Exact Isomorphism Research Articles

Coevolution of rostrum and brain in pig species.

Domestic pigs have a prominent cortical gyrus (the rostrum gyrus) isomorphic to the contralateral hemirostrum. It is unclear, however, if the size and shape of the rostrum gyrus are of evolutionary/functional relevance. Here, we address this question by assessing the relationship of rostrum and rostrum gyrus across eight pig species. To this end, we quantified rostrum morphology in fresh and alcohol-preserved pig specimens by surface scans, microfocus computed tomography scans, and photography. We establish that the size and shape of the rostrum gyrus can be precisely inferred from endocasts. We then took advantage of the accessibility of pig skulls and endocasts to assess features of the rostrum gyrus across species. Ourinvestigationledtothefollowingresults: (i) The rostra of pig species show basic similarities. (ii) A cortical rostrum gyrus is apparent in all pigs. (iii) The size of the rostrum gyrus differs across species and outgroups of the evolutionary dominant suinae (i.e., peccaries and the babirusa) have a noticeably smaller rostrum gyrus. (iv) Warthogs have a derived rostrum morphology with an extra fold and a very wide rostrum; the warthog rostrum gyrus recapitulates these rostrum features. (v) Domestic pigs have relatively smaller rostrum gyrus than wild boars. We also provide indications for a conserved cytoarchitectonic patterning of the rostrum gyrus. We conclude that the rostrum gyrus is a neural module that was putatively present in the common ancestor of pigs and that this neural module is conserved across pig species. Natural selection maintains the rostrum gyrus' size and its exact isomorphism to the rostrum.

Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search

In ``Playing Pool with π'' \cite{Galperin}, Galperin invented an extraordinary method to learn the digits of π by counting the collisions of billiard balls. Here I demonstrate an exact isomorphism between Galperin's bouncing billiards and Grover's algorithm for quantum search. This provides an illuminating way to visualize Grover's algorithm.

Noncommutative standard table algebras with at most one nontrivial multiplicity

Standard table algebras that have an irreducible character of degree two, and with all other irreducible characters of multiplicity one, are classified to exact isomorphism.

Reformulation of Multifluid Plasma Equations in Terms of Spacetime Algebra

Using the exact isomorphism between multifluid plasma theory and classical electromagnetism, we apply the theorems and techniques of spacetime algebra to the multifluid equations of a plasma in order to present a new formulation in a compact and elegant way. A set of Maxwell equations for a plasma is combined into a single equation similar to their counterparts in electromagnetism, linear gravitation and fluid mechanics. The presented formulation also proposes a useful method for the generalization of the plasma wave equation.

A Methodology to Generate Attributed Graphs with a Bounded Graph Edit Distance for Graph-Matching Testing

This paper presents a methodology for generating pairs of attributed graphs with a lower and upper- bounded graph edit distance (GED). It is independent of the type of attributes on nodes and edges. The algorithm is composed of three steps: randomly generating a graph, generating another graph as a sub-graph of the first, and adding structural and semantic noise to both. These graphs, together with their bounded distances, can be used to manufacture synthetic databases of large graphs. The exact GED between large graphs cannot be obtained for runtime reasons since it has to be computed through an optimal algorithm with an exponential computational cost. Through this database, we can test the behavior of the known or new sub-optimal error-tolerant graph-matching algorithms against a lower and an upper bound GED on large graphs, even though we do not have the true distance. It is not clear how the error induced by the use of sub-optimal algorithms grows with problem size. Thus, with this methodology, we can generate graph databases and analyze if the current assumption that we can extrapolate algorithms’ behavior from matching small graphs to large graphs is correct or not. We also show that with some restrictions, the methodology returns the optimal GED in a quadratic time and that it can also be used to generate graph databases to test exact sub-graph isomorphism algorithms.

Noncommutative reality-based algebras of rank 6

ABSTRACTWe show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. Four in the list are primitive, but we show three of them cannot be realized as adjacency algebras of association schemes. In the last section of the paper, we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.

Challenging the Time Complexity of Exact Subgraph Isomorphism for Huge and Dense Graphs with VF3.

Graph matching is essential in several fields that use structured information, such as biology, chemistry, social networks, knowledge management, document analysis and others. Except for special classes of graphs, graph matching has in the worst-case an exponential complexity; however, there are algorithms that show an acceptable execution time, as long as the graphs are not too large and not too dense. In this paper we introduce a novel subgraph isomorphism algorithm, VF3, particularly efficient in the challenging case of graphs with thousands of nodes and a high edge density. Its performance, both in terms of time and memory, has been assessed on a large dataset of 12,700 random graphs with a size up to 10,000 nodes, made publicly available. VF3 has been compared with four other state-of-the-art algorithms, and the huge experimentation required more than two years of processing time. The results confirm that VF3 definitely outperforms the other algorithms when the graphs become huge and dense, but also has a very good performance on smaller or sparser graphs.

Filtering graphs to check isomorphism and extracting mapping by using the Conductance Electrical Model

This paper presents a new method of filtering graphs to check exact graph isomorphism and extracting their mapping. Each graph is modeled by a resistive electrical circuit using the Conductance Electrical Model (CEM). By using this model, a necessary condition to check the isomorphism of two graphs is that their equivalent resistances have the same values, but this is not enough, and we have to look for their mapping to find the sufficient condition. We can compute the isomorphism between two graphs in O(N3), where N is the order of the graph, if their star resistance values are different, otherwise the computational time is exponential, but only with respect to the number of repeated star resistance values, which usually is very small. We can use this technique to filter graphs that are not isomorphic and in case that they are, we can obtain their node mapping. A distinguishing feature over other methods is that, even if there exists repeated star resistance values, we can extract a partial node mapping (of all the nodes except the repeated ones and their neighbors) in O(N3). The paper presents the method and its application to detect isomorphic graphs in two well know graph databases, where some graphs have more than 600 nodes.

A Sub-graph Matching Method based on Calibration of Characteristics of Topological Footprint

Approximate sub-graph matching is important in many graph data mining fields. At present, current solutions can be difficult to implement, have an expensive pre-processing phase, or only work for given types of graph. In this paper a novel generic approach is presented which addresses these issues. An approximate sub-graph matcher (A-SGM) calculates the distance between the topological characteristics (footprint) of the sub-graphs to be matched, applying a weighting to the different sub-graph characteristics and those of neighbor nodes. The weights are calibrated for each dataset with a simulated annealing process using sample sets of graph nodes to reduce computational cost, and an exact isomorphism matcher as a fitness function which takes into account how well the match maintains the neighboring node degree distributions. Benchmarking is performed on several state of the art methods and real and synthetic graph datasets to evaluate the precision, recall and computational cost. The results show that the A-SGM is competitive with state of the art methods in terms of precision, recall and execution time. General Terms Machine Intelligence (Data and Web Mining), Applications of Computer Science in Modeling, Data and Information Systems.

On convex relaxation of graph isomorphism

We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving n2 equality and n2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic.

Composite fermion-boson mapping for fermionic lattice models

We present a mapping of elementary fermion operators onto a quadratic form of composite fermionic and bosonic cluster operators. The mapping is an exact isomorphism as long as the physical constraint of one composite particle per cluster is satisfied. This condition is treated on average in a composite particle mean-field approach, which consists of an ansatz that decouples the composite fermionic and bosonic sectors. The theory is tested on the 1D and 2D Hubbard models. Using a Bogoliubov determinant for the composite fermions and either a coherent or Bogoliubov state for the bosons, we obtain a simple and accurate procedure for treating the Mott insulating phase of the Hubbard model with mean-field computational cost.

Isomorphism testing of Boolean functions computable by constant-depth circuits

Given two n-variable Boolean functions f and g, we study the problem of computing an ε-approximate isomorphism between them. An ε-approximate isomorphism is a permutation π of the n Boolean variables such that f(x1,x2,…,xn) and g(xπ(1),xπ(2),…,xπ(n)) differ on at most an ε fraction of all Boolean inputs {0,1}n. We give a randomized 2O(nlog⁡(n/ε)O(d)) time algorithm that computes an ε-approximate isomorphism between two isomorphic Boolean functions f and g that are given by depth d circuits of poly(n) size, where d is a constant independent of n, for any positive ε. In contrast, the best known algorithm for computing an exact isomorphism between n-ary Boolean functions has running time 2O(n)[12] even for functions computed by poly(n) size DNF formulas. Our algorithm is based on a result for hypergraph isomorphism with bounded edge size [4] and the classical Linial–Mansour–Nisan result on approximating small depth and size Boolean circuits by small degree polynomials using Fourier analysis [11].

Top-k Similar Graph Matching Using TraM in Biological Networks

Many emerging database applications entail sophisticated graph-based query manipulation, predominantly evident in large-scale scientific applications. To access the information embedded in graphs, efficient graph matching tools and algorithms have become of prime importance. Although the prohibitively expensive time complexity associated with exact subgraph isomorphism techniques has limited its efficacy in the application domain, approximate yet efficient graph matching techniques have received much attention due to their pragmatic applicability. Since public domain databases are noisy and incomplete in nature, inexact graph matching techniques have proven to be more promising in terms of inferring knowledge from numerous structural data repositories. In this paper, we propose a novel technique called TraM for approximate graph matching that off-loads a significant amount of its processing on to the database making the approach viable for large graphs. Moreover, the vector space embedding of the graphs and efficient filtration of the search space enables computation of approximate graph similarity at a throw-away cost. We annotate nodes of the query graphs by means of their global topological properties and compare them with neighborhood biased segments of the datagraph for proper matches. We have conducted experiments on several real data sets, and have demonstrated the effectiveness and efficiency of the proposed method

Classical field isomorphisms in two-fluid plasmas

Previous work recognized a new framework for the equations of a multifluid plasma, wherein each species can be described by a set of equations remarkably similar to the Maxwell equations of classical electrodynamics. This paper extends the previous effort to form an exact isomorphism between the multifluid theory and classical electrodynamics. The major benefits of the new formulation are that the explicit coupling between different species is minimized, and theorems and techniques of classical electrodynamics can be immediately applied to the new multifluid formulation. We introduce the exact isomorphism and investigate some of the immediate consequences from classical electrodynamics. To provide a visualization of the isomorphism, previous 1D and 2D numerical simulations are postprocessed and presented to illustrate the generalized fields and source terms.

On pseudocyclic table algebras and applications to pseudocyclic association schemes

Pseudocyclic table algebras are generalizations of Bose-Mesner algebras of pseudocyclic association schemes. Constructions of pseudocyclic association schemes have been studied among others by Hollmann [Hh], Hollmann and Xiang [HX] and Mathon [M]. In this paper we prove some general properties of pseudocyclic association schemes via the study of pseudocyclic table algebras. In particular, we will show that subschemes and quotient schemes of pseudocyclic association schemes are also pseudocyclic association schemes, and combinatorial morphisms between any two t-pseudocyclic association schemes induce algebra homomorphisms between their Bose-Mesner algebras. Furthermore, we will prove that an algebra isomorphism between two t-pseudocyclic table algebras under some weak condition is an exact isomorphism. Properties of pseudocyclic P-polynomial table algebras will also be studied.

LD: A Polynomial Time Algorithm for Tree Isomorphism

Graph isomorphism problem has always been mathematics and engineering technology community concern, the reason mainly from two aspects: First, in theory, is generally believed that the problem is NP-complete problem; Second, the graph isomorphism the problem with good prospects, in chemistry, operations research, computer science, electronics, network theory has applications in many fields, but the exponential complexity of the algorithm and the algorithm itself makes the limitations applicable to the object involved with complex graphics the application of structure is difficult to determine the start. In this paper, class tree is proposed based on the node to delete the exact graph isomorphism problem, you can quickly determine the graph isomorphism problem, and theoretical analysis and experiments show that the algorithm can determine the class in polynomial time tree isomorphism problem.

Piecewise linear loop quantum gravity

We define a modification of loop quantum gravity (LQG) in which graphs are required to consist of piecewise linear edges, which we call piecewise linear LQG (plLQG). At the diffeomorphism-invariant level, we prove that plLQG is equivalent to standard LQG, as long as one chooses the class of diffeomorphisms appropriately. That is, we exhibit a unitary map between the diffeomorphism-invariant Hilbert spaces that maps physically equivalent operators into each other. In addition, using the same ideas as in standard LQG, one can define a Hamiltonian and master constraint in plLQG, and the unitary map between plLQG and LQG then provides an exact isomorphism of dynamics in the two frameworks. Furthermore, loop quantum cosmology (LQC) can be exactly embedded into plLQG. This allows a prior program of the author to embed LQC into LQG at the dynamical level to proceed. In particular, this allows a formal expression for a physically motivated embedding of LQC into LQG at the diffeomorphism-invariant level to be given.

On Homogeneous Standard Integral Table Algebras of Degree 4

ABSTRACT In this paper we classify, up to exact isomorphism, the algebras in the title which contain a faithful element and in which every non-trivial table subset has dimension at least five.

Solving inexact graph isomorphism problems using neural networks

We present a neural network approach to solve exact and inexact graph isomorphism problems for weighted graphs. In contrast to other neural heuristics or related methods this approach is based on a neural refinement procedure to reduce the search space followed by an energy-minimizing matching process. Experiments on random weighted graphs in the range of 100–5000 vertices and on chemical molecular structures are presented and discussed.

The classification of a class of homogeneous integral table algebras of degree five

The purpose of this paper is to give the classification of homogeneous integral table algebras of degree 5 containing a faithful real element of width 2. In fact, these algebras are classified to exact isomorphism, that is the sets of structure constants which arise from the given basis are completely determined. This is work towards classifying homogeneous integral table algebras of degree 5.