Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number

Balanced signed graphs appear in the context of social groups with symmetric relations between individuals where a positive edge represents friendship and a negative edge represents enmities between the individuals. The frustration number f of a signed graph is the size of the minimal set F of vertices whose removal results in a balanced signed graph; hence, a connected signed graph G˙ is balanced if and only if f=0. In this paper, we consider the balance of G˙ via the relationships between the frustration number and eigenvalues of the symmetric Laplacian matrix associated with G˙. It is known that a signed graph is balanced if and only if its least Laplacian eigenvalue μn is zero. We consider the inequalities that involve certain Laplacian eigenvalues, the frustration number f and some related invariants such as the cut size of F and its average vertex degree. In particular, we consider the interplay between μn and f.

On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

Cliques in high-dimensional random geometric graphs

Random geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The n vertices of a random geometric graph are points in d-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when d is fixed. However, the case of growing dimension d is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when $$n\rightarrow \infty$$ n → ∞ , and $$d \rightarrow \infty$$ d → ∞ , and average vertex degree grows significantly slower than n. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a. s. if only $$d \gg \log ^{1 + \epsilon } n$$ d ≫ log 1 + ϵ n . As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $$\log ^2 n \ll d \ll \log ^3 n$$ log 2 n ≪ d ≪ log 3 n that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small n.

Constraining cosmology with big data statistics of cosmological graphs

ABSTRACT By utilizing large-scale graph analytic tools implemented in the modern big data platform, apache spark, we investigate the topological structure of gravitational clustering in five different universes produced by cosmological N-body simulations with varying parameters: (1) a WMAP 5-yr compatible ΛCDM cosmology, (2) two different dark energy equation of state variants, and (3) two different cosmic matter density variants. For the big data calculations, we use a custom build of standalone Spark/Hadoop cluster at Korea Institute for Advanced Study and Dataproc Compute Engine in Google Cloud Platform with sample sizes ranging from 7 to 200 million. We find that among the many possible graph-topological measures, three simple ones: (1) the average of number of neighbours (the so-called average vertex degree) α, (2) closed-to-connected triple fraction (the so-called transitivity) $\tau _\Delta$, and (3) the cumulative number density ns ≥ 5 of subgraphs with connected component size s ≥ 5, can effectively discriminate among the five model universes. Since these graph-topological measures are directly related with the usual n-points correlation functions of the cosmic density field, graph-topological statistics powered by big data computational infrastructure opens a new, intuitive, and computationally efficient window into the dark Universe.

Quantitative description and classification of protein structures by a novel robust amino acid network: interaction selective network (ISN)

To quantitatively categorize protein structures, we developed a quantitative coarse-grained model of protein structures with a novel amino acid network, the interaction selective network (ISN), characterized by the links based on interactions in both the main and side chains. We found that the ISN is a novel robust network model to show the higher classification probability in the plots of average vertex degree (k) versus average clustering coefficient (C), both of which are typical network parameters for protein structures, and successfully distinguished between “all-α” and “all-β” proteins. On the other hand, one of the typical conventional networks, the α-carbon network (CAN), was found to be less robust than the ISN, and another typical network, atomic distance network (ADN), failed to distinguish between these two protein structures. Considering that the links in the CAN and ADN are defined by the interactions only between the main chain atoms and by the distance of the closest atom pair between the two amino acid residues, respectively, we can conclude that reflecting structural information from both secondary and tertiary structures in the network parameters improves the quantitative evaluation and robustness in network models, resulting in a quantitative and more robust description of three-dimensional protein structures in the ISN.

Development of a Novel Shaft Dryer for Coal-Based Green Needle Coke Drying Process

The industry of coal-based green needle coke develops rapidly in recent years. The green coke produced by the delayed coking process usually has a moisture content of 10%–25%, which damages the calcining kiln and needle coke quality. A standing dehydration tank is currently used to reduce the moisture content of green coke. However, this process has several weaknesses such as unstable operation, large land area occupation, and low productivity. To solve this issue, a novel drying system with a shaft dryer proposed in this work is suitable for green coke drying. Moreover, the performances of the green coke are investigated to design the proposed shaft dryer. The experimental result shows that the average vertex angle of the pile of green cokes is 109.2°. The pressure drop of the dryer increases linearly with the green coke bed height, and the green coke with a larger size has a smaller pressure drop. The specific pressure drops are 5714, 5554, 5354, and 5114 Pa/m, with median green coke sizes of 26.85, 29.00, 30.45, and 31.80 mm, respectively. Tooth spacing is another important parameter which influences the mass of green coke leakage. The optimal tooth spacing and rotary speed of the rollers are determined by the required production yield.

3560 Using Research Performance Progress Report data to Explore CTSI-Stakeholder Engagement through Network Analysis

OBJECTIVES/SPECIFIC AIMS: To develop a social network model of collaborations within and external to the University of Rochester Medical Center (URMC) CTSI using data from the annual Research Performance Progress Report (RPPR) as well as other sources, to provide longitudinal evaluation of the CTSI’s engagement with key stakeholder groups. METHODS/STUDY POPULATION: The annually submitted RPPR follows a specific format with well-defined sections. The Highlights, Milestones and Challenges Report includes areas in which CTSI function leaders provide details about program integration and innovation, including collaborations with other functions or external groups. The Highlights, Milestones and Challenges Report was qualitatively coded to identify function-collaborator dyads. Each entity in the dyad became a node in the network. Nodes were connected by edges named by the dyads. The network included two types of nodes. The first were CTSI internal functions/programs, i.e. the entities that submitted RPPR sections and formed an interconnected sub-network. The second type of nodes were entities external to the CTSI (collaborators, internal or external to the CTSI site). These entities were named by functions submitting RPPR narratives. External nodes with similar meanings were consolidated. Duplicate edges were removed. CTSI-external nodes were grouped into five stakeholder categories: URMC, University of Rochester (UR), community, other CTSA institutions, CTSA consortium. Thus, these nodes were connected to the CTSI internal nodes, but not to each other. A second source of collaboration data was function-reported internal metrics. As part of the internal metric data collection, functions list partners who play a role in improving metric data or who are responsible for providing data. Partners identified in the internal metrics data, but not specified in the RPPR, were added to the network. RESULTS/ANTICIPATED RESULTS: Twenty-three internal CTSI functions submitted an RPPR and represent the CTSI internal nodes. Internal CTSI functions identified 235 collaborations (edges): 125 collaborations with other CTSI internal functions, 57 collaborations with URMC entities, 14 with UR entities, 15 with the external community, 15 with other institutions (CTSA hubs and other universities), and 9 with CTSA consortium entities. Thirty-eight of the collaborations were identified in the internal metrics partners section. In total, the network comprised 104 nodes. Graph density was.022 for full network and.21 for the CTSI internal sub-network. The global clustering coefficient, a measure of connectivity, for the CTSI internal sub-network was.252. DISCUSSION/SIGNIFICANCE OF IMPACT: The RPPR provides an underutilized source of data for annually repeated analyses of internal and external CTSI collaborations and is a way to enhance use of this routinely collected information. Analyses of the network yield metrics for measuring CTSI reach and impact on stakeholder groups over time. For example, measures such as number of nodes representing entities external to CTSI and average vertex degree of the CTSI Internal nodes track aspects of CTSI collaborations. Visualizations using different layouts or highlighting different sub-networks provide a representation of CTSI engagement with the communities of stakeholders as well as insights to relationships between functions, regions of collaboration, and areas of gaps. These data also provide an important new mechanism to engage the CTSI leadership and function leads in understanding how their work contributes to the overall network and synergies they have with each other.

Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

We consider the ensemble of [Formula: see text] real random symmetric matrices [Formula: see text] obtained from the determinant form of the Ihara zeta function associated to random graphs [Formula: see text] of the long-range percolation radius model with the edge probability determined by a function [Formula: see text]. We show that the normalized eigenvalue counting function of [Formula: see text] weakly converges in average as [Formula: see text], [Formula: see text] to a unique measure that depends on the limiting average vertex degree of [Formula: see text] given by [Formula: see text]. This measure converges in the limit of infinite [Formula: see text] to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

Graph Theoretical Representation, Analysis and Synthesis of Amorphous Metal Oxide Networks

ABSTRACTWith the advent of amorphous oxide semiconductors (AOS) like amorphous indium gallium zinc oxide (a-IGZO), the analysis and prediction of amorphous structures has regained importance, more so, since first principles based studies are being increasingly employed to explain device behavior. Negative bias illumination stress in a-IGZO thin film transistors is one such example. However, the amorphous atomic structure is complex and defect or dopant studies require each site to be modeled independently and this leads to significant computational costs. Therefore, a simplification in the representation of the amorphous oxide network is effected so that it may lead to identifying similar atomic sites. The amorphous network is visualized as a network of polyhedra. The polyhedra has at its center a cation with the bonded oxygen atoms at the vertices and it comprises the short range interactions characterized by bond lengths and bond angles. Based on a first principles study of 10 a-IGZO models containing 36 cations each, it was found that the 360 polyhedra of the a-IGZO models can actually be described with only ten polyhedral motifs. These polyhedra are then connected to each other via a shared vertex or an edge; face-sharing was not observed in these models. Graph theory is used to map this network using either a graph of cationic polyhedra as the nodes or a bipartite graph (composed of cations and anions as individual nodes), each of which is described using the respective adjacency matrix. The second nearest interactions are characterized by the degree of each vertex and each atomic site is now characterized by a polyhedron and network metrics; and hence, can be compared with same-element sites. The changes in network itself, are quantified as the composition changes, when varying the ratio of In:Ga:Zn in a-IGZO. For example, the average vertex connectivity of a pair of indium sites reflects on the continuity of overlap between the In-5s orbitals which compose the conduction band minimum in a-IGZO, which, in turn, affects the transport properties of the semiconductor. Thus, the long range interactions of the physical amorphous network are described by the graph metrics. Moreover, evolutionary algorithm in conjunction with this graph theoretic representation can be used to generate new amorphous models. Two parent graph are chosen and then spliced and then bred. The new graph is then reverse-engineered to form an amorphous model which then undergoes ionic and volume relaxation in the framework of density functional theory. The resulting graph is the child and the new amorphous model, with the energy as the fitness criteria.

An application of graph theory to linguistic complexity

This article introduces a new measure of linguistic complexity which is based on the dual nature of the linguistic sign. Complexity is analyzed as consisting of three components, namely the conceptual complexity (complexity of the signified), the formal complexity (complexity of the signifier) and the form-meaning correspondence complexity. I describe a way of plotting the form-meaning relationship on a graph with two tiers (the form tier and the meaning tier) and apply a complexity measure from graph theory (average vertex degree) to assess the complexity of such graphs. The proposed method is illustrated by estimating the complexity of full noun phrases (determiner + adjective + noun) in English, Swedish, and German. I also mention the limitations and the problems which might arise when using this method.