Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph $G$, a vertex pair $(s,t) \in V(G)\times V(G)$, and a set of edge faults $F \subseteq E(G)$, a replacement path $P(s,t,F)$ is an $s$-$t$ shortest path in $G \setminus F$. For integer parameters $L,f$, a replacement path covering (RPC) is a collection of subgraphs of $G$, denoted by $\mathcal{G}_{L,f}=\{G_1,\ldots, G_r \}$, such that for every set $F$ of at most $f$ faults (i.e., $|F|\le f$) and every replacement path $P(s,t,F)$ of at most $L$ edges, there exists a subgraph $G_i\in \mathcal{G}_{L,f}$ that contains all the edges of $P$ and does not contain any of the edges of $F$. The covering value of the RPC $\mathcal{G}_{L,f}$ is then defined to be the number of subgraphs in $\mathcal{G}_{L,f}$. We present efficient deterministic constructions of $(L,f)$-RPCs whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). We also provide an almost matching lower bound for the value of these coverings. A key application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of the our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon, Chechik and Cohen (ICALP 2019)

Vertex Fault Tolerant Additive Spanners

A {\em fault-tolerant} structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. In this paper, we address the problem of designing a {\em fault-tolerant} additive spanner, namely, a subgraph $H$ of the network $G$ such that subsequent to the failure of a single vertex, the surviving part of $H$ still contains an \emph{additive} spanner for (the surviving part of) $G$, satisfying $dist(s,t,H\setminus \{v\}) \leq dist(s,t,G\setminus \{v\})+\beta$ for every $s,t,v \in V$. Recently, the problem of constructing fault-tolerant additive spanners resilient to the failure of up to $f$ \emph{edges} has been considered by Braunschvig et. al. The problem of handling \emph{vertex} failures was left open therein. In this paper we develop new techniques for constructing additive FT-spanners overcoming the failure of a single vertex in the graph. Our first result is an FT-spanner with additive stretch $2$ and $\widetilde{O}(n^{5/3})$ edges. Our second result is an FT-spanner with additive stretch $6$ and $\widetilde{O}(n^{3/2})$ edges. The construction algorithm consists of two main components: (a) constructing an FT-clustering graph and (b) applying a modified path-buying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for {\em fault-tolerant multi-source additive spanners}, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in $S \times V$ for a given subset of sources $S\subseteq V$. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges)

Fractional quantum revivals in the atomic gravitational cavity

In this paper we discuss the quantum dynamics and fractional quantum revivals of an integrable nonlinear system, consisting of an atom bouncing vertically from an evanescent field, for two cases with the simplified infinite-potential and the more practical exponential potential, respectively. We study the two cases separately, then contrast and compare the results and reach the conclusion that provided the starting position of the atoms is not too close to the reflecting surface supporting the evanescent wave (this condition is always satisfied in present experiments in this field), the two cases will produce the same results. This means that the idealized infinite potential is a good approximation to the more realistic exponential potential. Because the quantum analysis of the infinite-potential case is quite simple and straighforward (since its Schrödinger equation has analytical solutions), this will greatly simplify the quantum analysis of the more complicated exponential potential case and hence has practical significance

Structural correlations in bacterial metabolic networks

<p>Abstract</p> <p>Background</p> <p>Evolution of metabolism occurs through the acquisition and loss of genes whose products acts as enzymes in metabolic reactions, and from a presumably simple primordial metabolism the organisms living today have evolved complex and highly variable metabolisms. We have studied this phenomenon by comparing the metabolic networks of 134 bacterial species with known phylogenetic relationships, and by studying a neutral model of metabolic network evolution.</p> <p>Results</p> <p>We consider the 'union-network' of 134 bacterial metabolisms, and also the union of two smaller subsets of closely related species. Each reaction-node is tagged with the number of organisms it belongs to, which we denote organism degree (OD), a key concept in our study. Network analysis shows that common reactions are found at the centre of the network and that the average OD decreases as we move to the periphery. Nodes of the same OD are also more likely to be connected to each other compared to a random OD relabelling based on their occurrence in the real data. This trend persists up to a distance of around five reactions. A simple growth model of metabolic networks is used to investigate the biochemical constraints put on metabolic-network evolution. Despite this seemingly drastic simplification, a 'union-network' of a collection of unrelated model networks, free of any selective pressure, still exhibit similar structural features as their bacterial counterpart.</p> <p>Conclusions</p> <p>The OD distribution quantifies topological properties of the evolutionary history of bacterial metabolic networks, and lends additional support to the importance of horizontal gene transfer during bacterial metabolic evolution where new reactions are attached at the periphery of the network. The neutral model of metabolic network growth can reproduce the main features of real networks, but we observe that the real networks contain a smaller common core, while they are more similar at the periphery of the network. This suggests that natural selection and biochemical correlations can act both to diversify and to narrow down metabolic evolution.</p

Evolutionary connectionism: algorithmic principles underlying the evolution of biological organisation in evo-devo, evo-eco and evolutionary transitions

The mechanisms of variation, selection and inheritance, on which evolution by natural selection depends, are not fixed over evolutionary time. Current evolutionary biology is increasingly focussed on understanding how the evolution of developmental organisations modifies the distribution of phenotypic variation, the evolution of ecological relationships modifies the selective environment, and the evolution of reproductive relationships modifies the heritability of the evolutionary unit. The major transitions in evolution, in particular, involve radical changes in developmental, ecological and reproductive organisations that instantiate variation, selection and inheritance at a higher level of biological organisation. However, current evolutionary theory is poorly equipped to describe how these organisations change over evolutionary time and especially how that results in adaptive complexes at successive scales of organisation (the key problem is that evolution is self-referential, i.e. the products of evolution change the parameters of the evolutionary process). Here we first reinterpret the central open questions in these domains from a perspective that emphasises the common underlying themes. We then synthesise the findings from a developing body of work that is building a new theoretical approach to these questions by converting well-understood theory and results from models of cognitive learning. Specifically, connectionist models of memory and learning demonstrate how simple incremental mechanisms, adjusting the relationships between individually-simple components, can produce organisations that exhibit complex system-level behaviours and improve the adaptive capabilities of the system. We use the term “evolutionary connectionism” to recognise that, by functionally equivalent processes, natural selection acting on the relationships within and between evolutionary entities can result in organisations that produce complex system-level behaviours in evolutionary systems and modify the adaptive capabilities of natural selection over time. We review the evidence supporting the functional equivalences between the domains of learning and of evolution, and discuss the potential for this to resolve conceptual problems in our understanding of the evolution of developmental, ecological and reproductive organisations and, in particular, the major evolutionary transitions

Facilitated Variation: How Evolution Learns from Past Environments To Generalize to New Environments

One of the striking features of evolution is the appearance of novel structures in organisms. Recently, Kirschner and Gerhart have integrated discoveries in evolution, genetics, and developmental biology to form a theory of facilitated variation (FV). The key observation is that organisms are designed such that random genetic changes are channeled in phenotypic directions that are potentially useful. An open question is how FV spontaneously emerges during evolution. Here, we address this by means of computer simulations of two well-studied model systems, logic circuits and RNA secondary structure. We find that evolution of FV is enhanced in environments that change from time to time in a systematic way: the varying environments are made of the same set of subgoals but in different combinations. We find that organisms that evolve under such varying goals not only remember their history but also generalize to future environments, exhibiting high adaptability to novel goals. Rapid adaptation is seen to goals composed of the same subgoals in novel combinations, and to goals where one of the subgoals was never seen in the history of the organism. The mechanisms for such enhanced generation of novelty (generalization) are analyzed, as is the way that organisms store information in their genomes about their past environments. Elements of facilitated variation theory, such as weak regulatory linkage, modularity, and reduced pleiotropy of mutations, evolve spontaneously under these conditions. Thus, environments that change in a systematic, modular fashion seem to promote facilitated variation and allow evolution to generalize to novel conditions

Origin of structural difference in metabolic networks with respect to temperature

<p>Abstract</p> <p>Background</p> <p>Metabolism is believed to adaptively shape-shift with changing environment. In recent years, a structural difference with respect to temperature, which is an environmental factor, has been revealed in metabolic networks, implying that metabolic networks transit with temperature. Subsequently, elucidatation of the origin of these structural differences due to temperature is important for understanding the evolution of life. However, the origin has yet to be clarified due to the complexity of metabolic networks.</p> <p>Results</p> <p>Consequently, we propose a simple model with a few parameters to explain the transitions. We first present mathematical solutions of this model using mean-field approximation, and demonstrate that this model can reproduce structural properties, such as heterogeneous connectivity and hierarchical modularity, in real metabolic networks both qualitatively and quantitatively. We next show that the model parameters correlate with optimal growth temperature. In addition, we present a relationship between multiple cyclic properties and optimal growth temperature in metabolic networks.</p> <p>Conclusion</p> <p>From the proposed model, we find that such structural properties are determined by the emergence of a short-cut path, which reduces the minimum distance between two nodes on a graph. Furthermore, we investigate correlations between model parameters and growth temperature; as a result, we find that the emergence of the short-cut path tends to be inhibited with increasing temperature. In addition, we also find that the short-cut path bypasses a relatively long path at high temperature when the emergence of the new path is not inhibited. Even further, additional network analysis provides convincing evidence of the reliability of the proposed model and its conclusions on the possible origins of differences in metabolic network structure.</p

Detecting Network Communities: An Application to Phylogenetic Analysis

This paper proposes a new method to identify communities in generally weighted complex networks and apply it to phylogenetic analysis. In this case, weights correspond to the similarity indexes among protein sequences, which can be used for network construction so that the network structure can be analyzed to recover phylogenetically useful information from its properties. The analyses discussed here are mainly based on the modular character of protein similarity networks, explored through the Newman-Girvan algorithm, with the help of the neighborhood matrix . The most relevant networks are found when the network topology changes abruptly revealing distinct modules related to the sets of organisms to which the proteins belong. Sound biological information can be retrieved by the computational routines used in the network approach, without using biological assumptions other than those incorporated by BLAST. Usually, all the main bacterial phyla and, in some cases, also some bacterial classes corresponded totally (100%) or to a great extent (>70%) to the modules. We checked for internal consistency in the obtained results, and we scored close to 84% of matches for community pertinence when comparisons between the results were performed. To illustrate how to use the network-based method, we employed data for enzymes involved in the chitin metabolic pathway that are present in more than 100 organisms from an original data set containing 1,695 organisms, downloaded from GenBank on May 19, 2007. A preliminary comparison between the outcomes of the network-based method and the results of methods based on Bayesian, distance, likelihood, and parsimony criteria suggests that the former is as reliable as these commonly used methods. We conclude that the network-based method can be used as a powerful tool for retrieving modularity information from weighted networks, which is useful for phylogenetic analysis