Generating connected acyclic digraphs uniformly at random

We describe a simple algorithm based on a Markov chain process to generate simply connected acyclic directed graphs over a fixed set of vertices. This algorithm is an extension of a previous one, designed to generate acyclic digraphs, non necessarily connected.Comment: 6 page

Genetic Representations for Evolutionary Minimization of Network Coding Resources

We demonstrate how a genetic algorithm solves the problem of minimizing the resources used for network coding, subject to a throughput constraint, in a multicast scenario. A genetic algorithm avoids the computational complexity that makes the problem NP-hard and, for our experiments, greatly improves on sub-optimal solutions of established methods. We compare two different genotype encodings, which tradeoff search space size with fitness landscape, as well as the associated genetic operators. Our finding favors a smaller encoding despite its fewer intermediate solutions and demonstrates the impact of the modularity enforced by genetic operators on the performance of the algorithm.Comment: 10 pages, 3 figures, accepted to the 4th European Workshop on the Application of Nature-Inspired Techniques to Telecommunication Networks and Other Connected Systems (EvoCOMNET 2007

Evolutionary Approaches to Minimizing Network Coding Resources

We wish to minimize the resources used for network coding while achieving the desired throughput in a multicast scenario. We employ evolutionary approaches, based on a genetic algorithm, that avoid the computational complexity that makes the problem NP-hard. Our experiments show great improvements over the sub-optimal solutions of prior methods. Our new algorithms improve over our previously proposed algorithm in three ways. First, whereas the previous algorithm can be applied only to acyclic networks, our new method works also with networks with cycles. Second, we enrich the set of components used in the genetic algorithm, which improves the performance. Third, we develop a novel distributed framework. Combining distributed random network coding with our distributed optimization yields a network coding protocol where the resources used for coding are optimized in the setup phase by running our evolutionary algorithm at each node of the network. We demonstrate the effectiveness of our approach by carrying out simulations on a number of different sets of network topologies.Comment: 9 pages, 6 figures, accepted to the 26th Annual IEEE Conference on Computer Communications (INFOCOM 2007

Is Hyper-extensionality Preservable Under Deletions of Graph Elements?

Any hereditarily finite set S can be represented as a finite pointed graph \u2013dubbed membership graph\u2013 whose nodes denote elements of the transitive closure of {S} and whose edges model the membership relation. Membership graphs must be hyper-extensional, that is pairwise distinct nodes are not bisimilar and (uniquely) represent hereditarily finite sets. We will see that the removal of even a single node or edge from a membership graph can cause \u201ccollapses\u201d of different nodes and, therefore, the loss of hyper-extensionality of the graph itself. With the intent of gaining a deeper understanding on the class of hyper-extensional hereditarily finite sets, this paper investigates whether pointed hyper-extensional graphs always contain either a node or an edge whose removal does not disrupt the hyper-extensionality property

Uniform random generation of large acyclic digraphs

Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory networks, not only the estimation of model parameters but the reconstruction of the structure itself is of great interest. As well as for the assessment of different structure learning algorithms in simulation studies, a uniform sample from the space of directed acyclic graphs is required to evaluate the prevalence of certain structural features. Here we analyse how to sample acyclic digraphs uniformly at random through recursive enumeration, an approach previously thought too computationally involved. Based on complexity considerations, we discuss in particular how the enumeration directly provides an exact method, which avoids the convergence issues of the alternative Markov chain methods and is actually computationally much faster. The limiting behaviour of the distribution of acyclic digraphs then allows us to sample arbitrarily large graphs. Building on the ideas of recursive enumeration based sampling we also introduce a novel hybrid Markov chain with much faster convergence than current alternatives while still being easy to adapt to various restrictions. Finally we discuss how to include such restrictions in the combinatorial enumeration and the new hybrid Markov chain method for efficient uniform sampling of the corresponding graphs.Comment: 15 pages, 2 figures. To appear in Statistics and Computin

An effective genetic algorithm for network coding

The network coding problem (NCP), which aims to minimize network coding resources such as nodes and links, is a relatively new application of genetic algorithms (GAs) and hence little work has so far been reported in this area. Most of the existing literature on NCP has concentrated primarily on the static network coding problem (SNCP). There is a common assumption in work to date that a target rate is always achievable at every sink as long as coding is allowed at all nodes. In most real-world networks, such as wireless networks, any link could be disconnected at any time. This implies that every time a change occurs in the network topology, a new target rate must be determined. The SNCP software implementation then has to be re-run to try to optimize the coding based on the new target rate. In contrast, the GA proposed in this paper is designed with the dynamic network coding problem (DNCP) as the major concern. To this end, a more general formulation of the NCP is described. The new NCP model considers not only the minimization of network coding resources but also the maximization of the rate actually achieved at sinks. This is particularly important to the DNCP, where the target rate may become unachievable due to network topology changes. Based on the new NCP model, an effective GA is designed by integrating selected new problem-specific heuristic rules into the evolutionary process in order to better diversify chromosomes. In dynamic environments, the new GA does not need to recalculate target rate and also exhibits some degree of robustness against network topology changes. Comparative experiments on both SNCP and DNCP illustrate the effectiveness of our new model and algorithm

Partition MCMC for inference on acyclic digraphs

Acyclic digraphs are the underlying representation of Bayesian networks, a widely used class of probabilistic graphical models. Learning the underlying graph from data is a way of gaining insights about the structural properties of a domain. Structure learning forms one of the inference challenges of statistical graphical models. MCMC methods, notably structure MCMC, to sample graphs from the posterior distribution given the data are probably the only viable option for Bayesian model averaging. Score modularity and restrictions on the number of parents of each node allow the graphs to be grouped into larger collections, which can be scored as a whole to improve the chain's convergence. Current examples of algorithms taking advantage of grouping are the biased order MCMC, which acts on the alternative space of permuted triangular matrices, and non ergodic edge reversal moves. Here we propose a novel algorithm, which employs the underlying combinatorial structure of DAGs to define a new grouping. As a result convergence is improved compared to structure MCMC, while still retaining the property of producing an unbiased sample. Finally the method can be combined with edge reversal moves to improve the sampler further.Comment: Revised version. 34 pages, 16 figures. R code available at https://github.com/annlia/partitionMCM