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Journal articles on the topic 'Differential Graph'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Chen, Yuanmei, Haizhen Ren, Lei Zhang, and Yang Zhao. "Computation of Differential in Strong Product Network of Paths via Recursive Iteration." Journal of Physics: Conference Series 2025, no. 1 (September 1, 2021): 012050. http://dx.doi.org/10.1088/1742-6596/2025/1/012050.

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Abstract The set differential can be attributed to some influence maximization problems, which are widely used. For example, it is a basic algorithm problem of information diffusion in social networks. In 2004, Lewis et al. extended this concept to the graph, defined the differential of graph. The present article mainly studies the differential in strong product network of paths. Using the basic block recursive iterative method, we obtain closed form formulas for the differential on the strong product network of paths. The method used here is also effective for the differentials in square graphs.

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2

B, Vembu, and Loghambal S. "Pseudo-Graph Neural Networks On Ordinary Differential Equations." Journal of Computational Mathematica 6, no. 1 (March 22, 2022): 117–23. http://dx.doi.org/10.26524/cm.125.

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In this paper, we extend the idea of continuous-depth models to pseudo graphs and present pseudo graph ordinary differential equations (PGODE), which are inspired by the neural ordinary differential equation (NODE) for data in the Euclidean domain. All existing graph networks have discrete depth. A pseudo graph neural network (PGNN) is used to parameterize the derivative of hidden node states, and the output states are the solution to this ordinary differential equation (ODE). A memory-efficient framework with precise gradient estimates is then proposed for free-form ODEs. We also introduce the framework of continuous–depth pseudo graph neural networks (PGNNs) on ODE by blending discrete structures and differential equations

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3

Chen, Xihui, Sjouke Mauw, and Yunior Ramírez-Cruz. "Publishing Community-Preserving Attributed Social Graphs with a Differential Privacy Guarantee." Proceedings on Privacy Enhancing Technologies 2020, no. 4 (October 1, 2020): 131–52. http://dx.doi.org/10.2478/popets-2020-0066.

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AbstractWe present a novel method for publishing differentially private synthetic attributed graphs. Our method allows, for the first time, to publish synthetic graphs simultaneously preserving structural properties, user attributes and the community structure of the original graph. Our proposal relies on CAGM, a new community-preserving generative model for attributed graphs. We equip CAGM with efficient methods for attributed graph sampling and parameter estimation. For the latter, we introduce differentially private computation methods, which allow us to release communitypreserving synthetic attributed social graphs with a strong formal privacy guarantee. Through comprehensive experiments, we show that our new model outperforms its most relevant counterparts in synthesising differentially private attributed social graphs that preserve the community structure of the original graph, as well as degree sequences and clustering coefficients.

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4

Pushpam, P. Roushini Leely, and D. Yokesh. "Differentials in certain classes of graphs." Tamkang Journal of Mathematics 41, no. 2 (June 30, 2010): 129–38. http://dx.doi.org/10.5556/j.tkjm.41.2010.664.

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Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

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5

Konyrkulzhayeva, М. N. "THE DIRICHLET PROBLEM ON THE ORIENTED GRAPHS." BULLETIN Series of Physics & Mathematical Sciences 70, no. 2 (June 30, 2020): 84–90. http://dx.doi.org/10.51889/2020-2.1728-7901.12.

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Differential operators on graphs often arise in mathematics and different fields of science such as mechanics, physics, organic chemistry, nanotechnology, etc. In this paper the solutions of the Dirichlet problem for a differential operator on a star-shaped graph are deduced. And the differential operator with standard matching conditions in the internal vertices and the Dirichlet boundary conditions at boundary vertices are studied. Task is a model the oscillation of a simple system of several rods with an adjacent end. In work the formula of the Green function of the Dirichlet problem for the second order equation on directed graph is showed. Spectral analysis of differential operators on geometric graphs is the basic mathematical apparatus in solving modern problems of quantum mechanics.

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6

Basilio, Ludwin, Sergio Bermudo, Jesús Leaños, and José Sigarreta. "β-Differential of a Graph." Symmetry 9, no. 10 (September 30, 2017): 205. http://dx.doi.org/10.3390/sym9100205.

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7

Wang, Weiya, Geng Yang, Lin Bao, Ke Ma, Hao Zhou, and Yunlu Bai. "Travel Trajectory Frequent Pattern Mining Based on Differential Privacy Protection." Wireless Communications and Mobile Computing 2021 (August 5, 2021): 1–14. http://dx.doi.org/10.1155/2021/6379530.

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Now, many application services based on location data have brought a lot of convenience to people’s daily life. However, publishing location data may divulge individual sensitive information. Because the location records about location data may be discrete in the database, some existing privacy protection schemes are difficult to protect location data in data mining. In this paper, we propose a travel trajectory data record privacy protection scheme (TMDP) based on differential privacy mechanism, which employs the structure of a trajectory graph model on location database and frequent subgraph mining based on weighted graph. Time series is introduced into the location data; the weighted trajectory model is designed to obtain the travel trajectory graph database. We upgrade the mining of location data to the mining of frequent trajectory graphs, which can discover the relationship of location data from the database and protect location data mined. In particular, to improve the identification efficiency of frequent trajectory graphs, we design a weighted trajectory graph support calculation algorithm based on canonical code and subgraph structure. Moreover, to improve the data utility under the premise of protecting user privacy, we propose double processes of adding noises to the subgraph mining process by the Laplace mechanism and selecting final data by the exponential mechanism. Through formal privacy analysis, we prove that our TMDP framework satisfies ε -differential privacy. Compared with the other schemes, the experiments show that the data availability of the proposed scheme is higher and the privacy protection of the scheme is effective.

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8

Rezapour, Shahram, Chernet Tuge Deressa, Azhar Hussain, Sina Etemad, Reny George, and Bashir Ahmad. "A Theoretical Analysis of a Fractional Multi-Dimensional System of Boundary Value Problems on the Methylpropane Graph via Fixed Point Technique." Mathematics 10, no. 4 (February 12, 2022): 568. http://dx.doi.org/10.3390/math10040568.

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Few studies have investigated the existence and uniqueness of solutions for fractional differential equations on star graphs until now. The published papers on the topic are based on the assumption of existence of one junction node and some boundary nodes as the origin on a star graph. These structures are special cases and do not cover more general non-star graph structures. In this paper, we state a labeling method for graph vertices, and then we prove the existence results for solutions to a new family of fractional boundary value problems (FBVPs) on the methylpropane graph. We design the chemical compound of the methylpropane graph with vertices specified by 0 or 1, and on every edge of the graph, we consider fractional differential equations. We prove the existence of solutions for the proposed FBVPs by means of the Krasnoselskii’s and Scheafer’s fixed point theorems, and further, we study the Ulam–Hyers type stability for the given multi-dimensional system. Finally, we provide an illustrative example to examine our results.

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9

Basilio, Ludwin A., Sergio Bermudo, Juan C. Hernández-Gómez, and José M. Sigarreta. "General Properties on Differential Sets of a Graph." Axioms 10, no. 4 (October 20, 2021): 265. http://dx.doi.org/10.3390/axioms10040265.

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Let G=(V,E) be a graph, and let β∈R. Motivated by a service coverage maximization problem with limited resources, we study the β-differential of G. The β-differential of G, denoted by ∂β(G), is defined as ∂β(G):=max{|B(S)|−β|S|suchthatS⊆V}. The case in which β=1 is known as the differential of G, and hence ∂β(G) can be considered as a generalization of the differential ∂(G) of G. In this paper, upper and lower bounds for ∂β(G) are given in terms of its order |G|, minimum degree δ(G), maximum degree Δ(G), among other invariants of G. Likewise, the β-differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have.

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10

Remiddi, E. "Differential equations for Feynman graph amplitudes." Il Nuovo Cimento A 110, no. 12 (December 1997): 1435–52. http://dx.doi.org/10.1007/bf03185566.

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11

Krömer, Pavel, Michal Prílepok, Jan Platoš, and Václav Snášel. "GRAPH VISUALISATION BY CONCURRENT DIFFERENTIAL EVOLUTION." Neural Network World 25, no. 4 (2015): 369–86. http://dx.doi.org/10.14311/nnw.2015.25.019.

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12

Markl, Martin. "Natural differential operators and graph complexes." Differential Geometry and its Applications 27, no. 2 (April 2009): 257–78. http://dx.doi.org/10.1016/j.difgeo.2008.10.008.

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13

Ignatyev, Mikhail. "Inverse scattering problem for Sturm-Liouville operator on non-compact A-graph. Uniqueness result." Tamkang Journal of Mathematics 46, no. 4 (December 22, 2015): 401–22. http://dx.doi.org/10.5556/j.tkjm.46.2015.1806.

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We consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering problem for Sturm--Liouville differential operator with standard matching conditions in the internal vertices. Transport, spectral and scattering problems for differential operators on graphs appear frequently in mathematics, natural sciences and engineering. In particular, direct and inverse problems for such operators are used to construct and study models in mechanics, nano-electronics, quantum computing and waveguides. The most complete results on (both direct and inverse) spectral problems were achieved in the case of Sturm--Liouville operators on compact graphs, in the noncompact case there are no similar general results. In this paper, we establish some properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the scattering data. A uniqueness theorem for such inverse problem is proved.

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14

Huang, Xueqin, Xianqiang Zhu, Xiang Xu, Qianzhen Zhang, and Ailin Liang. "Parallel Learning of Dynamics in Complex Systems." Systems 10, no. 6 (December 15, 2022): 259. http://dx.doi.org/10.3390/systems10060259.

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Dynamics always exist in complex systems. Graphs (complex networks) are a mathematical form for describing a complex system abstractly. Dynamics can be learned efficiently from the structure and dynamics state of a graph. Learning the dynamics in graphs plays an important role in predicting and controlling complex systems. Most of the methods for learning dynamics in graphs run slowly in large graphs. The complexity of the large graph’s structure and its nonlinear dynamics aggravate this problem. To overcome these difficulties, we propose a general framework with two novel methods in this paper, the Dynamics-METIS (D-METIS) and the Partitioned Graph Neural Dynamics Learner (PGNDL). The general framework combines D-METIS and PGNDL to perform tasks for large graphs. D-METIS is a new algorithm that can partition a large graph into multiple subgraphs. D-METIS innovatively considers the dynamic changes in the graph. PGNDL is a new parallel model that consists of ordinary differential equation systems and graph neural networks (GNNs). It can quickly learn the dynamics of subgraphs in parallel. In this framework, D-METIS provides PGNDL with partitioned subgraphs, and PGNDL can solve the tasks of interpolation and extrapolation prediction. We exhibit the universality and superiority of our framework on four kinds of graphs with three kinds of dynamics through an experiment.

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15

Tang, Chuan Yin, Xin Yu Hou, Hua Yin, and Ying Zhang. "The Application of Bond Graph Theory on Automobiles Modeling." Applied Mechanics and Materials 120 (October 2011): 339–42. http://dx.doi.org/10.4028/www.scientific.net/amm.120.339.

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Based on the bond graph theory, the acquisition of state equation of vehicle suspension is presented. Set an example to a five degrees of freedom vehicle suspension model ,the simulated results are obtained with the aid of Matlab/Simulink software. Bond graphs are equation based , and are superior to traditional differential equations, they can provide the dynamic digital simulation in time and frequency domain, they can present the static simulation and omit the transition and class-decreasing process which is needed for traditional differential equations.

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16

Basilio, Ludwin, Jair Simon, Jesús Leaños, and Omar Cayetano. "The Differential on Graph Operator Q(G)." Symmetry 12, no. 5 (May 6, 2020): 751. http://dx.doi.org/10.3390/sym12050751.

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If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

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17

LEFLOCH, PHILIPPE G. "GRAPH SOLUTIONS OF NONLINEAR HYPERBOLIC SYSTEMS." Journal of Hyperbolic Differential Equations 01, no. 04 (December 2004): 643–89. http://dx.doi.org/10.1142/s0219891604000287.

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For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves. In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertical" parts of the graphs) with the traveling wave equation of the augmented model (in the "vertical" parts). We consider the Cauchy problem within the class of graph solutions. The graph solution to the Cauchy problem is constructed by completion of the discontinuities of the entropy solution. The uniqueness is established by applying a general uniqueness theorem due to Baiti, LeFloch, and Piccoli. The proposed geometric framework illustrates the importance of the uniform distance between graphs to deal with solutions of nonlinear hyperbolic problems.

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18

Vasilyeva, Elena, Maik Thiele, Christof Bornhövd, and Wolfgang Lehner. "Considering User Intention in Differential Graph Queries." Journal of Database Management 26, no. 3 (July 2015): 21–40. http://dx.doi.org/10.4018/jdm.2015070102.

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Empty answers are a major problem by processing pattern matching queries in graph databases. Especially, there can be multiple reasons why a query failed. To support users in such situations, differential queries can be used that deliver missing parts of a graph query. Multiple heuristics are proposed for differential queries, which reduce the search space. Although they are successful in increasing the performance, they can discard query subgraphs relevant to a user. To address this issue, the authors extend the concept of differential queries and introduce top-k differential queries that calculate the ranking based on users' preferences and significantly support the users' understanding of query database management systems. A user assigns relevance weights to elements of a graph query that steer the search and are used for the ranking. In this paper the authors propose different strategies for selection of relevance weights and their propagation. As a result, the search is modelled along the most relevant paths. The authors evaluate their solution and both strategies on the DBpedia data graph.

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19

Wang, Yuye, Jing Yang, and Jianpei Zhan. "Differentially Private Attributed Network Releasing Based on Early Fusion." Security and Communication Networks 2021 (July 26, 2021): 1–13. http://dx.doi.org/10.1155/2021/9981752.

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Vertex attributes exert huge impacts on the analysis of social networks. Since the attributes are often sensitive, it is necessary to seek effective ways to protect the privacy of graphs with correlated attributes. Prior work has focused mainly on the graph topological structure and the attributes, respectively, and combining them together by defining the relevancy between them. However, these methods need to add noise to them, respectively, and they produce a large number of required noise and reduce the data utility. In this paper, we introduce an approach to release graphs with correlated attributes under differential privacy based on early fusion. We combine the graph topological structure and the attributes together with a private probability model and generate a synthetic network satisfying differential privacy. We conduct extensive experiments to demonstrate that our approach could meet the request of attributed networks and achieve high data utility.

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20

Caga-anan, Randy L., and Sergio R. Canoy, Jr. "On the I-differential of a graph." Applied Mathematical Sciences 8 (2014): 4397–404. http://dx.doi.org/10.12988/ams.2014.46402.

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21

Zhang, Sen, and Weiwei Ni. "Graph Embedding Matrix Sharing With Differential Privacy." IEEE Access 7 (2019): 89390–99. http://dx.doi.org/10.1109/access.2019.2927365.

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22

Andreotti, Eleonora, Dominik Edelmann, Nicola Guglielmi, and Christian Lubich. "Constrained Graph Partitioning via Matrix Differential Equations." SIAM Journal on Matrix Analysis and Applications 40, no. 1 (January 2019): 1–22. http://dx.doi.org/10.1137/17m1160987.

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23

Basilio-Hernández, Ludwin A., Walter Carballosa, Jesús Leaños, and José M. Sigarreta. "On the Differential Polynomial of a Graph." Acta Mathematica Sinica, English Series 35, no. 3 (December 14, 2018): 338–54. http://dx.doi.org/10.1007/s10114-018-7307-3.

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24

Živković, Marko. "Second extra differential on odd graph complexes." Higher Structures 6, no. 1 (July 15, 2022): 439–49. http://dx.doi.org/10.21136/hs.2022.10.

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25

TANG, Dewei. "Differential Property of Tri-axial Differential Mechanism Based on Bond Graph." Journal of Mechanical Engineering 47, no. 23 (2011): 14. http://dx.doi.org/10.3901/jme.2011.23.014.

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26

Chicha, Elie, Bechara Al Bouna, Mohamed Nassar, Richard Chbeir, Ramzi A. Haraty, Mourad Oussalah, Djamal Benslimane, and Mansour Naser Alraja. "A User-Centric Mechanism for Sequentially Releasing Graph Datasets under Blowfish Privacy." ACM Transactions on Internet Technology 21, no. 1 (February 2021): 1–25. http://dx.doi.org/10.1145/3431501.

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In this article, we present a privacy-preserving technique for user-centric multi-release graphs. Our technique consists of sequentially releasing anonymized versions of these graphs under Blowfish Privacy. To do so, we introduce a graph model that is augmented with a time dimension and sampled at discrete time steps. We show that the direct application of state-of-the-art privacy-preserving Differential Private techniques is weak against background knowledge attacker models. We present different scenarios where randomizing separate releases independently is vulnerable to correlation attacks. Our method is inspired by Differential Privacy (DP) and its extension Blowfish Privacy (BP). To validate it, we show its effectiveness as well as its utility by experimental simulations.

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27

Rajesh Kanna, M. R., R. Pradeep Kumar, Soner Nandappa, and Ismail Naci Cangul. "On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method." Applied Mathematics and Nonlinear Sciences 5, no. 2 (August 10, 2020): 85–98. http://dx.doi.org/10.2478/amns.2020.2.00017.

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AbstractThree main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modifications of the notion of energy have been introduced since then. Most of them are defined by means of graph matrices constructed by vertex degrees. In this paper we define another type of energy called q-distance energy by means of distances and matrices. We study some fundamental properties and also establish some upper and lower bounds for this new energy type.

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28

Li, Xiaolin, Li Xu, Hongyan Zhang, and Qikui Xu. "Differential privacy preservation for graph auto-encoders: A novel anonymous graph publishing model." Neurocomputing 521 (February 2023): 113–25. http://dx.doi.org/10.1016/j.neucom.2022.11.083.

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29

Farman, Mariam R., Ivo L. Hofacker, and Fabian Amman. "MSF: Modulated Sub-graph Finder." F1000Research 7 (March 22, 2019): 1346. http://dx.doi.org/10.12688/f1000research.16005.2.

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High throughput techniques such as RNA-seq or microarray analysis have proven tobe invaluable for the characterization of global transcriptional gene activity changesdue to external stimuli or diseases. Differential gene expression analysis (DGEA) is the first step in the course of data interpretation, typically producing lists of dozens to thousands of differentially expressed genes. To further guide the interpretation of these lists, different pathway analysis approaches have been developed. These tools typically rely on the classification of genes into sets of genes, such as pathways, based on the interactions between the genes and their function in a common biological process. Regardless of technical differences, these methods do not properly account for cross talk between different pathways and rely on binary separation into differentially expressed gene and unaffected genes based on an arbitrarily set p-value cut-off. To overcome this limitation, we developed a novel approach to identify concertedly modulated sub-graphs in the global cell signaling network, based on the DGEA results of all genes tested. To this end, expression patterns of genes are integrated according to the topology of their interactions and allow potentially to read the flow of information and identify the effectors. The described software, named Modulated Sub-graph Finder (MSF) is freely available at https://github.com/Modulated-Subgraph-Finder/MSF.

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30

Diblík, Josef, and Mária Kúdelčíková. "Positive Solutions of Advanced Differential Systems." Scientific World Journal 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/613832.

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We study asymptotic behavior of solutions of general advanced differential systemsy˙(t)=F(t,yt), whereF:Ω→ℝnis a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument andΩis a subset inℝ×Crn,Crn:=C([0,r],ℝn),yt∈Crn, andyt(θ)=y(t+θ),θ∈[0,r]. A monotone iterative method is proposed to prove the existence of a solution defined fort→∞with the graph coordinates lying between graph coordinates of two (lower and upper) auxiliary vector functions. This result is applied to scalar advanced linear differential equations. Criteria of existence of positive solutions are given and their asymptotic behavior is discussed.

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31

Zheng, Yao, Dong Hua Zhou, Zhi Lun Ouyang, Jing Lin Zhu, and Chun Xiu Han. "Nomo-Graph for Determining the Effective Length of Columns with the Leaning Columns." Applied Mechanics and Materials 610 (August 2014): 88–93. http://dx.doi.org/10.4028/www.scientific.net/amm.610.88.

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In this paper the elastic load-carrying capacity of the stability columns, which provide lateral support to leaning columns, is investigated. The solution to the effective length of the columns is obtained by using of differential equations. The influences of the leaning columns on the stabilizing columns are studied and the nomo-graphs for determining effective length of the supporting columns are made for practical using. The effective length of columns can be quickly determined by using the nomo-graph, which shows intuitively the trends of the curves in the nomo-graph owing to some relevant parameters.

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32

Farman, Mariam R., Ivo L. Hofacker, and Fabian Amman. "MSF: Modulated Sub-graph Finder." F1000Research 7 (August 29, 2018): 1346. http://dx.doi.org/10.12688/f1000research.16005.1.

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High throughput techniques such as RNA-seq or microarray analysis have proven to be invaluable for the characterization of global transcriptional gene activity changes due to external stimuli or diseases. Differential gene expression analysis (DGEA) is the first step in the course of data interpretation, typically producing lists of dozens to thousands of differentially expressed genes. To further guide the interpretation of these lists, different pathway analysis approaches have been developed. These tools typically rely on the classification of genes into sets of genes, such as pathways, based on the interactions between the genes and their function in a common biological process. Regardless of technical differences, these methods do not properly account for cross talk between different pathways and rely on binary separation into differentially expressed gene and unaffected genes based on an arbitrarily set p-value cut-off. To overcome this limitation, we developed a novel approach to identify concertedly modulated sub-graphs in the global cell signaling network, based on the DGEA results of all genes tested. Thereby, expression patterns of genes are integrated according to the topology of their interactions and allow potentially to read the flow of information from the perturbation source to the effectors. The described software, named Modulated Sub-graph Finder (MSF) is freely available at https: //github.com/Modulated-Subgraph-Finder/MSF.

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33

Farman, Mariam R., Ivo L. Hofacker, and Fabian Amman. "MSF: Modulated Sub-graph Finder." F1000Research 7 (April 14, 2019): 1346. http://dx.doi.org/10.12688/f1000research.16005.3.

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High throughput techniques such as RNA-seq or microarray analysis have proven to be invaluable for the characterizing of global transcriptional gene activity changes due to external stimuli or diseases. Differential gene expression analysis (DGEA) is the first step in the course of data interpretation, typically producing lists of dozens to thousands of differentially expressed genes. To further guide the interpretation of these lists, different pathway analysis approaches have been developed. These tools typically rely on the classification of genes into sets of genes, such as pathways, based on the interactions between the genes and their function in a common biological process. Regardless of technical differences, these methods do not properly account for cross talk between different pathways and most of the methods rely on binary separation into differentially expressed gene and unaffected genes based on an arbitrarily set p-value cut-off. To overcome this limitation, we developed a novel approach to identify concertedly modulated sub-graphs in the global cell signaling network, based on the DGEA results of all genes tested. To this end, expression patterns of genes are integrated according to the topology of their interactions and allow potentially to read the flow of information and identify the effectors. The described software, named Modulated Sub-graph Finder (MSF) is freely available at https://github.com/Modulated-Subgraph-Finder/MSF.

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34

Zeid, A. A., and J. L. Overholt. "Singularly Perturbed Bond Graph Models for Simulation of Multibody Systems." Journal of Dynamic Systems, Measurement, and Control 117, no. 3 (September 1, 1995): 401–10. http://dx.doi.org/10.1115/1.2799131.

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This paper develops a bond graph-based formalism for modeling multibody systems in a singularly perturbed formulation. As opposed to classical multibody modeling methods, the singularly perturbed formulation is explicit, which makes it suitable for modular simulation. Kinematic joints that couple rigid bodies are described by a set of differential equations with an order of magnitude smaller time scale than that of the system. Singularly perturbed models of joints can be used to investigate nonlinear properties of joints, such as clearance and friction. The main restriction of this approach is that the simulation may need to be computed using 64 bits precision because of the two-time scale nature of the solution. The formalism is based on developing bond graph models of an elementary set of graphical velocity-based constraint functions. This set can be used to construct bond graphs of any type of mechanical joint. Here, this set is used to develop bond graphs of several joints used in multibody systems and spatial mechanisms. Complex models of multibody systems may now be built by graphically concatenating bond graphs of rigid bodies and bond graphs of joints. The dynamic equations of the system are automatically generated from the resulting bond graph model. The dynamic equation derived from the bond graph are in explicit state space form, ready for numerical integration, and exclude the computationally intensive terms that arise from acceleration analysis.

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35

Basilio, Ludwin A., Sergio Bermudo, Jesús Leaños, and José M. Sigarreta. "The differential of the line graph L(G)." Discrete Applied Mathematics 321 (November 2022): 82–89. http://dx.doi.org/10.1016/j.dam.2022.05.004.

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Choi, Jeongwhan, Hwangyong Choi, Jeehyun Hwang, and Noseong Park. "Graph Neural Controlled Differential Equations for Traffic Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 6 (June 28, 2022): 6367–74. http://dx.doi.org/10.1609/aaai.v36i6.20587.

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Traffic forecasting is one of the most popular spatio-temporal tasks in the field of machine learning. A prevalent approach in the field is to combine graph convolutional networks and recurrent neural networks for the spatio-temporal processing. There has been fierce competition and many novel methods have been proposed. In this paper, we present the method of spatio-temporal graph neural controlled differential equation (STG-NCDE). Neural controlled differential equations (NCDEs) are a breakthrough concept for processing sequential data. We extend the concept and design two NCDEs: one for the temporal processing and the other for the spatial processing. After that, we combine them into a single framework. We conduct experiments with 6 benchmark datasets and 20 baselines. STG-NCDE shows the best accuracy in all cases, outperforming all those 20 baselines by non-trivial margins.

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ElGhawalby, Hewayda, and Edwin Hancock. "Heat Kernel Embeddings, Differential Geometry and Graph Structure." Axioms 4, no. 3 (July 21, 2015): 275–93. http://dx.doi.org/10.3390/axioms4030275.

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Yurko, V. A. "Recovering Differential Operators on a Bush-Type Graph." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 9, no. 2 (2009): 59–65. http://dx.doi.org/10.18500/1816-9791-2009-9-2-59-65.

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Amini, L., C. Jutten, S. Achard, O. David, H. Soltanian-Zadeh, G. A. Hossein-Zadeh, P. Kahane, L. Minotti, and L. Vercueil. "Directed Differential Connectivity Graph of Interictal Epileptiform Discharges." IEEE Transactions on Biomedical Engineering 58, no. 4 (April 2011): 884–93. http://dx.doi.org/10.1109/tbme.2010.2099227.

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Macwan, Kamalkumar R., and Sankita J. Patel. "Node Differential Privacy in Social Graph Degree Publishing." Procedia Computer Science 143 (2018): 786–93. http://dx.doi.org/10.1016/j.procs.2018.10.388.

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Borges de Sousa, João, and Fernando Lobo Pereira. "A Differential Game with Graph Constrained Switching Strategies." IFAC Proceedings Volumes 34, no. 6 (July 2001): 887–91. http://dx.doi.org/10.1016/s1474-6670(17)35292-8.

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Bermudo, S., and H. Fernau. "Lower bounds on the differential of a graph." Discrete Mathematics 312, no. 22 (November 2012): 3236–50. http://dx.doi.org/10.1016/j.disc.2012.07.021.

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Hegde, Sharan, Anjum Khan, and Vinay Prasad T. "First Redefined Zagreb Index of Generalized Transformation Graph." International Journal of Science, Engineering and Management 9, no. 3 (March 16, 2022): 28–32. http://dx.doi.org/10.36647/ijsem/09.03.a006.

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Approximation of the solution of the differential equations is done by Bernoulli polynomial. Bernoulli polynomial and operational matrix of differentiation were used in reducing differential equations into algebraic equations. The method and its application is demonstrated through illustrative examples and found that the method is computationally attractive. The Bernoulli polynomial method has been applied to compare the numerical solution of differential equations with the existing method of Rationalized Haar Function.

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44

Walther, Hans-Otto. "On the solution manifold of a differential equation with a delay which has a zero." Electronic Journal of Qualitative Theory of Differential Equations, no. 31 (2022): 1–10. http://dx.doi.org/10.14232/ejqtde.2022.1.31.

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For a differential equation with a state-dependent delay we show that the associated solution manifold X f of codimension 1 in the space C 1 ( [ − r , 0 ] , R ) is an almost graph over a hyperplane, which implies that X f is diffeomorphic to the hyperplane. For the case considered previous results only provide a covering by 2 almost graphs.

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FREIDLIN, MARK, and MATTHIAS WEBER. "ON STOCHASTICITY OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH A SMALL DELAY." Stochastics and Dynamics 05, no. 03 (September 2005): 475–86. http://dx.doi.org/10.1142/s0219493705001547.

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We show that solutions of a class of differential equations with a small delay can be approximated, in a sense, by a stochastic process on a graph associated with the equation. This process moves as a deterministic motion inside any edge of the graph, but, after reaching an interior vertex of the graph, the process chooses one of the other adjacent edges to proceed there with a certain probability. These probabilities are calculated explicitly. The stochasticity is an intrinsic property of the differential equation with small delay.

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Chu, Chia-Chun. "STRUCTURAL SYNTHESIS OF NOVEL BASIC TWO-DOF DIFFERENTIALS." Transactions of the Canadian Society for Mechanical Engineering 37, no. 3 (September 2013): 713–22. http://dx.doi.org/10.1139/tcsme-2013-0059.

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Differentials are very important in many crucial applications, but almost all differentials are geared differential mechanisms. Therefore, the main purpose of this paper is to synthesize non-gear or novel differentials. All kinds of basic differentials can be contained in the results by the method. First, a modified graph representation is provided for differential mechanisms with different type. Second, the design concept of composition and decomposition the geometric constraint is presented. Then, ten fundamental entities and seven properties of differentials are collated. Finally, eight feasible results with two degrees-of-freedom and two basic loops are obtained. About the eight results, there are two are existing designs and the other six are novel.

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Provotorov, Vyacheslav V., Sergey M. Sergeev, and Hoang Van Nguyen. "Point control of a differential-difference system with distributed parameters on the graph." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 17, no. 3 (2021): 277–86. http://dx.doi.org/10.21638/11701/spbu10.2021.305.

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The article considers the problem of point control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. The differential- difference system is closely related to the evolutionary differential system and moreover the properties of the differential system are preserved. This connection is established by the universal method of semi-discretization in a time variable for a differential system, which provides an effective tool in order to find conditions for unique solvability and continuity on the initial data for the differential-difference system. For this differential-difference system, a special case of the optimal control problem is studied: the problem of point control action on the controlled differential-difference system is considered by the control, concentrated at all internal nodes of the graph. At the same time, the restrictive set of permissible controls is set by the means of conditions depending on the nature of the applied tasks. In this case, the controls are concentrated at the end points of the edges adjacent to each inner node of the graph. This is a characteristic feature of the study presented, quite often used in practice when building a mechanism for managing the processes of transportation of different kinds of masses over network media. The study essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — obtained ratios that determine optimal point control. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on the graph, which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

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RAMOS, ÁLVARO KRÜGER. "THE MEAN CURVATURE EQUATION ON SEMIDIRECT PRODUCTS : HEIGHT ESTIMATES AND SCHERK-LIKE GRAPHS." Journal of the Australian Mathematical Society 101, no. 1 (February 26, 2016): 118–44. http://dx.doi.org/10.1017/s1446788715000713.

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In the ambient space of a semidirect product $\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$, we consider a connected domain ${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$. Given a function $u:{\rm\Omega}\rightarrow \mathbb{R}$, its ${\it\pi}$-graph is $\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$. In this paper we study the partial differential equation that $u$ must satisfy so that $\text{graph}(u)$ has prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a ${\it\pi}$-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on ${\rm\Omega}$ and on the supremum the function attains on the boundary of ${\rm\Omega}$. When $\text{trace}(A)>0$, we prove that the oscillation of a minimal graph, assuming the same constant value $n$ along the boundary, tends to zero when $n\rightarrow +\infty$ and goes to $+\infty$ if $n\rightarrow -\infty$. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal ${\it\pi}$-graphs assuming the value zero along a piecewise smooth curve ${\it\gamma}$ with endpoints $p_{1},\,p_{2}$ and having as boundary ${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$.

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Dorff, Michael, Ryan Viertel, and Magdalena Wołoszkiewicz. "Convex Combinations of Minimal Graphs." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/724268.

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Given a collection of minimal graphs,M1,M2,…,Mn, with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions onM1,M2,…,Mnso that a convex combination of them will be a minimal graph. We will then provide two examples, taking a convex combination of Scherk's doubly periodic surface with the catenoid and Enneper's surface, respectively.

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WILDS, ROY, and LEON GLASS. "AN ATLAS OF ROBUST, STABLE, HIGH-DIMENSIONAL LIMIT CYCLES." International Journal of Bifurcation and Chaos 19, no. 12 (December 2009): 4055–96. http://dx.doi.org/10.1142/s0218127409025225.

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We present a method for constructing dynamical systems with robust, stable limit cycles in arbitrary dimensions. Our approach is based on a correspondence between dynamics in a class of differential equations and directed graphs on the n-dimensional hypercube (n-cube). When the directed graph contains a certain type of cycle, called a cyclic attractor, then a stable limit cycle solution of the differential equations exists. A novel method for constructing regulatory systems that we call minimal regulatory networks from directed graphs facilitates investigation of limit cycles in arbitrarily high dimensions. We identify two families of cyclic attractors that are present for all dimensions n ≥ 3: cyclic negative feedback and sequential disinhibition. For each, we obtain explicit representations for the differential equations in arbitrary dimension. We also provide a complete listing of minimal regulatory networks, a representative differential equation, and a bifurcation analysis for each cyclic attractor in dimensions 3–5. This work joins discrete concepts of symmetry and classification with analysis of differential equations useful for understanding dynamics in complex biological control networks.

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