Final Report Summary - OPTIMAL NETWORKS (Optimal interconnection networks)
In modern society, networks are increasingly more important, such as communication networks, computer networks, transportation networks, the Internet, social networks. In this project we will investigate the topology of networks in terms of its underlying model using graph theory. Our aims are to obtain new knowledge of the structure of networks which are in some sense optimal, for example, in terms of the number of nodes given restrictions on other parameters of the network, such as the number of connections attached at each node, or the maximum allowed distance between any two nodes in the network.
In this project we have obtained several new original results concerning the existence and topology of networks which are in some sense optimal.
Research approaches related to the problem that we investigate fall into two main directions. On one hand, we form hypotheses and proofs regarding the non-existence of graphs (directed or undirected) of order close to the general theoretical upper bounds, and so we improve, that is, lower the upper bounds. On the other hand, we design new constructions to produce large graphs (directed or undirected), furnishing better lower bounds on the maximum possible number of vertices in a network which is optimal with respect to the number of nodes, given the restriction on the maximum possible number of connections at any particular node and given the requirement of any two nodes being within a prescribed distance (in terms of the number of the intermediate nodes that need to be traversed to reach any one node of the network from any other node.
In this project we have found improved solutions for the upper bounds in the general case, as well as new largest known graphs in the case the graphs are bipartite (that is, consisting of two sets of vertices such that there is no direct connection between these two sets).
One of the aims of the project was to establish an on-going collaboration between King's College London and The University of Newcastle, Australia. This has been established and the researchers actively involved so far include Prof. Costas Iliopoulos, Dr Jackie Daykin, Dr Solon Pissis, Mr Michalis Christou (all from King's College) and Prof. Mirka Miller, Dr Joe Ryan, Mr Oudone Phanalasy (all from the University of Newcastle). Further collaborations are likely to eventuate over the next few years.
Another outcome of this project is the contribution to the research training of three Doctor of Philosophy (PhD) students Solon Pissis, Michalis Christou and Oudone Phanalasy. Some of that has been devoted to the degree / diameter problem and some to other related problems in graph theory. These activities also resulted in new original results which have been published or submitted for publication.
Other outcome of interest to the European Union (EU) flowing on from this project is the collaboration of Prof. Mirka Miller with researchers from University of West Bohemia, Czech Republic, and UPC, Spain. This has already produced two publications in refereed journals in graph theory. Currently there are several other spin-off research collaborations starting as a result of this project.
To summarise, this project has been highly successful, resulting in:
1. new fundamental knowledge that has been published in seven refereed journal papers and several papers in press or submitted;
2. contributions to the training of three research students;
3. established ongoing collaboration between researchers from Australia and EU countries.
Summary description of project context and objectives
This project aims to obtain new knowledge concerning the topology of interconnection networks. The concept of an interconnection network goes back at least to the 1950's. An interconnection network is a system consisting of entities (also called nodes or points), and connections (links) amongst the entities. In such a system both the entities and the connections may be very diverse. For example, the entities can be computers, microprocessors, other networks, people while the connections can be physical (such as physical wires) or conceptual (such as relationships between two people).
Processing and distribution of data has become an essential element in the development and the day-to-day functioning of our society. In many systems the communication among the system's entities are key factors in the performance of the network. Such systems include very large-scale integration (VLSI) circuits, telephone networks, computer networks, image processing, to name just a few.
A current problem in modern society is our need for ever-increasing computing power. One way to supply this is by using parallel computing. Parallel computing requires high volume of communication and interconnection of processors and peripherals. In the design of computer networks, especially peer-to-peer networks (each network connects as both client and server), e.g. Gnutella; and in parallel processing architectures using node-to-node links for inter-central processing unit (CPU) data exchange, e.g. Beouff clusters, the following question is very important.
If each node connects to no more than d other nodes, how many nodes can there be in a network such that no two nodes are separated by more than k steps?
Except for a very few cases of small graphs, the answer to this question is unknown.
An interconnection network is usually modelled by a graph in which vertices represent nodes and undirected or directed edges stand for the connections. Such a graph corresponding to an interconnection network is also called the topology of the network. Although this model does not take into account implementation factors, it does provide an effective means of abstraction which allows us to study many relevant network properties. There are many requirements that influence the design of an interconnection network, including in particular the following factors which will be investigated in this project.
Small maximum degree: The degree of a node is the number of connections attached to that node. This is restricted by hardware constraints. Large interconnection networks require large amounts of wiring which in the case of optical fibre is costly. Moreover, the performance of a system is spent largely on operating the wiring. Therefore, it is desirable to keep the maximum degree in a network as small as possible.
Small diameter: The distance between two vertices is the length of a shortest path between them. The diameter of the network is the maximum distance between any two nodes in the network. A network with a large diameter faces many problems, such as long communications delays, coupling problems and the existence of various types of uncontrolled noise (that is, distorted part of a signal). Therefore, it is desirable to have diameter as small as possible.
Node symmetry or regularity: A network is said to have the property of node symmetry if all nodes behave alike. Such a feature is highly desirable because it allows the use of the same algorithms at each node. We say that a network is regular if each node has the same degree. Network regularity is one of the requirements of node symmetry.
The available budget often imposes additional constraints on network design, so that an 'ideal' network may be too expensive. Additionally, even disregarding the budget, it is not always possible to achieve all the desired requirements in an interconnection network due to the fact that some requirements may conflict with each other. For example, a complete graph satisfies the requirements of a small diameter (just 1) and node symmetry (maximum possible). However, in a complete network of n nodes, each node would have to be connected to all the other n-1 nodes, and large number of connections would be required for such a topology. Therefore, using complete graphs to model large interconnection networks is impractical and network designers need to find a compromise between the desirable requirements and the feasible factors.
The topology of an interconnection network depends on the particular application and so there are many interpretations of what an 'optimal' network is. One possible general interpretation can be expressed as follows.
An optimal network contains the maximum possible number of nodes, given a limitation on the number of connections attached to a node and a limitation on the number of traversed links between any two farthest nodes.
What then is the largest number of nodes in a network, sometimes referred to as the order of the network, with a constraint on degree and diameter? If links are modelled by undirected edges, this leads to the degree / diameter problem: Given natural numbers and D, find the largest possible number of vertices n(D) in a graph of maximum degree and diameter at most D.
The study of this theoretical problem and its applications forms the heart of this project. The research objectives are to obtain new fundamental results in the various aspects of network topology.
The project is in graph theory, which is a scientific branch that straddles mathematics and theoretical computer science. Although most of our results are theoretical, they will be applicable to the design of various interconnection networks, including computer networks and social networks, amongst others.
The main objectives of the project were to obtain new fundamental knowledge concerning a problem in extremal graph theory, in particular, to create new largest graphs for the given parameters, and to obtain new results concerning the upper bound on the largest possible number of nodes in networks with given bound on the number of connections at each node. Other objectives of this project were to establish an on-going collaboration the Newcastle University's Graph Theory and Applications (GTA) research group led by Prof. Mirka Miller with the King's College London's Algorithm Design Lab led by Prof. Costas Iliopoulos and other research groups in other countries of the EU.
Description of the main scientific and technical results / foregrounds
Below we give an overview of the results that have been obtained within this project.
1. Iliopoulos, C., Miller, M., Pissis, S.P. Parallel algorithms for mapping short degenerate and weighted sequences, International Journal of Foundations of Computer Science, Vol.23 Issue 2, pp.249-259 2012.
2. Miller, M., Perez-Roses, H., Ryan, J., The maximum degree&diameter-bounded subgraph in the mesh, Discrete Applied Mathematics, pp.1782-1790 2012.
3. Miller, M., Ryan, J., Ryjacek, Z., Teska, J., Vrana, P., Stability of hereditary graph classes under closure operations, Journal of Graph Theory, accepted August 2012.
4. Miller, M., Ryan, J., Ryjacek, Z., Distance-locally disconnected graphs, submitted to Discussiones Math.DM-GT accepted November 2012.
5. Daykin, J. W., Iliopoulos, C. S., Miller, M., Phanalasy, O., Rylands, L., Antimagicness of generalised corona and snowflake graphs, submitted to JCMCC, 2012.
6. Christou, M., Iliopoulos, C., Miller, M., Degree/diameter problem for trees and pseudotrees, submitted to AKCE, 2012.
7. Christou, M., Iliopoulos, C., Miller, M., Bipartite Ramsey numbers involving stars, stripes and trees, submitted to El. J. Combinatorics, 2012.
8. Christou, M., Iliopoulos, C., Miller, M., Maximising the size of planar graphs under girth constraints, submitted to JCMCC, accepted June 2012.
9. Marshall, K., Miller, M., Ryan, J., Extremal Graphs without Cycles of Length 8 or Less, Electronic Notes in Discrete Mathematics, pp.615-620 2011.
10. Feria-Puron, R., Miller, M., Pineda-Villavicencio, G., On graphs of defect at most 2, Discrete Applied Mathematics, Vol.159 Issue 13, pp.1331-1344 2011.
11. Miller, M., Nonexistence of graphs with cyclic defect, Electronic J. Combin. Vol.18 P71, 2011.
12. Loz, E., Macaj, M., Miller, M., Siagiova, J., Siran, J., Tomanova, J., Small vertex-transitive and Cayley graphs of girth six and given degree: An algebraic approach, Journal of Graph Theory, Vol.68 (4), pp.265-348 2011.
13. Phanalasy, O., Miller, M., Iliopoulos, C. S., Pissis, C. P., Vaezpour, E., Construction of antimagic labeling for the Cartesian product of regular graphs, Mathematics in Computer Science Vol 5(1), pp.81-87 2011.
Potential impact (including the socio-economic impact and the wider societal implications of the project so far) and the main dissemination activities and exploitation of results
The day-to-day running of modern society is largely dependent on interconnection networks such as transportation, communication, computer networks, networks for the distribution of goods, social networks. The theoretical analysis of such networks has become a subject of fundamental importance that has scientific, technological and socio-economic applications and consequences.
Optimal graphs are largest possible graphs, subject to constraints of degree (number of connections per node) and diameter (largest distance between any two nodes in the graph). It is astounding that, except for a few small cases, the size of these optimal graphs is largely unknown. The aims of this project are to discover new optimal graphs, to uncover their underlying structures and to investigate exploitable properties of such graphs.
Graphs are used increasingly more as models for various types of networks. The significance of this project is twofold: it contributes towards our understanding and the usefulness of such models, as well as provides fundamental new results in a problem which is arguably one of the pivotal unsolved problems in graph theory.
The intellectual merits of this project include new discoveries, developing novel concepts and designing new algorithms. The outcomes have been fundamental significant new results in extremal graph theory, development of new improved methods, constructing new graphs and lowering the upper bounds on the largest possible number of nodes that a network of given parameters may contain. The outcomes are likely to have lasting impact on the state of the art in extremal graph theory, as well as a possible practical impact in applications concerned with the structure and design of interconnection networks.
In this project we have obtained several new original results concerning the existence and topology of networks which are in some sense optimal.
Research approaches related to the problem that we investigate fall into two main directions. On one hand, we form hypotheses and proofs regarding the non-existence of graphs (directed or undirected) of order close to the general theoretical upper bounds, and so we improve, that is, lower the upper bounds. On the other hand, we design new constructions to produce large graphs (directed or undirected), furnishing better lower bounds on the maximum possible number of vertices in a network which is optimal with respect to the number of nodes, given the restriction on the maximum possible number of connections at any particular node and given the requirement of any two nodes being within a prescribed distance (in terms of the number of the intermediate nodes that need to be traversed to reach any one node of the network from any other node.
In this project we have found improved solutions for the upper bounds in the general case, as well as new largest known graphs in the case the graphs are bipartite (that is, consisting of two sets of vertices such that there is no direct connection between these two sets).
One of the aims of the project was to establish an on-going collaboration between King's College London and The University of Newcastle, Australia. This has been established and the researchers actively involved so far include Prof. Costas Iliopoulos, Dr Jackie Daykin, Dr Solon Pissis, Mr Michalis Christou (all from King's College) and Prof. Mirka Miller, Dr Joe Ryan, Mr Oudone Phanalasy (all from the University of Newcastle). Further collaborations are likely to eventuate over the next few years.
Another outcome of this project is the contribution to the research training of three Doctor of Philosophy (PhD) students Solon Pissis, Michalis Christou and Oudone Phanalasy. Some of that has been devoted to the degree / diameter problem and some to other related problems in graph theory. These activities also resulted in new original results which have been published or submitted for publication.
Other outcome of interest to the European Union (EU) flowing on from this project is the collaboration of Prof. Mirka Miller with researchers from University of West Bohemia, Czech Republic, and UPC, Spain. This has already produced two publications in refereed journals in graph theory. Currently there are several other spin-off research collaborations starting as a result of this project.
To summarise, this project has been highly successful, resulting in:
1. new fundamental knowledge that has been published in seven refereed journal papers and several papers in press or submitted;
2. contributions to the training of three research students;
3. established ongoing collaboration between researchers from Australia and EU countries.
Summary description of project context and objectives
This project aims to obtain new knowledge concerning the topology of interconnection networks. The concept of an interconnection network goes back at least to the 1950's. An interconnection network is a system consisting of entities (also called nodes or points), and connections (links) amongst the entities. In such a system both the entities and the connections may be very diverse. For example, the entities can be computers, microprocessors, other networks, people while the connections can be physical (such as physical wires) or conceptual (such as relationships between two people).
Processing and distribution of data has become an essential element in the development and the day-to-day functioning of our society. In many systems the communication among the system's entities are key factors in the performance of the network. Such systems include very large-scale integration (VLSI) circuits, telephone networks, computer networks, image processing, to name just a few.
A current problem in modern society is our need for ever-increasing computing power. One way to supply this is by using parallel computing. Parallel computing requires high volume of communication and interconnection of processors and peripherals. In the design of computer networks, especially peer-to-peer networks (each network connects as both client and server), e.g. Gnutella; and in parallel processing architectures using node-to-node links for inter-central processing unit (CPU) data exchange, e.g. Beouff clusters, the following question is very important.
If each node connects to no more than d other nodes, how many nodes can there be in a network such that no two nodes are separated by more than k steps?
Except for a very few cases of small graphs, the answer to this question is unknown.
An interconnection network is usually modelled by a graph in which vertices represent nodes and undirected or directed edges stand for the connections. Such a graph corresponding to an interconnection network is also called the topology of the network. Although this model does not take into account implementation factors, it does provide an effective means of abstraction which allows us to study many relevant network properties. There are many requirements that influence the design of an interconnection network, including in particular the following factors which will be investigated in this project.
Small maximum degree: The degree of a node is the number of connections attached to that node. This is restricted by hardware constraints. Large interconnection networks require large amounts of wiring which in the case of optical fibre is costly. Moreover, the performance of a system is spent largely on operating the wiring. Therefore, it is desirable to keep the maximum degree in a network as small as possible.
Small diameter: The distance between two vertices is the length of a shortest path between them. The diameter of the network is the maximum distance between any two nodes in the network. A network with a large diameter faces many problems, such as long communications delays, coupling problems and the existence of various types of uncontrolled noise (that is, distorted part of a signal). Therefore, it is desirable to have diameter as small as possible.
Node symmetry or regularity: A network is said to have the property of node symmetry if all nodes behave alike. Such a feature is highly desirable because it allows the use of the same algorithms at each node. We say that a network is regular if each node has the same degree. Network regularity is one of the requirements of node symmetry.
The available budget often imposes additional constraints on network design, so that an 'ideal' network may be too expensive. Additionally, even disregarding the budget, it is not always possible to achieve all the desired requirements in an interconnection network due to the fact that some requirements may conflict with each other. For example, a complete graph satisfies the requirements of a small diameter (just 1) and node symmetry (maximum possible). However, in a complete network of n nodes, each node would have to be connected to all the other n-1 nodes, and large number of connections would be required for such a topology. Therefore, using complete graphs to model large interconnection networks is impractical and network designers need to find a compromise between the desirable requirements and the feasible factors.
The topology of an interconnection network depends on the particular application and so there are many interpretations of what an 'optimal' network is. One possible general interpretation can be expressed as follows.
An optimal network contains the maximum possible number of nodes, given a limitation on the number of connections attached to a node and a limitation on the number of traversed links between any two farthest nodes.
What then is the largest number of nodes in a network, sometimes referred to as the order of the network, with a constraint on degree and diameter? If links are modelled by undirected edges, this leads to the degree / diameter problem: Given natural numbers and D, find the largest possible number of vertices n(D) in a graph of maximum degree and diameter at most D.
The study of this theoretical problem and its applications forms the heart of this project. The research objectives are to obtain new fundamental results in the various aspects of network topology.
The project is in graph theory, which is a scientific branch that straddles mathematics and theoretical computer science. Although most of our results are theoretical, they will be applicable to the design of various interconnection networks, including computer networks and social networks, amongst others.
The main objectives of the project were to obtain new fundamental knowledge concerning a problem in extremal graph theory, in particular, to create new largest graphs for the given parameters, and to obtain new results concerning the upper bound on the largest possible number of nodes in networks with given bound on the number of connections at each node. Other objectives of this project were to establish an on-going collaboration the Newcastle University's Graph Theory and Applications (GTA) research group led by Prof. Mirka Miller with the King's College London's Algorithm Design Lab led by Prof. Costas Iliopoulos and other research groups in other countries of the EU.
Description of the main scientific and technical results / foregrounds
Below we give an overview of the results that have been obtained within this project.
1. Iliopoulos, C., Miller, M., Pissis, S.P. Parallel algorithms for mapping short degenerate and weighted sequences, International Journal of Foundations of Computer Science, Vol.23 Issue 2, pp.249-259 2012.
2. Miller, M., Perez-Roses, H., Ryan, J., The maximum degree&diameter-bounded subgraph in the mesh, Discrete Applied Mathematics, pp.1782-1790 2012.
3. Miller, M., Ryan, J., Ryjacek, Z., Teska, J., Vrana, P., Stability of hereditary graph classes under closure operations, Journal of Graph Theory, accepted August 2012.
4. Miller, M., Ryan, J., Ryjacek, Z., Distance-locally disconnected graphs, submitted to Discussiones Math.DM-GT accepted November 2012.
5. Daykin, J. W., Iliopoulos, C. S., Miller, M., Phanalasy, O., Rylands, L., Antimagicness of generalised corona and snowflake graphs, submitted to JCMCC, 2012.
6. Christou, M., Iliopoulos, C., Miller, M., Degree/diameter problem for trees and pseudotrees, submitted to AKCE, 2012.
7. Christou, M., Iliopoulos, C., Miller, M., Bipartite Ramsey numbers involving stars, stripes and trees, submitted to El. J. Combinatorics, 2012.
8. Christou, M., Iliopoulos, C., Miller, M., Maximising the size of planar graphs under girth constraints, submitted to JCMCC, accepted June 2012.
9. Marshall, K., Miller, M., Ryan, J., Extremal Graphs without Cycles of Length 8 or Less, Electronic Notes in Discrete Mathematics, pp.615-620 2011.
10. Feria-Puron, R., Miller, M., Pineda-Villavicencio, G., On graphs of defect at most 2, Discrete Applied Mathematics, Vol.159 Issue 13, pp.1331-1344 2011.
11. Miller, M., Nonexistence of graphs with cyclic defect, Electronic J. Combin. Vol.18 P71, 2011.
12. Loz, E., Macaj, M., Miller, M., Siagiova, J., Siran, J., Tomanova, J., Small vertex-transitive and Cayley graphs of girth six and given degree: An algebraic approach, Journal of Graph Theory, Vol.68 (4), pp.265-348 2011.
13. Phanalasy, O., Miller, M., Iliopoulos, C. S., Pissis, C. P., Vaezpour, E., Construction of antimagic labeling for the Cartesian product of regular graphs, Mathematics in Computer Science Vol 5(1), pp.81-87 2011.
Potential impact (including the socio-economic impact and the wider societal implications of the project so far) and the main dissemination activities and exploitation of results
The day-to-day running of modern society is largely dependent on interconnection networks such as transportation, communication, computer networks, networks for the distribution of goods, social networks. The theoretical analysis of such networks has become a subject of fundamental importance that has scientific, technological and socio-economic applications and consequences.
Optimal graphs are largest possible graphs, subject to constraints of degree (number of connections per node) and diameter (largest distance between any two nodes in the graph). It is astounding that, except for a few small cases, the size of these optimal graphs is largely unknown. The aims of this project are to discover new optimal graphs, to uncover their underlying structures and to investigate exploitable properties of such graphs.
Graphs are used increasingly more as models for various types of networks. The significance of this project is twofold: it contributes towards our understanding and the usefulness of such models, as well as provides fundamental new results in a problem which is arguably one of the pivotal unsolved problems in graph theory.
The intellectual merits of this project include new discoveries, developing novel concepts and designing new algorithms. The outcomes have been fundamental significant new results in extremal graph theory, development of new improved methods, constructing new graphs and lowering the upper bounds on the largest possible number of nodes that a network of given parameters may contain. The outcomes are likely to have lasting impact on the state of the art in extremal graph theory, as well as a possible practical impact in applications concerned with the structure and design of interconnection networks.