Euclidean Networks Research Articles

Manifold-aware synthesis of high-resolution diffusion from structural imaging.

The physical and clinical constraints surrounding diffusion-weighted imaging (DWI) often limit the spatial resolution of the produced images to voxels up to eight times larger than those of T1w images. The detailed information contained in accessible high-resolution T1w images could help in the synthesis of diffusion images with a greater level of detail. However, the non-Euclidean nature of diffusion imaging hinders current deep generative models from synthesizing physically plausible images. In this work, we propose the first Riemannian network architecture for the direct generation of diffusion tensors (DT) and diffusion orientation distribution functions (dODFs) from high-resolution T1w images. Our integration of the log-Euclidean Metric into a learning objective guarantees, unlike standard Euclidean networks, the mathematically-valid synthesis of diffusion. Furthermore, our approach improves the fractional anisotropy mean squared error (FA MSE) between the synthesized diffusion and the ground-truth by more than 23% and the cosine similarity between principal directions by almost 5% when compared to our baselines. We validate our generated diffusion by comparing the resulting tractograms to our expected real data. We observe similar fiber bundles with streamlines having <3% difference in length, <1% difference in volume, and a visually close shape. While our method is able to generate diffusion images from structural inputs in a high-resolution space within 15 s, we acknowledge and discuss the limits of diffusion inference solely relying on T1w images. Our results nonetheless suggest a relationship between the high-level geometry of the brain and its overall white matter architecture that remains to be explored.

Efficient Processing of All Nearest Neighbor Queries in Dynamic Road Networks

The increasing trend of GPS-enabled smartphones has led to the tremendous usage of Location-Based Service applications. In the past few years, a significant amount of studies have been conducted to process All nearest neighbor (ANN) queries. An ANN query on a road network extracts and returns all the closest data objects for all query objects. Most of the existing studies on ANN queries are performed either in Euclidean space or static road networks. Moreover, combining the nearest neighbor query and join operation is an expensive procedure because it requires computing the distance between each pair of query objects and data objects. This study considers the problem of processing the ANN queries on a dynamic road network where the weight, i.e., the traveling distance and time varies due to various traffic conditions. To address this problem, a shared execution-based approach called standard clustered loop (SCL) is proposed that allows efficient processing of ANN queries on a dynamic road network. The key concept behind the shared execution technique is to exploit the coherence property of road networks by clustering objects that share common paths and processing the cluster as a single path. In an empirical study, the SCL method achieves significantly better performance than competitive methods and efficiently reduces the computational cost to process ANN queries in various problem settings.

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus Nℓ of a plane Euclidean network N by inserting iteratively a finite set of segments, called shortcut set, while reducing the diameter of the locus of the resulting network. There are two main differences with the classical augmentation problems: the endpoints of the segments are allowed to be points of Nℓ as well as points of the previously inserted segments (instead of only vertices of N), and the notion of diameter is adapted to the fact that we deal with Nℓ instead of N. This increases enormously the hardness of the problem but also its possible practical applications to network design. Among other results, we characterize the existence of shortcut sets, compute them in polynomial time, and analyze the role of the convex hull of Nℓ when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of a shortcut set is NP-hard, one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter.

Dialectical Logic K-Model: A Mathematical Model for Machine

An axiom system for dialectical logic K-model which based on Kirchhoff energy-method is established by author in the paper. The author describes that subjective-laws is the mirror imagine reflected from objective-laws and defines that the three-step which named by sensation, abstraction and thinking in artificial intelligence. At same time, describes that axiom system for dialectical logic K-model, in which contains such as logic-variable energy conservation law, Mozi’s principle( mini-max principle) and forbidden law, etc. In the axiom system also contain such as a continuous true-value-function system valued on interval and the K-graph for logic-variable. And describes the giving value method by matrix based on K-graph satisfied Kirchhoff laws to the logic variable. The author describes simply the linear and nonlinear logic variable system. And describes simply the logic variable involved three-dimension Euclidean space and topology networks space separately. Dialectical logic K-model would supply an computation algorithm idea for machine so that the machine is able to think by dialectical logic method, thus an important information-treated method maybe the dialectical logic.

Shortcut sets for plane Euclidean networks (Extended abstract)

We study the problem of augmenting the locus Nℓ of a plane Euclidean network N by inserting iteratively a finite set of segments, called shortcut set, while reducing the diameter of the locus of the resulting network. We first characterize the existence of shortcut sets, and compute shortcut sets in polynomial time providing an upper bound on their size. Then, we analyze the role of the convex hull of Nℓ when inserting a shortcut set. As a main result, we prove that one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter.

On the effect of the deployment setting on broadcasting in Euclidean radio networks

The paper studies broadcasting in radio networks whose stations are represented by points in the Euclidean plane (each station knows its own coordinates). In any given time step, a station can either receive or transmit. A message transmitted from station v is delivered to every station u at distance at most 1 from v, but u successfully hears the message if and only if v is the only station at distance at most 1 from u that transmitted in this time step. A designated source station has a message that should be disseminated throughout the network. All stations other than the source are initially idle and wake up upon the first time they hear the source message. It is shown in [17] that the time complexity of deterministic broadcasting algorithms depends on two parameters of the network, namely, its diameter (in hops) D and a lower bound d on the Euclidean distance between any two stations. The inverse of d is called the granularity of the network, denoted by g. Specifically, the authors of [17] present a deterministic broadcasting algorithm that works in time O (Dg) and prove that every broadcasting algorithm requires $$\varOmega \left( D \sqrt{g} \right) $$ΩDg time. In this paper, we distinguish between the arbitrary deployment setting, originally studied in [17], in which stations can be placed everywhere in the plane, and the new grid deployment setting, in which stations are only allowed to be placed on a d-spaced grid. Does the latter (more restricted) setting provide any speedup in broadcasting time complexity? Although the O (Dg) broadcasting algorithm of [17] works under the (original) arbitrary deployment setting, it turns out that the $$\varOmega \left( D \sqrt{g} \right) $$ΩDg lower bound remains valid under the grid deployment setting. Still, the above question is left unanswered. The current paper answers this question affirmatively by presenting a provable separation between the two deployment settings. We establish a tight lower bound on the time complexity of deterministic broadcasting algorithms under the arbitrary deployment setting proving that broadcasting cannot be completed in less than $$\varOmega (D g)$$Ω(Dg) time. For the grid deployment setting, we develop a deterministic broadcasting algorithm that runs in time $$O \left( D g^{5 / 6} \log g \right) $$ODg5/6logg, thus breaking the linear dependency on g.

Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars

A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the “last mile problem.” Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

Ab-initio reconstruction of complex Euclidean networks in two dimensions.

Reconstruction of complex structures is an inverse problem arising in virtually all areas of science and technology, from protein structure determination to bulk heterostructure solar cells and the structure of nanoparticles. We cast this problem as a complex network problem where the edges in a network have weights equal to the Euclidean distance between their endpoints. We present a method for reconstruction of the locations of the nodes of the network given only the edge weights of the Euclidean network. The theoretical foundations of the method are based on rigidity theory, which enables derivation of a polynomial bound on its efficiency. An efficient implementation of the method is discussed and timing results indicate that the run time of the algorithm is polynomial in the number of nodes in the network. We have reconstructed Euclidean networks of about 1000 nodes in approximately 24 h on a desktop computer using this implementation. We also reconstruct Euclidean networks corresponding to polymer chains in two dimensions and planar graphene nanoparticles. We have also modified our base algorithm so that it can successfully solve random point sets when the input data are less precise.

Heuristics for the Stochastic Eulerian Tour Problem

The Stochastic Eulerian Tour Problem (SETP) seeks the Eulerian tour of minimum expected length on an undirected Eulerian graph, when demand on the arcs that have to be serviced is probabilistic. The SETP is NP-hard and in this paper, we develop three constructive heuristics for this problem. The first two are greedy tour construction heuristics while the third is a sub-tour concatenation heuristic. Our experimental results show that for grid networks, the sub-tour concatenation heuristic performs well when the probability of service of each edge is greater than 0.1. For Euclidean networks, as the number of edges increases, the second heuristic performs the best among the three. Also, the expected length of our overall best solution is lower than the expected length of a random tour by up to 10% on average for grid networks and up to 2% for Euclidean networks.

Enumerating periodic patterns of synchrony via finite bidirectional networks

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n -dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n -dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

Disorder and decision cost in spatial networks

We introduce the concept of decision cost of a spatial graph, which measures the disorder of a given network taking into account not only the connections between nodes but their position in a two-dimensional map. The influence of the network size is evaluated and we show that normalization of the decision cost allows us to compare the degree of disorder of networks of different sizes. Under this framework, we measure the disorder of the connections between airports of two different countries and obtain some conclusions about which of them is more disordered. The introduced concepts (decision cost and disorder of spatial networks) can easily be extended to Euclidean networks of higher dimensions, and also to networks whose nodes have a certain fitness property (i.e., one-dimensional).

Ab-initio construction of some crystalline 3D Euclidean networks

We describe a technique for construction of 3D Euclidean (E 3) networks with partially-prescribed rings. The algorithm starts with 2D hyperbolic (H 2) tilings, whose symmetries are commensurate with the intrinsic 2D symmetries of triply periodic minimal surfaces (or infinite periodic minimal surfaces, IPMS). The 2D hyperbolic pattern is then projected from H 2 to E 3, forming 3D nets. Examples of cubic and tetragonal 3-connected nets with up to 288 vertices per unit cell, each linking a pair of 6-rings and a single 8-ring, are derived by projection onto the P, D, Gyroid and I-WP IPMS. A single example of a projection from close-packed trees in H 2 to E 3 (via the D surface) is also shown, that leads to a quartet of interwoven equivalent chiral nets. The configuration describes the channel system of a novel quadracontinuous branched minimal surface that is a chiral foam with four identical, open bubbles.

Phase Transitions in Euclidean Networks: A Mini-Review

In networks defined in Euclidean spaces, the probability of connection between two nodes depends on the distance between them. Various kinds of phase transitions occur in these networks when the distance dependent factor is governed by suitable tunable parameters. In this short review, we discuss the present status of the studies of the phase transitions in both static and growing Euclidean networks.

Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings

We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to generate three-dimensional Euclidean (E3) networks. The technique involves projection of edges of tilings of the hyperbolic plane (H2) onto three-periodic minimal surfaces, embedded in E3. Given the extraordinary wealth of symmetries commensurate with H2, we can generate networks in E3 that are difficult to construct otherwise. In particular, we form four-, five- and seven-connected (E3) nets containing three- and five-rings, viz. (3, 7), (5, 4) and (5, 5) tilings in H2.

Assigning codes in wireless networks: bounds and scaling properties

In the Code Division Multiple Access (CDMA) framework, collisions that can occur in wireless networks are eliminated by assigning orthogonal codes to stations, a problem equivalent to that of coloring graphs associated to the physical network. In this paper we present new upper and lower bounds for two versions of the problem (hidden and primary collision avoidance – HP-CA – or hidden collision avoidance only – H-CA). In particular, optimal assignments for special topologies and heuristics for general topologies are proposed. The schemes show better average results with respect to existing alternatives. Furthermore, the gaps between the upper bound given by the heuristic solution, the lower bound obtained from the maximum-clique problem, and the optimal solution obtained by branch and bound are investigated in the different settings. A scaling law is then proposed to explain the relations between the number of codes needed in Euclidean networks with different station densities and connection distances. The substantial difference between the two versions HP-CA and H-CA of the problem is investigated by studying the probabilistic distribution of connections as a function of the distance, and the asymptotic size of the maximum cliques.

Maximum number of independent paths and radio connectivity

Methods for determining the network reliability of a multihop packet radio network in the presence of hostile jammers are discussed. A new connectivity parameter called radio connectivity is defined as the maximum number of disjoint communication paths that are still usable between given nodes s and d after the jammer is on or, more generally, the minimum number of jammers needed to disconnect s and d. A lower bound on the radio connectivity is computed by studying the number of jamming independent paths. The time complexity of obtaining the radio connectivity is analyzed and shown to be NP-hard except for some special cases. Greedy heuristics for developing approximate answers for general networks are described. Euclidean networks, in which the nodes and links correspond to points and line segments in the Euclidean geometry and satisfy Euclid's four fundamental axioms, are also discussed. It is found that the maximum number of independent paths between a pair of source and destination nodes that can possibly exist is five. An extension in which there is a protected zone of known size around the sender and receiver is studied.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computing k‐shortest path lengths in euclidean networks

AbstractIn this paper, we examine the problem of finding k‐shortest paths between an origin and destination pair when distances are Euclidean. Two versions of a generalized Dijkstra algorithm are compared. Computational results show that the advantage of the adaptive version (measured by total number of permanent labels) grows with both k and the network size. For large networks, the adaptive algorithm exhibits a 100:1 advantage in the number of permanent labels set.

Shortest paths with euclidean distances: An explanatory model

AbstractThis paper considers the problem of finding the shortest path between an origin and destination pair in networks whose arc lengths are Euclidean distances. Dijkstra's algorithm and a modified version of Dijkstra's algorithm which is more adaptive to network topology are compared. We demonstrate on the infinite lattice network with diagonal arcs (a prototype of more general sparse Euclidean networks) that on the average the adaptive algorithm expands less than 8.3% the area that would be expanded by the Dijkstra algorithm and in the worst case it expands less than 10.7%. In addition, we present computational results for more general networks.