Modular Graph Research Articles

Elliptic modular graph forms. Part I. Identities and generating series

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.

EASS of MMD graphs

Modular multiplicative divisor graph G with n vertices is a bijection g from V(G) to {1,2,…, n} and induced function h from edges of G to {0,1,…,n-1}, h(vw)=g(v)g(w) mod n for every vw in edges of G then adding the labels of all the edges is a multiple of n. Here studies MMD labeling of an even arbitrary supersubdivision (EASS) of cycle Cn .

Wiener index and Steiner 3-Wiener index of graphs

Let [Formula: see text] be a set of vertices of a connected graph [Formula: see text]. The Steiner distance of [Formula: see text] is the minimum size of a connected subgraph of [Formula: see text] containing all the vertices of [Formula: see text]. The sum of all Steiner distances on sets of size [Formula: see text] is called the Steiner [Formula: see text]-Wiener index. A graph [Formula: see text] is modular if for every three vertices [Formula: see text] there exists a vertex [Formula: see text] that lies on the shortest path between every two vertices of [Formula: see text]. The Steiner 3-Wiener index of a modular graph is obtained in terms of its Wiener index. As concrete examples, we discuss the case of Fibonacci, Lucas cubes and the Cartesian product of modular graphs. The Steiner Wiener index of block graphs is also studied.

On the modular Mckay graph of SLn(p) with respect to its standard representation

Let F be an algebraically closed field of prime characteristic p. The modular McKay graph of G:=SLn(p) with respect to its standard FG-module W is the connected, directed graph whose vertices are the irreducible FG-modules and for which there is an edge from a vertex V1 to V2 if V2 occurs as a composition factor of the tensor product V1⊗W. We show that the diameter of this modular McKay graph is 12(p−1)(n2−n).

Two-loop superstring five-point amplitudes. Part II. Low energy expansion and S-duality

In an earlier paper, we constructed the genus-two amplitudes for five external massless states in Type II and Heterotic string theory, and showed that the α′ expansion of the Type II amplitude reproduces the corresponding supergravity amplitude to leading order. In this paper, we analyze the effective interactions induced by Type IIB superstrings beyond supergravity, both for U(1)R-preserving amplitudes such as for five gravitons, and for U(1)R-violating amplitudes such as for one dilaton and four gravitons. At each order in α′, the coefficients of the effective interactions are given by integrals over moduli space of genus-two modular graph functions, generalizing those already encountered for four external massless states. To leading and sub-leading orders, the coefficients of the effective interactions D2ℛ5 and D4ℛ5 are found to match those of D4ℛ4 and D6ℛ4, respectively, as required by non-linear supersymmetry. To the next order, a D6ℛ5 effective interaction arises, which is independent of the supersymmetric completion of D8ℛ4, and already arose at genus one. A novel identity on genus-two modular graph functions, which we prove, ensures that up to order D6ℛ5, the five-point amplitudes require only a single new modular graph function in addition to those needed for the four-point amplitude. We check that the supergravity limit of U(1)R-violating amplitudes is free of UV divergences to this order, consistently with the known structure of divergences in Type IIB supergravity. Our results give strong consistency tests on the full five-point amplitude, and pave the way for understanding S-duality beyond the BPS-protected sector.

Poisson equations for elliptic modular graph functions

We obtain Poisson equations satisfied by elliptic modular graph functions with four links. Analysis of these equations leads to a non–trivial algebraic relation between the various graphs.

A modular graph transformation rule set for IFC‐to‐CityGML conversion

AbstractConversion of Industry Foundation Classes (IFC) building models into CityGML city models is one of the operational scenarios for BIM–GIS integration, with a variety of applications producing and consuming data on either side. Given the in‐depth cross‐domain knowledge required to specify such conversions, the heterogeneity of the IFC input data and the use cases for the resulting CityGML, flexible and configurable solutions are needed that make conversion details accessible to domain specialists. Graph transformation as a conversion method fulfils these requirements. We propose to extend the modularity given by single transformation rules at a more coarse‐grained level and identify four layers with modules of associated rules. We describe a self‐contained set of rules across these modules and demonstrate its application to a range of building models.

Massive deformations of Maass forms and Jacobi forms

We define one-parameter "massive" deformations of Maass forms and Jacobi forms. This is inspired by descriptions of plane gravitational waves in string theory. Examples include massive Green's functions (that we write in terms of Kronecker-Eisenstein series) and massive modular graph functions.

On new algorithmic techniques for the weighted vertex coloring problem

The classical NP-hard weighted vertex coloring problem consists in minimizing the number of colors in colorings of vertices of a given graph so that, for each vertex, the number of its colors equals a given weight of the vertex and adjacent vertices receive distinct colors. The weighted chromatic number is the smallest number of colors in these colorings. There are several polynomial-time algorithmic techniques for designing efficient algorithms for the weighted vertex coloring problem. For example, standard techniques of this kind are the modular graph decomposition and the graph decomposition by separating cliques. This article proposes new polynomial-time methods for graph reduction in the form of removing redundant vertices and recomputing weights of the remaining vertices so that the weighted chromatic number changes in a controlled manner. We also present a method of reducing the weighted vertex coloring problem to its unweighted version and its application. This paper contributes to the algorithmic graph theory.

Relations between elliptic modular graphs

We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non-separating node of genus two string invariants that appear in the integrand of the D8ℛ4 interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti-holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given only by the Green function and not its derivatives. This results in a reduction in the number of basis elements in the space of elliptic modular graphs.

Asymptotics of lowest unitary SL(2,C) invariants on graphs

We describe a technique to study the asymptotics of SL(2,C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible.

Modular Graph Forms and Scattering Amplitudes in String Theory

In dieser Dissertation untersuchen wir die Niedrigenergieentwicklung von Streuamplituden geschlossener Strings auf Einschleifenniveau (d.h. auf Genus eins) in einem zehndimensionalen Minkowski-Hintergrund mit Hilfe einer speziellen Klasse von Funktionen, den sogenannten modularen Graphenformen. Diese erlauben eine systematische Berechnung der Niedrigenergieentwicklung und erfullen viele nicht-triviale algebraische- und Differentialgleichungen. Wir studieren diese Relationen detailliert und leiten Basiszerlegungen fur eine grose Zahl modularer Graphenformen her. Eines der Ergebnisse dieser Dissertation ist ein Mathematica-Paket, welches diese Vereinfachungen automatisiert. Wir benutzen diese Techniken, um die fuhrenden Niedrigenergieordnungen der Streuamplitude von vier Gluonen im heterotischen String auf Einschleifenniveau zu berechnen. Fur Stringamplituden auf Baumniveau bildet die Einwertigkeitsabbildung multipler Zetawerte offene Stringamplituden auf geschlossene Stringamplituden ab. Wir zeigen, dass ein bestimmter Vorschlag fur die Definition einer geeigneten einschleifen-Verallgemeinerung, der sogenannten elliptische Einwertigkeitsabbildung, nicht alle Terme im heterotischen String reproduzieren kann. Ferner studieren wir eine Erzeugendenfunktion, die vermutlich die Torusintegrale aller perturbativen Theorien geschlossener Strings enthalt. Wir bestimmen eine Differentialgleichung, die von dieser Erzeugendenfunktion erfullt wird und losen sie mit Hilfe von pfadgeordneten Exponentialen, was auf iterierte Integrale von holomorphen Eisensteinreihen fuhrt. Da eine ahnliche Konstruktion im offenen String zur Verfugung steht, eroffnet dies auserdem eine neue Perspektive auf die elliptische Einwertigkeitsabbildung.%%%%In this thesis, we investigate the low-energy expansion of scattering amplitudes of closed strings at one-loop level (i.e. at genus one) in a ten-dimensional Minkowski background using a special class of functions called modular graph forms. These allow for a systematic evaluation of the low-energy expansion and satisfy many non-trivial algebraic and differential relations. We study these relations in detail, leading to basis decompositions for a large number of modular graph forms which greatly reduce the complexity of the expansions of the integrals appearing in the amplitude. One of the results of this thesis is a Mathematica package which automatizes these simplifications. We use these techniques to compute the leading low-energy orders of the scattering amplitude of four gluons in the heterotic string at one-loop level. For tree-level string amplitudes, the single-valued map of multiple zeta values maps open-string amplitudes to closed-string amplitudes. The definition of a suitable one-loop generalization, a so-called elliptic single-valued map, is an active area of research and we show that a certain conjectural definition for this map, which was successfully applied to maximally supersymmetric amplitudes, cannot reproduce all terms in the heterotic string which has half-maximal…

Zero mode of the Fourier series of some modular graphs from Poincaré series

We consider specific linear combinations of two loop modular graph functions on the toroidal worldsheet with 2s links for s=2,3 and 4. In each case, it satisfies an eigenvalue equation with source terms involving E2s and Es2 only. On removing certain combinations of E2s and Es2 from it, we express the resulting expression as an absolutely convergent Poincaré series. This is used to calculate the power behaved terms in the asymptotic expansion of the zero mode of the Fourier expansion of these graphs in a simple manner.

Detecting Hierarchical and Overlapping Network Communities Based on Opinion Dynamics

It is common for communities in real-world networks to possess hierarchical and overlapping structures, which make community detection even more challenging. In this paper, by investigating consensus process of the classical DeGroot model in opinion dynamics, we propose a novel method based on the cumulative opinion distance (COD) to discover hierarchical and overlapping communities. It is shown that this method is different from those classical algorithms relying on static fitness metrics that depict the inhomogeneous connectivity across the network. The proposed method is validated from two aspects. First, by estimating the eigenvectors of adjacency matrices, we investigate the detectability limit of our algorithms on random networks, which together with the results concerning the convergence speed of consensus guarantees the performance of our method theoretically. Second, experiments on both \tcb{large scale} real-world networks and artificial benchmarks show that our method is very effective and competitive on hierarchical modular graphs. In particular, it outperforms the state-of-the-art algorithms on overlapping community detection.

Generating series of all modular graph forms from iterated Eisenstein integrals

We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes. By analysing the differential equation of the generating series we construct a solution for their low-energy expansion to all orders in the inverse string tension α′. Our solution is expressed through initial data involving multiple zeta values and certain real-analytic functions of the modular parameter of the torus. These functions are built from real and imaginary parts of holomorphic iterated Eisenstein integrals and should be closely related to Brown’s recent construction of real-analytic modular forms. We study the properties of our real-analytic objects in detail and give explicit examples to a fixed order in the α′-expansion. In particular, our solution allows for a counting of linearly independent modular graph forms at a given weight, confirming previous partial results and giving predictions for higher, hitherto unexplored weights. It also sheds new light on the topic of uniform transcendentality of the α′-expansion.

Learning Cross-Modal Context Graph for Visual Grounding

Visual grounding is a ubiquitous building block in many vision-language tasks and yet remains challenging due to large variations in visual and linguistic features of grounding entities, strong context effect and the resulting semantic ambiguities. Prior works typically focus on learning representations of individual phrases with limited context information. To address their limitations, this paper proposes a language-guided graph representation to capture the global context of grounding entities and their relations, and develop a cross-modal graph matching strategy for the multiple-phrase visual grounding task. In particular, we introduce a modular graph neural network to compute context-aware representations of phrases and object proposals respectively via message propagation, followed by a graph-based matching module to generate globally consistent localization of grounding phrases. We train the entire graph neural network jointly in a two-stage strategy and evaluate it on the Flickr30K Entities benchmark. Extensive experiments show that our method outperforms the prior state of the arts by a sizable margin, evidencing the efficacy of our grounding framework. Code is available at https://github.com/youngfly11/LCMCG-PyTorch.

Indoor haze particulate control using knowledge graphs within self-optimizing HVAC control systems

Transboundary haze pollution in South East Asia is posing a threat to conventional design of buildings yet indoor air pollution from haze particulate infiltration has still received less attention in Malaysia compared to haze pollution outdoors. Because of this minimal research effort, indoor building environments have increasingly become very complex environments for facility managers to monitor and control due to the corresponding growth in heterogeneity within building behavioural information and monitoring (sensory) systems. As a solution to this and part of an ongoing study, this paper presents the preliminary process of modelling heterogeneous building information related to indoor air quality (IAQ) (building envelope, sensors, contaminant properties, geometry, occupancy schedules and weather data) within modular and extensible semantic web knowledge graphs (KG). This work argues that this data model can preserve the existential and latent parametric relationships within such information therafter availing an accurate representation of the heterogeneous state-space in machine learning workflows of self-learning building monitors and controllers. Compared to the conventional homogenous feature vectors, KGs hold sufficient context-aware semantics for an algorithmic building control system to smartly monitor the IAQ and autonomously learn to adapt air handling units towards occupant comfort in an energy efficient manner. Specifically, this paper highlights the high-level implementation process of KGs within the deep Q-learning process of the aforementioned control systems. Finally, a brief discussion is provided on how this process reduces the complexity that facility managers face while operating their IAQ control systems followed by the conclusions and future work to be carried out in this study.

Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

SummaryWe propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process and naturally generalizes existing probabilistic models with overlapping block structure to the sparse regime. Our construction builds on vectors of completely random measures and has interpretable parameters, each node being assigned a vector representing its levels of affiliation to some latent communities. We develop methods for efficient simulation of this class of random graphs and for scalable posterior inference. We show that the approach proposed can recover interpretable structure of real world networks and can handle graphs with thousands of nodes and tens of thousands of edges.

From little string free energies towards modular graph functions

We study the structure of the non-perturbative free energy of a one-parameter class of little string theories (LSTs) of A-type in the so-called unrefined limit. These theories are engineered by N M5-branes probing a transverse flat space. By analysing a number of examples, we observe a pattern which suggests to write the free energy in a fashion that resembles a decomposition into higher-point functions which can be presented in a graphical way reminiscent of sums of (effective) Feynman diagrams: to leading order in the instanton parameter of the LST, the N external states are given either by the fundamental building blocks of the theory with N = 1, or the function that governs the counting of BPS states of a single M5-brane coupling to one M2-brane on either side. These states are connected via an effective coupling function which encodes the details of the gauge algebra of the LST and which in its simplest (non-trivial) form is captured by the scalar Greens function on the torus. More complicated incarnations of this function show certain similarities with so-called modular graph functions, which have appeared in the study of Feynman amplitudes in string- and field theory. Finally, similar structures continue to exist at higher instanton orders, which, however, also contain contributions that can be understood as the action of (Hecke) operators on the leading instanton result.

All-order differential equations for one-loop closed-string integrals and modular graph forms

We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.