Shortest paths in nearly conservative digraphs

We introduce the following notion: a digraph D = (V, A) with arc weights c: A → R is called nearly conservative if every negative cycle consists of two arcs. Computing shortest paths in nearly conservative digraphs is NP-hard, and even deciding whether a digraph is nearly conservative is coNP-complete. We show that the “All Pairs Shortest Path” problem is fixed parameter tractable with various parameters for nearly conservative digraphs. The results also apply for the special case of conservative mixed graphs

Functional Integration of Ecological Networks through Pathway Proliferation

Large-scale structural patterns commonly occur in network models of complex systems including a skewed node degree distribution and small-world topology. These patterns suggest common organizational constraints and similar functional consequences. Here, we investigate a structural pattern termed pathway proliferation. Previous research enumerating pathways that link species determined that as pathway length increases, the number of pathways tends to increase without bound. We hypothesize that this pathway proliferation influences the flow of energy, matter, and information in ecosystems. In this paper, we clarify the pathway proliferation concept, introduce a measure of the node--node proliferation rate, describe factors influencing the rate, and characterize it in 17 large empirical food-webs. During this investigation, we uncovered a modular organization within these systems. Over half of the food-webs were composed of one or more subgroups that were strongly connected internally, but weakly connected to the rest of the system. Further, these modules had distinct proliferation rates. We conclude that pathway proliferation in ecological networks reveals subgroups of species that will be functionally integrated through cyclic indirect effects.Comment: 29 pages, 2 figures, 3 tables, Submitted to Journal of Theoretical Biolog

Are there any good digraph width measures?

Many width measures for directed graphs have been proposed in the last few years in pursuit of generalizing (the notion of) treewidth to directed graphs. However, none of these measures possesses, at the same time, the major properties of treewidth, namely, 1. being algorithmically useful , that is, admitting polynomial-time algorithms for a large class of problems on digraphs of bounded width (e.g. the problems definable in MSO1MSO1); 2. having nice structural properties such as being (at least nearly) monotone under taking subdigraphs and some form of arc contractions (property closely related to characterizability by particular cops-and-robber games). We investigate the question whether the search for directed treewidth counterparts has been unsuccessful by accident, or whether it has been doomed to fail from the beginning. Our main result states that any reasonable width measure for directed graphs which satisfies the two properties above must necessarily be similar to treewidth of the underlying undirected graph

Clearing Contamination in Large Networks

In this work, we study the problem of clearing contamination spreading through a large network where we model the problem as a graph searching game. The problem can be summarized as constructing a search strategy that will leave the graph clear of any contamination at the end of the searching process in as few steps as possible. We show that this problem is NP-hard even on directed acyclic graphs and provide an efficient approximation algorithm. We experimentally observe the performance of our approximation algorithm in relation to the lower bound on several large online networks including Slashdot, Epinions and Twitter. The experiments reveal that in most cases our algorithm performs near optimally

Boolean delay equations on networks: An application to economic damage propagation

We introduce economic models based on Boolean Delay Equations: this formalism makes easier to take into account the complexity of the interactions between firms and is particularly appropriate for studying the propagation of an initial damage due to a catastrophe. Here we concentrate on simple cases, which allow to understand the effects of multiple concurrent production paths as well as the presence of stochasticity in the path time lengths or in the network structure. In absence of flexibility, the shortening of production of a single firm in an isolated network with multiple connections usually ends up by attaining a finite fraction of the firms or the whole economy, whereas the interactions with the outside allow a partial recovering of the activity, giving rise to periodic solutions with waves of damage which propagate across the structure. The damage propagation speed is strongly dependent upon the topology. The existence of multiple concurrent production paths does not necessarily imply a slowing down of the propagation, which can be as fast as the shortest path.Comment: Latex, 52 pages with 22 eps figure

A Brightwell-Winkler type characterisation of NU graphs

In 2000, Brightwell and Winkler characterised dismantlable graphs as the graphs $H$ for which the Hom-graph ${\rm Hom}(G,H)$, defined on the set of homomorphisms from $G$ to $H$, is connected for all graphs $G$. This shows that the reconfiguration version ${\rm Recon_{Hom}}(H)$ of the $H$-colouring problem, in which one must decide for a given $G$ whether ${\rm Hom}(G,H)$ is connected, is trivial if and only if $H$ is dismantlable. We prove a similar starting point for the reconfiguration version of the $H$-extension problem. Where ${\rm Hom}(G,H;p)$ is the subgraph of the Hom-graph ${\rm Hom}(G,H)$ induced by the $H$-colourings extending the $H$-precolouring $p$ of $G$, the reconfiguration version ${\rm Recon_{Ext}(H)}$ of the $H$-extension problem asks, for a given $H$-precolouring $p$ of a graph $G$, if ${\rm Hom}(G,H;p)$ is connected. We show that the graphs $H$ for which ${\rm Hom}(G,H;p)$ is connected for every choice of $(G,p)$ are exactly the ${\rm NU}$ graphs. This gives a new characterisation of ${\rm NU}$ graphs, a nice class of graphs that is important in the algebraic approach to the ${\rm CSP}$-dichotomy. We further give bounds on the diameter of ${\rm Hom}(G,H;p)$ for ${\rm NU}$ graphs $H$, and show that shortest path between two vertices of ${\rm Hom}(G,H;p)$ can be found in parameterised polynomial time. We apply our results to the problem of shortest path reconfiguration, significantly extending recent results.Comment: 17 pages, 1 figur