Deterministic Graph Research Articles

Path Optimality Conditions for Minimum Spanning Tree Problem with Uncertain Edge Weights

This paper investigates the uncertain minimum spanning tree (UMST) problem where the edge weights are assumed to be uncertain variables. In order to propose effective solving methods for the UMST problem, path optimality conditions as well as some equivalent definitions for two commonly used types of UMST, namely, uncertain expected minimum spanning tree (expected UMST) and uncertain α-minimum spanning tree (α-UMST), are discussed. It is shown that both the expected UMST problem and the α-UMST problem can be transformed into an equivalent classical minimum spanning tree problem on a corresponding deterministic graph, which leads to effective algorithms with low computational complexity. Furthermore, the notion of uncertain most minimum spanning tree (most UMST) is initiated for an uncertain graph, and then the equivalent relationship between the α-UMST and the most UMST is proved. Numerical examples are presented as well for illustration.

Structural ‘connectomic’ alterations in the limbic system of multiple sclerosis patients with major depression

Background: Major depression (MD) is a common psychiatric disorder in multiple sclerosis (MS). Despite the negative impact of MD on the quality of life of MS patients, little is known about its underlying brain mechanisms. Objective: We studied the whole-brain connectivity patterns that were associated with MD in MS. Alterations were mainly expected within limbic circuits. Methods: Diffusion tensor imaging data were collected in 20 MS patients with MD, 22 non-depressed MS patients and 16 healthy controls. We used deterministic tractography and graph analysis to study the white-matter connectivity patterns that characterized MS patients with MD. Results: We found that MD in MS was associated with increased local path length in the right hippocampus and right amygdala. Further analyses revealed that these effects were driven by an increased shortest distance between both the right hippocampus and right amygdala and a series of regions including the dorsolateral and ventrolateral prefrontal cortex, orbitofrontal cortex, sensory-motor cortices and supplementary motor area. Conclusion: Our data provide strong support for neurobiological accounts positing that MD in MS is mediated by abnormal ‘communications’ within limbic circuits. We also found evidence that MD in MS may be linked with connectivity alterations at the limbic-motor interface, a group of regions that translates emotions into survival-oriented behaviors.

Graph similarity search on large uncertain graph databases

Many studies have been conducted on seeking an efficient solution for graph similarity search over certain (deterministic) graphs due to its wide application in many fields, including bioinformatics, social network analysis, and Resource Description Framework data management. All prior work assumes that the underlying data is deterministic. However, in reality, graphs are often noisy and uncertain due to various factors, such as errors in data extraction, inconsistencies in data integration, and for privacy-preserving purposes. Therefore, in this paper, we study similarity graph containment search on large uncertain graph databases. Similarity graph containment search consists of subgraph similarity search and supergraph similarity search. Different from previous works assuming that edges in an uncertain graph are independent of each other, we study uncertain graphs where edges' occurrences are correlated. We formally prove that subgraph or supergraph similarity search over uncertain graphs is $$\#$&#P-hard; thus, we employ a filter-and-verify framework to speed up these two queries. For the subgraph similarity query, in the filtering phase, we develop tight lower and upper bounds of subgraph similarity probability based on a probabilistic matrix index (PMI). PMI is composed of discriminative subgraph features associated with tight lower and upper bounds of subgraph isomorphism probability. Based on PMI, we can filter out a large number of uncertain graphs and maximize the pruning capability. During the verification phase, we develop an efficient sampling algorithm to validate the remaining candidates. For the supergraph similarity query, in the filtering phase, we propose two pruning algorithms, one lightweight and the other strong, based on maximal common subgraphs of query graph and data graph. We run the two pruning algorithms against a probabilistic index that consists of powerful graph features. In the verification, we design an approximate algorithm based on the Horvitz---Thompson estimator to fast validate the remaining candidates. The efficiencies of our proposed solutions to the subgraph and supergraph similarity search have been verified through extensive experiments on real uncertain graph datasets.

Ranking paths in statistical energy analysis models with non-deterministic loss factors

Finding the contributions of transmission paths in statistical energy analysis (SEA) models has become an established valuable tool to detect and remedy vibro-acoustic problems. Paths are identified in SEA according to Craik׳s definition and recently, very efficient methods have been derived to rank them in the framework of graph theory. However, up to date classification schemes have only considered the mean values of loss factors for path comparison, their variance being ignored. This can result in significant errors in the final results. In this work it is proposed to address this problem by defining stochastic biparametric SEA graphs whose edges are assigned both, mean and variance values. Paths between subsystems are then compared according to a proposed cost function that accounts for the stochastic nature of loss factors. For an efficacious ranking of paths, the stochastic SEA graph is converted to an extended deterministic SEA graph where fast classification deterministic algorithms can be applied. The importance of nonneglecting the influence of the variance in path ranking is illustrated by means of some academic numerical examples.

Free energy of RNA-counterion interactions in a tight-binding model computed by a discrete space mapping.

The thermodynamic stability of a folded RNA is intricately tied to the counterions and the free energy of this interaction must be accounted for in any realistic RNA simulations. Extending a tight-binding model published previously, in this paper we investigate the fundamental structure of charges arising from the interaction between small functional RNA molecules and divalent ions such as Mg(2+) that are especially conducive to stabilizing folded conformations. The characteristic nature of these charges is utilized to construct a discretely connected energy landscape that is then traversed via a novel application of a deterministic graph search technique. This search method can be incorporated into larger simulations of small RNA molecules and provides a fast and accurate way to calculate the free energy arising from the interactions between an RNA and divalent counterions. The utility of this algorithm is demonstrated within a fully atomistic Monte Carlo simulation of the P4-P6 domain of the Tetrahymena group I intron, in which it is shown that the counterion-mediated free energy conclusively directs folding into a compact structure.

Uncertain Graph Classification Based on Extreme Learning Machine

The problem of graph classification has attracted much attention in recent years. The existing work on graph classification has only dealt with precise and deterministic graph objects. However, the linkages between nodes in many real-world applications are inherently uncertain. In this paper, we focus on classification of graph objects with uncertainty. The method we propose can be divided into three steps: Firstly, we put forward a framework for classifying uncertain graph objects. Secondly, we extend the traditional algorithm used in the process of extracting frequent subgraphs to handle uncertain graph data. Thirdly, based on Extreme Learning Machine (ELM) with fast learning speed, a classifier is constructed. Extensive experiments on uncertain graph objects show that our method can produce better efficiency and effectiveness compared with other methods.

Diagnosis and opacity problems for infinite state systems modeled by recursive tile systems

The analysis of discrete event systems under partial observation is an important topic, with major applications such as the detection of information flow and the diagnosis of faulty behaviors. These questions have, mostly, not been addressed for classical models of recursive systems, such as pushdown systems and recursive state machines. In this paper, we consider recursive tile systems, which are recursive infinite systems generated by a finite collection of finite tiles, a simplified variant of deterministic graph grammars (slightly more general than pushdown systems). Since these systems are infinite-state in general powerset constructions for monitoring do not always apply. We exhibit computable conditions on recursive tile systems and present non-trivial constructions that yield effective computation of the monitors. We apply these results to the classic problems of state-based opacity and diagnosability (off-line verification of opacity and diagnosability, and also run-time monitoring of these properties). For a decidable subclass of recursive tile systems, we also establish the decidability of the problems of state-based opacity and diagnosability.

Deterministic vs non-deterministic graph property testing

A graph property \(\mathcal{P}\) is said to be testable if one can check whether a graph is close or far from satisfying \(\mathcal{P}\) using few random local inspections. Property \(\mathcal{P}\) is said to be non-deterministically testable if one can supply a “certificate” to the fact that a graph satisfies \(\mathcal{P}\) so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovasz and Vesztergombi [9], who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemeredi’s regularity lemma.

Detecting Protein Complexes Based on Uncertain Graph Model.

Advanced biological technologies are producing large-scale protein-protein interaction (PPI) data at an ever increasing pace, which enable us to identify protein complexes from PPI networks. Pair-wise protein interactions can be modeled as a graph, where vertices represent proteins and edges represent PPIs. However most of current algorithms detect protein complexes based on deterministic graphs, whose edges are either present or absent. Neighboring information is neglected in these methods. Based on the uncertain graph model, we propose the concept of expected density to assess the density degree of a subgraph, the concept of relative degree to describe the relationship between a protein and a subgraph in a PPI network. We develop an algorithm called DCU (detecting complex based on uncertain graph model) to detect complexes from PPI networks. In our method, the expected density combined with the relative degree is used to determine whether a subgraph represents a complex with high cohesion and low coupling. We apply our method and the existing competing algorithms to two yeast PPI networks. Experimental results indicate that our method performs significantly better than the state-of-the-art methods and the proposed model can provide more insights for future study in PPI networks.

Decoding Binary Node Labels from Censored Edge Measurements: Phase Transition and Efficient Recovery

We consider the problem of clustering a graph G into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> x ⊕ Z, where B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior), and Z is a noise vector with Bernoulli (ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of x is possible if and only the graph G is connected, with a sharp threshold at the edge probability log (n)/n for Erdos-Renyi random graphs. The first goal of this paper is to determine how the edge probabilityp needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by α = np/log(n), it is shown that exact recovery is possible if and only if α > 2/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + o(1/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). In other words, 2/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph G, defining the degree rate as α = d/log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α > 4((1 + λ)/(1 - λ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + o(1/(1 - 2ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where 1-λ is the spectral gap of the graph G.

Combinatorial methods in netwotk reliability analysis

In this paper, a stochastic network is an undirected graph with unreliable edges and absolutely reliable nodes. Its connectedness probability is determined by reliability preserving network reduction. The principle of this method consists in splitting the underlying deterministic graph of the stochastic network into two edge-disjoint subgraphs via a separating node set. One of the subgraphs is replaced with a simpler structured graph (replacement graph) in such a way that the interesting reliability criterion of the original stochastic network is retained. Special attention is given to the construction of suitable replacement graphs. The case of a 3-point separating node set is considered in more detail.

Efficient Keyword Search on Uncertain Graph Data

As a popular search mechanism, keyword search has been applied to retrieve useful data in documents, texts, graphs, and even relational databases. However, so far, there is no work on keyword search over uncertain graph data even though the uncertain graphs have been widely used in many real applications, such as modeling road networks, influential detection in social networks, and data analysis on PPI networks. Therefore, in this paper, we study the problem of top-k keyword search over uncertain graph data. Following the similar answer definition for keyword search over deterministic graphs, we consider a subtree in the uncertain graph as an answer to a keyword query if 1) it contains all the keywords; 2) it has a high score (defined by users or applications) based on keyword matching; and 3) it has low uncertainty. Keyword search over deterministic graphs is already a hard problem as stated in [1], [2], [3]. Due to the existence of uncertainty, keyword search over uncertain graphs is much harder. Therefore, to improve the search efficiency, we employ a filtering-and-verification strategy based on a probabilistic keyword index, PKIndex. For each keyword, we offline compute path-based top-k probabilities, and attach these values to PKIndex in an optimal, compressed way. In the filtering phase, we perform existence, path-based and tree-based probabilistic pruning phases, which filter out most false subtrees. In the verification, we propose a sampling algorithm to verify the candidates. Extensive experimental results demonstrate the effectiveness of the proposed algorithms.

Analytic expression for the mean time to absorption for a random walker on the Sierpinski fractal. III. The effect of non-nearest-neighbor jumps

We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d≥2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearest-neighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.

On Watts' cascade model with random link weights

We study an extension of D. Watts 2002 model of information cascades in social networks where edge weights are taken to be random, an innovation motivated by recent applications of cascade analysis to systemic risk in financial networks. The main result is a probabilistic analysis that characterizes the cascade in an infinite network as the fixed point of a vector-valued mapping, explicit in terms of convolution integrals that can be e ciently evaluated numerically using the fast Fourier transform algorithm. A second result gives an approximate probabilistic analysis of cascades on “real world networks”, finite networks based on a fixed deterministic graph. Extensive cross testing with Monte Carlo estimates shows that this approximate analysis performs surprisingly well, and provides a flexible “microscope” that can be used to investigate properties of information cascades in real world networks over a wide range of model parameters.

A LEARNING AUTOMATA-BASED ALGORITHM TO THE STOCHASTIC MIN-DEGREE CONSTRAINED MINIMUM SPANNING TREE PROBLEM

Min-degree constrained minimum spanning tree (md-MST) problem is an NP-hard combinatorial optimization problem seeking for the minimum weight spanning tree in which the vertices are either of degree one (leaf) or at least degree d ≥ 2. md-MST problem is new to the literature and very few studies have been conducted on this problem in deterministic graph. md-MST problem has several appealing real-world applications. Though in realistic applications the graph conditions and parameters are stochastic and vary with time, to the best of our knowledge no work has been done on solving md-MST problem in stochastic graph. This paper proposes a decentralized learning automata-based algorithm for finding a near optimal solution to the md-MST problem in stochastic graph. In this work, it is assumed that the weight associated with the graph edge is random variable with a priori unknown probability distribution. This assumption makes the md-MST problem incredibly harder to solve. The proposed algorithm exploits an intelligent sampling technique avoiding the unnecessary samples by focusing on the edges of the min-degree spanning tree with the minimum expected weight. On the basis of the Martingale theorem, the convergence of the proposed algorithm to the optimal solution is theoretically proven. Extensive simulation experiments are performed on the stochastic graph instances to show the performance of the proposed algorithm. The obtained results are compared with those of the standard sampling method in terms of the sampling rate and solution optimality. Simulation experiments show that the proposed method outperforms the standard sampling method.

Deterministic quantum non-locality and graph colorings

One of the most fascinating consequences of quantum theory are non-local correlations: Two–possibly distant–parts of a system can have a behavior under measurements unexplainable by shared information. A manifestation thereof is so-called pseudo-telepathy: Tasks that can be performed by two parties who share a quantum state, whereas classically, communication would be necessary to always succeed. We show that pseudo-telepathy games can often be modeled by graphs: The classical strategy to win the game is a coloring of this graph with a given number of colors. We discuss these parallels and study the class of graphs corresponding to the first two-party pseudo-telepathy game, proposed by Brassard, Cleve, and Tapp in 1999. This leads to a proof that the game indeed has the desired property.

Efficient subgraph similarity search on large probabilistic graph databases

Many studies have been conducted on seeking the efficient solution for subgraph similarity search over certain (deterministic) graphs due to its wide application in many fields, including bioinformatics, social network analysis, and Resource Description Framework (RDF) data management. All these works assume that the underlying data are certain. However, in reality, graphs are often noisy and uncertain due to various factors, such as errors in data extraction, inconsistencies in data integration, and privacy preserving purposes. Therefore, in this paper, we study subgraph similarity search on large probabilistic graph databases. Different from previous works assuming that edges in an uncertain graph are independent of each other, we study the uncertain graphs where edges' occurrences are correlated. We formally prove that subgraph similarity search over probabilistic graphs is #P-complete, thus, we employ a filter-and-verify framework to speed up the search. In the filtering phase, we develop tight lower and upper bounds of subgraph similarity probability based on a probabilistic matrix index, PMI. PMI is composed of discriminative subgraph features associated with tight lower and upper bounds of subgraph isomorphism probability . Based on PMI, we can sort out a large number of probabilistic graphs and maximize the pruning capability. During the verification phase, we develop an efficient sampling algorithm to validate the remaining candidates. The efficiency of our proposed solutions has been verified through extensive experiments.

DEGREE-CONSTRAINED MINIMUM SPANNING TREE PROBLEM IN STOCHASTIC GRAPH

A degree-constrained minimum spanning tree (DCMST) problem is an NP-hard combinatorial optimization problem in graph theory seeking the minimum cost spanning tree with the additional constraint on the vertex degree. Several different approaches have been proposed in the literature to solve this problem using a deterministic graph. However, to the best of the author's knowlege, no work has been performed on solving the problem using stochastic edge-weighted graphs. In this article, a learning automata–based algorithm is proposed to find a near optimal solution of the DCMST problem using a stochastic graph, where the cost associated with the graph edge is a random variable with a priori unknown probability distribution. The convergence of the proposed algorithm to the optimal solution is theoretically proved based on the Martingale theorem. To show the performance of the proposed algorithm, several simulation experiments are conducted on stochastic Euclidean graph instances. Numerical results are compared with those of the standard sampling method (SSM). The numerical results confirm the superiority of the proposed sampling technique over the SSM both in terms of the sampling rate and solution optimality.

Analysis of partially observed recursive tile systems

The analysis of discrete event systems under partial observation is an important topic, with major applications such as the detection of information flow and the diagnosis of faulty behaviors. We consider recursive tile systems, which are infinite systems generated by a finite collection of finite tiles, a simplified variant of deterministic graph grammars. Recursive tile systems are expressive enough to capture classical models of recursive systems, such as the pushdown systems and the recursive state machines. They are infinite-state in general and therefore standard powerset constructions for monitoring do not always apply. We exhibit computable conditions on recursive tile systems and present non-trivial constructions that yield effective computation of the monitors. We apply these results to the classic problems of opacity and diagnosability.

Time series of scientific growth in Spanish doctoral theses (1848–2009)

This article analyses scientific growth time series using data for Spanish doctoral theses from 1848 to 2009, retrieved from national databases and an in-depth archive search. Data are classified into subseries by historical periods. The analytical techniques employed range from visual analysis of deterministic graphs to curve-fitting with exponential smoothing and AutoRegressive Integrated Moving Average models. Forecasts are made using the best model. The main finding is that Spanish output of doctoral theses appears to fit a quasi-logistic growth model in line with Price's predictions. An additional control variable termed year-on-year General Welfare is shown to modulate scientific growth, especially in the historical period from 1899 to 1939.