High-dimensional Network Research Articles

Three-Dimensional Associative Memory Using Exterior Product

Several models of high dimensional neural networks have been proposed. Complex neural networks are two dimensional neural networks. They has been investigated more than other high dimensional neural networks. Three-dimensional neural networks are Nitta’s 3D model, a model using quaternion numbers and an exterior model. Nitta’s 3D model and the model using quaternion numbers have rotational properties. The inherent properties of exterior model, however, have not been clearfied. We propose Hopfield network using exterior product to investigate exterior model. Real and complex numbers holds commutative and connection laws, but exterior product does not. Therefore it can not use similar discussions. In this paper, we show that the learning method similar to hebbian rule is useful.

Higher dimensional hexagonal networks

We define the higher dimensional hexagonal graphs as the generalization of a triangular plane tessellation, and consider it as a multiprocessor interconnection network. Nodes in a k-dimensional ( k-D) hexagonal network are placed at the vertices of a k-D triangular tessellation, so that each node has up to 2 k+2 neighbors. In this paper, we propose a simple addressing scheme for the nodes, which leads to a straightforward formula for computing the distance between nodes and a very simple and elegant routing algorithm. The number of shortest paths between any two nodes and their description are also provided in this paper. We then derive closed formulas for the surface area (volume) of these networks, which are defined as the number of nodes located at a given distance (up to a given distance, respectively) from the origin node. The number of nodes and the network diameter under a more symmetrical border conditions are also derived. We show that a k-D hexagonal network of size t has the same degree, the same or lower diameter, and fewer nodes than a ( k+1) -D mesh of size t. Simple embeddings between two networks are also described. That is, we show how to reduce the dimension of a mesh by removing some nodes, and converting it into a hexagonal network, while preserving the simplicity of basic data communication schemes such as routing and broadcasting.

Synthesis, structure and magnetic properties of a novel 4,4′-azopyridine-bridged cobalt coordination polymer

A novel coordination polymer [Co(azpy)(pht)(H 2O) 3] n 1 (azpy=4,4′-Azopyridine, pht= o-phtalate), has been synthesized and characterized by X-ray diffraction. 1 crystallizes in the monoclinic space group P2(1)/ c, a=10.469(5) Å, b=24.607(13) Å, c=7.818(4) Å, β=96.506(9)°, V=2001.0(18) Å 3, Z=4. Its structure consists of one-dimensional chains formed by the connecting of the adjacent cobalt(II) ions via azpy ligand bridges. The one-dimensional chains are braced by interchain hydrogen bonds and C–H⋯O interactions to construct a compact high-dimensional network structure. The most striking feature of 1 is that the adjacent cobalt(II) ions bridged by azpy ligand are in different distorted octahedral coordination environment. Magnetic susceptibility measurements in the range 77–300 K show very weak antiferromagnetic exchange between the cobalt(II) ions ( zJ=−8.40 cm −1).

New high-dimensional networks based on polyoxometalate and crown ether building blocks.

Three novel supramolecular assemblies constructed from polyoxometalate and crown ether building blocks, [(DB18C6)Na(H(2)O)(1.5)](2)Mo(6)O(19).CH(3)CN, 1, and [(Na(DB18C6)(H(2)O)(2))(3)(H(2)O)(2)]XMo(12)O(40).6DMF.CH(3)CN (X = P, 2, and As, 3; DB18C6 = dibenzo-18-crown-6; DMF = N,N-dimethylfomamide), have been synthesized and characterized by elemental analyses, IR, UV-vis, EPR, TG, and single crystal X-ray diffraction. Compound 1 crystallizes in the tetragonal space group P4/mbm with a = 16.9701(6) A, c = 14.2676(4) A, and Z = 2. Compound 2 crystallizes in the hexagonal space group P6(3)/m with a = 15.7435(17) A, c = 30.042(7) A, gamma = 120 degrees, and Z = 2. Compound 3 crystallizes in the hexagonal space group P6(3)/m with a = 15.6882(5) A, c = 29.9778(18) A, gamma = 120 degrees, and Z = 2. Compound 1 exhibits an unusual three-dimensional network with one-dimensional sandglasslike channels based on the extensive weak forces between the oxygen atoms on the [Mo(6)O(19)](2)(-) polyoxoanions and the CH(2) groups of crown ether molecules. Compounds 2 and 3 are isostructural, and both contain a novel semiopen cagelike trimeric cation [(Na(DB18C6)(H(2)O)(2))(3)(H(2)O)(2)](3+). In their packing arrangement, an interesting 2-D "honeycomblike" "host" network is formed, in which the [XMo(12)O(40)](3)(-) (X = As and P) polyoxoanion "guests" resided.

Zn(BDC)(H 2O) 2] n: a novel blue luminescent coordination polymer constructed from BDC-bridged 1-D chains via interchain hydrogen bonds (BDC = 1,4-benzenedicarboxylate)

A novel coordination polymer [Zn(BDC)(H 2O) 2] n 1 (where BDC = 1,4-benzenedicarboxylate), has been synthesized and characterized by X-ray diffraction. 1 crystallizes in the monoclinic space group C2/ c, a=15.09(2) Å, b=5.058(7) Å, c=12.196(16) Å, β=103.62(2)°, V=904(2) A ̊ 3 , Z=4. The most striking feature of 1 is that it consists of a high-dimensional network structure constructed from BDC-bridged 1-D chains via interchain hydrogen bonds. The coordination sphere of the zinc(II) ion is a distorted tetrahedron completed by four oxygen atoms from two water molecules and two BDC ligands. BDC adopts the bis-monodentated ( syn– anti) coordination mode linking two adjacent zinc(II) ions. 1 shows strong blue photoluminescence as the result of the fluorescence from the intraligand emission excited state.

HIGHER DIMENSIONAL HONEYCOMB NETWORKS

We define the higher dimensional honeycomb graphs as a generalization of hexagonal plane tessellation, and consider it as a multiprocessor interconnection network. A 3-D honeycomb mesh network with n nodes has degree 4 and diameter approximately 3.63n. The network cost, defined as the product of degree and diameter, is about 20 percents better for the 3-D honeycomb than for the 3-D mesh. We describe the addressing scheme, the routing and broadcasting algorithms for three-dimensional and higher dimensional honeycombs. Furthermore, a formula for the diameter of a higher dimensional honeycomb network of given size is determined.

An Algorithm for the 2-Median Problem on Two-Dimensional Meshes

The mesh (and its variant, the torus) is a popular topology for processor interconnection in parallel computers. It has practical advantages such as low degree and perfectly compact layout when compared to other well-known topologies, for example the hypercube. A notable example of parallel computers based on the mesh topology is the iWarp system [1]. Dally has shown that low-dimensional networks have lower latency and higher hot-spot throughput than high-dimensional networks [2]. In this paper, we study the problem of finding a 2-median set in a two-dimensional mesh. The p-median problem is a well-known problem in location theory, which is to locate p identical facilities in a network so that the sum of the distance between every client in the network and its nearest facility is minimized. This has practical application in real mesh-connected parallel computers, which is to locate servers in selected nodes so that service requests from client nodes can be directed to different servers to achieve load balancing, and the traffic so generated can be more spread out over the links. The problem has been proved to be NP-hard for general networks. For a survey on the NP-hardness and algorithms developed in recent years, one can refer to [3]. Mirchandani [4] gives a strong mathematical background for the problem, which also gathers many of the results before 1989. An early survey [5, 6] by Tansel et al. is still very useful, especially for beginners. Shmoys et al. [7] includes a short summary of some of the results. Recently, Guha and Khuller used the greedy approach to handle the problem [8]. Hamacher et al. dealt with a variation of the problem with multicriteria, instead of single criteria [9]. Lai and Chang proposed a

A new fluoroaluminophosphate chain with an Al/P ratio of unity.

A new compound, Al2P2O8F2.[(CH3)2CHNH2CH2CH2NH2CH(CH3)2] (denoted AlPO-CJ8), with a 1-dimensional fluoroaluminophosphate chain and an Al/P ratio of unity has been synthesized solvothermally by using isopropylamine as an organic additive. It is characterized by X-ray powder diffraction (XRD), inductively coupled plasma (ICP), ion selective electrodes (ISE), and TGA-DTA analyses and structurally determined by single-crystal X-ray diffraction analysis. AlPO-CJ8 crystallizes in the triclinic space group P1 with a = 5.0306(8) A, b = 9.3626(15) A, c = 10.6131(17) A, alpha = 65.949(4) degrees, beta = 88.218(4) degrees, gamma = 77.19 degrees, and Z = 2. Its structure is built up by alternation of tetrahedral PO3(=O) and AlO3F units to form infinite 1-D Al2P2O8F2(2-) macroanionic chains. The inorganic chains are held together by diprotonated N,N'-diisopropylethylenediamine through H-bonds. The organic species N,N'-diisopropylethylenediamine is believed to be formed through solvothermal reaction of the organic additive isopropylamine and the solvent ethylene glycol. The existence of terminal P=O and Al-F bond groups in the 1-D chain indicates that it has potential to further set up higher dimensional networks through condensations.

Improving Associative Memory Function in the Steepest Descending Asynchronous Network with Tunneling Rule

In this paper effects of a new evolutionary rule added to the dynamics of the steepest descending asynchronous network model were studied. By numerical simulations, we found that the neural network operates very efficiently to improve the fault-tolerance when ,its capacity a under the new rule. The simulations were conducted on both the low- and the high-dimensional networks. A modified training scheme is also introduced.

Performance-based constraints for multidimensional networks

A stochastic analysis of multidimensional networks with unidirectional or bidirectional links between nodes is presented. The analysis allows the development of an accurate model for examining the performance and cost trade-offs of different network configurations. The model is validated through simulation and does not rely on the simplifying assumptions of previous models. In addition, the model is valid for the hypercube network. Two new performance-based design constraints are introduced: constant maximum throughput and constant unity queue. These new constraints are fundamentally different than previous constraints, which are based on some characterization of hardware implementation costs. Both of the new constraints allow performance and cost comparisons of different network configurations to be made on the basis of an equal ability to handle a range of traffic load. Results under the new constraints clearly show that a low dimensional network, while offering the lowest message latency, must be significantly more expensive than a comparable high dimensional network and, in some cases, may be impractical to implement. In addition, the constraints demonstrate that performance is highly dependent on offered load.

One-dimensional chains consisting of copper(II) ions and an orthogonal anthracene-pyrimidine derivative: hierarchical formation of high-dimensional networks and their magnetic properties

Recrystallization of 9-(5-pyrimidinyl)anthracene (1) with copper(II) nitrate trihydrate formed three kinds of complexes: 1·Cu(NO3)2·H2O (2), 1·Cu(NO3)2·MeOH·H2O (3) and 1·Cu(NO3)2·2EtOH (4) by changing the recrystallization solvent. The crystal structure of the three complexes showed cooperation of metal coordination, hydrogen bonding, and aromatic stacking interactions in the hierarchical formation of one-, two- and three-dimensional networks. Complexes 2 and 3 formed zigzag one-dimensional chains consisting of 1 and Cu2+ ions by coordination of pyrimidine to Cu2+ ion in equatorial trans fashion. Complex 4 formed a spiral one-dimensional chain by coordination in equatorial cis fashion. The magnetic behavior of 2, 3, and 4 were measured on a SQUID susceptometer to reveal that there existed strong antiferromagnetic interactions between Cu2+ ions. The magnetic data of one-dimensional antiferromagnetic chains were analyzed by the Bonner–Fisher model. The antiferromagnetic interaction between Cu2+ ions was around −55 K, which is the largest value observed in pyrimidine-bridged Cu2+ ions.

Supramolecular Isomerism and Polymorphism in Dianion Salts of Pyromellitic Acid

The X-ray crystal structures of several salts of pyromellitic acid, H 4PMA, have been determined. The structures of the pyridinium, [C 6H 5NH] 2[H 2PMA], 1, 3-methylpyridinium, [3-CH 3C 6H 4NH] 2[H 2PMA], 2a, [3-CH 3C 6H 4NH] 2-[H 2PMA]·[3-CH 3C 6H 4N], 2b, 4-methylpyridinium, [(4-CH 3C 6H 4N) 2H][4-CH 3C 6H 4NH][H 2PMA], 3, 2-ethylpyridinium, [2-CH 3CH 2C 6H 4N] 2[H 2PMA], 4, 2,4,6-trimethylpyridinium, [2,4,6-(CH 3) 3C 6H 4NH] 2[H 2PMA], 5a and 5b, and quinolinium, [C 9H 7NH] 2[H 2PMA], 6, salts were characterized and analyzed in terms of their networking and crystal packing. The structures of 1– 6 reveal the existence of 0D, 1D, 2D, or 3D O-H···O − hydrogen bonded networks that are sustained and controlled by self-assembly of H 2PMA 2− anions. Five structures, 1, 2a, 3, 4, and 6 form 0D complexes, whereas 1D, 2D, and 3D networks occur in 5a, 2b, and 5b, respectively. The relatively high occurrence of the 0D species might be attributable to a greater likelihood that more efficient crystal packing should generally occur in 0D complexes when compared to higher dimensional networks, in which open spaces are more likely. The formation of such a variety of networks by a single tecton, supramolecular isomerism, resembles that which has been observed in coordination polymer networks in which 4,4′-bipyridine or related ligands are used as spacer ligands.

Chaos in high-dimensional neural and gene networks

Neural and gene networks are often modeled by differential equations. If the continuous threshold functions in the differential equations are replaced by step functions, the equations become piecewise linear (PL equations). The flow through the state space is represented schematically by paths and directed graphs on am n-dimensional hypercube. Closed pathways, called cycles, may reflect periodic orbits with associated fixed points in a chosen Poincaré section. A return map in the Poincaré section can be constructed by the composition of fractional linear maps. The stable and unstable manifolds of the fixed points can be determined analytically. These methods allow us to analyze the dynamics in higher-dimensional networks as exemplified by a four-dimensional network that displays chaotic behavior. The three-dimensional Poincaré map is projected to a two-dimensional plane. This much simpler piecewise linear two-dimensional map conserves the important qualitative features of the flow.

SYNCHRONOUS CHAOS IN HIGH-DIMENSIONAL MODULAR NEURAL NETWORKS

The relationship between certain types of high-dimensional neural networks and low-dimensional prototypical equations (neuromodules) is investigated. The high-dimensional systems consist of finitely many pools containing identical, dissipative and nonlinear single-units operating in discrete time. Under the assumption of random connections inside and between pools, the system can be reduced to a set of only a few equations, which — asymptotically in time and system size — describe the behavior of every single unit arbitrarily well. This result can be viewed as synchronization of the single units in each pool. It is stated as a theorem on systems of nonlinear coupled maps, which gives explicit conditions on the single unit dynamics and the nature of the random connections. As an application we compare a 2-pool network with the corresponding two-dimensional dynamics. The bifurcation diagrams of both systems become very similar even for moderate system size (N=50) and large disorder in the connection strengths (50% of mean), despite the fact, that the systems exhibit fairly complex behavior (quasiperiodicity, chaos, coexisting attractors).

Planar-adaptive routing

Network throughput can be increased by allowing multipath, adaptive routing. Adaptive routing allows more freedom in the paths taken by messages, spreading load over physical channels more evenly. The flexibility of adaptive routing introduces new possibilities of deadlock. Previous deadlock avoidance schemes in k -ary n -cubes require an exponential number of virtual channels. We describe a family of deadlock-free routing algorithms, called planar-adaptive routing algorithms, that require only a constant number of virtual channels, independent of networks size and dimension. Planar-adaptive routing algorithms reduce the complexity of deadlock prevention by reducing the number of choices at each routing step. In the fault-free case, planar-adaptive networks are guaranteed to be deadlock-free. In the presence of network faults, the planar-adaptive router can be extended with misrouting to produce a working network which remains provably deadlock free and is provably livelock free. In addition, planar-adaptive networks can simultaneously support both in-order and adaptive, out-of-order packet delivery. Planar-adaptive routing is of practical significance. It provides the simplest known support for deadlock-free adaptive routing in k -ary n -cubes of more than two dimensions (with k >2). Restricting adaptivity reduces the hardware complexity, improving router speed or allowing additional performance-enhancing network features. The structure of planar-adaptive routers is amenable to efficient implementation. Simulation studies show that planar-adaptive routers can increase the robustness of network throughput for nonuniform communication patterns. Planar-adaptive routers outperform deterministic routers with equal hardware resources. Further, adding virtual lanes to planar-adaptive routers increases this advantage. Comparisons with fully adaptive routers show that planar-adaptive routers, limited adaptive routers, can give superior performance. These results indicate the best way to allocate router resources to combine adaptivity and virtual lanes. Planar-adaptive routers are a special case of limited adaptivity routers. We define a class of adaptive routers with f degrees of routing freedom. This class, termed f-flat adaptive routers , allows a direct cost-performance tradeoff between implementation cost (speed and silicon area) and routing freedom (channel utilization). For a network of a particular dimension, the cost of adaptivity grows linearly with the routing freedom. However, the rate of growth is a much larger constant for high-dimensional networks. All of the properties proven for planar-adaptive routers, such as deadlock and livelock freedom, also apply to f -flat adaptive routers.

On bifurcations and chaos in random neural networks

Chaos in nervous system is a fascinating but controversial field of investigation. To approach the role of chaos in the real brain, we theoretically and numerically investigate the occurrence of chaos inartificial neural networks. Most of the time, recurrent networks (with feedbacks) are fully connected. This architecture being not biologically plausible, the occurrence of chaos is studied here for a randomly diluted architecture. By normalizing the variance of synaptic weights, we produce a bifurcation parameter, dependent on this variance and on the slope of the transfer function, that allows a sustained activity and the occurrence of chaos when reaching a critical value. Even for weak connectivity and small size, we find numerical results in accordance with the theoretical ones previously established for fully connected infinite sized networks. The route towards chaos is numerically checked to be a quasi-periodic one, whatever the type of the first bifurcation is. Our results suggest that such high-dimensional networks behave like low-dimensional dynamical systems.

Some more numerical experiments with LSNNO, a Fortran subroutine for solving large-scale nonlinear network optimization problems

This paper presents some more numerical experiments with LSNNO, a new FORTRAN subroutine for solving large-scale nonlinear network optimization problems. The implemented algorithm applies the concepts of partial separability and partitioned quasi-Newton updating to high-dimensional nonlinear network optimization problems. The experiments reported involve both academic and practical problems.

The impact of pipelined channels on k-ary n-cube networks

In a pipelined-channel interconnection network, multiple bits may be simultaneously in flight on a single wire. This allows the cycle time of the network to be independent of the wire lengths, significantly affecting the network design trade-offs. This paper investigates the design and performance of pipelined channel k-ary n-cube networks, with particular emphasis on the choice of dimensionality and radix. Networks are investigated under the constant link width, constant node size and constant bisection constraints. We find that the optimal dimensionality of pipelined-channel networks is higher than that of nonpipelined-channel networks, with the difference being greater under looser wiring constraints. Their radix should remain roughly constant as network size is grown, decreasing slightly for some unidirectional tori and increasing slightly for some bidirectional meshes. Pipelined-channel networks are shown to provide lower latency and higher bandwidth than their nonpipelined-channel counterparts, especially for high-dimensional networks. The paper also investigates the effects of switching overhead and message lengths, indicating where results agree with and differ from previous results obtained for nonpipelined-channel networks.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computer-simulation of the multiple-quantum dynamics of one-, two- and three-dimensional spin distributions

Multiple-quantum NMR has previously been used to characterize the distribution of spins, whether cluster or homogenously distributed, in solids. In this work, through computer simulation of the multiple-quantum coherence dynamics of nine spin networks, we examine the potential of this spectroscopy to differentiate between one-, two- and three-dimensional distributions of spins. In order to compute the multiple-quantum dynamics of networks of spins large enough to vary in dimensionality, the solution of the Liouville-von Neumann equation was written as an explicit, single time-step equation for the density matrix with pairwise, spatially averaged dipole interactions. In these nine spin simulations, differentiation of the multiple-quantum dynamics of a linear system in comparison with two- and three-dimensional networks was observed indicating the influence of the dimensionality of the network on spin correlation dynamics. While no differentiation between higher dimensional networks was observed, this can be attributed to the small number of spins in the simulation. Based on the occupancy of various lattice types, differences between spin correlation rates for surface and bulk, homogeneous distributions are expected when the number of correlated spins exceeds 20.

Social disintegration and the spread of AIDS: Thresholds for propagation along ‘sociogeographic’ networks

Previous work on the asymptotic spread of HIV infection along a low dimensional 'sociogeographic' network--a social network characteristically embedded within a limited geographic area--is extended to explore threshold conditions under which the infection extends widely beyond an initial set of infected individuals or communities. Results for one dimension suggest that threshold behavior is analogous to a chain reaction with criticality determined conjointly by the susceptibility of individuals within a community to a nexus of behavior conducive to rapid HIV spread and by the probability of transmission between susceptible communities. Once threshold is exceeded, a stochastic reformulation finds the asymptotic rate of transmission between communities may be markedly raised by positive correlation between susceptibility to rapid disease spread within a community and the transmissibility between communities, for example outmigration driven by social disintegration or residential instability arising from inherent structural factors associated with community susceptibility, as with male prostitution. Examination of threshold conditions for higher dimensional sociogeographic networks most likely characteristic of disease spread beyond the 'deep ghettos' now suffering the highest burden of infection suggests it is at least as, and likely more, effective to decrease the fraction of population susceptible to the high risk behavioral nexus as it is to lower the probability of disease transmission between susceptible individuals or communities.(ABSTRACT TRUNCATED AT 250 WORDS)