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COMPLEX SYSTEMS: ANALYSIS METHODS

Young Researchers MeetingYRM 2010

2-9-2010 Rome

Stefano De Porcellinis

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•Complex systems•Structural analyses•Functional analyses•Holistic models•Decomposition and reductionistic models•Integrated approaches•Conclusions

Summary

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•Complexity of a system derives from its structure (often ‘complicated’), from the dynamics occurring and from the processes that lead to the formation and evolution of the system.

•The analysis of complex system starts from the identification of the interactions existing among the elements of those systems and among the system and its surrounding environment.

Complex systems

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Biological systemsTechnological systems

Natural systems

Is there any common feature (or nature???) ?

Complex systems

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Complexity“The quality of being complex”I.e. “A whole made up of complicated or interrelated parts” (Merriam Webster )

The main “feature” of complexity relates with the interconnection and interdependeencies of several (many) elements.

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Topological analysesThe analysis of the topology is a first step of investigation.

– Nodes with the highest degree, hubs. (k>7)

•Trino

•Montalto

di Castro

Aurelia

Turbigo

Brindisi

Rossano

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Topological analysesTopological indices may be related to:•Diameter•Average distance among nodes•Degree of connection (mean, max,etc)•Centrality (number of minimal paths among all the nodes)•Clustering coefficents•etc…

Clustering coefficent for a node.

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Topological analysesFirst models of complex systems, Random Graphs (Erdos Renyi model, 70’s)

(a) Random graph. (b) Directed random graph

Shortest path lenghts follow the logarithm of the number of nodes. Low clustering coefficent.

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Actual very large systems show very small shortest paths (less than log) length, and high clustering coeffs.

Small-world models (Watts & Strogatz, 1998).A small world is obtained starting from a regular lattice and randomly rewiring some links.

Topological analyses

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Nodes in random and small-worls systems show quite constant degrees… Real systems don’t.

Often, degrees of real systems are dominated by a power law.

Scale-free models (Albert-Làszlò Barabàsi, 1999)

Topological analyses

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Characteristics of scale-free systems and dynamics:• Dynamic systems

• Evolution• Preferential attachment

• Environmental constraints

(a) the Internet, where nodes are routers and edges show physicalnetwork connections. (b) an ecosystem (c) professional collaborationnetworks between doctors(d) rail network of Barcelona, where nodesare subway stations and edges represent rail connections.

Topological analyses

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Scale-free systemsPeculiarities:•Highly connected nodes (hubs) are closely followed by smaller degree nodes, followed by even smaller degree ones… and so on.•Highly connected nodes are thought to serve specific purposes (usually)•Resilient to random faults•Sensible to targeted attacks•Nodes are organized in clusters, connected each other by hubs

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Scale-free systemsThe characteristics of scale-free systems can be exploited to augment the effects of external actions on them. Examples:1. Social networks, including collaboration networks. -> Targeted marketing

campaignes (Viral marketing); Targeted vaccination campaignes; 2. Protein-Protein interaction networks. -> Most effective treatments for systemic

deseases3. Sexual partners in humans, which affects the dispersal of sexually transmitted

deseases-> Comunication campaignes, focused on most exposed (“connected”) categories.

4. Many kinds of computer networks, including the World Wide Web -> High-bandwidth devices for hubs, targeted hacker attacks

5. Semantic networks -> Improved OCRs or data mining tools.

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Functional analysesTopology is only one of the possible analysis dimension.Indeed…

…underliyng topology is not the system itself!

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Functional analyses• Complexity Functional properties of the elements• Complexity Functional interactions among the elements• Complexity Mutual interactions, interdependencies

A B

C

Feedbacks are usual in real systems

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Stochastic traffic model

Functional analyses

DC Power-Flow

Functional analysis of coupled technological systems

Interactions

V. Rosato, L. Issacharoff, F. Tiriticco, S. Meloni, S. De Porcellinis, R. Setola. “Modelling interdependent infrastructures using interacting dynamical models”. International Journal of Critical Infrastructures. 2008 vol. 4, pp. 63-79 ISSN: 1475-3219. doi:10.1504/IJCIS.2008.016092.

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Functional analysesFunctional analysis of coupled technological systems

V. Rosato, L. Issacharoff, F. Tiriticco, S. Meloni, S. De Porcellinis, R. Setola. “Modelling interdependent infrastructures using interacting dynamical models”. International Journal of Critical Infrastructures. 2008 vol. 4, pp. 63-79 ISSN: 1475-3219. doi:10.1504/IJCIS.2008.016092.

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Methodology first developed by Hiames & c., based on the well-known Leontief economic models for market equilibrium

Briefly:

The model tries to figure out the impact of external perturbations on a system, taking into account cascade and feedback phenomena

Macro elements of the systems are modelled as monolithic sectors

Macro elements are described in term of inoperability xi (the inability of a system to perform its intended functions)

IIM (Input-output Inoperability Model)

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A

B

C0,3

0,2

0,4 0,6

Elements is an entity described by its inoperability level

Dependency (or Leontief) coefficient (aij)

u are external induced

inoperability

aij is the inoperability on i induced by the inoperability of j

Under some “reasonable” hypotheses

X(k+1)=Ax(k)+Bu(k)

IIM (Input-output Inoperability Model)

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0 * * *

* 0 * *

* * 0 *

* * * 0

Ai ij

j

a represents a measure of “weakness”

j iji

a

dependency index

influence gainThe influence that a specific element has on the global system

IIM (Input-output Inoperability Model)

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Hardest task: quantification of the dependency coefficientsDependency coefficients are generally identified using economical or statistical data.Data may also be collected under the form of “experience”, in terms of knowledge of peculiar behavior of the elements.

To manage such information, it is possible to exploit the expressiveness of fuzzy numbers, e.g.:

R. Setola and S. De Porcellinis. “Complex Networks and Critical Infrastructures”, Modelling, Estimation and Control of Networked Complex Systems in Understanding Complex Systems, Ed. A. Chiuso, L. Fortuna, M. Frasca, A. Rizzo, L. Schenato and S. Zampieri, Springer-Verlag, Berlin, 2009.

IIM (Input-output Inoperability Model)

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S. De Porcellinis, M. Sforna and R. Setola. “Critical infrastructure dependency assessment using input-output inoperability model. ”, International Journal of Critical Infrastructure Protection, Vol 2, Num 4. 2009, Elsevier.

R. Setola and S. De Porcellinis. “A Methodology to Estimate Input-output Inoperability Model Parameters”, Critical Information Infrastructures Security 2007, Lecture Notes in Computer Science, Springer-Verlag, Berlin, pp. 149 – 160, 2008.

IIM (Input-output Inoperability Model)

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Reductionistic approachesHolistic models tend to extremely simplify the simulated

dynamics.To have a more detailed description of our systems, we can

decompose them into “small” (numerous) elements, implementing a complex behaviour for each one.

Reductionistic approaches are often undertaken togheter with massive simulation.

The aim is to identify emerging behaviours deriving from the high number of interdependencies and non linearities within the system.

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Each element can have a quite complex functional model, like:

…also the nature of dependencies can be multiple, e.g. chemical, electrical,physical, sociological, etc.

S. Panzieri, S. De Porcellinis and R. Setola. “Modelling critical infrastructure via a mixed holistic reductionistic approach”. International Journal of Critical Infrastructures. 2009. vol. 5, pp. 86-99 ISSN: 1475-3219.

Reductionistic approaches

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Single scale analyses• Each one of the above mentioned methodologies is able

to focus the modeling and analysis on specific aspects of the complexity

• Each one, at the same time, tends to omit some aspects of the same phenomenon

Abstraction level

Physical

Logical

Detail Concepts

Holistic, system level

Interactions

Reductionistic,component

level

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Multiscale analyses, mixed holistic-reductionistic

Abstraction level

Physical

Logical

Detail Concepts

Mixed Holistic ReductionisticInteractions

Reductionistic,component

level

Holistic, system level

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Component level

System level

Component level

System level

•Interdependencies are described also between elements at different levels of abstraction

Multiscale analyses, MHR

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S. Panzieri, S. De Porcellinis and R. Setola. “Modelling critical infrastructure via a mixed holistic reductionistic approach”. International Journal of Critical Infrastructures. 2009. vol. 5, pp. 86-99 ISSN: 1475-3219.

•Models can take concurrently into account component dynamics, aggregate dynamics and whole system characteristics.

Multiscale analyses, MHR

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CISIA (terminato)

Minuto 10 Livelli: . ooo

0 200 400 600 800 1000 1200

0

200

400

600

800

1000

1200

System Components & macro

components

Electric Grid 35

Urban areas 6

Air-ports 2

Sea-ports 2

Railway 27

Highways 23

TLC 141

Simulation of complex systems, using an MHR simulation engine (CISIA, soon on www.cisia.net) and our multiplatform modeling tools (for more info, contact us at info@biomatica.it)

Multiscale analyses, MHR

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S. De Porcellinis, S. Panzieri, R. Setola and G. Ulivi. “Simulation of heterogeneous and interdependent critical infrastructures”. International Journal of Critical Infrastructures. 2008 vol. 4, pp. 110-128 ISSN: 1475-3219.

Multiscale analyses, MHRSimulation of complex systems, using an MHR simulation engine (CISIA, soon on www.cisia.net) and our multiplatform modeling tools tools (for more info, contact us at info@biomatica.it)

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Conclusions•Complexity derives from structural properties

•Complexity derives, also, from interactions and interdependencies

•Several approaches can be followed to address the issues related with complexity:

topological; functional; holistic; reductionistic

•Those methodologies have to deal with the lack of information and their

uncertainty

•Multiscale approaches seem to be promising, but have to be supported by the

development of innovative modelling tools.

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Thanks!

s.deporcellinis@biomatica.it

COMPLEX SYSTEMS: ANALYSIS METHODS