On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound

A communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound

An overview of the degree/diameter problem for directed, undirected and mixed graphs

A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.Peer Reviewe

Connectivity-Aware Semi-Decentralized Federated Learning over Time-Varying D2D Networks

Semi-decentralized federated learning blends the conventional device to-server (D2S) interaction structure of federated model training with localized device-to-device (D2D) communications. We study this architecture over practical edge networks with multiple D2D clusters modeled as time-varying and directed communication graphs. Our investigation results in an algorithm that controls the fundamental trade-off between (a) the rate of convergence of the model training process towards the global optimizer, and (b) the number of D2S transmissions required for global aggregation. Specifically, in our semi-decentralized methodology, D2D consensus updates are injected into the federated averaging framework based on column-stochastic weight matrices that encapsulate the connectivity within the clusters. To arrive at our algorithm, we show how the expected optimality gap in the current global model depends on the greatest two singular values of the weighted adjacency matrices (and hence on the densities) of the D2D clusters. We then derive tight bounds on these singular values in terms of the node degrees of the D2D clusters, and we use the resulting expressions to design a threshold on the number of clients required to participate in any given global aggregation round so as to ensure a desired convergence rate. Simulations performed on real-world datasets reveal that our connectivity-aware algorithm reduces the total communication cost required to reach a target accuracy significantly compared with baselines depending on the connectivity structure and the learning task.Comment: 10 pages, 5 figures. This paper has been accepted to ACM-MobiHoc 202

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs

Structural properties and labeling of graphs

The complexity in building massive scale parallel processing systems has re- sulted in a growing interest in the study of interconnection networks design. Network design affects the performance, cost, scalability, and availability of parallel computers. Therefore, discovering a good structure of the network is one of the basic issues. From modeling point of view, the structure of networks can be naturally stud- ied in terms of graph theory. Several common desirable features of networks, such as large number of processing elements, good throughput, short data com- munication delay, modularity, good fault tolerance and diameter vulnerability correspond to properties of the underlying graphs of networks, including large number of vertices, small diameter, high connectivity and overall balance (or regularity) of the graph or digraph. The first part of this thesis deals with the issue of interconnection networks ad- dressing system. From graph theory point of view, this issue is mainly related to a graph labeling. We investigate a special family of graph labeling, namely antimagic labeling of a class of disconnected graphs. We present new results in super (a; d)-edge antimagic total labeling for disjoint union of multiple copies of special families of graphs. The second part of this thesis deals with the issue of regularity of digraphs with the number of vertices close to the upper bound, called the Moore bound, which is unobtainable for most values of out-degree and diameter. Regularity of the underlying graph of a network is often considered to be essential since the flow of messages and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element. This means that the in-degree and out-degree of each processing element must be the same or almost the same. Our new results show that digraphs of order two less than Moore bound are either diregular or almost diregular.Doctor of Philosoph

Graphs and subgraphs with bounded degree

"The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modelled by a graph in which vertices represent 'nodes' (stations or processors) while undirected or directed edges stand for 'links' or other types of connections, physical or virtual. A cycle that contains every vertex of a graph is called a hamiltonian cycle and a graph which contains a hamiltonian cycle is called a hamiltonian graph. The problem of the existence of a hamiltonian cycle is closely related to the well known problem of a travelling salesman. These problems are NP-complete and NP-hard, respectively. While some necessary and sufficient conditions are known, to date, no practical characterization of hamiltonian graphs has been found. There are several ways to generalize the notion of a hamiltonian cycle. In this thesis we make original contributions in two of them, namely k-walks and r-trestles." --Abstract.Doctor of Philosoph

Edge manipulation techniques for complex networks with applications to communicability and triadic closure.

Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks. Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges. Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process. Extensive numerical tests are presented to validate our approaches

Improved Cardinality Bounds for Rectangle Packing Representations

Axis-aligned rectangle packings can be characterized by the set of spatial relations that hold for pairs of rectangles (west, south, east, north). A representation of a packing consists of one satisfied spatial relation for each pair. We call a set of representations complete for n ∈ ℕ if it contains a representation of every packing of any n rectangles. Both in theory and practice, fastest known algorithms for a large class of rectangle packing problems enumerate a complete set R of representations. The running time of these algorithms is dominated by the (exponential) size of R. In this thesis, we improve the best known lower and upper bounds on the minimum cardinality of complete sets of representations. The new upper bound implies theoretically faster algorithms for many rectangle packing problems, for example in chip design, while the new lower bound imposes a limit on the running time that can be achieved by any algorithm following this approach. The proofs of both results are based on pattern-avoiding permutations. Finally, we empirically compute the minimum cardinality of complete sets of representations for small n. Our computations directly suggest two conjectures, connecting well-known Baxter permutations with the set of permutations avoiding an apparently new pattern, which in turn seem to generate complete sets of representations of minimum cardinality

Edge manipulation techniques for complex networks with applications to communicability and triadic closure.

Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks. Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges. Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process. Extensive numerical tests are presented to validate our approaches