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propagators formalism. The equation defines the field operator for the field variable of the functional interaction, i.e. insulin concentration or electric potential.
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Complex Systems Modeling Christophe Lecerf , Thi Minh Luan Nguyen [email protected], [email protected]

Abstract—This paper addresses the simulation of the dynamics of complex systems by using hierarchical graph and multi-agent system. A complex system is composed of numerous interacting parts that can be described recursively. First we summarize the hierarchical aspect of the complex system. We then present a description of hierarchical graph as a data structure for structural modeling in parallel with dynamics simulation by agents. This method can be used by physiological modelers, ecological modelers, etc as well as in other domains that are considered as complex systems. An example issued from physiology will illustrate this approach. Index Terms—Complex system, hierarchical graph, multi-agent system, modeling and simulation

A system is considered as complex when a large number of elements, arranged in hierarchical structures are involved and, more important, when the interactions take place between elements belonging to different levels (hierarchy, space, time) of the organization. For example, information paths and neurons in the nervous system have each multiple inner levels that we can model while keeping in sight the interactions between the different levels. The interactions between numerous parts of the system produce a emergent behavior that is difficult to predict from the collection of elementary specifications. To model this type of system, we have to take into account not only individual entities features, but also interactions among components of different levels that happened in a real system. Traditional modeling and simulation methods offer just the vision of macro level behaviours, they do not provide insight views of micro level. Agent-based modeling appears as a suitable tool for simulating complex systems behaviour, in which several thousand of interacting entities can reproduce the real system properties. Modelers can then observe, analyze and study the evolution of particular components and the emergence of collective behaviour of the system under study. In this paper we will study the use of hierarchical graphs for modeling complex systems. More precisely, we will develop the theoretical aspects related to these modeling problems; in particular we will try to solve the questions related to the mapping of these hierarchical graphs with the structural organization of the represented systems on one hand and its dynamics on the other hand. The paper is organized as follows. Section 1 is dedicated to a short presentation of a complex system, its properties and the need of simulation in order to observe the evolution of its behaviour when facing to environmental changes. In section 2, the use of hierarchical graph in modeling the structural organiC. Lecerf, Centre LGI2P Ecole des Mines d’Ales - Site EERIE Parc Scientifique Georges Besse F 30035 Nimes Cedex 1, France L. Nguyen, Laboratoire de Recherche en Informatique Avanc´ee, Universit´e Paris 8 et EPHE, 41 rue Gay Lussac, 75005 Paris, France

zation is described. In order to simulate the dynamics of biological systems, agent-based simulation is also proposed in this section. Hierarchical graph play an intermediate role between real complexes systems and its dynamics simulation model. An example issued from physiology is described in section 3 in order to illustrate this approach. We summarize here some important concepts in integrative physiology and the use of our modeling method for biological systems. Finally, conclusions are drawn and future works are outlined in section 4. We will begin this paper by a brief resume of complex system and its properties in the next section I. C OMPLEX SYSTEM A complex system is composed of a set of components, each of them being itself a set of sub-components, in which various interactions between different organization levels take place. Recurrence or hierarchy is the most fundamental characteristic of this kind of system. Their hierarchical aspects are summarized in ([?]) by Aiello: - Abstraction hierarchy - Description hierarchy - Time hierarchy We have many examples of complex system around us, for example ecological system, financial market, physiological system, etc. One of the most important tasks in studying complex systems is the understanding of their complexity. We would like to present three techniques summarized by Jennings in [?]: - Decomposition: it means to divide the main problem in several smaller problems. Thanks to the decomposition, a complex system can be divided into many sub-systems at different level of the structural organization. In the first analyze, each of them can be treated as if it was independent. - Abstraction: this is the process to define a simplified model in order to represent the system which emphasizes some details or properties, while suppressing others. - Organization: this is the process to define and control the relationship among the various components. It allows us to group together certain number of basic components and then treat them like a single unit of a higher level. Using these techniques, modeling a complex system become more tractable and a hierarchical graph (detailed in the next section) can be used. It helps us to represent the structure and the communication inside these systems and therefore to predict their behaviour over time. The common properties of complex systems allow us to develop a unified resolution method. By using this modeling method, the biologists, ecologists, etc will

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benefit from the results of the parallel research on the complex systems in other fields than their own. II. M ODELING WITH DISTRIBUTED COMPUTING In general, a complex system is viewed as a continuous system since it receives the activation products from others components continuously. In order to illustrate this point, we can take the example of the neuron in the nervous system. Actually, a neuron cell works through the threshold principle, which means that it captures all input signals without any macroscopic reaction until it reaches its threshold, and then it produces an output signal which is a response to the complete series of inputs. While its dynamics is a kind of a step function, at this point the discrete event system is more suitable to represent this type of models. Moreover, distributed system is based on the reception of messages. Thanks to this similarity, we can think about the use of distributed computing techniques in simulation of the physiological processes. A. Hierarchical graph We now introduce briefly the important points of hierarchical graph. The detailed definition of hierarchical graph is presented in [?] by G. Engels, A. Sch¨urr. In short, a hierarchical graph is a graph with two types of nodes: atomic and complex. • The atomic node is the leaf of the hierarchy and does not have any internal state. • On the contrary, the complex nodes have an internal state, which is another hierarchical graph. A complex node contains a hierarchical graph which may contain other complex nodes. In other words, any complex node in the hierarchical graph is a component of another complex node of a higher level. The definition of the hierarchical graphs and the complex nodes is recurrent. Thus, a unique complex node is used to represent the whole complex system, assumed that each component is referred to as a complex node: the entire system is just the highest level one in the hierarchy. We illustrate this point in the next figure.

Fig. 1. Hierarchical graph

Hierarchical graph G(N,E) - E: set of edge - N: set of node which is whether atomic node or complex node. Here we have a simplified description of atomics node: AN =< ID, L >, in which • ID : node identifier

L : node’s label. The complex node is described as follow CN =< ID, L, C > • C denoted the internal state which is a complex or atomic node. By studying the properties of hierarchical graph, we find out that there is a perfect coherence between the structural organization of complex systems and the hierarchical graph. It’s really a effective tool for manage hierarchical aspects of the such a system, for example structural organization level. The graph is a structure which is used in the modeling of various situations. Actually, a binary relation between objects of the same set is enough to define a structure of graph. Therefore, expressed as graphs, many usual problems could be brought back to traditional problems of the graph theory such as: shortest path; cycle detection; connected components; etc. As we are interested in the dynamics of the model, the graph appears as a coupling intermediate between the physical system and the associated mathematical model. Traditionally, biological processes are modeled mathematically by a set of complicated equations. The traditional models have generally intended to offer a global vision of the study phenomena. Since the biological system involves multiples elements that can be modeled in various levels, these equations depend not only on global variables but also on local variables. Consequently, this one is not suited for designing behaviours models used for simulating a system which involves a great number of interacting elements. This limitation restricts the efficiency of such models and encourages us to use agent-based models for problems where many entities exist and communicate with others in some environments. •

B. Multi-agent system simulation The purpose of agent-based simulation is to enlarge our understanding of the multiple processes that occur in biological systems. It consists mainly of representing parts of system or natural phenomena as a collection of interacting entities that work without a direct external influence. In this system, we are interested in the coordinated behaviour of the individual agent to deduce a system level one. But what is an agent? According to Wooldridge and Jennings ([?]), this is a computer system, situated in some environment, which is capable of flexible autonomous action in order to meet its design objectives. It is defined as active object that: • Perceptive: agents should perceive their environment and respond in a timely fashion to changes that occur in it. • Pro-active: agents should not simply act in response to their environment; they should be able to exhibit opportunistic, goal-directed behaviour and take the initiative where appropriate. • Social: agents should be able to interact, when appropriate with others in order to meet its designed objectives. Since any component of a complex system can be referred as an agent because it has autonomous actions in that environment [?]. Then it possesses all mathematical method to show up a behaviour facing to events which are received from their entourage environment. Every agent has a set of state. At any

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Fig. 2. Functional model of the regulation of the respiratory system

time, it is in state S. In the absence of external event, the system remains on the current state. On the contrary, it receives the external event by its input port. The associated mathematical function will then specify how agent change and react due to this effect. To illustrate this point, we can take an example of a neurone in nervous network. A neurone has a cell body, a set of input ports (the dendrites) and a set of output ports (the axons). A neuron has weighted inputs from other neurons and the input signals form a weighted sum. Moreover, each neuron has its own threshold value, if the activation level exceeds the threshold, the neuron ”fires” and produces output signals to answer to the whole series activation. So, complex system can be seen as a collection of interacting entities that work without direct external influence in order to meet its objective design. It is a necessary step for automating calculation of its dynamics. Therefore, the hierarchical graph appears as a coupling mean between real complex systems and the simulation of their dynamics by cooperative agents. Using hierarchical graph to model the structural orga-

Fig. 3. Relations between a complex system,hierarchical graph and agents

nization of a complex system helps us in the process of defining the agents at work in the model. Actually, exchanges between sub-systems can be modeled by relations between agents.

These functional relations come across the levels defined in the structural hierarchy. III. A N EXAMPLE IN BIOLOGY We now illustrate this approach by presenting an example in physiology. Consider the following schema (figure 2) that represents the different steps of the regulation of respiratory function: many subsystems are involved, with very different scales and structures (mechanical subsystems, chemical subsystems, neural subsystems), all interconnected. Modeling such a system from scratch will a very difficult task. Hopefully, integrative physiology brings both a conceptual framework and a mathematical tool. The next section is devoted to expose these points A. Integrative physiology Traditionally, models in physiology are built independently from each other, and they are built in an empirical manner. Although the models are being sustained by biological experiments and results, different models cannot be easily integrated because they do not share any formal theoretical frame. One could sum up physics as describing the interactions between objects in the physical world with symmetrical forces in a continuous space. Integrative physiology describes the interactions between biological entities in a source-to-sink frame, using functional interactions that are non-symmetrical because of the structural discontinuity of the biological space. Let us explain this fundamental point. Even a simple cell, and moreover a multi-cellular biological organism, is a complex set of substructures in which specific metabolic process occur. At the scale of a human body, one can see and feel different organs that make it obvious that the biological space is neither homogeneous nor isotropic. At the very

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smaller scale of the cell, the space is also divided in local microenvironments, leading to metabolite channelling and pooling phenomena at work in enzymatic metabolism. Since the biological structures interact with each other, whatever its physical scale, the factor supporting the interactions come across the structural discontinuities, materialized by cell membranes for instance. In the integrative physiology conceptual frame, an elementary functional interaction is formally defined by a triplet source, product, sink and an equation for a field variable (the product) driven by a time-space field operator that describes the action through time and space of the source on the sink. This action is directed from the source towards the sink, with no feedback from the sink on the source because the product is transformed in the sink. Since the action takes place over a structural discontinuity, whatever its scale, the action of the source on the sink is non-local. For instance, insulin is a hormone (product) produced by specific cells in the pancreas (source), spread directly in the blood flow (structural discontinuity) and used by a muscular cell (sink) to control its glucose inputs. In another tissue, the example would be one neurone (source) acting on another one (sink) by the release of a neurotransmitter (product) in a synapse (structural discontinuity).

Fig. 4. Functional interaction: the general case scheme

In order to describe a biological organism with this conceptual frame, one has to set up the graph of all the functional interactions that occur in the organism. This operation ends up in a (usually large) graph called an organized formal biological system (O-FBS). The O-FBS should describe both the logical link of the source on the sink and the geometrical properties of the biological space in order to make the time-space field operator usable. Since the speed of the interaction factor (the product) in the biological medium is finite, a functional interaction is non-symmetrical, non-local and non-instantaneous. It should be noted that this finite speed is the consequence of the specific dynamics of the biological medium (synaptic space, blood, conjunctive tissue, etc.) regarding the specific product considered. Each functional interaction has its own field variable with its own dynamics, formalized by an equation summing three terms and referring to source, sink, time and space (figure 4) in the Spropagators formalism. The equation defines the field operator for the field variable of the functional interaction, i.e. insulin concentration or electric potential. The S-propagators formal-

ism was introduced by G.A. Chauvet ([?]) to offer a mathematical representation that provides the integration of elementary physiological mechanisms with respect to their structural hierarchy in biological organisms, thus leading to the building of the observed physiological functions. Since there also exists a functional hierarchy in biological organisms, a clear distinction is made between structural and functional organizations.

Fig. 5. Graph and equation describing the S-Propagator

R ∂ψ =D∇2 ψ + Dr (r0 ) ρr (r0 )P (r0 )ΦP (r0 )ψ(r0 , t0 )dr0 + ∂t Γr (r0 , t) The first term of the equation describes the local diffusion of the product in the physical space around the source. This term may have a strong influence if the source and the sink are very close and if the medium can directly transfer the product. For instance, this would be the case for neuronal cells since an action potential induces a local electric field around it. Conversely, such a term would have no effect in a long distance hormonal interaction. The second term is the strictly speaking S-propagator, and represents the non-local interaction due to the structural discontinuity. Therefore, this term describes the hierarchy of the organizational structure in the physical space. Since the structural discontinuity is also a biological subsystem, it is also described by a functional interaction with a source, product, sink triplet and a time-space field operator. This is the most important and most complex part of the S-propagators formalism because it makes this theoretical frame homogeneous and standardized with respect to space scale. Most of the biological mechanisms are nowadays described by over-simplified mathematical models in which the different interactions between subsystems cannot be taken into account due to the absence of a description with the functional interaction framework. Therefore, these models do not benefit from the S-propagator formalism. The third term of the time-space field operator describing a functional interaction represents the source, i.e. the internal local mechanisms that lead to the generation of the product emitted by the source. For instance, in the central nervous system, should the first two terms have put the local membrane potential above the local threshold, this term describes the mechanisms generating the action potential in the cell. The S-propagators conceptual framework for describing all the interactions in a biological system is very powerful as it brings a standardized formalism and a ”one source - one sink one product” clue to start the description. Obviously this hint will help to compare biological and distributed systems. The main point is that the study of both distributed and physiological systems finally goes through simulations in a discreteevent framework: the state of the model changes at only a discrete set of simulated time points ([?], [?]). When achieved on

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a distributed system, such a simulation imposes the network to be logically synchronized. Mathematically speaking, though most of the specific mathematical models that are used have no analytical solutions at this time (mainly because they are sets of partial differential equations in which some of the variables are themselves described by such sets), our assumption stands because the theoretical frame describing every functional interaction is unified and homogeneous. As shown by Lecerf in his paper [?], there are some common features between distributed computing and physiological system. We will now summarize these points. 1) A finite state machines network. Both distributed systems and physiological systems are described as a structure made of connected components communicating through a point-to-point network. For each component, time is going on in one direction and the processes are not reversible. 2) Finite messages. Units/nodes communicate by means of messages, which size is finite. Though this feature has much more meaning for a distributed system than for a physiological one (it says that a unit cannot spend an infinite time sending/receiving a message), this assertion stands for both systems. 3) A cost sensitive network. The S-propagators formalism takes into account the geometry of the modeled biological system, the length of a space defining the duration of the product propagation. This feature has an equivalent in distributed computing with the cost sensitive asynchronous networks. In this kind of networks, each link has a weight property that is used to measure the communication cost of a particular pathway in the network. These similarities allow us to apply method presented in section 2 for physiological system. B. Hierarchical graph and agent-based method One should note that such a technological transfer finds a meaning only with the advent of the unified formalism for modelling physiological functions made by Chauvet ([?], [?], [?]). Today, graph algorithms and related techniques are obviously the common feature between these two poles apart. The very first step was defining a physiological graph (which we called a φ-graph) with special properties for nodes and links in order to match the needs of integrative physiology modeling, this data structure allowing us to use some classical graph algorithms on biological data. For instance, shortest-path algorithms can be used on a φ-graph to identify an effective regulation loop, thus helping to identify physiological subsystems in the model O-FBS. Another application we are working on is to extract from a φgraph a qualitative description of the regulation loops in order to undergo a formal qualitative analysis of the system’s dynamics using the results of R. Thomas ([?]) to find out typical dynamics (like fix points, saddle points, and limit cycles) through a formal matrix calculus. The way to represent the biological data (biological entities and the physiological processes) on the computer play an essential role due to the enormous quantities of information of

such a system. Such an attempt ”cannot be envisaged without the help of powerful computer systems capable of representing and analyzing the intricate networks of physical and functional interactions between the different cellular components” [?]. In general, this complex system forms a large complex network that is constructed by a set of biological entities and the interaction between these entities. 1) Relations between Integrative Physiology and hierarchical graphs: With the directed graph, the unique direction of the edges represents the non-symmetry of functional interaction. Edges connect nodes of different organization levels; they represent the non-locality of the functional interaction that occurs between different structural levels. The hierarchical level of the graph has a clear correspondence with the organization level of the system. As a result, this type of graph can undoubtedly be used for modeling biological systems. The common and the different features between distributed computing and physiological system in particular and complex system ([?]) allowing us to take the advantages of distributed system to improve the simulation capacity by distributing agents across computer network. 2) Relations between Integrative Physiology and cooperative agents: Both biological and distributed system can be seen as a collection of interacting entities that work without a direct external control. As shown by Jennings in his paper [?], the agent-oriented philosophy for modeling and managing organizational relationships is appropriate for dealing with the dependencies and interactions that exist in such a system. The S-propagator formalism gives us the possibility to simulate the dynamics of such a system thanks to the homogeneity of the mathematical equations. Once the structural hierarchy is set, we have to associate each node with an agent and give it the adequate methods (described by S-propagator) in order to describe its activities. They perceive changes of the outside environment through input ports. Then they effect to the environment by their output ports. The communication of each component with the external world is modeled by messages exchange. Therefore, the functional interaction between the components of the biological system can now be modeled by relations between agents. When the mathematical expression of the relations are tractable, the dynamics of the complex system is calculable by a computer. IV. C ONCLUSION Agent-based simulation method does not replace traditional method in biological field. It can be combined with equationbased method in the following way. Within an individual agent, behavioural decisions may be done by the evaluation of equations ([?]). The system level behaviour is then determined by interactions among these agents. We presented in this paper the advantages of using hierarchical graphs for the modeling of the structural organization of complex systems. Associating agents to complex nodes and describing functional relations between these agents give a way to simulate the dynamics of the system. Therefore, hierarchical graph, which is an abstract data type ready for programmation,

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appear as a coupling structure between the real system and the simulation of its dynamics. Even with 21st century computers, the simulation of physiological systems remains a difficult task, and the S-propagators formalism does not seem to facilitate it at first sight since hierarchical differential partial equations systems exceptionally have analytical solutions. But having a unified mathematical framework to integrate all the biological data will certainly help to make more realistic simulations that could be validated by new experiments. We do not expect any direct application of this work in a very short term. Though, should a reliable physiological model of some subsystems of the human body exist, immediate applications would come out: medical care, education of specialists (physicians, nurses), general education of populations about shock phenomena or cardiac rhythm alterations. On the other hand, we hope a feedback for distributed computing through the development of new parallel and distributed simulation techniques for complex systems. Studying and modeling the behaviours of a complex system still remains a difficult task. R EFERENCES [1] Aiello, A. (1997). Environnement Orient´e Objet de Mod´elisation et de Simulation a` Ev´enements Discrets de Syst`emes Complexes. U.F.R. Sciences et Techniques, L’UNIVERSITE DE CORSE: 141. [2] Jerry Banks (Editor), J. S. C., Barry L. Nelson, David M. Nicol (2000). Discrete-Event System Simulation, Prentice Hall. [3] Chauvet G.A., (1993a). Hierarchical functional organization of formal biological systems: a dynamical approach. I. An increase of complexity by self-association increases the domain of stability of a biological system. Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, 339, 425-444. [4] Chauvet G.A., (1993b). Hierarchical functional organization of formal biological systems: a dynamical approach. II.The concept of nonsymmetry leads to a criterion of evolution deduced from an optimum principle of the (O-FBS) sub-system, Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, 339, 445-461. [5] Chauvet G.A., (1993c). Hierarchical functional organization of formal biological systems: a dynamical approach. III. The concept of nonlocality leads to a field theory describing the dynamics at each level of organization of the (D-FBS) sub-system, Philosophical Transactions of the Royal Society of London Series B-Biological Sciences, 339, 463-481. [6] A.G. Chauvet, Theoretical systems in Biology: Hierarchical and Functional Integration. Volume II: Tissues and Organs, Oxford: Pergamon, II , 1996 [7] A.G. Chauvet, Theoretical systems in Biology: Hierarchical and Functional Integration. Volume III: Organization and regulation, Oxford: Pergamon, III , 1996 [8] A.G. Chauvet, S-propagators: a formalism for the hierarchical organization of physiological systems. Application to the nervous and the respiratory system, Inter Journal of General Systems, 28 (1)53-96, 1999 [9] Chandy K.M., Misra J. (1979). Distributed Simulation: A case study in Design and Verification of Distributed Programs, IEEE Transactions of Software Engineering, SE-5(5), 440-452. [10] H. Van Dyke Parunak, R. S., Rick L. Riolo (1998). Agent-Based Modeling vs. Equation-Based Modeling: A Case Study and Users’ Guide. Proceedings of Multi-agent systems and Agent-based Simulation, Springer. [11] G. Engels, A.S. (1995). Encapsulated Hierarchical Graphs, Graph Types, and Meta Types. [12] Helden J.V., Naim A., Mancuso R.,Eldridge M., Wernisch L., Gilbert D. and Wodak S.J. (2000), Representing and analysing molecular and cellular function in the computer , Biol Chem 381(9-10), 921-35. [13] Jennings, N. R. An Agent-based Approach for Building Complex Software Systems. ACM 44(4): 35-41. [14] Law A.M., Kelton W.D., (1994). Simulation, modeling and analysis, 3 rd ed., MacGraw Hill Intl series. [15] C. Lecerf, M. Bui. (2001). Modeling physiological function: parallels with distributed processing. Eurosim. [16] Rickel, J. (1997). Automated modeling of complex systems to answer prediction question. Artificial Intelligence 93: 201-260. [17] Thomas R., D’Ari R. (1990). Biological Feedback, CRC Press, Boca Raton Florida. Thomas R., (1996).

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