latter assumes that the underlying mechanisms of a system's behaviour can be understood by studying its parts independently. A complex system, however, is more than the sum of its parts - a holistic or systems approach is requied. Its overall behaviour emerges through interaction between the constituent parts. Although the term ``complex system'' has multiple uses, some of its properties which are generally agreed upon are as follows.
Examples of complex systems can be found in all areas of science - non-linear dynamical systems (mathematics), networks (statistics), non-equilibrium chemical reactions (chemistry), protein interactions (molecular biology), brain function (neuro-science), colonies, flocks, crowds (animal and social behaviour), cellular automata (computer science), market behaviour (economics), etc. Understanding these systems is becoming increasingly urgent and outcomes of the research potentially impact on life on Earth and the societies in which we live. The study of these systems requires combining methods from more than one discipline as well as the development of new mathematical and computational tools and is widely recognised as a challenge to traditional computational science.
© Science and Technology Facilities Council 2009-11. Neither the Council nor the Laboratory accept any responsibility for loss or damage arising from the use of information contained in any of their reports or in any communication about their tests or investigations.
Complex systems, from the World Wide Web to the human brain, remain a grand challenge to scientific understanding . They have arisen in an evolutionary fashion through the interaction of their components in a larger environment. Sophisticated and effective concepts and methods are required in order to tackle problems related to such systems.
The following description is based on Wikipedia. A complex system is any system featuring a large number of interacting components, whose aggregate activity is non-linear and typically exhibits self organisation under selective pressures. The system as a whole exhibits one or more properties, and behaviour among the possible properties, not obvious from the properties of the individual parts.
Some other definitions are from H.J. Jensen's compexity research Web site .
Other specific definitions have been given, for instance the following from a series of articles from a special issuf of Science magazine (1999).
A system's complexity may be in one of two forms: disorganised complexity or organised complexity. In essence, disorganised complexity involves a very large number of parts and organised complexity is concerned with the subject system, quite possibly with only a limited number of parts, exhibiting emergent properties. An apparently simple double pendulum is a complex system in which the detailed dynamics cannot be simply predicted and for certain conditions may be chaotic.
Complex systems are met in many areas of natural science, mathematics and social science. Fields that specialise in the inter-disciplinary study of complex systems include systems theory, complexity theory, systems ecology and cybernetics. Examples of complex systems with many parts include ant colonies, human economies and social structures, climate, nervous systems, cells and living organisms, including human beings, as well as modern energy and IT infrastructures. Indeed, many systems of most interest to humans are complex systems. A nice introduction, similar in scope to the current report, was written by Ottino .
In the area of mathematics, arguably the largest contribution to the study of complex systems was the discovery of chaos in deterministic systems (such as the double pendulum which is goverened by a set of ordinary differential equations). This is a feature of certain dynamical systems that is strongly related to non-linearity. The study of neural networks was also integral in advancing the mathematics needed to study complex systems.
The notion of self organising systems is tied up with work in non-equilibrium thermodynamics, including that pioneered by chemist and Nobel laureate Ilya Prigogine in his study of dissipative structures.
See separate report .
Many complex systems can exhibit chaotic behaviours. For a dynamical system to be classified as chaotic, it must typically have the following properties.
An arbitrarily small perturbation of the current trajectory of a chaotic system may lead to significantly different future behaviour. This has been referred to as the ``butterfly effect'' - a popular mis-conception is that the movement of a butterfly can affect the weather.
Chaos theory is now a well founded branch of mathematics. It is applicable to many areas of science including: the weather; planetary motion; turbulence; radio engineering; financial markets; biology; medicine; earthquakes; solar flares; landscape formation; forest fires; social unrest; mathematics; computer science; engineering; philosophy; politics; population dynamics; psychology; and robotics. One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model  have been used to show how population growth under density dependence can lead to chaotic dynamics. It is also being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.
Complex adaptive systems (CAS) are special cases of complex systems. They are complex in that they are diverse and made up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the stock market, social groups such as insect and ant colonies, the bio-sphere and the eco-system, the brain and the immune system, the cell and the developing embryo, manufacturing businesses and any human social group based endeavour in a cultural and social system such as political parties or communities.
The term complex adaptive systems (CAS) was coined at the inter-disciplinary Santa Fe Institute (SFI, see below) by Holland, Gell-Mann and others.
CAS ideas and models are essentially evolutionary, grounded in modern biological views on adaptation and evolution. The theory of complex adaptive systems bridges developments of systems theory with the ideas of generalised Darwinism, which suggests that Darwinian principles of evolution can explain a range of complex phenomena from cosmic to social objects. These can often be simulated using Agent Based Models.
A non-linear system is one whose behaviour can't be expressed as a sum of the behaviours of its parts or of their multiples. In technical terms, the behaviour of a non-linear system is not subject to the principle of superposition whereas linear systems are.
In practical terms, this means a small perturbation may cause a large effect (e.g. the butterfly effect), a proportional effect, or even no effect at all. In linear systems, effect is always directly proportional to cause.
Analysis of networks of all kinds is growing in importance and it has been realised that networks can be representations of complex systems and illustrate emergent behaviour, for instance clustering .
Whilst important, one should note that not all complex systems at first sight exhibit networks. A colony of ants collaborate to build a nest, but there is no formal connection between them. Traders, computers or phones do however have such links. Network science has been used to analyse pandemics, such as the spread of H1N1 swine flue .
Network analysis views relationships in terms of nodes and links (edges). Nodes are the individual actors within the networks, and links are the relationships between the actors. The resulting graph based structures are often very complex. There can be many kinds of links between the nodes. This is sometimes refered to as SNA, Social Network Analysis, and is most successful in small captive populations of obligate social species in a restricted territory. Nevertheless it has also been a successful tool in other circumstances, for instance in epidemiology, e.g. studying tuberculosis transmission in a wild meerkat population. SNA is also an important tool for mining intelligence information, for police or military purposes.
Network theory is an area of applied mathematics and a branch of graph theory. It can be applied to many disciplines including particle physics, computer science, biology, economics, operations research, and sociology. Indeed it may be possible that many if not all complex systems can be represented in some way as networks. Network theory concerns itself with the study of graphs as a representation of either symmetric relations or, more generally, of asymmetric relations between discrete objects. Examples of this include logistical networks, the World Wide Web, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc.
Social network analysis maps relationships between individuals in social networks. Such individuals are often persons, but may be groups (including cliques and cohesive blocks), organisations, nation states, Web sites, citations between scholarly publications (scientometrics), etc.
Network analysis, and its close relation traffic analysis, has significant use in intelligence. By monitoring the communication patterns between the network nodes, its structure can be established. This can be used for uncovering insurgent networks either hierarchical or leaderless in nature.
With the recent explosion of publicly available biological data obtained using high throughput techniques, the analysis of molecular networks has gained significant interest. The type of analysis in this context is closely related to social network analysis, but often focuses on localised patterns. For example network motifs are small sub-graphs that are present in the network more than expected. Activity motifs are similar patterns in the attributes of nodes and edges in the network that are over represented given the network structure.
Most social, biological and technological networks display substantial non-trivial topological features with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or dis-assortativity among vertices, community structure and hierarchical structure. In the case of directed networks these features also include reciprocity, triad significance profile and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.
Two well known and much studied classes of complex networks are scale free networks and small world networks (as in ``6 degrees of separation''), whose discovery and definition are canonical case studies in the field. Both are characterised by specific structural features - power law degree distributions for the former and short path lengths and high clustering for the latter. However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structure have attracted attention.
The field continues to develop rapidly and has brought together researchers from many areas including mathematics, physics, biology, computer science, sociology, epidemiology, etc. Ideas from network science have been applied to the analysis of metabolic and genetic regulatory networks, the design of robust and scalable communication networks both wired and wireless, the development of vaccination strategies for the control of disease, and a broad range of other practical benefits. Research on networks has seen regular publication in some of the most visible scientific journals and vigorous funding in many countries, has been the topic of conferences in a variety of different fields and has been the subject of numerous books. See for example .
The World Wide Web forms a large directed graph, whose vertices are documents and edges are hyper-links. Barabasi et al.  demonstrated that despite its apparently random character, the topology of this graph has a number of universal scale free characteristics. They introduced a model that interprets the Web as a free network, capturing in a minimal fashion the self organisation processes which govern it.
For information about network analysis software see http://en.wikipedia.org/wiki/Social_network_analysis_software. I've chosen the following for further investigation: igraph, socnetv, statnet and various R packages. See also http://rsat.ulb.ac.be/rsat/index_neat.html.
In biology, phylogenetics is the study of evolutionary relatedness among various groups of organisms, e.g. species, populations, which is discovered through molecular sequencing data and morphological data matrices. A phylogenetic tree is a particular kind of network with evolutionary or other implications about relatedness - it is a kind of directed graph. Taxonomy, the classification of organisms (or other artefacts) according to similarity, has been richly informed by phylogenetics but remains methodologically and logically distinct. The fields overlap however in the science of phylogenetic systematics or cladism, where only phylogenetic trees are used to delimit taxa, each representing a group of lineage connected individuals. Lineage is not restricted to evolution, but can be defined from another metric.
This is becoming a general methodology for information analysis and has already been applied to many branches of science, e.g. linguistics and cultural evolution, such as in the work of Mark Pagel et al. at Reading http://www.evolution.reading.ac.uk.
Phylogenetic analysis is computationally demanding and typically uses methods such as Bayesian Markov chain Monte Carlo or Metropolis coupled Markov chain Monte Carlo techniques to search for relatedness. Such techniques rely on appropriate data being available. This raises the question of how to cope with missing or incomplete data. This question is also addressed in the longitudinal modelling of social data, e.g. by Crouchley et al. who aim to derive quantitative statements relating aspects of society such as wealth and educational attainment.
Network analysis and phylogenetics may be different ways to represent and interpret aspects of a complex system. The former typically shows relationships between components of a similar type, the latter classifies different types of component according to some other relationship between their properties.
Luciano da Fontoura Costa et al. have been particularly active in the field of complex networks. Because of their generality , complex networks represent a natural alternative for representing, characterising and modelling the structure and non-linear dynamics of discrete complex systems . Recent developments have aimed at characterising and relating the structure and dynamics of complex systems represented by complex networks. This includes systematic characterisation of connectivity and dynamics in complex systems . The concepts illustrated in complex systems have an underlying geometry which includes morphological neuronal networks . The concept of super-edges has recently been introduced  - an approach which natural and comprehensively integrates the topology and dynamics of a given complex system, as well as the concept of avalanches of activations in complex systems  and how they can be modelled in terms of concentric representations of complex network. Among the results obtained to date, we have the fact that the occurrence of avalanches is intrinsically related to the hierarchical organisation of the network as well as to the presence of long range connections. It can be shown that the dynamics of avalanches can be used to identify modularity in complex networks . Dynamical complex systems with dynamical topology and moving nodes can also be addressed .
This section now describes some more dynamic characteristics of complex systems.
The behaviour of a system cannot easily be predicted simply by solving equations and therefore must be simulated in some way to reveal its full behaviour. Simulations may be hard to repeat as they are very sensitive to conditions, general behaviours must be extracted.
Functions that do not repeat values after some period - otherwise known as chaotic oscillations or chaos.
Systems alternate in an apparently random way between two or more exclusive states (bifurcation).
The system in some way follows a power law applicable to both fine and coarse scale views.
Self Organised Criticality (SOC) is considered to be one of the ways that complexity arises in nature. In physics, SOC is a property of some classes of dynamical system which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/ or temporal scale invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to precise values.
The phenomenon was possibly first identified by Turcotte, Smalley and Solla . SOC is typically observed in slowly driven non-equilibrium systems with extended degrees of freedom and a high level of non-linearity. Many individual examples have been identified but to date there is no known set of general characteristics that guarantee a system will display SOC. Its concepts have nevertheless been applied across fields as diverse as geophysics, physical cosmology, evolutionary biology and ecology, economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology, etc.
Characteristic of ``chaotic'' systems and governed by a Lyapunov exponent.
Characteristic of many non-linear systems which can switch behaviour patterns or phases.
Complex systems may exhibit behaviours that are emergent, which is to say that whilst the results may be deterministic, they may have properties which are only revealed by simulation or at a higher level of observation. For example, the termites in a mound have physiology, biochemistry and biological development that are at one level of analysis, but their collective social behaviour and mound building capability are properties that emerge from the collection of termites and needs to be analysed differently.
Complex adaptive systems are characterised by a high degree of adaptive capacity. One emergent property is that of resilience in the face of perturbation. This is highly important in understanding natural systems, such as the earth's eco-system.
Synchronicity is another emergent behaviour which has been identified in many systems. It was first identified in swarms, for instance insects, birds or fish which collectively respond to very small signals communicated between the individual agents.
It can be difficult to determine the boundaries of a complex system. The decision is ultimately made by the observer.
Complex systems are usually open systems, that is they exist in a thermodynamic gradient and dissipate energy. In other words, complex systems are frequently far from energetic equilibrium: but despite this flux, there may be pattern stability.
The history of a complex system may be important. Because complex systems are dynamical systems they change over time, and prior states may have an influence on present states. More formally, complex systems often exhibit hysteresis.
The components of a complex system may themselves be complex systems. For example, an economy is made up of organisations, which are made up of people, which are made up of cells - all of which are complex systems.
As well as coupling rules, the dynamic network of a complex system is important. Small world or scale free networks which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.
Both negative (damping) and positive (amplifying) feedback are often found in complex systems. The effects of an element's behaviour are fed back in such a way that the element itself is altered.
Something already said above.
Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him ``rule 30''.
See separate report .
The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand, for instance in following the solution of Ricker's equation mentioned above. Computers made these repeated calculations practical, while figures and images made it possible to visualise these systems and discover islands of stability. One of the earliest electronic digital computers, ENIAC, was for instance used to run simple weather forecasting models.
Many examples are given in the Atlas . We do not know what software was used to generate these images.
We are currently investigating the use of GraphViz (including dot), see http://www.graphviz.org. This is supported by AT&T Research.
The GaphViz on-line gallery contains a set of images of large sparse matrices created by Yifan Hu at AT&T Research (formerly at Daresbury) http://www.research.att.com/~yifanhu/GALLERY/GRAPHS.
These are generated using his own algorithm, described on http://www.research.att.com/~yifanhu/research_interest.html. Networks or graphs encapsulate complex relations in many application areas. Algorithms to analyse these networks, for example, in finding out community structures/ clusters, calculating PageRank, and visualizing the networks, play an important role in discovering structures and gaining knowledge from complex data. A novel multi-level force directed algorithm with fast force calculation using octree data structure and supernode force approximation was proposed and implemented, resulted in an efficient graph drawing algorithm with a complexity of O(|V| log(|V|) + |E|). This algorithm can handle graphs of millions of nodes, as well as large tree like graphs, such as the Mathematics Genealogy Trees, see http://www.research.att.com/~yifanhu/GALLERY/MATH_GENEALOGY
GraphViz can also be used to display phylogenetic trees, e.g. using the Newick notation.
One of the original applications of chaos theory where phenomena such as exponential divergence were discovered by studying computer simulations.
Sudies of epilepsy.
Observation has shown that earthquakes exhibit scale invariant behaviour such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori Law describing the frequency of after shocks.
Reference  of 2008 lists many applications of research into complex systems. It has a bibliography of 565 references and covers the following areas: social networks; communication; economy; financial market; computer science; Internet; World Wirde Web; citations; transportation; electric power transmission; bio-molecular networks; medicine; ecology; neuro-science; linguistics; earthquakes; physics; chemistry; mathematics; climate networks; security and surveillance; epedemiology; collaboration networks. This illustrates the level of world wide interest in this kind of study.
Systems are sets of entities, physical or abstract, comprising a whole where each component interacts with or is related to at least one other component and they all serve a common objective. The scientific research field which is engaged in the interdisciplinary study of universal system based properties of the world is general system theory, systems science and recently systemics. They investigate the abstract properties of the matter and mind, their organisation, searching concepts and principles which are independent of the specific domain, independent of their substance, type, or spatial or temporal scales of existence.
An example is systems ecology which is an interdisciplinary field of ecology, taking a holistic approach to the study of ecological systems, especially ecosystems. Systems ecology can be seen as an application of general systems theory to ecology. Central to the systems ecology approach is the idea that an ecosystem is a complex system exhibiting emergent properties.
Systems ecology focuses on interactions and transactions within and between biological and ecological systems and is especially concerned with the way the functioning of ecosystems can be influenced by human interventions. It uses and extends concepts from thermodynamics and develops other macroscopic descriptions of complex systems.
The Santa Fe Institute (SFI) is a non-profit research institute located in Santa Fe, New Mexico, USA and dedicated to the study of complex systems. Swarm, the first software tool created for ABMS was developed at the Santa Fe Institute in 1994, Swarm was specifically designed for Artificial Life applications, see .
SFI, founded in 1984, was effectively a spin-off from work at Los Alamos National Laboratory.
Joshua Epstein is director of the Brookings Institute Center for Social and Economic Dynamics in Washington DC http://www.brook.edu/ES/dynamics/models/history.htm. We note that Axtell and Epstein are well known for work on Agent Based Modelling and Simulation, for instance publishing a first large scale study of epidemics in 1996 .
Other research institutes engaged in complexity science include:
Jeff Schank's Wiki contains a list of researchers, organisations, centres and institutions which are using agent based modelling, many for the study of complex systems: http://www.agent-based-models.com/blog/resources/organizations/.
The 3rd workshop on Complex Systems Modelling and Simulation (CoSMoS) will take place on 12/8/2010 in Denmark as a satellite event of the AlifeXII: 12th International Conference on the Synthesis and Simulation of Living Systems. Previous CoSMoS workshops have provided a forum for research examining all aspects of the modelling and simulation of complex systems. The 2010 workshop will focus on the engineering aspects of modelling and simulating (artificial) living systems.
Constructing models and simulations of complex systems is a challenging and interdisciplinary task. Elements might include choice of modelling tools and techniques, simulation infrastructures, concurrency, the process of moving from models to simulations, arguing validity of simulations, and the identification of re-usable engineering techniques such as patterns. The CoSMoS workshop series is part of a four year initiative, based at the Universities of York and Kent, UK, to develop a framework and infrastructure for the construction of of generic complex systems simulations. See http://www.cosmos-research.org/workshops/cosmos-workshop-2010/.
An NSF report was published in 2007 entitled: Modeling and Simulation at the Exascale for Energy and the Environment. Report on the Advanced Scientific Computing Research Town Hall Meetings on Simulation and Modeling at the Exascale for Energy, Ecological Sustainability and Global Security (E3). http://www.sc.doe.gov/ascr/ProgramDocuments/ProgDocs.html. This notes as one of its goals to identify emerging domains of computation and computational science that could have dramatic impacts on economic development, such as agent based simulation, self assembly, and self organization and suggests this can be addressed by Math and Algorithms. Advancing mathematical and algorithmic foundations to support scientific computing in emerging disciplines such as molecular self assembly, systems biology, behavior of complex systems, agent based modeling, and evolutionary and adaptive computing.
The EPSRC funds complexity research through various mechanisms. The primary route for funding is through responsive modegrants, however there has been a variety of managed calls specifically aimed towards research in complexity.
The responsive mode scheme can be used to support a wide variety of proposals, including feasibility studies, mobility for postdoctoral researchers, overseas travel grants and visiting researchers, and long-term proposals to develop or maintain critical mass.
The Novel Computation call was an invitation to computer scientists, biologists, engineers, life scientists, mathematicians, neuroscientists, physicists, physical scientists and anyone interested in exploring new ways to create, control, model, understand or predict complex systems. The objectives included: complexity and computation; nature inspired computation; analysis and control of complex systems; hybrid systems; complexity and information uncertainty. There was up to £60k available, to fund a 6 or 12 month network, after which time a full research proposal was submitted. There was £10M available to fund the full proposals. Full proposals were submitted at the end of September 2004. Networks were awarded to: J. Efstathiou (Oxford) Coping with Complexity; C. Taylor (Bristol) Performance based engineering: The computational challenge; S. Saad (Aston) Inference and cooperative behaviour in large interacting systems; G. Ackland (Edinburgh) Novel approaches to networks of interacting agents; A. Adamatzky (West of England) Non-linear media based computers; S. Bullock (Leeds) Simple models of complex networks research cluster and H.J. Jensen (Imperial College) Evolutionary models of complex systems.
The Multidisciplinary Critical Mass call to encourage the development of novel mathematical and/ or statistical methods to address research challenges in other disciplines, with funding for centres of critical mass that will enable substantial research programmes in these areas to be established. A single grant worth £1M is available each year. The 2004 grant was awarded to: C.J. Budd (Bath) Establishment of the University of Bath Centre for Complex Systems - a 5 year programme to generate the critical mass necessary for the development of a world class maths/ engineering/ biology interdisciplinary centre, exploiting the blurred distinction between mathematics, statistics and computation, when analysing large complex systems.
The EPSRC is aiming to establish an international institute for Large Scale Complex IT to act as a cauldron to foster collaboration and develop tools and techniques through an international programme of study groups, similar to the Isaac Newton Institute programme. A total of £10M is available for the centre.
The Centres for Integrated Systems Biology call aims to establish a number of Centres for Integrative Systems Biology, possessing the vision, breadth of intellectual leadership and research resources to integrate traditionally separate disciplines such as biology, chemistry, computer science, engineering, mathematics and physics in a programme of international quality research in quantitative and predictive systems biology. Up to three centres to be funded per year, with awards of up to £5M over a five year duration.
Through the call for Taught Courses in Complexity Science and Complex Systems EPSRC investment will focus on the training environment which will supply the skilled people most able to address the emerging research problems in complexity science and complex systems.
The Isaac Newton Institute, Cambridge, aims to stimulate research in all areas of mathematical sciences by bringing together researchers from all over the world to focus on research programmes of extended duration. The EPSRC currently contributes over £3M towards the running of the institute. Complexity related programmes running over the next few years, include: Interactions and Growth in Complex Stochastic Systems; Statistical Mechanics of Molecular and Cellular Biological Systems; Principles of the Dynamics of Non-equilibrium Systems; Developments in Quantitative Finance.
Generative Social Science is widely regarded as one of the grand challenges of the social sciences. The term was popularised in 1996 by Epstein and Axtell  who define it as simulation that allows us to grow social structures in silico demonstrating that certain sets of micro-specifications are sufficient to generate the macro-phenomena of interest. It is consistent with the development of the complexity sciences, with the development of de-centralised and distributed agent based simulation and with ideas about social and spatial emergence. It requires large scale data bases for its execution as well as powerful techniques of visualisation for its understanding and dissemination. It provides experimental conditions under which key policy initiatives can be tested on large scale populations simulated at individual level. It is entirely coincident with the development of e-social science which provides the infrastructure on which such modelling must take place.
See Web site for the ESRC-funded GENESIS project http://www.genesis.ucl.ac.uk/.
This document was generated using the LaTeX2HTML translator Version 2008 (1.71)
Copyright © 1993, 1994, 1995, 1996,
Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.
The command line arguments were:
latex2html -local_icons -split 3 -html_version 4.0 Complex
The translation was initiated by Rob Allan on 2011-02-18