ANDREAS AABRANDT et al: TOPOLOGICAL RANKINGS IN COMMUNICATION NETWORKS DOI 10.5013/IJSSST.a.16.01.07 7.1 ISSN: 1473-804x online, 1473-8031 print Topological Rankings in Communication Networks Andreas Aabrandt, Vagn Lundsgaard Hansen, Chresten Træholt Department of Applied Mathematics and Computer Science Department of Electrical Engineering, CEE Technical University of Denmark Email: aabran@elektro.dtu.dk, vlha@dtu.dk, ctr@elektro.dtu.dk Abstract — In the theory of communication the central problem is to study how agents exchange information. This problem may be studied using the theory of connected spaces in topology, since a communication network can be modelled as a topological space such that agents can communicate if and only if they belong to the same path connected component of that space. In order to study combinatorial properties of such a communication network, notions from algebraic topology are applied. This makes it possible to determine the shape of a network by concrete invariants, e.g. the number of connected components. Elements of a network may then be ranked according to how essential their positions are in the network by considering the effect of removing them. Defining a ranking of a network which takes the individual position of each entity into account has the purpose of assigning different roles to the entities, e.g. agents, in the network. In this paper it is shown that the topology of a given network induces a ranking of the entities in the network. Furthermore, it is demonstrated how to calculate this ranking and thus how to identify weak sub-networks in any given network. Keywords - Ranking; Communication Networks; Topology I. INTRODUCTION The present paper supplements in sections VI and VII the investigation proposed in [1] with a description, in concrete examples, of the process of modelling communication networks with topological spaces. All communication relies on the assumption that a communication link is present for exchange of information between two or more agents. The problem is that this assumption may be broken in practice, though not usually for a longer duration causing disturbances. The assumption can be interpreted as a problem of determining path connected components of a topological space. In this paper we investigate how to describe and analyze failures in communication networks, and a concrete method for doing this will be proposed. To help convey the ideas introduced here to other applications (than communication failures), a notion of communication barrier will be formulated. The theory of connected spaces is part of the mathematical field of topology. We shall make use of notions from algebraic topology to describe communication networks in order to take advantage of algebraic structures which are apt for algorithmic description and hence more appropriate for applications. Specifically, problems will be formulated in the category of so-called abstract simplicial pairs, which are combinatorial by nature. The problem of determining the shape of such spaces is transferred to another category, namely the category of chain complexes. In this category the problem is reduced to solving a problem in linear algebra. This means that the shape of a communication network can be formulated as a rather simple problem in linear algebra by its very construction. The theoretical framework introduced in this paper will be used to construct a ranking of the communication nodes such that the more critical nodes, measured in terms of connectedness, will receive a higher ranking. In order to define the ranking, it is necessary that the shape of the space under consideration is known. Furthermore, all elements being ranked must be comparable and hence the ranking takes place separately in each of the path components of the space. Ranking has many applications. As an example consider assigning roles to agents with different levels of freedom according to their position in the network such that a high ranking in the network corresponds to a lower degree of freedom for the agent. If a set of agents can communicate in a meshed network topology, then each agent is assigned an intrinsic rank, which determines this agent’s importance for the network to be able to communicate as a whole. For a given agent, a higher rank means that it is more likely to make communication between other agents impossible if this agent were to fail. Thus agents with a high rank can be viewed as more critical for the network than agents with a low rank. In this paper we define such a ranking system. Section II surveys related work. In section III, the notions of categories, functors, (abstract) simplicial complexes and chain complexes are introduced. These notions will serve as the fundamental building blocks in the subsequent sections. In section IV, the theory of homology is introduced. In the following sections homology is used to determine the shapes of communication networks modelled by abstract simplicial complexes. To this end, the notion of a barrier is defined in