El Concreto Pensado

Communication networks have become essential part of the people’s daily routines and therefore modern society largely relies on its proper functioning. From the geographical point of view, the communication networks could be designed to provide services on a local as well on the global level. Generally, the larger networks (networks which cover wider geographical area and there- fore connecting more users) are considered more important (critical) than smaller ones. The most prominent example of most widely used communi- cation network is the global Internet. The more detailed justification of its importance nowadays become superfluous. It is worth mentioning that Inter- net is considered as a part of national critical infrastructure and today, when the probability and severity of natural disasters and other threatening events have increased, the protection of communication networks is on the national agenda in many countries [22]. On the other hand, an importance of numerous smaller networks on national or regional level shouldn’t be underestimated. There are many private or public owned networks independent from global Internet which provide various services like national telephone networks, mili- tary networks or communication networks connecting universities or scientific institutions.

The network could be generally considered as a system with its measurable system functions. The function of a communication network is transfer of

data, hence the most common measures of a system function of a communica- tion network are related to the data quantity transferred through the network. However, the network resilience in general is usually not assessed by a single variable. Multiple control and state parameters of a multi-dimensional com- plex system make the prediction of the system’s resilience difficult. However, there is an analytical framework which allows us to collapse the behaviour of different networks onto a single universal resilience function by systematically separating the roles of the system’s dynamics and topology. The formalism proposed in [23] reduces Aij into an 1D system. It is shown that the patterns

of the resilience depend only in system’s intrinsic dynamic, regardless of the specific topology or weights. All the parameters are condensed in a single βef f and indicate that density, heterogeneity and symmetry are the three key

factors to define the system’s resilience. This approach provides a tool for an accurate prediction for the system’s response to various perturbations. Here, some of the most important measures which could serve as a system functions of a communications networks are discussed.

Throughput. The amount of data which could be successfully delivered over the communication network per certain time slot is called throughput. Throughput is commonly measured in amount of data (bits) per second, or in data packets per time slot. The data may be delivered over the physical or logical link and it has to pass through certain number of links and nodes. Therefore, the throughput can be measured in the node or on the link. Ad- ditionally, one can measure the throughput of the full path from node n to node m or an average throughput of the whole network, etc.

Connectivity. A graph (network) is connected if there exists at least one possible path between any pair of nodes, which means that each node can communicate with any other node in the network. In disconnected graphs, this condition is not met. Disconnected graph is consisted of two or more independent connected graphs. The measure called connectivity is defined as a minimum number of elements (nodes or edges) which can be removed to make connected graph disconnected. This measure does not say much about

Network Resilience 19 the throughput or possible congestions in the network. The measure which largely coincides with connectivity was a subject of one of the earliest math- ematical proofs in the network theory. In year 1927 Karl Menger showed that the number of node-independent paths between two vertices is always exactly equal to the minimum number of other vertices in the network that must fail in order for those two vertices to become disconnected from each other [24, p. 424]. Let us consider a path P1(V1, E1) from node i to node j in

certain network. The path consists of sets of nodes V1 and links E1. If there

is another path P2(V2, E2) which does not share any common element with

path P1(V1, E1), except the first and last node, those paths are described as

node-independent. The number of node-independent paths in certain network could be used as a indicator of its robustness. This is almost exactly the same as a minimum number of vertices which has to fail in order to make a net- work disconnected, just applied to the particular pair of nodes. Sometimes, the simple connectivity is not enough to explain robustness or survivability of the real world networks. Therefore, some other measures are introduced which focus particularly on spatially correlated or region based failures within the network. The region based connectivity is introduced as a measure which shows the minimum number of nodes (links) that have to fail within any region of the network before it is disconnected. This measure takes into ac- count not only the topology, but the network’s geometry. As an extension of the region based connectivity, there is region-based component decomposition number (RBCDN) which measures the number of connected components in which the network decomposes once all the nodes of a region fail [25]. Network diameter. Another quantitative measure of topological robustness of the network is network diameter. Diameter of a network is defined as a length d = maxu,vd(u, v) of a longest shortest path between any two vertices

in the network. The average diameter is sometimes referred to as a diameter. Average diameter is described as an average length of the shortest paths between any two nodes in a network. If the diameter d is smaller, the nodes are able to communicate between each other more easily. Larger number of nodes does not necessarily mean the network has a large diameter.

Additional measures. Even the throughput and connectivity could be con- sidered as the most important measures of network’s operational function, there are other measures which could provide additional information about the network itself. These measures could be particularly important in the times of the disruption and undesired events. For example, the number of ac- tive nodes after the disruption, the average congestion in the network or the number of infected nodes in the case of malicious virus attack. Particularly important measure is a size of the largest connected component or the largest connected subgraph. If the node in the network is damaged (removed), it is usually considered that all associated links are broken. Sufficient number of such damages could make one initially connected graph disconnected, which means that the graph is split into two or more smaller pieces (subgraphs). The simplest measure of such an impact of the network is the relative size of the largest connected component remained from the network Sf/S0 where S0

is the original size of the network before the disruption. When Sf  S0 the

network has been broken into many small parts and therefore is not functional any more [14, p. 118].