Configuration Graph Research Articles

On the maximum vertex degree in a conditional configuration graph

We consider configuration graphs under the condition that the sum of vertex degrees is bounded from above by n. The vertex degrees are independent identically distributed random variables. The distribution of the vertex degree ξ is unknown and has the only limiting condition that P{ξ=k} пропорционально d/( k g ln h k), where d>0, g>1, h≥0. We obtained the limit distributions of the maximum vertex degree as the number of graph vertices and n tends to infinity.

On the connectivity condition for an internet graph with a variable parameter of the vertex degree distribution

We consider a configuration graph with N vertices, whose degrees are independent and identically distributed according to power-law distribution with the parameter τ = τ(N). They are equal to the number of each vertex’s numbered semi-edges. The graph is constructed by joining all of the semi-edges pairwise equiprobably to form edges. Such models can be used to adequately describe various communication networks and Internet topologies. The properties of the graph depend on the behaviour of the parameter τ. The paper investigates the conditions under which a random configuration graph is asymptotically connected as N → ∞.

A new perspective and method for understanding the relationship between characters and words in the Shang dynasty

Relationships between characters and words can be divided into two levels, the relationship among characters and the relationship among words – one of the core problems in Chinese philology. The relationships in Oracle Bone Inscriptions (OBIs) are very complicated due to the diversity of the configurations of graphs and the unclearness of phonetic information, which cause great difficulty in the study of these relationships, including whether characters are variants or have distinct meanings. OBIs have their own particularity, namely, a high degree of hieroglyphics. When studying the relationship between characters and words in OBIs, we are limited to actual divination materials and exploration of characters’ design intention. In this regard, relevant theories and methods in cognitive linguistics provide reference. In terms of characters, graphemes in the lao 牢-set, xian 陷-set, zhu 逐-set, kan 坎-set and che 车-set are visually distinct, but in terms of language are variants representing one word.

Optimized Selection of Reliable and Cost-Effective Safety-Critical System Architectures

We address the problem of synthesizing safety-critical embedded and cyber-physical system architectures to minimize a cost function while guaranteeing the desired reliability. We represent a system architecture as a configurable graph in which both the nodes (components) and edges (interconnections) may fail. We then propose a compact analytical formalism to efficiently reason about the reliability of the overall system based on the failure probabilities of the components, and provide expressions of the design constraints that avoid exhaustive enumeration of failure cases on all possible graph configurations. Based on these constraints, we cast the synthesis problem as an optimization problem and propose monolithic and iterative optimization schemes to decrease the problem complexity. We implement the proposed algorithms in the ArchEx framework, leveraging a pattern-based specification language to facilitate problem formulation. Design problems from aircraft electric power distribution networks and reconfigurable industrial manufacturing systems illustrate the effectiveness of our approach.

Highly degenerate photonic flat bands arising from complete graph configurations

Inspired by complete graph theory, we demonstrate that a metallic claw "meta-atom" structure can carry a high number of nearly degenerate resonant modes. A photonic meta-crystal composing of a lattice of such meta-atoms exhibits a large number of flat bands that are squeezed into a narrow frequency window, and these flat bands can be designed to locate in a wide complete 3D bandgap. The degeneracy dimension (Nf) of the flat bands is determined by the number of branches (Nb) of the metallic claw with Nf=Nb-3, which is geometrically related to the complete graph theory. Different from those flat bands emerging from special lattice arrangements (e.g., Kagome lattice), the isolated flat bands here are insensitive to lattice perturbations. The proposed mechanism offers a new platform for realizing various dispersion-less phenomena and a new paradigm to realize high density of states and spectra compressing.

Non-abelian Quantum Statistics on Graphs

We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

A Two-Stage Stochastic Optimization Approach for Detecting Structurally Stable Clusters in Randomly Changing Graphs

We introduce a stochastic extension for the problem of finding nonhereditary subgraphs of maximum size in randomly changing graphs. The proposed formulation utilizes a two-stage stochastic optimization framework for identifying subgraphs whose structural properties can be preserved and repaired whenever the underlying graph's topology changes randomly. Particular focus is placed on finding nonhereditary subgraphs that represent a diameter-based clique relaxation known as an $s$ -club. A combinatorial branch-and-bound algorithm is developed and demonstrated to be computationally effective on various configurations of random and real-life graphs.

Non-Abelian Anyons on Graphs from Presentations of Graph Braid Groups

The aim of this paper is to analyse algorithms for constructing presentations of graph braid groups from the point of view of anyonic quantum statistics on graphs. In the first part of this paper, we provide a comprehensive review of an algorithm for constructing so-called minimal Morse presentations of graph braid groups that relies on discrete Morse theory. Next, we introduce the notion of a physical presentation of a graph braid group as a presentation whose generators have a direct interpretation as particle exchanges. We show how to derive a physical presentation of a graph braid group from its minimal Morse presentation. In the second part of the paper, we study unitary representations of graph braid groups that are constructed from their presentations. We point out that algebraic objects called moduli spaces of flat bundles encode all unitary representations of graph braid groups. For $2$-connected graphs, we conclude the stabilisation of moduli spaces of flat bundles over graph configuration spaces for large numbers of particles. Moreover, we set out a framework for studying locally abelian anyons on graphs whose non-abelian properties are only encoded in non-abelian topological phases assigned to cycles of the considered graph.

ON THE DISTRIBUTION OF THE SECOND DEGREES OF CONFIGURATION GRAPHS VERTICES

The object is configuration graphs with N vertices, numbered from 1 to N, whose vertex degrees are independent identically distributed random variables. The second degree η(2) of an arbitrary vertex A of a configuration graph is equal to the sum of the degrees of the vertices adjacent to the vertex A excluding the edges going to A. As N →∞, the form of the generating function and the distribution of the random variable η(2) are found for graphs whose vertex degrees have the Poisson distribution. Also, as N →∞, the form of the generating function of a random variable η(2) is obtained for graphs with the vertex degree distribution pk > 0, k= 1, 2, . . . , such that pk ∼ d/kg( lnk)h, h 0, g>7/3, d>0, as k→∞.

An application of the theory of $${{\,\mathrm{FI}\,}}$$FI-algebras to graph configuration spaces

Recent work of An et al. (Subdivisional spaces and graph braid groups. arXiv:1708.02351, 2019), as well as Ramos (Algebraic Geom Topol. arXiv:1609.05611, 2018), has shown that the homology groups of configuration spaces of graphs can be equipped with the structure of a finitely generated graded module over a polynomial ring. In this work we study this module structure in certain families of graphs using the language of $${{\,\mathrm{FI}\,}}$$-algebras recently explored by Nagel and Romer (FI- and OI-modules with varying coefficients. arXiv:1710.09247, 2017). As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.

Collapsible subgraphs of a 4-edge-connected graph

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger’s result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connected graph, in Catlin et al. (2009), it is shown that every 4-edge-connected graph remains collapsible after removing any two edges. We prove the following. •[(i)] Every 4-edge-connected G contains two vertices x,y such that one of x and y has minimum degree in G and both G−x and G−y are collapsible.•[(ii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤3. Then G−X is collapsible if and only if X is not contained in a 4-edge-cut of G.•[(iii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤4. Then G−X is collapsible if and only if G−X is not contractible to a member in {K2c,K2,K2,2,K2,3,K2,4}. These extend former results of Jaeger (1979) and Catlin (1988).

Model-based clustering with sparse covariance matrices

Finite Gaussian mixture models are widely used for model-based clustering of continuous data. Nevertheless, since the number of model parameters scales quadratically with the number of variables, these models can be easily over-parameterized. For this reason, parsimonious models have been developed via covariance matrix decompositions or assuming local independence. However, these remedies do not allow for direct estimation of sparse covariance matrices nor do they take into account that the structure of association among the variables can vary from one cluster to the other. To this end, we introduce mixtures of Gaussian covariance graph models for model-based clustering with sparse covariance matrices. A penalized likelihood approach is employed for estimation and a general penalty term on the graph configurations can be used to induce different levels of sparsity and incorporate prior knowledge. Model estimation is carried out using a structural-EM algorithm for parameters and graph structure estimation, where two alternative strategies based on a genetic algorithm and an efficient stepwise search are proposed for inference. With this approach, sparse component covariance matrices are directly obtained. The framework results in a parsimonious model-based clustering of the data via a flexible model for the within-group joint distribution of the variables. Extensive simulated data experiments and application to illustrative datasets show that the method attains good classification performance and model quality. The general methodology for model-based clustering with sparse covariance matrices is implemented in the R package mixggm, available on CRAN.

A Retroactive Approach for Dynamic Shortest Path Problem

Dynamic shortest path algorithms modify the existing shortest path tree or graph, taking into account changes in the underlying graph configuration. In the premise of this paper, the dynamic Dijkstra algorithm is specifically considered which is used for the solution of single source shortest path problem in dynamic graphs. Dynamic Single Source Shortest Path (DSSSP) problem is considered from a completely different perspective. For incorporating the dynamic changes into the solution, the retroactive data structure has been used. Dynamics of retroactive data structures provide a natural order for propagating the desired changes in the underlying graph configuration. DSSSP has been solved in efficient way with worst case complexity O(m log n) and also proved the correctness of our proposed algorithm.

Respondent-driven sampling and the homophily configuration graph.

Respondent-Driven Sampling (RDS) is a popular method for surveying hard-to-reach populations, especially in the public health domain. Adjusting for the complex sampling mechanism of the RDS procedure is challenging. We propose a new model for the RDS mechanism motivated by a graph model, which we call the Homophily Configuration Graph. Under this model, we develop a new estimator for population proportions that is robust to seed bias, differential activity, differential recruitment and short recruitment chains. We also connect it to existing RDS theory by showing that, if the sample fraction is small, our estimator limits to the popular Salganik-Heckathorn estimator. We perform simulation studies on both empirically observed networks and networks with known statistical properties, suggesting that this new estimator has less bias than currently recommended estimators.

Evaluating practitioner cyber-security attack graph configuration preferences

Attack graphs and attack trees are a popular method of mathematically and visually representing the sequence of events that lead to a successful cyber-attack. Despite their popularity, there is no standardised attack graph or attack tree visual syntax configuration,and more than seventy self-nominated attack graph and twenty attack tree configurations have been described in the literature – each of which presents attributes such as preconditions and exploits in a different way. This research proposes a practitioner-preferred attack graph visual syntax configuration which can be used to effectively present cyber-attacks.Comprehensive data on participant (n=212) preferences was obtained through a choice based conjoint design in which participants scored attack graph configuration based on their visual syntax preferences. Data was obtained from multiple participant groups, which included lecturers, students and industry practitioners with cyber-security specific or general computer science backgrounds.The overall analysis recommends a winning representation with the following attributes. The flow of events is represented top-down as in a flow diagram – as opposed to a fault tree or attack tree where it is presented bottom-up, preconditions – the conditions required for a successful exploit, are represented as ellipses and exploits are represented as rectangles. These results were consistent across the multiple groups and across scenarios which differed according to their attack complexity. The research tested a number of bottom-up approaches – similar to that used in attack trees. The bottom-up designs received the lowest practitioner preference score indicating that attack trees – which also utilise the bottom-up method, are not a preferred design amongst practitioners – when presented with an alternative top-down design. Practitioner preferences are important for any method or framework to become accepted, and this is the first time that an attack modelling technique has been developed and tested for practitioner preferences.

Limit distributions of vertex degrees in a conditional configuration graph

The configuration graph where vertex degrees are independent identically distributed random variables is often used for modeling of complex networks such as the Internet. We consider a random graph consisting of N vertices. The random variables η 1 ,….,η N are equal to the degrees of vertices with the numbers 1,… ,N The probability P{η i =k}, i=1,…, N, k=1,2,…., is equivalent to h(k)/k τ as k→∞, where h(x) is a slowly varying function integrable in any finite interval, τ>1. We obtain the limit distributions of the maximum vertex degree and the number of vertices with a given degree under the condition that the sum of degrees is equal to n and N,n→∞.

Limit distributions of maximum vertex degree in a conditional configuration graph

We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables following the power-law distribution with positive parameter τ. We study the random graphs under the conditions that the sum of vertex degrees does not exceed n and the parameter τ is a random variable uniformly distributed on the interval [a,b], 1≤a<b<∞. We obtain the limit distributions of maximum vertex degree for for different relations between the parameters N and n tending to infinity.

The structure of a configuration graph with a normally distributed parameter of the power series distribution of vertex degrees

We consider configuration graphs with N vertices. The degrees of the vertices are independent random variables identically distributed according to the power law, with a positive parameter τ . They are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. We study the subset of such random graphs under the condition that the sum of vertex degrees is known and it is equal to n. Let τ be a random variable following a truncated normal distribution on an arbitrary fixed finite interval. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree for various zones of N and n tendency to infinity.

Imitation Upper Confidence Bound for Bandits on a Graph

We consider a graph of interconnected agents implementing a common policy and each playing a bandit problem with identical reward distributions. We restrict the information propagated in the graph such that agents can uniquely observe each other's actions. We propose an extension of the Upper Confidence Bound (UCB) algorithm to this setting and empirically demonstrate that our solution improves the performance over UCB according to multiple metrics and within various graph configurations.

Matter wave coupling of spatially separated and unequally pumped polariton condensates

Spatial quantum coherence between two separated driven-dissipative polariton condensates created non-resonantly and with a different occupation is studied. We identify the regions where the condensates remain coherent with the phase difference continuously changing with the pumping imbalance and the regions where each condensate acquires its own chemical potential with phase differences exhibiting time-dependent oscillations. We show that in the mutual coherence limit the coupling consists of two competing contributions: a symmetric Heisenberg exchange and the Dzyloshinskii-Moriya asymmetric interactions that enable a continuous tuning of the phase relation across the dyad and derive analytic expressions for these types of interactions. The introduction of non-equal pumping increases the complexity of the type of the problems that can be solved by polariton condensates arranged in a graph configuration. If equally pumped polaritons condensates arrange their phases to solve the constrained quadratic minimisation problem with a real symmetric matrix, the non-equally pumped condensates solve that problem for a general Hermitian matrix.