Homogeneous Graphs Research Articles

Modulus of continuity eigenvalue bounds for homogeneous graphs and convex subgraphs with applications to quantum Hamiltonians

We adapt modulus of continuity estimates to the study of spectra of combinatorial graph Laplacians, as well as the Dirichlet spectra of certain weighted Laplacians. The latter case is equivalent to stoquastic Hamiltonians and is of current interest in both condensed matter physics and quantum computing. In particular, we introduce a new technique which bounds the spectral gap of such Laplacians (Hamiltonians) by studying the limiting behavior of the oscillations of their solutions when introduced into the heat equation. Our approach is based on recent advances in the PDE literature, which include a proof of the fundamental gap theorem by Andrews and Clutterbuck.

Reducts of the Henson graphs with a constant

Let (Hn,E) denote the Henson graph, the unique countable homogeneous graph whose age consists of all finite Kn-free graphs. In this note the reducts of the Henson graphs with a constant are determined up to first-order interdefinability. It is shown that up to first-order interdefinability (H3,E,0) has 13 reducts and (Hn,E,0) has 16 reducts for n≥4.

Information Recovery From Pairwise Measurements

This paper is concerned with jointly recovering n node variables $\left \{{ x_{i}}\right \} _{1\leq i\leq n}$ from a collection of pairwise difference measurements. Imagine we acquire a few observations taking the form of $x_{i}-x_{j}$ ; the observation pattern is represented by a measurement graph $\mathcal {G}$ with an edge set $\mathcal {E}$ , such that $x_{i}-x_{j}$ is observed if and only if $(i,j)\in \mathcal {E}$ . To account for noisy measurements in a general manner, we model the data acquisition process by a set of channels with given input/output transition measures. Employing information-theoretic tools applied to channel decoding problems, we develop a unified framework to characterize the fundamental recovery criterion, which accommodates general graph structures, alphabet sizes, and channel transition measures. In particular, our results isolate a family of minimum channel divergence measures to characterize the degree of measurement corruption, which together with the size of the minimum cut of $\mathcal {G}$ dictates the feasibility of exact information recovery. For various homogeneous graphs, the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics; alternatively, the minimum sample complexity required for these graphs scales like $({n\log n})/({\mathsf {Hel}_{1/2}^{\min }})$ for certain information metric $\mathsf {Hel}_{1/2}^{\min }$ defined in the main text, as long as the alphabet size is not super-polynomial in n . We apply our general theory to three concrete applications, including the stochastic block model, the random corruption model, and the haplotype assembly problem. Our theory leads to orderwise tight recovery conditions for all these scenarios.

Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs

Abstract A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random graph, infinite disjoint unions of complete graphs K n {K_{n}} with n ∈ ℕ {n\in\mathbb{N}} vertices, the K n {K_{n}} -free graphs, finite unions of the infinite complete graph K ω {K_{\omega}} , and duals of such graphs. The groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} of automorphisms of such graphs Γ have a natural topology, which is compatible with multiplication and inversion, i.e. the groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} where Γ is ultrahomogeneous. We show that if Γ is ultrahomogeneous, then Aut ⁡ ( Γ ) {\operatorname{Aut}(\Gamma)} has 2-generated dense subgroups, and that under certain conditions given f ∈ Aut ⁡ ( Γ ) {f\in\operatorname{Aut}(\Gamma)} there exists g ∈ Aut ⁡ ( Γ ) {g\in\operatorname{Aut}(\Gamma)} such that the subgroup generated by f and g is dense. We also show that, roughly speaking, g can be chosen with a high degree of freedom. For example, if Γ is either an infinite disjoint union of K n {K_{n}} or a finite union of K ω {K_{\omega}} , then g can be chosen to have any given finite set of orbit representatives.

The poset of all copies of the random graph has the 2-localization property

Let G be a countable graph containing a copy of the countable universal and homogeneous graph, also known as the random graph. Let Emb(G) be the monoid of self-embeddings of G, P(G)={f[G]:f∈Emb(G)} the set of copies of G contained in G, and IG the ideal of subsets of G which do not contain a copy of G. We show that the poset 〈P(G),⊂〉, the algebra P(G)/IG, and the inverse of the right Green's pre-order 〈Emb(G),⪯R〉 have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequence [bnm:〈n,m〉∈ω×ω] of elements of B⋀n∈ω⋁m∈ωbnm=⋁T∈Bt(ω<ω)⋀n∈ω⋁φ∈T∩ωn+1⋀k≤nbkφ(k), where Bt(ω<ω) denotes the set of all binary subtrees of the tree ω<ω.

Analytical calculation of average fixation time in evolutionary graphs.

The ability of a mutant individual to overtake the whole of a population is one of the fundamental problems in evolutionary dynamics. Fixation probability and Average Fixation Time (AFT) are two important parameters to quantify this ability. In this paper we introduce an analytical approach for exact calculation of AFT. Using this method we obtain AFT for two types of evolutionary graphs: cycle graph, as a highly homogeneous graph and star graph, as a highly heterogeneous graph. We use symmetries of these graphs to calculate AFT. Analytical results are confirmed with simulation. We also examine the effect of adding some random edges to each of these structures.

D-Levels locally homogeneous graphs

This article introduces the concept of a D-levels locally homogeneous graph which forms a special type of self-centered graphs. Some results relating D-levels locally homogeneous graphs to diametrical graphs (which are also self-centered) are obtained. Diametrical graphs of small order (≤ 8) are shown to be D-levels locally homogeneous. When the diameter is 2, it is shown that the concept of a D-levels locally homogeneous graph and that of a diametrical graph coincide. Bipartite diametrical graphs of diameter 3 and regular S-graphs of diameter 4 are characterized as D-levels locally homogeneous graphs. We also prove that the vertex-transitive graphs are precisely the totally D-levels locally homogeneous graphs.

Fast sparsely synchronized brain rhythms in a scale-free neural network

We consider a directed version of the Barabási-Albert scale-free network model with symmetric preferential attachment with the same in- and out-degrees and study the emergence of sparsely synchronized rhythms for a fixed attachment degree in an inhibitory population of fast-spiking Izhikevich interneurons. Fast sparsely synchronized rhythms with stochastic and intermittent neuronal discharges are found to appear for large values of J (synaptic inhibition strength) and D (noise intensity). For an intensive study we fix J at a sufficiently large value and investigate the population states by increasing D. For small D, full synchronization with the same population-rhythm frequency fp and mean firing rate (MFR) fi of individual neurons occurs, while for large D partial synchronization with fp>〈fi〉 (〈fi〉: ensemble-averaged MFR) appears due to intermittent discharge of individual neurons; in particular, the case of fp>4〈fi〉 is referred to as sparse synchronization. For the case of partial and sparse synchronization, MFRs of individual neurons vary depending on their degrees. As D passes a critical value D* (which is determined by employing an order parameter), a transition to unsynchronization occurs due to the destructive role of noise to spoil the pacing between sparse spikes. For D<D*, population synchronization emerges in the whole population because the spatial correlation length between the neuronal pairs covers the whole system. Furthermore, the degree of population synchronization is also measured in terms of two types of realistic statistical-mechanical measures. Only for the partial and sparse synchronization do contributions of individual neuronal dynamics to population synchronization change depending on their degrees, unlike in the case of full synchronization. Consequently, dynamics of individual neurons reveal the inhomogeneous network structure for the case of partial and sparse synchronization, which is in contrast to the case of statistically homogeneous random graphs and small-world networks. Finally, we investigate the effect of network architecture on sparse synchronization for fixed values of J and D in the following three cases: (1) variation in the degree of symmetric attachment, (2) asymmetric preferential attachment of new nodes with different in- and out-degrees, and (3) preferential attachment between pre-existing nodes (without addition of new nodes). In these three cases, both relation between network topology (e.g., average path length and betweenness centralization) and sparse synchronization and contributions of individual dynamics to the sparse synchronization are discussed.

Logarithmic Harnack inequalities for homogeneous graphs

In this paper, we prove logarithmic Harnack inequalities for homogeneous graphs. As a consequence, we derive lower estimates for the log-Sobolev constant, extending previous results for Ricci flat graphs.

Ramsey Precompact Expansions of Homogeneous Directed Graphs

In 2005, Kechris, Pestov and Todorčević provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of homogeneous directed graphs. In this paper, we show that the age of any homogeneous directed graph allows a Ramsey precompact expansion. Moreover, we verify the relative expansion properties and consequently describe the respective universal minimal flows.

Addendum to ‘The Katětov construction modified for a T0-quasi-metric space’

It is known that if K is a compact subset of the (separable complete) metric Urysohn space (${\mathbb U}$, d) and f is a Katětov function on the subspace K of (${\mathbb U}$, d), then there is z ∈ ${\mathbb U}$ such that d(z, x) = f(x) for all x ∈ K.Answering a question of Normann, we show in this article that the supseparable bicomplete q-universal ultrahomogeneous T0-quasi-metric space (q${\mathbb U}$, D) recently discussed by the authors satisfies a similar property for Katětov function pairs on subsets that are compact in the associated metric space (q${\mathbb U}$, Ds).

TAMES: A Truthful Double Auction for Multi-Demand Heterogeneous Spectrums

To accommodate the soaring mobile broadband traffic, the Federal Communications Commission (FCC) in the U.S. sets out to retrieve under-utilized spectrum (e.g., TV Whitespace) and lay the groundwork for spectrum redistribution. Auction is an efficient way to allocate resources to those who value them the most. The large pool of spectrums to be released, especially the ones in TV Whitespace, consist of wide-range frequencies. Apart from spatial reuse, spectrum heterogeneity imposes new challenges for spectrum auction design: 1) Wireless service providers with different targeted cell coverages have different spectrum frequency preferences; 2) interference relationship is frequency-dependent due to frequency-selective signal fading. Unfortunately, existing spectrum auction mechanisms either assume spectrum valuation is homogeneous or use homogeneous interference graph to group buyers who can reuse the same spectrum. In this paper, we propose TAMES, an auction framework for heterogeneous spectrum transaction. We consider a multi-seller-multi-buyer double auction, in which every buyer submits a bid, consisting of the spectrum demand and a bidding profile of prices for spectrums contributed by all sellers. A novel buyer grouping approach is proposed to tackle the problem of heterogeneous interference graph. TAMES is proved to be truthful as well as individually rational. The simulation results show that TAMES significantly improves spectrum utilization, sellers' revenue and buyers' utility by making smart use of spectrum heterogeneity, while keeping low running time comparable with existing auction mechanisms. Moreover, via simulation, we show how to help buyers obtain continuous spectrums which further improves buyers' satisfaction.

Unsupervised spatial event detection in targeted domains with applications to civil unrest modeling.

Twitter has become a popular data source as a surrogate for monitoring and detecting events. Targeted domains such as crime, election, and social unrest require the creation of algorithms capable of detecting events pertinent to these domains. Due to the unstructured language, short-length messages, dynamics, and heterogeneity typical of Twitter data streams, it is technically difficult and labor-intensive to develop and maintain supervised learning systems. We present a novel unsupervised approach for detecting spatial events in targeted domains and illustrate this approach using one specific domain, viz. civil unrest modeling. Given a targeted domain, we propose a dynamic query expansion algorithm to iteratively expand domain-related terms, and generate a tweet homogeneous graph. An anomaly identification method is utilized to detect spatial events over this graph by jointly maximizing local modularity and spatial scan statistics. Extensive experiments conducted in 10 Latin American countries demonstrate the effectiveness of the proposed approach.

Mining Social Distance from Mobile Phone Communication Networks

Nowadays, smartphones have been widely deployed. Scientists can obtain position and proximity data as well as call and SMS log data easily. Rich information are hidden behind these mobile data. For example, social interaction patterns of smartphone users. The discovery of those patterns can benefit many important applications such as (1) Relationship inference; (2) Community detection in counter terrorism. However, mobile data owns many unique characteristics, such as (1) The amount of data is massive; (2) Timeliness is very important; (3) Different types of data are highly coupled. Traditional statistical methods have been used to analyze mobile data, which can not be applied to huge amount of mobile data streams in real-world situations. In this paper, we take mobile data as an example, to analyze the social distance and fulfill popular tasks such as (1) Relationship inference; (2) Community detection. We propose a time-constrained homogeneous graph model, which takes a mobile call log as input and models the social distance problem by a time-based sliding window scheme. We further design an efficient modified EdgeRank algorithm to compute the social distances among users. We propose an algorithm for community detection. Experimental results on real data sets show that the proposed approach can infer social distance efficiently and achieves an accuracy of 80-85% automatically. We apply the results of social distance relationships to community detection and an improvement of accuracy of 5% can be achieved.

Double adjacency graphs-based discriminant neighborhood embedding

Discriminant neighborhood embedding (DNE) is a typical graph-based dimensionality reduction method, and has been successfully applied to face recognition. By constructing an adjacency graph, aiming to keep the local structure for original data in the subspace, it is able to find the optimal discriminant direction effectively. Not for every sample does DNE set up a link between it and its heterogeneous samples when constructing the adjacency graph, which would result in a small between-class scatter. Motivated by this fact, we develop an extension of DNE, called double adjacency graphs-based discriminant neighborhood embedding (DAG-DNE) by introducing two adjacency graphs, or homogeneous and heterogeneous neighbor adjacency graphs. In DAG-DNE, neighbors belonging to the same class are compact while neighbors belonging to different classes become separable in the subspace. Thus, DAG-DNE could keep the local structure of a given data and find a good projection matrix for them. To investigate the performance of DAG-DNE, we compare it with the state-of-the-art dimensionality reduction techniques such as DNE and MFA on several publicly available datasets. Experimental results show the feasibility and effectiveness of the proposed method.

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

For a countable ultrahomogeneous graph G=〈G,ρ〉 let P(G) denote the collection of sets A⊂G such that 〈A,ρ∩[A]2〉≅G. The order types of maximal chains in the poset 〈P(G)∪{∅},⊂〉 are characterized as:(I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph Hn, for some n≥3;(II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of μ disjoint complete graphs of size ν, where μν=ω.

Voter model on the two-clique graph.

I examine the mean consensus time (i.e., exit time) of the voter model in the so-called two-clique graph. The two-clique graph is composed of two cliques interconnected by some links and considered as a toy model of networks with community structure or multilayer networks. I analytically show that, as the number of interclique links per node is varied, the mean consensus time experiences a crossover between a fast consensus regime [i.e., O(N)] and a slow consensus regime [i.e., O(N(2))], where N is the number of nodes. The fast regime is consistent with the result for homogeneous well-mixed graphs such as the complete graph. The slow regime appears only when the entire network has O(1) interclique links. The present results suggest that the effect of community structure on the consensus time of the voter model is fairly limited.

Evolutionary game dynamics in populations with heterogenous structures.

Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant's fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.

Towards a theory of random walk planning: Regress factors, fair homogeneous graphs and extensions

Random walks are a relatively new component used in several state of the art satisficing planners. Empirical results have been mixed: while the approach clearly outperforms more systematic search methods such as weighted A* on many planning domains, it fails in many others. So far, the explanations for these empirical results have been somewhat ad hoc. This paper proposes a formal framework for comparing the performance of random walk and systematic search methods. Fair homogenous and Infinitely Regressable homogenous graphs are proposed as graph classes that represents characteristics of the state space of prototypical planning domains, and is simple enough to allow a theoretical analysis of the performance of both random walk and systematic search algorithms. This gives well-founded insights into the relative strength and weaknesses of these approaches. The close relation of the models to some well-known planning domains is shown through simplified but semi-realistic planning domains that fulfill the constraints of the models. One main result is that in contrast to systematic search methods, for which the branching factor plays a decisive role, the performance of random walk methods is determined to a large degree by the Regress Factor, the ratio between the probabilities of progressing towards and regressing away from a goal with an action. The performance of random walk and systematic search methods can be compared by considering both branching and regress factors of a state space.

Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks

We investigate the effects of local population structure in reaction-diffusion processes representing a contact process (CP) on metapopulations represented as complex networks. Considering a model in which the nodes of a large scale network represent local populations defined in terms of a homogeneous graph, we show by means of extensive numerical simulations that the critical properties of the reaction-diffusion system are independent of the local population structure, even when this one is given by a ordered linear chain. This independence is confirmed by the perfect matching between numerical critical exponents and the results from a heterogeneous mean-field theory suited, in principle, to describe situations of local homogeneous mixing. The analysis of several variations of the reaction-diffusion process allows us to conclude the independence from population structure of the critical properties of CP-like models on metapopulations, and thus of the universality of the reaction-diffusion description of this kind of models.