Available online at www.sciencedirect.com

1877–0509 © 2011 Published by Elsevier Ltd. doi:10.1016/j.procs.2011.08.075

Procedia Computer Science 6 (2011) 402–407

Complex Adaptive Systems, Volume 1 Cihan H. Dagli, Editor in Chief

Conference Organized by Missouri University of Science and Technology 2011- Chicago, IL

Scale-free networks of collaborative processes to design distributed control systems

Francesco Ragoa, Pasquale Franzeseb aMegatris Comp. Llc, Newark, DE,USA

bUniveristà Federico II, Napoli, Italy

Abstract

Pervasive computing is offering a level of personalized control over complex and distributed systems.We have designed a system using a communication network that displays a high degree of scale free behavior described by power law. Scale-free systems guarantee aggregation that is useful in building automation, sensors grid management and power generator grids. Synchronizability and stability are main problems of different processes.In order to keep a system stable and easy to manage is useful to have a scale-free structure. A mathematical model was used to define synchronizability criteria. At last, using aweighting scheme, the system was able to create scale-free networks with scale-free aggregates. The schema is activated when processes exchange run in the network. The design approach is valuable because it is one of the first attempts to use scale-free approach to design control systems.There are functional advantages of a scale free network topology. A scale-free network displays high degree of tolerance against random failures as only a few prominent hubs dominate their topology. However, vulnerability of hub is a risk to system reliability. © 2011 Published by Elsevier B.V.

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1. Main text

Many real world complex networks show desirable properties like fast internal synchronizability, communication, robustness and stability. To build a distributed system, such properties are critical for communication efficiency and system reliability.

To achieve the goal of synchronizability of different collaborative processes is useful to design the system structure as a scale-free complex network. Complex flooding algorithms are necessary to manage information

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exchanges in lattice networks or random graphs; this means a heavy message exchange with a high communication overhead. Therefore a hierarchy is needed both in structure and in hardware elements of the system.

If hub nodes receive a higher power it is possible to build a system that is both scale-free and hierarchical. In this way it is possible to specify a central model that is updated by merging local data and that can control a distributed system of collaborative processes without keeping all elaboration in the central control devices.

Variations in the environment or in the controlled objects are therefore read, processed and eventually corrected rapidly, keeping the system in control and stable.

In this paper we introduce a distributed control architecture and we demonstrate the usefulness of scale-free model for collaborative distributed processes using the results of J. Lu and G. Chen [1]. We have been able to create scale-free networks by using a weighting scheme with scale-free aggregates. This paper reports a valuable design approach because in our knowledge is one of the first attempt to use scale-free approach to design control systems. Our goal is to test by design the functional advantages and disadvantage of a scale free network topology. A scale-free network displays high degree of tolerance against random failures as only a few prominent hubs dominate their topology. However, vulnerability of hub is a risk to system reliability.

2. Collaborative Distributed Architecture

Decentralized control is an environment where individual components simultaneously react to local conditions. These individual components interact with neighboring components to exhibit the desired adaptive behaviors. The complex behaviors are a resultant not only of property of the system of connections or architecture, but also of process modules that are part of feedbacks calculation. The decentralized nature of information in many large-scale or even little systems requires the control systems to be decentralized.

Distributed processes provide a way of linking process components of various types at possibly different locations to create a common environment.

The application areas can be, for example: Supply Chain Process, material and information flow management (from manufacturers through distributors to customers), Multiple Energy Plants Management processes.

One of the main interests of Collaborative Process Design is the development of complex processes that require collaborative effort. Analysts with different domain knowledge and expertise, possibly at different locations, develop process software agreeing on standard interface. Another motivation is related to the need to define different models of usage and reliability of a complex environment.

3. Distributed ProcessArchitecture

We present the conceptual architecture of a distributed control system as the framework where collaborative processes activities happen. We shortly introduce the structure and the behavior views of the system.

Events are collected in an automated system by a grid of sensors. Data are input to processors called controllers. Controllers have the mission to run processes and to control plants’ component. Each component has a model which permits to value the foreseen status of the component versus measurement input. The processes generatemessages to other processes situated in other nodes. The processes’ results can be feedbacks for actuators.

Process models can be based on classical differential equations, linguistic fuzzy models or qualitative physics model and stakeholders can reuse them instantiating new ones and reusing the already existing. Human agents can profile their preferred aspects of the models modifying basic variable values respecting rules or constraints which make system stable. This means that specific behavior of plant components are adapted to human stakeholder needs.

The system can be defined as collaborative because a model of a component or of a plant can be available to other stakeholders permitting reuse and sharing as vAppa.

The collaborative architecture has a set of rules that defines a unified and coherent structure consisting of constituent parts and connections that establish how those parts fit and work together. An architecture is primarily concerned about the internal standard interfaces among the system's components and the interface between the

aVirtual application areavailable for use and described in a catalogue.

404 Francesco Rago and Pasquale Franzese / Procedia Computer Science 6 (2011) 402–407

system and its external environment, especially the stakeholders.

4. Scale-Free Models

We are giving now some basic definitions [2].A complex network is a set of nodes linked by connectionsand can be easily modeled as a graph. Each connection canestablish a link between exactly two nodes. We define the degree of a node as the number of connections attached to it. By distance between two nodes we mean the length of the shortest path from one to another.

Common models of a complex network are the Erdös-Rényi and the Watts-Strogatz models. The former is realized by taking a set of nodes with no connections and then establishing a link between each couple of nodes with a fixed probability; the second one is realized starting from a regular lattice model, where the nodes are distributed at regular intervals and each node is connected with its k-nearest neighbors, in such a regular lattice each connection is rewired changing one of its ends with a fixed probability.

It was observed in the Watts-Strogatz model that the average distance between any couple of nodes dropped from a value linear in the number of nodes N to a value proportional to log(N) for a rewiring probability between 0 and 0.01.

A network have Small-world property when has a distance between any pair of nodes in the network which is small (i.e. logarithmic) when compared with the number of nodes.

A network is Scale-free when the distribution of degrees in the network follows an inverse power law in the form of x-α.A sound model for scale-free networks was introduced by Barabàsi and Albert.Such model starts with a set of nodes and then grows adding at each time step a new node, the new node establishes a number of connections with pre-existing nodes with a probability proportional to their degree. The law for determining new nodes links is called preferential attachment, consequently, the new node is most likely to be linked to the pre-existing nodes with the highest degree. Further descriptions of Scale-free models can be found in [3].

5. Advantages of Small-world and Scale-free Complex Networks

Inter-node communication benefits the short distance between nodes in a network with Small-world property, causing a reduced time and energy cost for message exchange and a fast convergence on the side of synchronization. For an extensive coverage of synchronization in Complex Networks see [4].

Ingredients of a network in a scale-free structure are growth and preferential attachment. Such attachments permit to process components the exchange of instantiated variables.

Connectivity distribution of the network is statistically homogenous, with peak at an average value and decay exponentially.

In our system, collaborative processes have a guided evolution toward scale-free structure and their connectivity distributions have a power-law form. The graph in Figure 1 represents processes at a specific time t. It is a typical scale-free network where each node represents a process instance[5],[6]. The hub nodes are shown in dark, while the other nodes are instances.

Processes networks are dynamically formed by continuous addition or subtraction of nodes to the network, and edges can be added or rearranged. Scale-free structure of processes grows with preferential attachment by adding new nodes which are preferentially attached to existing nodes with large numbers of connections.

The generation scheme of a scale-free model begins with a small number of nodes and at every time step a new node is introduced and is connected to already-existed nodes. It was verified that a new node is connected to a node depending on the degree of the node itself.

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Fig. 1 Process at specific time t

6. Scale-free Dynamical Network Synchronizability

As described in previous paragraphs, our system can be naturally described by a network with nodes representing the dynamical units and links representing interactions among them.

The study of collective behaviors of complex networks and their synchronous behavior has received a great deal of attention in the past years[7],[8],[9].

Is worth mentioning a synchronizability theorem provided by Pecora and Carroll[10] indicating the collective synchronous behavior of a network as completely determined by the network structure. The network synchronizability is completely determined by two factors: the first is the inner linking function and the second one is related to the eigenvalues of the network structural matrix. Consequently the synchronized region problems were analyzed and disconnected synchronized regions were found in [11].

As the synchronizability is correlated with topological properties of network, there is not a general set of theorems relating the synchronizability and topological properties. Donetti[12] pointed out that a network with optimized synchronizability should have an extremely homogeneous structure, i.e., the distributions of topological properties should be very narrow.

A network is completely determined by its outer-linking structure, i.e., the corresponding graph and manipulation of graphs can be very helpful for networks synchronization.

Consider a dynamical network consisting of N-coupled nodes each making processes processing, with each node being an-dimensional dynamical system. The exact topology is certain, but changes in time.

Consider a weighted complex dynamical network consisting with linearly couplings, which is characterized by:

���

� ����

�� � � �

���

�

���

��

where� �� �

�� �

��� �

��� � � �

���

� ��� �

� is the state vector of the ith node,�� ��

� �

� �� is a

nonlinear vector field, node dynamics is �� � � ���

� � � ���� is the inner-coupling matrix and � � �

������

� ����

is the weight configuration matrix. If there is a link from node i to node j, then���

� �, otherwise,���

� � . In this model, the inner coupling matrix A is not necessarily symmetric, and the weight configuration matrixC

needs not to be symmetric, irreducible and diffusive. These weights are defined by design. Our systems consider a controlled complex dynamical network as follows:

406 Francesco Rago and Pasquale Franzese / Procedia Computer Science 6 (2011) 402–407

���

� ����

�� � � �

������ ��

�

���

��

� ��

where����

� is the control. Further, suppose that if there is an edge between node i and node j, then aij=aji=−1, i.e., A is a Laplacian matrix. In

this setting, if the graph corresponding to A is connected, then 0 is an eigenvalue of A with multiplicity 1 and all the other eigenvalues of A are strictly positive, which are denoted by� �

� �

� � �

�.

The dynamical network is said to achieve (asymptotically) synchronization if ����� � �

���� � � � �

���� �

������� � � Because of the diffusive coupling configuration, the synchronous state ����!"#$%&%s(t) is a solution of an individual node, i.e., Δs˙(t)= f(s(t)).

The local stability of the synchronized solution can be determined by analyzing the so-called master stability equation, even iff functions are not always differentiable. The master equation has to be used considering finite difference approach. Itis well known that the synchronized solution of dynamical network is locally asymptotically stable if��

� "� ' � (�)� � � and if the synchronized region S is bounded, *+

�� +

�,, then the eigenratio &��� �

��

-��

of the network structural matrix A characterizes the synchronizability. The consequence is that synchronization property of the time-varying dynamical network is completely determined by its inner-coupling matrix A(t), and the eigenvalues �

���and the corresponding eigenvectors φ

are functions of the coupling

configuration matrix C(t).

7. The processes of scale-free net creation process

If we guarantee synchronization defining constraints on network structural matrix A, we can create a processes network with scale-free characteristics.

Starting from the first activated process node, a weight is added to edges every time they are visited by messages sent from one node to another.

The general power-law function follows a polynomial form. Starting with first activated node,each neighbor edge is weighted according to:

���

� ��

where� is the normalized number of visits the node has received so far, and R is a parameter of the model.

Figure 2: log-log cumulative probability versus degree k

Nodes within a sub network are visited frequently while nodes outside a sub network, even high-degree nodes,

remain unvisited. This effect is critical to this model's ability to recreate dynamic centrality. The above weighting is able to create scale-free network aggregates. The experimental data confirm that the networks with weighting constant R> 0 have a more pronounced scale-free behavior. It is possible in Figure 2to see a longer tails, which

Francesco Rago and Pasquale Franzese / Procedia Computer Science 6 (2011) 402–407 407

suggests a more pronounced scale-free behavior. This means the created sub networks are scale-free sub networks.The sub networks have a number of nodes which appear among the top nodes in both networks, but subnet hubs are largely different because the non-homogeneity of the nodes functionality in the plant. This is consistent with empirical dynamic scale-free networks [11],[13].

8. Conclusion

In this paper, we have applied network synchronization theorems to guarantee the synchronization of collaborative processes. The time-varying dynamical network is completely determined by its inner-coupling matrix, by eigenvalues and by the corresponding eigenvectors of the coupling configuration matrix.

Using weighting scheme our system was able to create scale-free networks with scale-free aggregate. This result is totally based on a scheme that updates weights when processes messages exchange is running in the network. The weighting has deep technical meaning because the schema permits to design specific type of networks with the advantage of a high degree of tolerance. The experience opens a new area of study as small-world, scale-free and other models that can drive engineering tools to design more and more complex systems.

9. References

[1] J. Lü, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” in IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 51, no. 4, pp. 787–796, Apr. 2004.

[2] X. Wang and G. Chen, “Small-World, Scale-Free and Beyond” in IEEE Circuits and Systems Mtechnicalagazine 3, Pages 6–20.

[3] A. László Barabási, R. Alberta and H. Jeong “Mean-field theory for scale-free random networks” in Physica A: Statistical Mechanics and its Applications Volume 272, Issues 1-2, 1 October 1999, Pages 173-187.

[4] C. Li, G. Chen, “Synchronization in general complex dynamical networks with coupling delays” in Physica A: Statistical Mechanics and its Applications, Volume 343, 15 November 2004, Pages 263-278.

[5] S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, pp.268–276, 2001. [6] R. Albert and A.-L. Barabási, “Statistical mechanics of complex networks,” Rev. Modern Phys., vol. 74, pp.

47–97, 2002. [7] A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, pp. 509–512,

1999. [8] A. Coppel, Stability and Asymptotic Behavior of Differential Equation. Boston, MA: D. C. Heath, 1965. [9] H. K. Khalil, Nonlinear Systems, Upper Saddle River, NJ: Prentice-Hall, 1996. [10] L. M. Pecora and T. L. Carroll, “Synchronizationy in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821–

824, 1990. [11] S. Hill, D. Braha, "A dynamic model of time-dependent complex networks", New England Complex Systems

Institute, Cambridge MAarXiv:0901.4407v2 [physics.soc-ph] 14 Jul 2010 [12] L. Donetti, P. I. Hurtado, and M. A. Muñoz, Phys. Rev. Lett. 95, 188701,2005. [13] R. Albert and A.-L. Barabasi, Rev. Mod. Phys., 74, 47 (2002). [14] W.Liu, J. Wu and H.Shen, "Architecture Design of an Integrated Communication and Broadcasting Network",

International Symposium on Parallel and Distributed Processing with Applications, 2010. [15] F. Kuhn and R. Oshman, "Dynamic Networks: Models and Algorithms", SIGACT News Volume 42, Number

1, March 2011. [16] B.Bollobás and O. M. Riordan, “Mathematical Results on Scale-free Random Graphs”, Handbook of Graphs

and Networks' (S.Bockholdt and H. Schuster (Eds.)), Wiley VCH, Weinheim, 2003. [17] R. Albert, H. Jeong, and A.L. Barab´asi, "Attack and error tolerance of complex networks", Nature 406, 378-

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