II. Examination of a Distributed NetworkSince destruction of a small number of nodes in a decentralized network can destroy communications, the properties, problems, and hopes of building "distributed" communications networks are of paramount interest.
The term "redundancy level" is used as a measure of connectivity, as defined in Fig. 2. A minimum span network, one formed with the smallest number of links possible, is chosen as a reference point, and is called "a network of redundancy level one." If two times as many links are used in a gridded network than in a minimum span network, the network is said to have a redundancy level of two. Figure 2 defines connectivity of levels 1, 1.5, 2, 3, 4, 6, and 8. Redundancy level is equivalent to link-to-node ratio in an infinite size array of stations. Obviously, at levels above three there are alternate methods of constructing the network. However, it was found that there is little difference regardless of which method is used. Such an alternate method is shown for levels three and four, labelled R'. This specific alternate mode is also used for levels six and eight.
Each node and link in the array of Fig. 2 has the capacity and the switching flexibility to allow transmission between any ith station and any jth station, provided a path can be drawn from the ith to the jth station.
Node DestructionFigure 4 indicates network performance as a function of the probability of destruction for each separate node. If the expected "noise" was destruction caused by conventional hardware failure, the failures would be randomly distributed through the network. But, if the disturbance were caused by enemy attack, the possible "worst cases" must be considered.
To bisect a 32-link network requires direction of 288 weapons each with a probability of kill, pk = 0.5, or 160 with a pk = 0.7, to produce over an 0.9 probability of successfully bisecting the network. If hidden alternative command is allowed, then the largest single group would still have an expected value of almost 50 per cent of the initial stations surviving intact. If this raid misjudges complete availability of weapons, or complete knowledge of all links in the cross section, or the effects of the weapons against each and every link, the raid fails. The high risk of such raids against highly parallel structures causes examination of alternative attack policies. Consider the following uniform raid example. Assume that 2,000 weapons are deployed against a 1000-station network. The stations are so spaced that destruction of two stations with a single weapon is unlikely. Divide the 2,000 weapons into two equal 1000-weapon salvos. Assume any probability of destruction of a single node from a single weapon less than 1.0; for example, 0.5. Each weapon on the first salvo has a 0.5 probability of destroying its target. But, each weapon of the second salvo has only a 0.25 probability, since one-half the targets have already been destroyed. Thus, the uniform attack is felt to represent a known worst-case configuration in the following analysis.
Such worst-case attacks have been directed against an 18x18-array network model of 324 nodes with varying probability of kill and redundancy level, with results shown in Fig. 4. The probability of kill was varied from zero to unity along the abscissa while the ordinate marks survivability. The criterion of survivability used is the percentage of stations not physically destroyed and remaining in communications with the largest single group of surviving stations. The curves of Fig. 4 demonstrate survivability as a function of attack level for networks of varying degrees of redundancy. The line labeled "best possible line" marks the upper bound of loss due to the physical failure component alone. For example, if a network underwent an attack of 0.5 probability destruction of each of its nodes, then only 50 per cent of its nodes would be expected to survive--regardless of how perfect its communications. We are primarily interested in the additional system degradation caused by failure of communications. Two key points are to be noticed in the curves of Fig. 4. First, extremely survivable networks can be built using a moderately low redundancy of connectivity level. Redundancy levels on the order of only three permit withstanding extremely heavy level attacks with negligible additional loss to communications. Secondly, the survivability curves have sharp break-points. A network of this type will withstand an increasing attack level until a certain point is reached, beyond which the network rapidly deteriorates. Thus, the optimum degree of redundancy can be chosen as a function of the expected level of attack. Further redundancy buys little. The redundancy level required to survive even very heavy attacks is not great--on the order of only three or four times that of the minimum span network.
Link DestructionIn the previous example we have examined network performance as a function of the destruction of the nodes (which are better targets than links). We shall now re-examine the same network, but using unreliable links. In particular, we want to know how unreliable the links may be without further degrading the performance of the network.
Figure 5 shows the results for the case of perfect nodes; only the links fail. There is little system degradation caused even using extremely unreliable links--on the order of 50 per cent down-time--assuming all nodes are working.
Combination Link and Node DestructionThe worst case is the composite effect of failures of both the links and the nodes. Figure 6 shows the effect of link failure upon a network having 40 per cent of its nodes destroyed. It appears that what would today be regarded as an unreliable link can be used in a distributed network almost as effectively as perfectly reliable links. Figure 7 examines the result of 100 trial cases in order to estimate the probability density distribution of system performance for a mixture of node and link failures. This is the distribution of cases for 20 per cent nodal damage and 35 per cent link damage.
See Craig, L. J., and I. S. Reed, "Overlapping Tessellated Communications Networks," IRE Trans. Comm.. Sys., CS-10 (1962) 125-129.