WO2008129312A1

WO2008129312A1 - Analysis of path diversity structure in networks using recursive abstraction

Info

Publication number: WO2008129312A1
Application number: PCT/GB2008/050232
Authority: WO
Inventors: Constantinos Christofi Constantinou; Alexander Sergeevich Stepanenko
Original assignee: Ideas Network Ltd
Priority date: 2007-04-20
Filing date: 2008-03-31
Publication date: 2008-10-30
Also published as: GB0707666D0

Abstract

An analysis method, termed Logical Network Abridgement, for identifying the full inherent path diversity present within a graph representation of a network, of any dimension, consisting of a plurality of vertices and edges. The method generally comprises identifying bi-connected parts of the network, and identifying, with respect to predetermined criteria, a basis set of cycles in the bi- connected parts and assigning a logical vertex of first type at a next higher level to each of the cycles. Alogical vertex of second type is associated with each edge common to two or more cycles in the basis set and a new bipartite graph is defined by interconnecting every logical vertex of second type to all logical vertices of first type whose cycles contain the corresponding edge of the vertex of second type. These steps are repeated forhigher levels of abstraction until a level is reached where no cycles exist and path diversity isminimised. This method of analysing the intrinsic path diversity of a network enables diversity analysis to be applied more accurately and readily to any type of network (social, transport, biological, communication) so that the knowledge gained can be acted on.

Description

ANALYSIS OF PATH DIVERSITY STRUCTURE IN NETWORKS USING RECURSIVE ABSTRACTION

Field of the Invention

This invention relates to a method for analyzing the path diversity of a graph representation of a network using a recursive abstraction procedure which is referred to herein as logical network abridgement and to the apparatus for implementing this method.

A graph representation of a network, which comprises a plurality of vertices and edges, is illustrated in Figure 1.

For example, the network might represent a data communication network where each vertex corresponds to a switching node and each edge corresponds to a transmission line interconnecting the switching nodes. In the case of a data communication network, the network of Figure 1 depicts a plurality of switching nodes VO.1 to VO.10 interconnected by transmission lines EO.1 to EO.16. Alternatively, the network might represent a road transport network where each vertex represents a road junction and each edge a road segment between road junctions. In the case of a road transport network, the network of Figure 1 depicts a plurality of road junctions VO.1 to VO.10 interconnected by road segments EO.1 to EO.16.

Description of the Related Art

In WO2004/086692, the whole contents of which is incorporated herein by reference, a method of constructing a recursive abstraction of the graph of a data communication network into a series of logical levels is described which shows how, through knowledge of the network's inherent path diversity, it is possible to provide resilient communication through exploiting all these paths. However, in WO2004/086692 the graph describing the physical switching nodes and transmission links of the data communication network is arbitrarily embedded in a two dimensional plane in order to define the first level of the recursive abstraction. Because of the requirement for the network to be embedded in a two dimensional plane, an exception process is then required in order to first remove and then, at a later stage, restore any non-planar edges present in the original network.

As illustrated in Figure 2, with the method described in WO2004/086692 once the physical nodes and links have been embedded in a plane (level 0), cycles are identified (C1.1 to C1.7 in Figure 2) to coincide with the faces of the chosen planar embedding and then abstracted to form the virtual vertices of the next higher level of abstraction (V1.1 to V1.7). Each cycle, providing two alternative paths on the cycle between each pair of vertices, thus represents a basic unit of first type of diversity present in the original physical network. Common edges between pairs of cycles in the initial network level are also abstracted to form the virtual edges (E1.1 to E1.8) at the next higher level. Each common edge, having at least two shared vertices between the neighbouring cycles, thus represents diverse connectivity between these cycles, this being a second type of diversity unit. As illustrated in Figure 3, the abstraction process is then repeated until a logical level is reached where there are no longer any cycles and, therefore, no further diverse paths between vertices. Each logical vertex contains sets of alternative path pairs between vertices on the corresponding cycle at the previous logical level. Each logical edge contains at least a pair of common vertices between two adjacent cycles at the previous logical level. The lack of such path diversity is represented by the presence of disjoint vertices at the next logical level, as is the case in Figure 3. Therefore, the logical network abridgement encodes path diversity, by construction, in such a way that greater intrinsic path diversity maps into a bigger number of levels of abstraction.

Summary of the Invention

In a first aspect the present invention provides a network analysis method for identifying path diversity present within a network, said network analysis method comprising the steps of: a) at an initial level, identifying one or more vertices and one or more edges of at least part of a physical network; b) with respect to predetermined criteria, determining a basis set of cycles in the current level, based on the identified vertices and edges; c) assigning a logical vertex of first type, at a next higher level, to each of said cycles of the basis set of cycles determined in step b); d) associating a logical vertex of second type, at said next higher level, with respect to each edge common to two or more cycles in the basis set determined in step b); e) defining a new bipartite graph, by assigning edges interconnecting every logical vertex of second type to all logical vertices of first type whose cycles contain the corresponding edge of the said vertex of second type; f) repeating method steps b), c), d) and e) in respect of the vertices and edges of at least one next higher level.

In a preferred embodiment the network analysis method further comprises the initial step of forming a graph representation of said part of the physical network.

More preferably, the network analysis method further comprises the step of, at each and every level of abstraction, forming a graph representation of said part of the network and identifying bi-connected parts of the graph, which do not share any cycles, and performing each of the subsequent method steps for each bi-connected part of the graph independently.

More preferably still, two or more logical vertices of second type each connected to same pair of logical vertices of first type and corresponding to lower-level edges that are adjacent may be merged into a single vertex of second type which is connected to the union of said logical vertices of first type.

More preferably still, the network analysis method further comprises the step of unifying the two types of diversity by eliminating the distinction between vertices of first and second types. Ideally, the method steps b), c), d) and e) are repeated until a next higher level is reached where no cycles exist.

Furthermore, the number of cycles in the basis set is determined in accordance with the following equation: b = e - v + c where b is the number of cycles in the basis set, e is the number of edges, v is the number of vertices and c is the number of separate sets of connected components and the basis set of cycles is selected in dependence upon predetermined criteria.

One example only of the predetermined criteria henceforth are:

(i) each cycle of the basis set is independent from the remaining cycles in the basis set;

(ii) when the number of edges constituting each cycle in the basis set are added together, the total number is minimised; and

(iii) when ambiguity in the choice of cycles exists according to criterion (ii), the cycles whose constituent vertices have the lowest lexicographical order are selected preferentially.

In a separate aspect the present invention provides a computerised network analysis system for identifying path diversity present within a network, said network analysis system comprising a processor and a memory in which is stored software instructions to be performed by the processor, the software instructions comprising: a) at an initial level, identifying one or more vertices and one or more edges of at least part of a physical network; b) with respect to predetermined criteria, determining a basis set of cycles in the current level, based on the identified vertices and edges; c) assigning a logical vertex of first type, at a next higher level, to each of said cycles of the basis set of cycles determined in step b); d) associating a logical vertex of second type, at said next higher level, with respect to each edge common to two or more cycles in the basis set determined in step b); e) defining a new bipartite graph, by assigning edges interconnecting every logical vertex of second type to all logical vertices of first type whose cycles contain the corresponding edge of the said vertex of second type; f) repeating method steps b), c), d) and e) in respect of the vertices and edges of at least one next higher level.

The present invention is directed to a more general and precise method of analysing the intrinsic path diversity of any network that can be represented as a graph, irrespective of the number of dimensions of the network. This method dispenses with the need to select a planar graph embedding for the graph representation of the physical level or any further higher level. This removes the earlier asymmetric treatment of planar and non-planar links and the arbitrary nature of their selection. These enhancements allow such an approach to diversity analysis to be applied more accurately and readily to any type of network (social, transport, biological, communication) so that the knowledge gained can be acted on, including but not limited to the formation of routing algorithms for the control of data communication networks (as described in WO2004/086692 ).

Independently of the nature, type or characteristics of the network, in each case with the present invention it is possible to use the process of logical network abridgement to determine the inherent path diversity of the network. This knowledge can then be employed in the context of the specific nature of the network to assess the capabilities of the network to satisfy specific operational requirements. For example, in the case of a road transport network, knowledge of the path diversity can be used to compare alternative road layouts and assess the robustness of their traffic carrying capacity to specific road or junction closures. In the case of a data communication network, knowledge of the network path diversity can be used, for example, to provide a structured method for identifying all the different paths that can be used to connect any given pair of nodes across the network. Brief Description of the Drawings

In order that the invention and its applications may be elucidated further, application-specific embodiments thereof will now be described, by way of example only, with reference to the accompanying drawings in which:-

Figure 1 is a graph representation of an arbitrary physical network, level 0, which for the purposes of this example corresponds to a road transportation network where a plurality of road junctions VO.1 to VO.10 are interconnected by roads EO.1 to EO.16;

Figure 2 is a graph representation of level 0 (in black) and the next higher logical level, level 1 , (in grey) using the recursive abstraction method of WO2004/086692 as applied to the network illustrated in Figure 1 ;

Figure 3 is a graph representation of level 1 (in black) and the next higher logical level, level 2, (in grey) using the recursive abstraction method of WO2004/086692 as applied to the network illustrated in Figure 2;

Figure 4 is a diagram to show how the level 0 network (in black) of

Figure 1 may be divided into a basis set of cycles C1.1 to C1.7 and then abstracted to a level 1 graph (in grey) consisting of cycle vertices V_C1.1 to

V_C1.7 represented by circles and hyperedge vertices V_hc1.1 to V_hc1.8 represented by crosses, using the method of the present invention;

Figure 5 is a diagram showing how the level 1 network (in black and only partially labelled for the sake of simplicity) of Figure 4 is modified in removing the distinction between the two types of vertices and now consists of two cycles C2.1 and C2.2 which can now in turn be abstracted into the level 2 network (in grey and disjointed) comprising of vertices V_C2.1 and V_C2.2;

Figures 6 to 14 are diagrams showing how the method of logical network abridgement of the present invention provides an enhanced and accurate view of the network diversity of two alternative improvements to the road transportation network portrayed in Figure 1 , namely

Figure 6 is a diagram of the network of Figure 1 with two extra roads (in bold) added represented by EO.17 and EO.18; Figure 7 is a diagram of the network of Figure 1 with two extra roads added (in bold), different to the new roads of Figure 6, represented by EO.18 and EO.19 and one extra road junction represented by VO.11 which also split the road represented by EO.7 into EO.7 and EO.17;

Figure 8 is a diagram to show how the physical level 0 network (in black) of Figure 6 may be divided into a basis set of cycles and then abstracted to a level 1 graph (in grey) consisting of cycle vertices V_C1.1 to V_C1.9 and hyperedge vertices V_hc1.1 to V_hc1.10, using the method of the present invention;

Figure 9 is a diagram showing how the level 1 network (in black) of Figure 8 is modified in removing the distinction between the two types of vertices and now consists of two cycles which can in turn be abstracted into the level 2 network (in grey) comprising the vertices V_C2.1 and V_C2.2;

Figure 10 is a diagram to show how the physical level 0 network (in black) of Figure 7 may be divided into a basis set of cycles C1.1 to C1.9, using the method of the present invention; and then abstracted to a level 1 graph (in grey) consisting of cycle vertices and hyperedge vertices using the method of the present invention (Note, for clarity, the cycle of C1.5 which includes the edge EO.10 is identified using dots whilst the other cycles are identified using dashes);

Figure 11 is a diagram to show the completion of the abstraction of the basis set of cycles selected in Figure 10 into a level 1 graph consisting of cycle vertices V_C1.1 to V_C1.9 and hyperedge vertices V_hc1.1 to V_hc1.11 , using the method of the present invention;

Figure 12, is a diagram to show how the level 1 network of Figure 11 may then be modified (in black) to remove transient vertices and the distinction between vertices and then divided into a basis set of cycles C2.1 to C2.5, using the method of the present invention and then abstracted to a level 2 graph (in grey) consisting of cycle vertices and hyperedge vertices using the method of the present invention (Note that hyperedges V_hc1.5 and V_hc1.6 have become vertices V1.3 and V1.5 causing a fundamental change to the structure of the graph that would not have occurred with the abstraction method of WO2004/086692);

Figure 13 is a diagram to show the completion of the abstraction of the basis set of cycles selected in Figure 12 into a further level 2 bipartite graph consisting of cycle vertices V_C2.1 to V_C2.5 and hyperedge vertices V_hc2.1 to V_hc2.5,, using the method of the present invention;

Figure 14 is a graph representation of how the level 2 network (in black) of Figure 13 may then be modified to remove transient vertices and the distinction between vertices and the single remaining cycle at level 2 abstracted to a logical level 3 graph (in grey) where the path diversity falls to a single cycle vertex V_C3.1 ; and

Figures 15 to 21 are diagrams showing how the method of logical network abridgement of the present invention provides an enhanced and accurate view of the robustness of the two alternative improvements to a road closure, namely

Figure 15 is a diagram of the network of Figure 6 with a road closure (E0.9);

Figure 16 is a diagram illustrating how the road closure in the network (in black) of Figure 15 alters the abstraction of the network at a logical level 1 of abstraction (in grey) - in comparison to Figure 8 - using the method of the present invention;

Figure 17 is a diagram illustrating how the road closure in the network (in black) of Figure 15 alters the network abstraction at a logical level 2 of abstraction (in grey) - in comparison to Figure 9 - using the method of the present invention;

Figure 18 is a diagram of the network of Figure 7 with the same road closure (E0.9);

Figure 19 is a diagram illustrating how the road closure in the network (in black) of Figure 18 alters the basis set of cycles (in grey) - in comparison to Figure 10 - using the method of the present invention (Note, for clarity, the cycle of C1.5 which includes the edge EO.10 is identified using dots whilst the other cycles are identified using dashes); Figure 20 is a diagram illustrating how the road closure in the network of Figure 18 alters the abstraction of the network at a logical level 1 of abstraction (in comparison to Figure 11 ), using the method of the present invention; and

Figure 21 is a diagram illustrating how the road closure in the network (in black) of Figure 18 alters the abstraction of the network at a logical level 2 of abstraction (in grey) - in comparison to Figure 12 - using the method of the present invention.

Detailed Description of Exemplary Embodiments

Reference is made to Figures 1 and 4 to 14 which illustrate the method of recursive abstraction, referred to herein as logical network abridgement, by its application to the analysis of three closely related road transportation networks portrayed in Figures 1 , 6 and 7.

Referring first to Figure 1 , the diversity present in the graphical representation of the physical road transportation network (level 0) is analysed first.

The number of cycles required to fully represent a network, i.e. the number of cycles in the basis set, can be determined in accordance with the following equation: b = e - v + c equation 1 where b is the basis number of cycles, e is the number of edges in the network, v is the number of vertices and c is the number of separate sets of connected components (where all vertices are connected together there is a single set of connected vertices and c = 1 ). In the case of the network shown in Figure 1 the number of cycles in the basis set is 7 (= 16 - 10 + 1 ).

In Figure 4 a basis set of cycles (C1.1 , C1.2, C1.3, C1.4, C1.5, C1.6 and C1.7) is identified for the graph representation of the road transport network. Specific criteria are required to select the most appropriate basis set from all the various possible basis sets. In the basis set selected in this example the following criteria were used: (i) each cycle of the basis set is independent from the remaining cycles in the basis set; (ii) when the number of edges constituting each cycle in the basis set are added together, the total number is minimised and (iii) when ambiguity in the choice of cycles exists according to criterion (ii) the cycles whose constituent vertices have the lowest lexicographical order are selected preferentially. In the context of other applications, analogous but different criteria for selecting a cycle basis set will apply.

Every cycle in the basis set at level 0 is then identified with a logical vertex of type 1 (depicted as gray circles in Figure 4: V_C1.1 , V_C1.2, V_C1.3, Vc 1.4, V_C15, V_C1.6 and V_C1.7) and every edge at level 0 common to two more cycles in the basis set is identified as a logical vertex of type 2 (depicted as gray crosses in Figure 4: V_hc1.1 , V_hc1.2, V_hc1.3, V_hc1.4, V_hc1.5, V_hc1.6, V_hc1.7 and V_hc1.8). Every logical vertex of type 2 is interconnected to all logical vertices of type 1 whose cycles contain the corresponding edge of the vertex of type 2. Thus a level 1 abstracted bi-partite graph is arrived at as shown in gray in Figure 4.

A modified level 1 abstracted graph is then derived by removing the distinction between vertices of type 1 and 2 and labelling all vertices in Figure 5 as V1.1 , V1.2, V1.3, etc. up to V1.15 and all the edges as E1.1 , E1.2, E1.3, etc. up to E1.16. The number of cycles in the basis set of cycles is then calculated for the level 1 abstracted bi-partite graph and in this case is found to be b = 2 (e = 16, v = 15, c = 1 ). The procedure then repeats once more in this example to arrive at two disjoint vertices V_C2.1 and V_C2.2 at level 2, shown in gray in Figure 5. The disjoint vertices at level 2 are a manifestation of the fact that the original road network of Figure 1 consists of two better connected parts than the parts are connected between themselves. This is also manifest at level 1 as a unique path V_C1.3 to V_C1.5 between two cycles at level 1 (shown in gray in Figure 4). Therefore, a road planner can evaluate alternative proposals for one or more new roads, not just according to cost, but also according to the resulting road transport network path diversity and therefore resiliency to congestion, road closures or road junction closures. It should be noted that preferably two or more logical vertices of type 2, each connected to same pair of logical vertices of type 1 and corresponding to lower-level edges that are adjacent, may be merged into a single vertex of type 2 which is connected to the union of the logical vertices of type 1.

Two alternative proposals for improving the road transportation network of Figure 1 are shown in Figures 6 and 7. The first proposal involves building a new road in two segments depicted by edges EO.17 and EO.18 and upgrading three road junctions, VO.3, VO.6 and VO.8 as shown in Figure 6. The second proposal similarly involves building a new road in two segments depicted by edges EO.18 and EO.19, upgrading two road junctions VO.3 and VO.8 and building a new road junction VO.11 and a new bridge where edges EO.10 and EO.19 would have otherwise intersected. The second proposal is more expensive than the first in terms of cost, but as can be seen in Figures 9 and 14, it results in one more level of abstraction and thus corresponds to a more diverse road network. Furthermore, road improvement alternative number 1 has a graph that becomes disconnected at level 2, shown in Figure 9, which shows that the new road proposal has not improved on the path diversity of the original road transportation network as can be seen in Figure 5.

In Figure 10 the number of cycles of the basis set of cycles is determined to be b = 9 (e = 19, v = 11 , c = 1 ). Thus, unlike the previous examples of graphs, not all of the circular interconnections of vertices are required to fully define the diversity of the network. Using the cycle selection criteria stated earlier, the circular interconnection bounded by E0.11 , EO.14, EO.12 and EO.10 is redundant and is omitted in favour of the other three four-hop cycles C1.5, C1.6 and C1.7 of lower lexicographical order.

To demonstrate the differing resiliency of the two alternative road network proposals to a road closure, say due to an accident, reference is now made to Figures 15, 16, 17, 18, 19, 20 and 21. Firstly Figures 15 and 18 show clearly that the two road improvement alternatives are not equally resilient after the road closure depicted by the removal of edge EO.9, since for the first alternative improved road layout two parts of the road network are only connected via a single road junction VO.6 that is likely to suffer from congestion. This is embodied in the logical network abridgment as a disconnected level 1 , as seen in Figure 17 in black. On the other hand, the second alternative improved road layout has a connected level 1 and a disconnected level 2 in its logical network abridgment. This can be easily interpreted as the fact that the road closure has reduced the available path diversity through the network, but has not created any obvious potential congestion hotspots.

As a result of using the logical network abridgement process of the current invention it is therefore possible, in this example, to more completely and accurately analyse the full depth of diversity inherent in the alternative improvement plans for the road transport network. Specifically, the process shows that the improvement alternative portrayed in Figure 7 is much more robust (reaching level 3) than the improvement alternative portrayed in Figure 6 (reaching only a disjoint level 2 graph). This in turn allows the road transportation planners to take more appropriate action by incorporating this knowledge into their decision making.

The method described above is one adapted for computerised implementation to enable fully automated analysis of path diversity in complex, multi-dimensional networks.

Claims

What is claimed:

1. A network analysis method for identifying path diversity present within a network, said network analysis method comprising the steps of: a) at an initial level, identifying one or more vertices and one or more edges of at least part of a physical network; b) with respect to predetermined criteria, determining a basis set of cycles in the current level, based on the identified vertices and edges; c) assigning a logical vertex of first type, at a next higher level, to each of said cycles of the basis set of cycles determined in step b); d) associating a logical vertex of second type, at said next higher level, with respect to each edge common to two or more cycles in the basis set determined in step b); e) defining a new bipartite graph, by assigning edges interconnecting every logical vertex of second type to all logical vertices of first type whose cycles contain the corresponding edge of the said vertex of second type; f) repeating method steps b), c), d) and e) in respect of the vertices and edges of at least one next higher level.

2. The network analysis method of claim 1 , further comprising the initial step of forming a graph representation of said part of the physical network.

3. The network analysis method of claim 2, further comprising the step of, for each and every level of abstraction, forming a graph representation of said part of the network and identifying bi-connected parts of the graph which do not share any cycles, and performing each of the subsequent method steps for each bi- connected part of the graph independently.

4. The network analysis method of claim 3, further comprising the step of merging two or more logical vertices of second type each connected to same pair of logical vertices of first type and corresponding to lower-level edges that are adjacent into a single vertex of second type which is connected to the union of said logical vertices of first type.

5. A network analysis method as claimed in claim 4, further comprising the step of unifying the two types of diversity by eliminating the distinction between vertices of first and second types.

6. A network analysis method as claimed in claim 1 , further comprising repeating method steps b), c), d) and e) until a next higher level is reached where no cycles exist.

7. A network analysis method as claimed in claim 1 , wherein the number of cycles in the basis set is determined in accordance with the following equation: b = e - v + c where b is the number of cycles in the basis set, e is the number of edges, v is the number of vertices and c is the number of separate sets of connected components.

8. A network analysis method as claimed in claim 7, wherein the basis set of cycles is selected in dependence upon predetermined criteria.

9. A computerised network analysis system for identifying path diversity present within a network, said network analysis system comprising a processor and a memory in which is stored software instructions to be performed by the processor, the software instructions comprising: a) at an initial level, identifying one or more vertices and one or more edges of at least part of a physical network; b) with respect to predetermined criteria, determining a basis set of cycles in the current level, based on the identified vertices and edges; c) assigning a logical vertex of first type, at a next higher level, to each of said cycles of the basis set of cycles determined in step b); d) associating a logical vertex of second type, at said next higher level, with respect to each edge common to two or more cycles in the basis set determined in step b); e) defining a new bipartite graph, by assigning edges interconnecting every logical vertex of second type to all logical vertices of first type whose cycles contain the corresponding edge of the said vertex of second type; f) repeating method steps b), c), d) and e) in respect of the vertices and edges of at least one next higher level.