Exploring the “Middle Earth” of network spectra via a Gaussian matrix function

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrodinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index - an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. We also show that in real-world networks this index is related to the existence of important structural patters, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs

Gaussianization of the spectra of graphs and networks : theory and applications

Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra. In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them. In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra. In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them. In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs

Double gaussianization of graph spectra

The graph spectrum is the set of eigenvalues of a simple graph with n vertices. Here we fold this graph spectrum at a given pair of reference eigenvalues and then exponentiate the resulting folded graph spectrum. This process produces double Gaussianized functions of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidences from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. In the examples that we have examined, they enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove here several results for the trace of the double Gaussianized adjacency matrix of simple graphs–the double Gaussianized Estrada index. Finally we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or inactive ones. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. Such indices reflect the electron population of the HOMO/LUMO and eigenvalues close to them, in the so-called K and L regions of PAHs

Computational Methods for Cognitive and Cooperative Robotics

In the last decades design methods in control engineering made substantial progress in the areas of robotics and computer animation. Nowadays these methods incorporate the newest developments in machine learning and artificial intelligence. But the problems of flexible and online-adaptive combinations of motor behaviors remain challenging for human-like animations and for humanoid robotics. In this context, biologically-motivated methods for the analysis and re-synthesis of human motor programs provide new insights in and models for the anticipatory motion synthesis. This thesis presents the author’s achievements in the areas of cognitive and developmental robotics, cooperative and humanoid robotics and intelligent and machine learning methods in computer graphics. The first part of the thesis in the chapter “Goal-directed Imitation for Robots” considers imitation learning in cognitive and developmental robotics. The work presented here details the author’s progress in the development of hierarchical motion recognition and planning inspired by recent discoveries of the functions of mirror-neuron cortical circuits in primates. The overall architecture is capable of ‘learning for imitation’ and ‘learning by imitation’. The complete system includes a low-level real-time capable path planning subsystem for obstacle avoidance during arm reaching. The learning-based path planning subsystem is universal for all types of anthropomorphic robot arms, and is capable of knowledge transfer at the level of individual motor acts. Next, the problems of learning and synthesis of motor synergies, the spatial and spatio-temporal combinations of motor features in sequential multi-action behavior, and the problems of task-related action transitions are considered in the second part of the thesis “Kinematic Motion Synthesis for Computer Graphics and Robotics”. In this part, a new approach of modeling complex full-body human actions by mixtures of time-shift invariant motor primitives in presented. The online-capable full-body motion generation architecture based on dynamic movement primitives driving the time-shift invariant motor synergies was implemented as an online-reactive adaptive motion synthesis for computer graphics and robotics applications. The last chapter of the thesis entitled “Contraction Theory and Self-organized Scenarios in Computer Graphics and Robotics” is dedicated to optimal control strategies in multi-agent scenarios of large crowds of agents expressing highly nonlinear behaviors. This last part presents new mathematical tools for stability analysis and synthesis of multi-agent cooperative scenarios.In den letzten Jahrzehnten hat die Forschung in den Bereichen der Steuerung und Regelung komplexer Systeme erhebliche Fortschritte gemacht, insbesondere in den Bereichen Robotik und Computeranimation. Die Entwicklung solcher Systeme verwendet heutzutage neueste Methoden und Entwicklungen im Bereich des maschinellen Lernens und der künstlichen Intelligenz. Die flexible und echtzeitfähige Kombination von motorischen Verhaltensweisen ist eine wesentliche Herausforderung für die Generierung menschenähnlicher Animationen und in der humanoiden Robotik. In diesem Zusammenhang liefern biologisch motivierte Methoden zur Analyse und Resynthese menschlicher motorischer Programme neue Erkenntnisse und Modelle für die antizipatorische Bewegungssynthese. Diese Dissertation präsentiert die Ergebnisse der Arbeiten des Autors im Gebiet der kognitiven und Entwicklungsrobotik, kooperativer und humanoider Robotersysteme sowie intelligenter und maschineller Lernmethoden in der Computergrafik. Der erste Teil der Dissertation im Kapitel “Zielgerichtete Nachahmung für Roboter” behandelt das Imitationslernen in der kognitiven und Entwicklungsrobotik. Die vorgestellten Arbeiten beschreiben neue Methoden für die hierarchische Bewegungserkennung und -planung, die durch Erkenntnisse zur Funktion der kortikalen Spiegelneuronen-Schaltkreise bei Primaten inspiriert wurden. Die entwickelte Architektur ist in der Lage, ‘durch Imitation zu lernen’ und ‘zu lernen zu imitieren’. Das komplette entwickelte System enthält ein echtzeitfähiges Pfadplanungssubsystem zur Hindernisvermeidung während der Durchführung von Armbewegungen. Das lernbasierte Pfadplanungssubsystem ist universell und für alle Arten von anthropomorphen Roboterarmen in der Lage, Wissen auf der Ebene einzelner motorischer Handlungen zu übertragen. Im zweiten Teil der Arbeit “Kinematische Bewegungssynthese für Computergrafik und Robotik” werden die Probleme des Lernens und der Synthese motorischer Synergien, d.h. von räumlichen und räumlich-zeitlichen Kombinationen motorischer Bewegungselemente bei Bewegungssequenzen und bei aufgabenbezogenen Handlungs übergängen behandelt. Es wird ein neuer Ansatz zur Modellierung komplexer menschlicher Ganzkörperaktionen durch Mischungen von zeitverschiebungsinvarianten Motorprimitiven vorgestellt. Zudem wurde ein online-fähiger Synthesealgorithmus für Ganzköperbewegungen entwickelt, der auf dynamischen Bewegungsprimitiven basiert, die wiederum auf der Basis der gelernten verschiebungsinvarianten Primitive konstruiert werden. Dieser Algorithmus wurde für verschiedene Probleme der Bewegungssynthese für die Computergrafik- und Roboteranwendungen implementiert. Das letzte Kapitel der Dissertation mit dem Titel “Kontraktionstheorie und selbstorganisierte Szenarien in der Computergrafik und Robotik” widmet sich optimalen Kontrollstrategien in Multi-Agenten-Szenarien, wobei die Agenten durch eine hochgradig nichtlineare Kinematik gekennzeichnet sind. Dieser letzte Teil präsentiert neue mathematische Werkzeuge für die Stabilitätsanalyse und Synthese von kooperativen Multi-Agenten-Szenarien

Metal oxide nanocrystals for light-driven energy storage

In a world struggling to face the disruptive consequences of global warming, developing new energy conversion and storage solutions is of fundamental importance. This PhD thesis focuses on emerging heterostructures based on Indium Tin Oxide nanocrystals (ITO NCs) and two-dimensional Transition Metal Dichalcogenides (2D TMDs) for innovative light-driven optoelectronic nanodevices and energy storage solutions, combining the harvesting, conversion and storage aspects into a unique hybrid nanomaterial. Doped Metal Oxide (MO) NCs are attracting growing interest as nano-supercapacitors due to their ability to store extra charges in their electronic structure with record-high values of capacitance. Remarkably, these materials can be charged with light (i.e., photodoping), a process at the core of this project and so far not understood electronically. Here, the fundamental features involved in the charge accumulation process are investigated and the physics of photodoping explained. Complete control over energetic band bending and depletion layer engineering is demonstrated, exposing the key role of electronically depleted layers in core-shell NCs. Light-induced depletion layer modulation and band bending is the main mechanism responsible for the storage of extra charges in doped MO supercapacitors. Moreover, multi-electron transfer reversible reactions were observed in photodoped NCs when exposed to a frequently used electron acceptor. The coupling between ITO NCs and 2D TMDs allowed the implementation of a novel all-optical localized charge injection scheme for the manipulation of unperturbed 2D materials. Hybrid 0D-2D heterostructures proved all-solid-state photodoping possible, with promising charging dynamics and capacitance values. Theoretical modeling tools were developed, leading to the optimization of the charge storage capacity of 0D NCs. This work is of particular interest for the fabrication of the next-generation of nanostructured light-driven supercapacitors

Hope College Abstracts: 15th Annual Celebration of Undergraduate Research and Creative Performance

The 15th Annual Celebration of Undergraduate Research and Creative Performance was held on April 15, 2016 in the Richard and Helen DeVos Fieldhouse at Hope College and featured student-faculty collaborative research projects. This program is a record reflective of those projects between the 2015-2016 academic year

Association of Christians in the Mathematical Sciences Proceedings 2019

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 29-June 1, 2019 at Indiana Wesleyan University

Comparative investigation of eigenvalue-based molecular descriptors

Molekulski deskriptori su brojevi ili nizovi brojeva koji se koriste za kvantifikovanje molekulske strukture. Posebna klasa molekulskih deskriptora su grafovske invarijante. Poznate su i kao topoloski molekulski deskriptori. Izvo ˇ denje ovih deskriptora omoguceno je zamenom ´ molekula molekulskim grafom. Mnoge korisne matematicke veli ˇ cine mogu se izra ˇ cunati iz ˇ molekulskog grafa, npr. sopstvene vrednosti. Stoga je postalo moguce konstruisati molekulske ´ deskriptore koji se zasnivaju na sopstvenim vrednostima. Oni se nazivaju topoloski molekulski ˇ deskriptori zasnovani na sopstvenim vrednostima. Danas ih ima mnostvo. Samo nekoliko ˇ njih koristi sopstvene vrednosti dobijene iz klasicne matrice susedstva. Me ˇ du njima se isticuˇ energija grafa, Estradin indeks i rezolventna energija. U okviru ove doktorske disertacije izvrseno je uporedno ispitivanje ovih deskriptora. ˇ Prvi deo poglavlja Rezultati i diskusija izvestava o rezultatima u vezi sa istra ˇ zivanjem rela- ˇ cija izmedu energije grafa, Estradinog indeksa i rezolventne energije. Tri topoloska molekulska ˇ deskriptora zasnovana na sopstvenim vrednostima uporedena su pomocu nekoliko skupova ´ alkana i benzenoidnih ugljovodonika. Otkrivene su i diskutovane relacije medu njima. Identifikovani su strukturni parametri koji upravljaju ovim odnosima i dobijene su odgovarajuce´ formule zasnovane na visestrukoj linearnoj regresiji. Pokazalo se da sva tri istra ˇ zena indeksa ˇ kodiraju gotovo iste strukturne informacije o molekulu. Oni se razlikuju samo po stepenu osetljivosti na grananje molekula i po broju nevezivnih molekulskih orbitala. Dalja analiza energije grafa, Estradinog indeksa i rezolventne energije vezana je za degenerativnost ovih deskriptora. Da bi se testirao diskriminativni potencijal ovih deskriptora, korisˇceno je nekoliko klasa izomera hemijskih stabala. U ovim skupovima broj atoma ugljeni- ´ ka se kretao od 9 do 20. Kvantifikovanje degenerativnosti je uradeno pomocu dobro utvr ´ dene velicine. Rezultati pokazuju da energija grafa i Estradin indeks imaju sli ˇ can nivo degenera- ˇ tivnosti. Nagla promena degenerativnosti rezolventne energije u slucaju hemijskih stabala ˇ zahtevala je dodatno ispitivanje. Dobijeni rezultati su pokazali da postoji mnogo hemijskih stabala sa istom rezolventnom energijom. Ona se zovu 푟–ekvienergetska hemijska stabla. Zatim su predstavljeni podaci vezani za pretrazivanje rezolventnih ekvienergetskih hemijskih ˇ stabala. Treci deo poglavlja Rezultati i diskusija donosi rezultate o strukturnoj osetljivosti ener- ´ gije grafa, Estradinog indeksa i rezolventne energije na nekoliko serija katakondenzovanih i perikondenzovanih izomernih benzenoidnih ugljovodonika. Strukturna osetljivost je jedno od najvaznijih i najmanje istra ˇ zenih svojstava grafovskih invarijanti. Nedavno je predstavlje- ˇ na nova metoda za procenu strukturne osetljivosti topoloskih molekulskih deskriptora. Ovaj ˇ algoritam se sastoji od nekoliko razlicitih koraka. Zasnovan je na Tanimoto indeksu i Morga- ˇ novim kruznim fingerprintovima. Utvr ˇ deno je da energija grafa, Estradin indeks i rezolventna energija imaju slicnu strukturnu osetljivost na katakondenzovane izomere. Energija grafa je ˇ najosetljivija na male promene u perikondenzovanim benzenoidnim ugljovodonicima. Pored toga, osetljivost ovih deskriptora testirana je na katakondenzovanim izomerima sa razlicitim ˇ brojem zaliva, uvala i fjordova. Otkriveno je da se vrednost ovih deskriptora postepeno menja sa postepenim povecanjem broja ovih strukturnih detalja. Estradin indeks i rezolventna ´ energija se slicno pona ˇ saju, i u nekim slu ˇ cajevima pokazuju istu strukturnu osetljivost. To se ˇ moze pripisati visokoj korelaciji izme ˇ du njih. U cetvrtom delu poglavlja Rezultati i diskusija predstavljeni su rezultati ispitivanja uticaja ˇ cikla na vrednost energije grafa, Estradinog indeksa i rezolventne energije. Naime, pokazano je da indeksi dobro opisuju fine strukturne detalje, te se moze pretpostaviti da ukoliko znamo ˇ kako je deskriptor koreliran sa strukturom onda mozemo da saznamo i kako osobine zavise ˇ od strukture. U cilju ispitivanja uticaja cikla na vrednost molekulskih deskriptora zasnovanih na sopstvenim vrednostima dizajnirana su tri in silicio eksperimenta . Poslednji deo ovog poglavlja predstavlja rezultate potencijalne hemijske primenljivosti nasih deskriptora. Ta ˇ cnije, ispitan je potencijal predvi ˇ danja fizicko–hemijskih osobina. Ener- ˇ gija grafa, Estradin indeks i rezolventna energija testirani su kao orude za predvidanje tacke ˇ kljucanja, toplote obrazovanja i koeficijenta raspodele oktanol/voda alkana. Pokazano je da ˇ se molekulski deskriptor zasnovan na sopstvenim vrednostima ne moze pojedina ˇ cno koristiti ˇ za uspesno predvi ˇ danje ovih fizicko–hemijskih osobina. Prvi zagreba ˇ cki indeks, broj nula u ˇ spektru i broj metil grupa, takode, moraju biti ukljuceni u modele. Dobijene statisti ˇ cke ve- ˇ licine pokazuju da su modeli konstruisani pomo ˇ cu Estradinog indeksa i rezolventne energije ´ znatno bolji od modela sa energijom grafa. Takav trend je jos izra ˇ zeniji u slu ˇ caju koeficijenta ˇ raspodele oktanol/voda alkanaMolecular descriptors are numbers or series of numbers used for quantification of molecular structure. A special class of molecular descriptors are graph invariants. They are also known as topological molecular descriptors. The derivation of these descriptors has been enabled by the substitution of molecule by a molecular graph. Many useful mathematical quantities may be calculated from a molecular graph, e.g., eigenvalues. Therefore, it became possible to construct molecular descriptors that are using eigenvalues. These are called eigenvalue–based topological molecular descriptors. Today, there are plethora of them. Only few of them are using eigenvalues obtained from the ordinary adjacency matrix. The graph energy, Estrada index, and resolvent energy are the most prominent among them. Within this doctoral dissertation comparative investigation of these descriptors have been performed. The first part of Results and discussion chapter reports results regarding investigation of relationships among graph energy, Estrada index, and resolvent energy. Three eigenvalue– based topological molecular descriptors are compared using several datasets of alkanes and benzenoid hydrocarbons. The relations among them are found and discussed. Structural parameters that govern these relations are identified and the corresponding formulae based on multiple linear regression have been obtained. It has been shown that all three investigated indices are encoding almost the same structural information of a molecule. They differ only by the extent of the sensitivity on a structural branching of a molecule and on the number of non–bonding molecular orbitals. Further analysis of the graph energy, the Estrada index, and the resolvent energy is concerned with the degeneracy of these descriptors. To test discriminative potential of these descriptors, several classes of chemical-tree-isomers have been employed. In these sets number of carbon atoms ranged from 9 up to 20. The quantification of degeneracy has been done using well–established measure. The results show that graph energy and Estrada index exert similar degeneracy level. The specious degeneracy of the resolvent energy in the case of chemical trees is discussed. Obtained results indicated that there are many chemical trees with the same resolvent energy. These are called 푟–equienergetic chemical trees. Then, the results on searching for resolvent equienergetic chemical trees are given. The third part of Results and discussion chapter brings results on structural sensitivity of the graph energy, the Estrada index, and the resolvent energy on several series of catacondensed and pericondensed isomeric benzenoid hydrocarbons. Structural sensitivity is one of the most important and the least investigated property of graph invariants. Recently, a novel method for assessing the structural sensitivity of topological molecular descriptors was put forward. This algorithm consists of several different steps. It is based on Tanimoto index and Morgan circular fingerprints. It was found that graph energy, Estrada index, and resolvent energy exert similar structural sensitivity on catacondensed isomers. The graph energy showed best performance on pericondensed molecules. Additionally, the sensitivities of these descriptors were tested on the catacondensed isomers with the increasing number of bays, coves, and fjords. It was revealed that these descriptors gradually change with the increasing number of these structural features. The Estrada index and resolvent energy perform similarly and in some cases with the same structural sensitivity. This may be attributed to the high correlation between them. The fourth part of the Results and discussion chapter presents the results of the examination of the influence of the cycle on the value of the graph energy, the Estrada index, and the v resolvent energy. Namely, it has been shown that indices describe fine structural details well, so it can be assumed that if we know how the descriptor correlates with the structure then we can also find out how the properties depend on the structure. In order to examine the influence of a cycle on the value of molecular descriptors based on the eigenvalues, three 푖푛 푠푖푙푖푐표 experiments were designed. The last part of this chapter presents results of potential chemical applicability of our descriptors. More precisely, predictive potential of eigenvalue–based topological molecular descriptors was examined. The graph energy, the Estrada index, and the resolvent energy were tested as parameters for the prediction of boiling points, heats of formation, and octanol/water partition coefficients of alkanes. It was shown that an eigenvalue–based molecular descriptor cannot be individually used for successful prediction of these physico–chemical properties. The first Zagreb index, the number of zeros in the spectrum and the number of methyl groups must be also involved in the models. Performed statistics showed that the models constructed using the Estrada index and the resolvent energy are significantly better than ones with the graph energy. Such trend is even more noticeable in the case of octanol/water partition coefficients of alkanes