Hierarchical Random Graph Research Articles

Motif based hierarchical random graphs: structural properties and critical points of an Ising model

A class of random graphs is introduced and studied. The graphs are constructed in an algorithmic way from five motifs which were found in [Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science, 2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski M., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the short-range bonds are non-random, whereas the long-range bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, small-world property. For one of the motifs, the critical point of the Ising model defined on the corresponding graph has been studied.

Exploratory analysis of protein translation regulatory networks using hierarchical random graphs

BackgroundProtein translation is a vital cellular process for any living organism. The availability of interaction databases provides an opportunity for researchers to exploit the immense amount of data in silico such as studying biological networks. There has been an extensive effort using computational methods in deciphering the transcriptional regulatory networks. However, research on translation regulatory networks has caught little attention in the bioinformatics and computational biology community.ResultsIn this paper, we present an exploratory analysis of yeast protein translation regulatory networks using hierarchical random graphs. We derive a protein translation regulatory network from a protein-protein interaction dataset. Using a hierarchical random graph model, we show that the network exhibits well organized hierarchical structure. In addition, we apply this technique to predict missing links in the network. ConclusionsThe hierarchical random graph mode can be a potentially useful technique for inferring hierarchical structure from network data and predicting missing links in partly known networks. The results from the reconstructed protein translation regulatory networks have potential implications for better understanding mechanisms of translational control from a system’s perspective.

Percolation in a hierarchical random graph

We study asymptotic percolation as N ! 1 in an infinite ran- dom graph GN embedded in the hierarchical group of order N, with con- nection probabilities depending on an ultrametric distance between vertices. GN is structured as a cascade of finite random subgraphs of (approximate) Erd '' os-Renyi type. However, the results are different from those of classical random graphs, e.g., the average length of paths in the giant component of an ultrametric ball is much longer than in the classical case. We give a criterion for percolation, and show that percolation takes place along giant compo- nents of giant components at the previous level in the cascade of subgraphs for all consecutive hierarchical distances. The proof involves a hierarchy of doubly stochastic random graphs with vertices having an internal structure and random connection probabilities.

Hierarchical random graph representation of handwritten characters and its application to Hangul recognition

A hierarchical random graph (HRG) representation for handwritten character modeling is presented. Based on the HRG, a Hangul, Korean scripts, recognition system also has been developed. In the HRG, the bottom layer is constructed with extended random graphs to describe various strokes, while the next upper layers are constructed with random graphs (Wong and Ghahraman, IEEE Trans. Pattern Anal. Mach. Intell. 2(4) (1980) 341) to model spatial and structural relationships between strokes and between sub-characters. As the proposed HRG is a stochastic model, the recognition is formulated into the problem that chooses a model producing maximum probability given an input data. In this context, a matching score is acquired not by any heuristic similarity function, but by a probabilistic measure. The recognition process starts from converting an input character image into an attributed graph through the preprocessing and the graph representation. Matching between an attributed graph and the hierarchical graph model is performed bottom-up. Since the hierarchical structure in an attributed graph is decided after the recognition ends depending on the best interpretation of the graph matching, we can avoid incorrect sub-character segmentation. Model parameters of the hierarchical graph have been estimated automatically from the training data by EM algorithm (Dempster et al., J. Roy. Stat. Soc. 39 (1977) 1) and embedded training. The recognition experiments conducted with unconstrained handwritten Hangul characters show the usefulness and the effectiveness of the proposed HRG.

Texture synthesis and pattern recognition for partially ordered Markov models

The uses of texture in image analysis are widespread, ranging from remotely sensed data to medical imaging to military applications. Image processing tasks that use texture characteristics include classification, region segmentation, and synthesis of data. While there are several approaches available for texture modeling, the research presented here is concerned with stochastic texture models. Stochastic approaches view a texture as the realization of a random field and are most useful when the texture appears noisy or when it lacks smooth geometric features. The model introduced in this paper is a subclass of Markov random fields (MRFs) called partially ordered Markov models (POMMs). Markov random fields are a class of stochastic models that incorporate spatial dependency between data points. One major disadvantage of MRFs is that, in general, an explicit form of the joint probability of the random variables describing the model is not obtainable. However, a popular subclass of MRFs, called Markov mesh models (MMMs), allows the explicit description of the joint probability in terms of spatially local conditional probabilities. We show how POMMs are a generalization of MMMs and demonstrate the versatility of POMMs to texture synthesis and pattern recognition in imaging. Specifically, we give a fast, one-pass algorithm for simulating textures using POMMs, and introduce examples of heterogeneous models that suggest potential applications for object recognition purposes. Then we address an inverse problem, where we present results from a series of statistical experiments designed to estimate parameters of stochastic texture models for both binary and gray value data. Although the applications in this paper focus on imaging, in their most general form, POMMs can be found in areas such as probabilistic expert systems, Bayesian hierarchical modeling, influence diagrams, and random graphs and networks.

Formal definition and entropy calculation of hierarchical attributed random graph

Abstract For the representation of a complex object, the object is decomposed into several parts, and it is described by these decomposed parts and their relations. In general, the parts can be the primitive elements that can not be decomposed further, or still can be decomposed into their subparts. Therefore, the hierarchical description method is very natural and the hierarchical description is represented by a hierarchical graph whose vertices represent either primitive elements or graphs. These graohs also have vertices which contain primitive elements or graphs. When some uncertainty exists in the hierarchical description of a complex object either due to noise or minor deformation, a probabilistic description of the object ensemble is necessary. For this purpose, in this paper, we formally define the hierarchical attributed random graph and derive the equations for the entropy calculation of the hierarchical random graph.