El matrimonio o la prostitución

In order to compare the robustness properties of different graphs, we need to find the network criticality of different graphs. In the following we provide a list of some important networks accompanied with an experimental plan to assess the robustness of different networks and compare the behavior of the network criticality and algebraic connectivity. This study considers the following networks types.

1. Type 1: Networks with given degree distribution (power law is an example of these networks)

2. Type 2: Networks with given link weight distributions (we assume that the network topology has a fixed number of nodes and randomness is only in weight distribution).

3. Type 3: Networks with given betweenness distribution

4. Type 4: Structured construction of recursive networks using Cartesian and Kro-necker products.

5. Type 5: Erdos-Renyi random graphs 6. Type 6: Scale-free networks

7. Type 7: Small-world networks

8. Type 8: Networks with given node/link connectivity (and minimum degree) distribution.

9. Type 9: Hierarchy of networks (type 1-7)

Here is an experimental plan to assess the robustness of these networks.

1. Find and plot the distribution of criticality and algebraic connectivity for net-works of type 1-7.

2. Find the distribution of criticality and algebraic connectivity for hierarchical networks. Is there any relationship between criticality and algebraic connectivity of the final network and its ingredients?

3. Investigate the behavior of criticality and algebraic connectivity when the node/link connectivity changes (type 8).

4. Investigate the scalability issues in structured growing networks.

5. Investigate the distance variations in different network types and its effect on criticality and algebraic connectivity.

6. Compare different networks types (their behavior with respect to criticality and algebraic connectivity). This results in a better understanding of the properties of algebraic connectivity and network criticality.

7. Investigate the behavior of network criticality versus other Laplacian eigenvalues of the graph.

7.3 Conclusions

In this thesis, we addressed issues related to the management of backbone (core) transport networks, namely the algorithmic architecture of such network management systems. A main concern in core networks is the existence of unanticipated events in the system, such as traffic surges, a sudden topology change, or changes in active sources (sinks) for traffic. A core network should be able to smoothly react to such unwanted events so as to provide reliable data delivery service for the customers according to the service level agreements (SLA).

The majority of traffic control systems in use by service providers so far are man-ually configured by human intervention. The continual growth in traffic volume, di-versity, and heterogenous requirements make it impossible to continue working with

the present network management systems. Automated service and network manage-ment are essential to creating and maintaining a flexible and agile service/application delivery infrastructure that also has much lower operations expense than existing systems.

In this thesis we proposed the AutoNet, a management framework for data delivery over core networks. We argued that the traffic engineering system requirements can be met by a self-management system based on autonomic computing. We gave a conceptual design of our autonomic traffic engineering system, AutoNet, but we focused on a set of essential graph theoretic algorithms that provide the means for adaptive management required by the autonomic system.

The theory of evolution motivated the conceptual idea underlying AutoNet. Evo-lutionary processes are among self-organizing systems by nature. Darwin’s theory describes the process of natural selection by which slight variations, if useful, are pre-served. Every process has a survival value as a result of natural selection that quantifies its overall sensitivity or robustness to the external variations. In this thesis we looked for an appropriate survival value for communication networks that indicates how adaptable a system is to unexpected events.

In any network, from small designed networks, to large-scale social networks, and even to the Internet, connectivity is a crucial factor as it is essential for communication.

Therefore, the first parameter to consider as a candidate for ”survival value” is the connectivity of the graph. In the first part of this thesis we proposed some metrics, by extending some ideas from graph-theory, to model the robustness of a network.

This is the first step toward defining an appropriate survival value for communication networks. We defined Link Criticality Index (LCI) as the deterministic betweenness of a link per unit of available link bandwidth (betweenness of the link over its available bandwidth). We proposed Path Criticality Routing (PCR) algorithm based on evalu-ating the LCI of different links of the network. This algorithm tries to find the least

critical path. The success of PCR algorithm encouraged us to study the behavior of LCI analytically. This study is the subject of the second part of this thesis.

We extended the idea of LCI and defined node/link criticality as the probabilistic betweenness of the node/link per unit of node/link weight (random-walk betweenness of the node/link over its weight). We showed that the criticality of a node/link is independent of the position of the node/link in the network, and it is only a function of link weights. We termed the fraction of node/link betweenness over the node/link weight ”network criticality” and investigated its properties. The network criticality is a global metric quantifying the robustness of the network, therefore, we used network criticality as the survival value for the networks. Any communication network should evolve in a way that minimizes the network criticality.

While Darwin’s theory does not consider any ”final target” for the evolutionary changes in the nature, viewing survival as the network management goal can lead to an implicit optimization problem. The optimization must address the real-time efficiency and performance of the whole network as a short-term goal, while striving to maintain and improve the survival value of the network as a long-term goal. This is the subject of the third part of the thesis. We investigated the problem of optimizing the survival value (network criticality) under some constraints. We proved that network criticality is a strictly convex function of link weights, and investigated the problem of minimizing network criticality as a convex optimization problem. This led to some guidelines for designing more precise traffic management methods (improving PCR algorithm), as well as directions for network planning problem.

In the final part of the thesis, we returned to the architecture of AutoNet and provided details of its necessary building blocks. Our emphasis in this section of the thesis was to show how the concepts of autonomic computing and virtual networks can be used to build autonomic networks capable of self-optimizing, self-configuring and self-healing.

This research work is just in its early stage of development. There are definitely much more research questions that need to be addressed. While most of the work over the last years has focused on frameworks and models, this thesis attempted to tackle the algorithmic aspect of the autonomic promise in the telecommunications world. An abundance of work remains to be done on these issues, and will definitely constitute a major research area for years to come.