Minimum Number Of Links Research Articles

Inbound traffic capture link-design problem independent of assumptions on users’ route choices

In some traffic management situations, a cordon (a set of points at which traffic flows into a given area) is set in a road network to establish a reference for the location of equipment to implement traffic measurements and controls (e.g., traffic volume surveys and congestion charging). However, few studies have focused on the optimum location of a cordon. We devise a problem denoted the inbound traffic capture link-design problem to select the optimum combination of links for inclusion in a cordon. We regard this combination as the minimum number of links that can capture traffic on all routes, under the condition that there is a path between nodes inside the cordon that is not captured. We formulate this model by employing the graph theory concept of the minimum cut, and use the concept of a Steiner tree with auxiliary network flows to express the constraint of ensuring that there is an uncaptured path inside a cordon. After a basic formulation, to obtain an identical cordon, we devise two subsidiary schemes. In addition, we perform a linear relaxation of our method to reduce its computational cost. The results of computational experiments confirm that our model selects the optimal cordon location formed by a combination of capturing links and also outputs an identical cordon as a boundary line of an area. As the model is computationally feasible, even when applied on a large network, we believe it will have a wide range of practical applications.

Embedded Edge-Connectivity Reliability Evaluation of Augmented Hypercube Interconnection Networks

In the contemporary world, to meet the increasing need to deal with massive data, the interconnection networks for large-scale parallel and distributed systems need to expand for stronger scalability requirements. Let [Formula: see text] be a recursive [Formula: see text]-dimensional network. Since the presence of faulty links or processors may disconnect the entire network, one hopes that every remaining processor lies in an undamaged lower-dimensional subnetwork. Under this circumstance, in 2012, Yang and Wang first proposed the conception of [Formula: see text]-embedded edge-connectivity [Formula: see text] of [Formula: see text], which is defined as the minimum number of links whose removal results in several disconnected components, and each processor is contained in an [Formula: see text]-dimensional subnetwork [Formula: see text]. The augmented cube, denoted by [Formula: see text], proposed by Choudum and Sunitha, is a momentous variant of the hypercube as an interconnection topology of parallel computing. It retains many favorable properties of the hypercube and possesses several embedded properties that the hypercube and other variations do not have. For [Formula: see text] and [Formula: see text], this paper determines [Formula: see text]-embedded edge-connectivity of [Formula: see text]-dimensional augmented cube, [Formula: see text], and shows exact values [Formula: see text], where [Formula: see text] if [Formula: see text], and [Formula: see text] otherwise. The parameters can provide more accurate measurements for the reliability and fault-tolerance of the corresponding systems.

Calibration of Microscopic Traffic Flow Simulation Models considering Subsets of Links and Parameters

This study proposes a methodology for the calibration of microscopic traffic flow simulation models by enabling simultaneous selection of traffic links and associated parameters. The analyst selects any number and combination of links and model parameters for calibration. Most calibration methods consider the entire network and use ad hoc approaches without enabling a specific selection of location and associated parameters. In practice, only a subset of links and parameters is used for calibration based on several factors such as expert knowledge of the system or constraints imposed by local governance. In this study, the calibration problem for the simultaneous selection of links and parameters was formulated using a mathematical programming approach. The proposed methodology is capable of calibrating model parameters considering multiple time periods and performance measures simultaneously. Traffic volume and speed are the performance measures used in this study, and the methodology is developed without considering the characteristics of a specific traffic flow model. A genetic algorithm was implemented to find a solution to the proposed mathematical program. In the experiments, two traffic models were calibrated: the first set of experiments included selection of links only, while all associated parameters were considered for calibration. The second set of experiments considered simultaneous selection of links and parameters. The implications of these experiments indicate that the models were calibrated successfully subject to selection of a minimum number of links. As expected, the more links and parameters that are used for calibration, the more time it takes to find a solution, but the overall results are better. All parameter values were reasonable and within constraints after successful calibration.

On approximations to minimum link visibility paths in simple polygons

We investigate a practical variant of the well-known polygonal visibility path (watchman) problem. For a polygon P, a minimum link visibility path is a polygonal visibility path in P that has the minimum number of links. The problem of finding a minimum link visibility path is NP-hard for simple polygons. If the link-length (number of links) of a minimum link visibility path (tour) is Opt for a simple polygon P with n vertices, we provide an algorithm with runtime that produces polygonal visibility paths (or tours) of link-length at most (or ), where k is a parameter dependent on P, is an output sensitive parameter and γ is the approximation factor of an time approximation algorithm for the geometric travelling salesman problem (path or tour version).

Deterministic and stochastic control of kirigami topology

Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami.

A hybrid link protection scheme for ensuring network service availability in link-state routing networks

The internet is playing an increasingly crucial role in both personal and business activities. In addition, with the emergence of real-time, delay sensitive and mission-critical applications, stringent network availability requirement is put forward for internet service providers (ISPs). However, commonly deployed intradomain link-state routing protocols react to link failures by globally exchanging link state advertisements and recalculating routing table, inevitably causing significant forwarding discontinuity after a failure. Therefore, the loop-free criterion (LFC) approach has been widely deployed by many ISPs for coping with the single network component failure scenario in large internet backbones. The success of LFC lies in its inherent simplicity, but this comes at the expense of letting certain failure scenarios go unprotected. To achieve full failure coverage with LFC without incurring significant extra overhead, we propose a novel link protection scheme, hybrid link protection (HLP), to achieve failure resilient routing. Compared to previous schemes, HLP ensures high network availability in a more efficient way. HLP is implemented in two stages. Stage one provides an efficient LFC based method (MNP-e). The complexity of the algorithm is less than that of Dijkstra's algorithm and can provide the similar network availability with LFC. Stage two provides backup path protection (BPP) based on MNP-e, where only a minimum number of links need to be protected, using special paths and packet headers, to meet the network availability requirement. We evaluate these algorithms in a wide spread of relevant topologies, both real and synthetic, and the results reveal that HLP can achieve high network availability without introducing conspicuous overhead. HLP not only needs around 10% time of that of full protection, but also provides full protection capabilities that full protection provide.

Optimization of Communication Network Topology in Distributed Control Systems Subject to Prescribed Decay Rate.

In this paper, we propose a simple cohesive framework to find an optimal directed control network topology with minimum number of links while a prescribed decay rate is satisfied in the transient response of a distributed control system. In order to guarantee the system's decay rate to be faster than a prespecified value, a constraint on the dominant eigenvalue of the system is required to be considered. This results in a nonconvex optimization problem as eigenvalue of a parametric nonsymmetric matrix is a nonconvex, nonsmooth, and even non-Lipschitz function. Here, we present a convex equivalent optimization problem whose minimizer also solves this eigenvalue optimization problem. This optimization problem proposes a state-feedback matrix which results in a decay rate faster than a given value while input signal costs are considered. The equivalent optimization problem in combination with sparsity-promoting optimal control constitutes a combinatorial optimization problem. Using alternating direction method of multipliers, the problem is decomposed into a chain of analytically solvable subproblems which are differentiable and separable. The proposed optimization framework includes relative preference between the topology of the control network and the decay rate of the system. The simulation results show the effectiveness of the proposed framework.

Rectilinear path problems in restricted memory setup

We study the rectilinear path problem in the presence of disjoint axis parallel rectangular obstacles in the read-only and in-place setup. The input to the problem is a set R of n axis-parallel rectangular obstacles in R2. The objective is to answer the following query efficiently. Path-Query(p,q):Given a pair of points p and q, report an axis-parallel path from p to q avoiding the obstacles in R. In the read-only setup, we show that Path-Query(p,q) problem can be solved in O(n2s+nlogs) time using O(s) extra space. We also show that the existence of an x-monotone path and reporting it, if it exists, can be done with the same asymptotic time complexity. If the objective is to test the existence of an xy-monotone path between the given pair of points p and q avoiding the obstacles, and report it if exists, then our proposed algorithm needs O(n2s+nlogs+Mslogn) time with O(s) extra space, where Ms is the time complexity for computing the median of n elements in the read-only setup using O(s) extra space. Finally, we show that when the obstacles are unit squares instead of rectangles of arbitrary size, then there always exists a path of O(n) links between a pair of query points, and the path can be reported in O(nn) time using O(1) extra work-space. It is also shown that there is an instance where the minimum number of links in a path between a pair of specified points is O(n).The objective of the Path-Query(p,q) in the in-place setup is to preprocess the input rectangles in a data structure in the input array itself such that for any pair of query points p and q, a rectilinear path can be reported efficiently. Here we propose an algorithm with O(nlogn) preprocessing time and O(n3/4+χ) query time, where χ is the number of links (bends) in the path. Both the preprocessing and query answering need O(1) extra space.

Resource optimization scheme for multimedia-enabled wireless mesh networks.

Wireless mesh networking is a promising technology that can support numerous multimedia applications. Multimedia applications have stringent quality of service (QoS) requirements, i.e., bandwidth, delay, jitter, and packet loss ratio. Enabling such QoS-demanding applications over wireless mesh networks (WMNs) require QoS provisioning routing protocols that lead to the network resource underutilization problem. Moreover, random topology deployment leads to have some unused network resources. Therefore, resource optimization is one of the most critical design issues in multi-hop, multi-radio WMNs enabled with multimedia applications. Resource optimization has been studied extensively in the literature for wireless Ad Hoc and sensor networks, but existing studies have not considered resource underutilization issues caused by QoS provisioning routing and random topology deployment. Finding a QoS-provisioned path in wireless mesh networks is an NP complete problem. In this paper, we propose a novel Integer Linear Programming (ILP) optimization model to reconstruct the optimal connected mesh backbone topology with a minimum number of links and relay nodes which satisfies the given end-to-end QoS demands for multimedia traffic and identification of extra resources, while maintaining redundancy. We further propose a polynomial time heuristic algorithm called Link and Node Removal Considering Residual Capacity and Traffic Demands (LNR-RCTD). Simulation studies prove that our heuristic algorithm provides near-optimal results and saves about 20% of resources from being wasted by QoS provisioning routing and random topology deployment.

Communication-optimal eventually perfect failure detection in partially synchronous systems

Since Chandra and Toueg introduced the failure detector abstraction for crash-prone systems, several algorithms implementing failure detectors in partially synchronous systems have been proposed. Their performance can be measured by their Communication efficiency, defined as the number of links used forever. In this regard, in a communication-efficient algorithm only n links are used forever, n being the number of processes in the system. In this paper, we present communication optimality, a communication efficiency degree reached when only c links are used forever, c being the number of correct processes. We show that c is the minimum number of links used forever required to implement ◇P and that c is also optimal for ◇S and Ω when c<n. Finally, we propose two communication-optimal ◇P algorithms following respectively one-to-all and one-to-one communication patterns to manage suspicions, showing that there is a trade-off between detection latency and sporadic communication overhead.

Network Formation: Neighborhood Structures, Establishment Costs, and Distributed Learning

We consider the problem of network formation in a distributed fashion. Network formation is modeled as a strategic-form game, where agents represent nodes that form and sever unidirectional links with other nodes and derive utilities from these links. Furthermore, agents can form links only with a limited set of neighbors. Agents trade off the benefit from links, which is determined by a distance-dependent reward function, and the cost of maintaining links. When each agent acts independently, trying to maximize its own utility function, we can characterize “stable” networks through the notion of Nash equilibrium. In fact, the introduced reward and cost functions lead to Nash equilibria (networks), which exhibit several desirable properties such as connectivity, bounded-hop diameter, and efficiency (i.e., minimum number of links). Since Nash networks may not necessarily be efficient, we also explore the possibility of “shaping” the set of Nash networks through the introduction of state-based utility functions. Such utility functions may represent dynamic phenomena such as establishment costs (either positive or negative). Finally, we show how Nash networks can be the outcome of a distributed learning process. In particular, we extend previous learning processes to so-called “state-based” weakly acyclic games, and we show that the proposed network formation games belong to this class of games.

IMDB Network Revisited: Unveiling Fractal and Modular Properties from a Typical Small-World Network

We study a subset of the movie collaboration network, http://www.imdb.com, where only adult movies are included. We show that there are many benefits in using such a network, which can serve as a prototype for studying social interactions. We find that the strength of links, i.e., how many times two actors have collaborated with each other, is an important factor that can significantly influence the network topology. We see that when we link all actors in the same movie with each other, the network becomes small-world, lacking a proper modular structure. On the other hand, by imposing a threshold on the minimum number of links two actors should have to be in our studied subset, the network topology becomes naturally fractal. This occurs due to a large number of meaningless links, namely, links connecting actors that did not actually interact. We focus our analysis on the fractal and modular properties of this resulting network, and show that the renormalization group analysis can characterize the self-similar structure of these networks.

Complex networks embedded in space: Dimension and scaling relations between mass, topological distance, and Euclidean distance

Many real networks are embedded in space, and often the distribution of the link lengths $r$ follows a power law, $p(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\delta}}$. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations of model networks with a narrow degree distribution, embedded in lattices of dimensions ${d}_{e}=1$ and ${d}_{e}=2$. For networks with $\ensuremath{\delta}&lt;{d}_{e}$, $d$ is infinity, while for $\ensuremath{\delta}&gt;2{d}_{e}$, $d$ has the value of the embedding dimension ${d}_{e}$. In the intermediate regime of interest ${d}_{e}\ensuremath{\le}\ensuremath{\delta}&lt;2{d}_{e}$, our numerical results suggest that $d$ decreases continuously from $d=\ensuremath{\infty}$ to ${d}_{e}$, with $d\ensuremath{-}{d}_{e}\ensuremath{\propto}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})/[{\ensuremath{\delta}}^{\ensuremath{'}}({\ensuremath{\delta}}^{\ensuremath{'}}\ensuremath{-}1)]$ and ${\ensuremath{\delta}}^{\ensuremath{'}}=\ensuremath{\delta}/{d}_{e}$. We also analyze how the mass $M$ and the Euclidean distance $r$ increase with the topological distance $\ensuremath{\ell}$ (minimum number of links between two sites in the network). Our results suggest that in the intermediate regime ${d}_{e}\ensuremath{\le}\ensuremath{\delta}&lt;2{d}_{e}$, $M(\ensuremath{\ell})$ and $r(\ensuremath{\ell})$ increase with $\ensuremath{\ell}$ as a stretched exponential, $M(\ensuremath{\ell})\ensuremath{\sim}\mathrm{exp}[Ad{\ensuremath{\ell}}^{{\ensuremath{\delta}}^{\ensuremath{'}}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})}]$ and $r(\ensuremath{\ell})\ensuremath{\sim}\mathrm{exp}[A{\ensuremath{\ell}}^{{\ensuremath{\delta}}^{\ensuremath{'}}(2\ensuremath{-}{\ensuremath{\delta}}^{\ensuremath{'}})}]$, such that $M(\ensuremath{\ell})\ensuremath{\sim}r{(\ensuremath{\ell})}^{d}$. For $\ensuremath{\delta}&lt;{d}_{e}$, $M$ increases exponentially with $\ensuremath{\ell}$ (as known for $\ensuremath{\delta}=0$), while $r$ is constant and independent of $\ensuremath{\ell}$. For $\ensuremath{\delta}\ensuremath{\ge}2{d}_{e}$, we find the expected power-law scaling, $M(\ensuremath{\ell})\ensuremath{\sim}{\ensuremath{\ell}}^{{d}_{\ensuremath{\ell}}}$ and $r(\ensuremath{\ell})\ensuremath{\sim}{\ensuremath{\ell}}^{1/{d}_{\mathrm{min}}}$, with ${d}_{\ensuremath{\ell}}{d}_{\mathrm{min}}=d$. In ${d}_{e}=1$, we find the expected result, ${d}_{\ensuremath{\ell}}={d}_{\mathrm{min}}=1$, while in ${d}_{e}=2$ we find surprisingly that although $d=2$, ${d}_{\ensuremath{\ell}}&gt;2$ and ${d}_{\mathrm{min}}&lt;1$, in contrast to regular lattices.

The existence of panchromatic colourings for uniform hypergraphs

The well-known extremal problem concerning panchromatic colouring of hypergraphs is investigated. An -colouring of the set of vertices of a hypergraph is said to be panchromatic if each link of the hypergraph is incident to vertices of all colours. The quantity defined as the minimum number of links in an -uniform hypergraph which admits no panchromatic -colourings is studied. New lower and upper bounds for are obtained, which improve earlier results for many of the relations between the parameters and . In addition, a sufficient condition for the existence of a panchromatic -colouring of an -uniform hypergraph is obtained.Bibliography: 18 items.

Boolean networks with reliable dynamics

We investigated the properties of Boolean networks that follow a given reliable trajectory in state space. A reliable trajectory is defined as a sequence of states, which is independent of the order in which the nodes are updated. We explored numerically the topology, the update functions, and the state space structure of these networks, which we constructed using a minimum number of links and the simplest update functions. We found that the clustering coefficient is larger than in random networks and that the probability distribution of three-node motifs is similar to that found in gene regulation networks. Among the update functions, only a subset of all possible functions occurs, and they can be classified according to their probability. More homogeneous functions occur more often, leading to a dominance of canalyzing functions. Finally, we studied the entire state space of the networks. We observed that with increasing systems size, fixed points become more dominant, moving the networks close to the frozen phase.

Traversing a Set of Points with a Minimum Number of Turns

Given a finite set of points S in źd, consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let Gnd be an nן×n grid in źd. Kranakis et al. (Ars Comb. 38:177---192, 1994) showed that L(Gn2)=2nź1 and $\frac{4}{3}n^{2}-O(n)\le L(G^{3}_{n})\le \frac{3}{2}n^{2}+O(n)$ and conjectured that, for all dź3, $L(G^{d}_{n})=\frac{d}{d-1}n^{d-1}\pm O(n^{d-2}).$ We prove the conjecture for d=3 by showing the lower bound for L(Gn3). For d=4, we prove that $L(G^{4}_{n})=\frac{4}{3}n^{3}\pm O(n^{5/2}).$ For general d, we give new estimates on L(Gnd) that are very close to the conjectured value. The new lower bound of $(1+\frac{1}{d})n^{d-1}-O(n^{d-2})$ improves previous result by Collins and Moret (Inf. Process. Lett. 68:317---319, 1998), while the new upper bound of $(1+\frac{1}{d-1})n^{d-1}+O(n^{d-3/2})$ differs from the conjectured value only in the lower order terms. For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in źd with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.

The encoding complexity of network coding

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying communication network G. The nodes of the multicast network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h/sup 3/k/sup 2/. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h/sup 3/k/sup 2/. We show that the number of encoding nodes may depend both on h and k by presenting acyclic coding networks that require /spl Omega/(h/sup 2/k) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2B+1)h/sup 3/k/sup 2/, where B is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an /spl Nscr/P-hard problem.

Operations on Rigid Formations of Autonomous Agents

This paper is concerned with the maintenance of rigid formations of mobile au- tonomous agents. A key element in all future multi-agent systems will be the role of sensor and communication networks as an integral part of coordination. Network topologies are critically im- portant for autonomous systems involving mobile underwater, ground and air vehicles and for sensor networks. This paper focuses on developing techniques and strategies for the analysis and design of sensor and network topologies required to achieve a rigid formation for cooperative tasks. Energy ef- ficiency and communication bandwidth are critically important in formations of mobile autonomous agents, and hence strategies that make efficient use of power and energy are beneficial. Therefore, we develop topologies for providing sensing and communications with the minimum number of links, and propose methods requiring the minimum number of changes in the set of links in dynamic mis- sions and maneuvers, including agent departure from a rigid formation, splitting a rigid formation and merging rigid sub-formations. To do this in a systematic manner, it is necessary to develop a framework for modeling agent formations that characterizes the sensing and communication links needed to maintain the formations. The challenge is that a comprehensive theory of such topologies of formations with sensing and communication limitations is in the earliest stage of development. Central to the development of these techniques and strategies will be the use of tools from rigidity theory, and graph theory.

Reliable multicast routing in mobile networks: a neural-network approach

Mobile networks that handle multicast communication services, such as video-on-demand, news-on-demand etc., require a kind of reliable and secure point-to-point, point-to-multipoint specific group communications for sophisticated organisation of multicast communications. A reliable multicast tree is an efficient connectivity between the source node and the multicast group members through the dependable hosts. Because the mobile host changes its access point over time, multicast routes must be updated. This poses several challenges to provide an efficient multicast routing. A neural-network-based multicast routing algorithm is proposed for constructing a reliable multicast tree that connects the participants of a multicast group. The problem is tackled by dividing a mobile network into clusters of nodes based on the adjacency relation between the nodes (mobile support stations), by considering a suitable neighbourhood distance. The centre cluster, whose nodes are almost equidistant from the multicast group members, is computed to construct a shortest multicast tree that passes through the centre cluster and reliable routers among all the group members. A Kohonens self-organising-map neural network has been used for clustering. Hopfield neural networks are used to construct a multicast tree with a minimum number of links which passes through the nodes that belong to centre cluster. In addition, the tree is constructed as and when the member(s) moves. This scheme should construct a reliable multicast tree and minimise recomputation time of the multicast tree as and when the multicast route is updated when the mobile hosts change their access point due to mobility. The computational power of the proposed neural-network-based multicast routing algorithm is demonstrated through simulation. The algorithm is also tested for mobility of the participating mobile hosts. The proposed work facilitates a possible multicast routing algorithm for future high-speed mobile networks.

Compounding of gossip graphs

Gossiping refers to the following task: In a group of individuals connected by a communication network, every node has a piece of information and needs to transmit it to all the nodes in the network. The networks are modeled by graphs, where the vertices represent the nodes, and the edges, the communication links. In this paper, we concentrate on minimum gossip graphs of even order, that is, graphs able to achieve gossiping in minimum time and with a minimum number of links. More precisely, we derive upper bounds for their number of edges from a compounding method, the k-way split method, previously introduced for broadcasting by Farley [Networks 9 (1979), 313–332]. We show that this method can be applied to gossiping in some cases and that this generalizes some compounding methods for gossip graphs given in [5]. We also show that, when applicable, this method gives the best-known upper bounds on the size of minimum gossip graphs in most cases, either improving or matching them. Notably, we present for the first time two families of regular gossip graphs of order n and of degree ⌈log2(n)⌉ − 3 and ⌈log2(n)⌉ − 4, respectively. We also give some lower bounds on the number of edges of gossip graphs which improve the ones given by Fertin [5]. Moreover, we show that the above compounding method also applies for minimum linear gossip graphs (or MLGGs) of even order, which corresponds to a variant of gossiping where the time of information transmission between two nodes depends on the amount of information exchanged. We also prove that this gives the best-known upper bounds for Gβ,τ(n)—the size of an MLGG of order n—in most cases. In particular, we derive from this method the exact value of Gβ,τ(72), which was previously unknown. © 2000 John Wiley & Sons, Inc.