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The authors have declared that no competing interests exist.

‡ These authors also contributed equally to this work.

We introduce a rapid deterministic algorithm for identification of the most critical links which are capable of causing network disruptions. The algorithm is based on searching for the shortest cycles in the network and provides a significant time improvement compared with a common brute-force algorithm which scans the entire network. We used a simple measure, based on standard deviation, as a vulnerability measure. It takes into account the importance of nodes in particular network components. We demonstrate this approach on a real network with 734 nodes and 990 links. We found the worst scenarios for the cases with and without people living in the nodes. The evaluation of all network breakups can provide transportation planners and administrators with plenty of data for further statistical analyses. The presented approach provides an alternative approach to the recent research assessing the impacts of simultaneous interruptions of multiple links.

Modern society is highly dependent on various types of networks, among which road networks occupy the most prominent place. People would not be able to utilize even the most basic services, such as medical care, without a functioning road network. An efficient road network thus ranks among the priorities for any society. Its serviceability can be affected, however, by various types of events which originate within the transport system (such as traffic accidents, congestions, technical failures, etc.) but also by events caused by external forces (such as floods, landslides, heavy snowfalls, storms, wildfires, earthquakes, etc.). The most challenging issue for road administrators is the development of methods which can help in dealing with and preventing critical situations when both types of events occur.

Identification of critical road links is part of vulnerability analysis of transportation networks. This analysis pays attention to particular links and evaluates their importance within the whole network. The manner in which reduced capacity of a link or its complete blockage will affect the functioning of the entire network is often studied. A large number of road links are sometimes interrupted concurrently for various reasons. Such a situation can lead to cascading effects when other links collapse and the overall impact on the network performance is enormous. It is therefore important to analyze impacts of as many as possible combinations of concurrently interrupted links. Such an analysis is, however, computationally demanding. It requires the application of additional restrictions on the set of analyzed links. These restrictions often encompass certain properties which are common for the set of the links in question. This means, for instance, that the links are located in the same region and are therefore close to one another.

In this paper, we focus our attention on disasters when the road links are completely interrupted and a road network is disintegrated into several isolated parts. Such a situation can result in a number of people cut-off from sources of food, water and medical treatment. When such events occur, a rescue effort related to the reconnection of isolated components with a high number of people has the highest priority. We thus introduce a simple but practical measure evaluating network disintegration based on the overall number of people isolated from the primary network.

We introduce, in this work, a novel deterministic algorithm, based on cycles in graphs, which enables the identification of the most critical links and reduces computational demands. The suggested algorithm thus identifies all possible road network break-ups caused by up to 9 concurrently interrupted links. Identification of all the decompositions of the network, for the defined number of interrupted links and their evaluation, is the aim of this work.

The importance of networks in everyday life leads to the need to study their properties. Serviceability, accessibility and vulnerability rank among the most prominent concepts at present [

The first issue is how to evaluate various combinations of interrupted road links. One can draw inspiration from the vast source of literature covering vulnerability measures. The work [

Another issue is the identification of the worst-case scenarios. This is difficult, however, to solve due to high computational demands. For instance, if a network consists of 1,000 road links and we plan to evaluate all combinations for 3 concurrently interrupted links, we have to process 166,167,000 combinations. The number of combinations rises to 41,417,124,750 for 4 links. These numbers suggest that the respective state space (all possible combinations of interrupted road links) is extremely large and that to evaluate any combination of disrupted road links using a brute-force examination is beyond the scope of current-day computers (see for instance [

In this paper, attention is thus paid to network disintegration into several parts caused by concurrent interruption of several road links. This issue of finding the most critical links is related to the problem of generating the partitions of a graph into a fixed number of cuts evaluated by a function. A

In our algorithm we do not take into account any flows in the network (see for instance [

In this section, a network vulnerability measure, represented by a suitable loss function, is introduced along with a new rapid and efficient deterministic algorithm.

For the purpose of this paper we modified the definitions of cuts and minimum cuts. By a

The proposed deterministic algorithm makes possible finding all disintegrations of a network for a given number of links in a reasonable time without complete examination of the large state space of the road network. This algorithm is able to examine all minimum k-cuts of the given network under the predefined numbers of cut-set links and components or further limitations.

To describe the algorithm, we use the standard notation in graph theory, namely _{i}∈_{j}∈_{0} = _{n}. A

The idea of the algorithm is as follows. Assume there is a link and its ending nodes. If a cycle containing the link exists for the nodes, the nodes belong to the same component. Otherwise, they belong to different components. Part of the cycle then indicates a detour. To the best of our knowledge, no algorithm based upon searching for cycles (see below for a more detailed description) has ever been used for an analysis of a disintegrated network.

We now provide a precise description of the algorithm using the following pseudocode. A

Input conditions:

_{graph}\

_{graph},

_{1} _{2}

_{1},_{2})

Algorithm 1 begins by determining all the cuts incorporating the links belonging to the graph’s spanning tree (since all the cuts have to incorporate a spanning-tree link). The cuts generated by more links (and containing the particular _{1}, _{2}_{1},_{2} using a graph algorithm which finds the shortest path in the graph

In the beginning of the process, a spanning tree of the original graph is generated. Since, in general, a graph may have several spanning trees, any of them can be used for the computation. There is a need to examine all the spanning tree links. One by one, links are inserted into the set of restricted links. This step is illustrated with link

In the end, a minimum cut appears because no alternative path exists for any of the nodes (nodes 1 and 2 in

The algorithm is applied recursively on the components which appear after the first run, in order to obtain the minimum k-cuts.

Our primary focus is on such events where a road network breaks up into several _{m} is a graph with _{i} is the number of people living with the

If there are only _{i} = 0 for

The values of _{i},_{i} equal to the number of nodes in _{i} can also represent the demand or more generally the importance of the

This is not, however, the only way to define the vulnerability measure. The measures in other papers can be used as well or new measures can be developed based upon the requirements of the contracting authorities. The measures affect the total time of computation but are not incorporated into the algorithm. The process of evaluation of minimum k-cuts proceeds as follows (assume

Put

If

Order all evaluated minimum k-cuts.

In this section, the performance of the algorithm on the real network of the Zlín region, which consists of 990 links and 734 nodes, is demonstrated.

It is apparent that there is no possibility to evaluate all the combinations for larger number of concurrently interrupted links. For many networks, the number of break-ups is, however, much smaller.

Number of concurrently interrupted links | Number of break-ups | Number of all combinations of links | Ratio |
---|---|---|---|

1 | 143 |
990 | 0.14000 |

2 | 10,376 | 489,555 | 0.02000 |

3 | 510,220 | 161,226,780 | 0.00300 |

4 | 19,154,308 | 39,782,707,965 | 0.00048 |

* This number indicates all dead-end links

The primary contribution of this paper is a proposal of a novel algorithm which is able to efficiently find and evaluate network cuts with a predefined number of concurrently interrupted network links. The principal difference between the proposed algorithm and the brute-force approach is the speed of the computation.

Number of concurrently interrupted links | Brute-force algorithm | Algorithm–Cycles |
---|---|---|

1 | 1 s | 1 s |

2 | 30 s | 14 s |

3 | 10.25 hrs | 4 min |

4 | 105 days | 11.5 hrs |

The shaded table cell represents an estimation of the expected running time of the brute-force approach determining all the cuts generated by 4 interrupted links as it was impossible to measure it precisely. To compute the estimation, we use the number of combinations of 3 and 4 links and the computational time for 3 links. It is apparent that it is impossible to evaluate the scenarios with more than 3 concurrently interrupted links using the brute-force approach. In addition, the Zlín region ranks among the smallest ones in the Czech Republic (only approximately 1,000 road links) and therefore computation of the same scenario for larger regions is not possible.

In this section, we present the results from the application of the proposed algorithm under various limitations for the Zlín region (Czech Republic) with a population of 587,624 people.

This section examines the internal disintegration of the network since actual networks are usually parts of a larger network (e.g., a network of a region is connected at its borders to the network of the entire country). In order to prevent access to the isolated parts from neighboring regions, we only admit interruption of internal links, i.e., the links which do not lie on the borders of the particular region.

All links which have to be open, in order to study the internal subnetwork, are marked. The algorithm then omits all results related to the combinations of links containing at least one of the marked links.

In the example we restrict our attention to the disintegrations of up to 5 components caused by up to 4 links in the Zlín region which seemed to be more interesting than other cases. The results are summarized in

A node diameter represents proportionally the number of inhabitants. The worst scenario is not shown here because it is represented by only one cut-off node, the center of the city of Zlín. This case is more illustrative.

Rank | Number of components | Value of the loss function | Inhabitants in components | Ratio of inhabitants cut off from the main component |
---|---|---|---|---|

1 | 2 | 245,391 | 555,800; 31,824 | 5.4% |

2 | 2 | 246,616 | 558,092; 29,532 | 5.0% |

3 | 2 | 247,323 | 559,410; 28,214 | 4.8% |

4 | 3 | 247,384 | 559,779; 21,511; 6,334 | 4.7% |

5 | 2 | 247,592 | 559,910; 27,714 | 4.7% |

6 | 2 | 247,594 | 559,914; 27,710 | 4.7% |

7 | 2 | 247,618 | 559,958; 27,666 | 4.7% |

8 | 3 | 247,800 | 560,543; 20,747; 6,334 | 4.6% |

9 | 2 | 248,280 | 561,190; 26,434 | 4.5% |

10 | 2 | 248,301 | 561,228; 26,396 | 4.5% |

This section assumes the same limitations as in the previous one but there is now only one individual living in a node. The results provide us with more information about the spatial structure of the networks than the previous ones. The results concerning the worst-case scenarios can be found in

This network disintegration leaves 42 from the 731 nodes (5.7%) out of connection.

Rank | Number of components | Value of the loss function | Number of nodes |
Ratio of nodes cut off the main component |
---|---|---|---|---|

1 | 2 | 304.8 | 691; 43 | 5.9% |

2 | 2 | 305.3 | 692; 42 | 5.7% |

3 | 3 | 305.8 | 693; 38; 3 | 5.6% |

4 | 3 | 305.8 | 693; 38; 3 | 5.6% |

5 | 3 | 305.8 | 693; 38; 3 | 5.6% |

6 | 3 | 305.8 | 693; 38; 3 | 5.6% |

7 | 3 | 305.8 | 693; 38; 3 | 5.6% |

8 | 3 | 305.8 | 693; 38; 3 | 5.6% |

9 | 2 | 305.9 | 693; 41 | 5.6% |

10 | 2 | 305.9 | 693; 41 | 5.6% |

If we focus on the number of nodes as inputs into the loss function, the results will look different. The area of the inaccessible part of the network would be larger (see

As can be seen in the examples above, the algorithm is able to compute the disintegration of the network under various restrictions such as the number of components, the number of interrupted links and the limitation on internal disintegrations. The loss function can also be easily modified because _{i} can be understood as weights of components, which measure their importance in the network.

To exemplify all the properties of the algorithm and to justify the used number of interrupted links we took data from the Zlín region for 917 days which indicate that the probability of occurrence of a scenario, when four and more road links are concurrently interrupted, is 20% (see

The vast majority (87%) of break-up scenarios are caused by interruptions of dead-end links and their combinations. The algorithm is also able, however, to find combinations of the links, which are only involved in hundreds of break-ups (see

The links which cause the worst break-ups only participate in hundreds of cases.

To demonstrate the necessity of a new fast algorithm we indicated the comparison of the computation time of our algorithm and the brute-force algorithm. As we have presented, the algorithm represents a significant improvement in the computation of network disintegrations. The proposed algorithm was able to compute all the break-ups over 11.5 hours compared with 105 days for the brute-force algorithm (see

Despite the ability of the algorithm to noticeably reduce the state space, it nevertheless has to analyze a vast number of combinations which have to be evaluated and saved. The disadvantage can be compensated by the fact that we can compare the current state of a network with all the combinations of interrupted links causing a break-up computed beforehand. In real time we can consequently obtain an alert if a worst-case scenario might occur in the network. We can also identify the links which have to be preserved as operational in order to avoid certain forms of traffic collapse. The links are not usually the same ones as in the worst-case scenarios but can still have a large impact on the network. The principal advantage of our algorithm is its deterministic nature. This means that it is able to precisely identify all the possible scenarios, unlike the stochastic approaches. The suggested approach can only be applied to network of a sufficient size. Despite the fact that it is able to identify all the break-ups, many combinations exist for larger networks and therefore there is also a certain limit related to computer performance. This limitation could be overcome using a stochastic approach.

We have further introduced a vulnerability measure based on the number of isolated people as a loss function which evaluates the impact of a given combination of interrupted links on the network (Results section). This approach shall be used during events which result in the disintegration of the network or in the phases of planning for the worst case scenarios. Additional measures evaluating the actual state of the network can be used as well. Several other vulnerability measures can also be used.

The main aim of the paper was to introduce a novel algorithm for computation of minimum k-cuts for a given number of interrupted links. This was the reason why we restricted our attention to the relatively simple vulnerability measure which can cover only several aspects of the impact of an event. It could be interesting in the next phase of the research to employ other vulnerability measures and analyze the disintegration of the network from the point of view of accessibility and connectivity.

This work adds to the current state of the art:

It introduces a new

The algorithm represents an alternative approach to the lower and upper estimates of transportation network vulnerability ([

The algorithm is also able to provide results for large transportation networks corresponding to administrative units in reasonable time (compare to [

Based on the arguments above, we believe that the incorporation of the algorithm into an online warning system as a tool for decision makers will have a significant positive impact on transportation security and could contribute to early warning before states of emergency.

An archive file with the program. It contains all the code files in java, libraries used and the Zlín region network files.

(ZIP)

The access to the CERIT-SC computing and storage facilities provided under the program Center CERIT Scientific Cloud, part of the Operational Program Research and Development for Innovations, reg. no. CZ.1.05/3.2.00/08.0144 is greatly appreciated. We would further like to thank M. Moriš for consultation on algorithms and J. Sedoník for help with preparation of the figures.

^{th}International IEEE Conference on Intelligent Transportation Systems. 2009: 1–5.