Undirected Ones Research Articles

Intrinsic Universality in Automata Networks III: On Symmetry versus Asynchrony

An automata network is a finite assembly of interconnected entities endowed with a set of local maps defined over a common finite alphabet. These relationships are represented through an interaction graph. Together with the local functions, an assignment known as an update scheme directs the evolution of the network by updating specific subsets of entities at discrete time steps. Despite the scrutiny of interaction graphs and update schemes, their profound impact on automata network dynamics remains largely open. This work investigates the intricate interplay between these aspects, with a focus on how update schemes can counterbalance constraints stemming from symmetric local interactions. This paper is the third of a series about intrinsic universality, a notion that assesses both dynamical and computational complexity, encompassing transient behaviors, attractors, and prediction or reachability problems. We consider four update modes—parallel, block-sequential, local clocks, and general periodic— along with several families of symmetric signed conjunctive boolean networks defined by local constraints on signs. Our main result is to show a diagonal complexity leap in this two-dimensional landscape: the stronger the local constraints the higher the level of asynchrony required to obtain intrinsic universality or increase in complexity. We also show how in some cases asynchronism allows to simulate directed interactions from undirected ones with the same local rules.

Contribution of directedness in graph spectra

In graph analyses, directed edges are often approximated to undirected ones so that the adjacency matrices may be symmetric. However, such simplification has not been thoroughly verified. In this study, we investigate how directedness affects the graph spectra by introducing random directization, which is an opposite operation of neglecting edge directions. We analytically reveal that uniformly random directization typically conserves the relative spectral structure of the adjacency matrix in the perturbative regime. The result of random directization implies that the spectrum of the adjacency matrix can be conserved after the directedness is ignored.

Performance comparison of some centrality measures used in detecting anomalies in directed social networks

Recently, researchers have shown an increased interest in combining Statistical Process Control/Monitoring and Social Network Analysis. One approach to detect anomalies in social networks is to monitor some summary statistics of the network structure using control charts. We aim in this study to conduct a performance comparison among some network centrality measures (betweenness, closeness, and degree) in terms of their ability in detecting anomalies using control charts. Although directed networks include more information on the nodes communication than the undirected ones, they are rarely considered and evaluated in literature. Hence, the performance comparison is conducted assuming weighted and unweighted directed networks. Two network-level measures are used; the average and standard deviation for each metric. Our simulation results revealed that the degree centrality measure in most cases is recommended to be used for both weighted and unweighted networks. Also, it can generally be concluded that the average of a centrality measure performs better than the standard deviation in detecting anomalies. However, if the structural change is major in weighted networks specifically, we recommend the standard deviation of the centrality measure to be used.

Gene Networks with Transcriptional Bursting Recapitulate Rare Transient Coordinated High Expression States in Cancer.

Non-genetic transcriptional variability is a potential mechanism for therapy resistance in melanoma. Specifically, rare subpopulations of cells occupy a transient pre-resistant state characterized by coordinated high expression of several genes and survive therapy. How might these rare states arise and disappear within the population? It is unclear whether the canonical models of probabilistic transcriptional pulsing can explain this behavior, or if it requires special, hitherto unidentified mechanisms. We show that a minimal model of transcriptional bursting and gene interactions can give rise to rare coordinated high expression states. These states occur more frequently in networks with low connectivity and depend on three parameters. While entry into these states is initiated by a long transcriptional burst that also triggers entry of other genes, the exit occurs through independent inactivation of individual genes. Together, we demonstrate that established principles of gene regulation are sufficient to describe this behavior and argue for its more general existence. A record of this paper's transparent peer review process is included in the Supplemental Information.

LMI conditions for contraction and synchronization

This paper deals with a systematic way to check if a certain class of nonlinear systems can define a contraction. The proposed approach combines the use of sector bound condition or monotonic condition satisfied by the nonlinearities with the use of an adequate control function matrix. The contraction analysis results are provided under the form of linear matrix inequalities (LMIs). The design of a nonlinear control law (that is a control law depending on nonlinearities) is proposed in order for the closed loop to be a contraction. The adaptation of the contraction results to the synchronization of n identical nonlinear systems is also investigated by considering two kinds of communication graphs (full and general undirected ones).

A fusion probability matrix factorization framework for link prediction

Link prediction is a fundamental research problem in network data analysis. Networks usually contain rich node-to-node topological metrics and their effective use is crucial to solve the link prediction problem. Despite significant advances, the existing metric-based link prediction methods usually only consider one single topological metric and thus show some limitations in different types of networks; the existing matrix factorization-based models mainly focus on modeling the adjacent matrix of a network, and this is hard to ensure the modeling of those topological metrics that can play an important role in link prediction. This study develops effective approaches by fusing the adjacent matrix and some key topological metrics in a unified probability matrix factorization framework. In these approaches, we consider not only the symmetric metrics but also the asymmetric metrics which are usually not taken into consideration in the related work. In our probability matrix factorization framework, we first present two fusion models by fusing two kinds of metrics respectively, and based on the fusion models, we put forward the final fusion models which fuse the two kinds of metrics simultaneously. To verify the performance of all the fusion models, we conduct the experiments with six directed networks and six undirected ones, and the extensive experiments show that the proposed models provide impressive predicting performance for link prediction.

Fractality and scale-free effect of a class of self-similar networks

In this paper, given an initial directed graph as a self-similar pattern and fix two nodes in the pattern, we can iterate the graph by replacing any directed edge with the initial graph of pattern and identifying the fixed nodes of pattern with the endpoints of directed edge. Using the iteration again and again, we obtain a family of growing self-similar networks. Modify these networks to be undirected ones, we obtain growing self-similar undirected networks. We obtain the fractality of our self-similar networks and find out the scale-free effect in terms of the matrix related to two fixed nodes in the initial graph.

An Empirical Comparison of Algorithms to Find Communities in Directed Graphs and Their Application in Web Data Analytics

Detecting communities in graphs is a fundamental tool to understand the structure of Web-based systems and predict their evolution. Many community detection algorithms are designed to process undirected graphs (i.e., graphs with bidirectional edges) but many graphs on the Web-e.g., microblogging Web sites, trust networks or the Web graph itself-are often directed . Few community detection algorithms deal with directed graphs but we lack their experimental comparison. In this paper we evaluated some community detection algorithms across accuracy and scalability. A first group of algorithms (Label Propagation and Infomap) are explicitly designed to manage directed graphs while a second group (e.g., WalkTrap) simply ignores edge directionality; finally, a third group of algorithms (e.g., Eigenvector) maps input graphs onto undirected ones and extracts communities from the symmetrized version of the input graph. We ran our tests on both artificial and real graphs and, on artificial graphs, WalkTrap achieved the highest accuracy, closely followed by other algorithms; Label Propagation has outstanding performance in scalability on both artificial and real graphs. The Infomap algorithm showcased the best trade-off between accuracy and computational performance and, therefore, it has to be considered as a promising tool for Web Data Analytics purposes.

A New Notion of Effective Resistance for Directed Graphs—Part I: Definition and Properties

The graphical notion of effective resistance has found wide-ranging applications in many areas of pure mathematics, applied mathematics and control theory. By the nature of its construction, effective resistance can only be computed in undirected graphs and yet in several areas of its application, directed graphs arise as naturally (or more naturally) than undirected ones. In Part I of this work, we propose a generalization of effective resistance to directed graphs that preserves its control-theoretic properties in relation to consensus-type dynamics. We proceed to analyze the dependence of our algebraic definition on the structural properties of the graph and the relationship between our construction and a graphical distance. The results make possible the calculation of effective resistance between any two nodes in any directed graph and provide a solid foundation for the application of effective resistance to problems involving directed graphs.

On Mixed Almost Moore Graphs of Diameter Two

Mixed almost Moore graphs appear in the context of the Degree/Diameter problem as a class of extremal mixed graphs, in the sense that their order is one less than the Moore bound for mixed graphs. The problem of their existence has been considered before for directed graphs and undirected ones, but not for the mixed case, which is a kind of generalization. In this paper we give some necessary conditions for the existence of mixed almost Moore graphs of diameter two derived from the factorization in $\mathbb{Q}[x]$ of their characteristic polynomial. In this context, we deal with the irreducibility of $\Phi_i(x^2+x-(r-1))$, where $\Phi_i(x)$ denotes the i-th cyclotomic polynomial.

Phase transitions in Ising models on directed networks.

We examine Ising models with heat-bath dynamics on directed networks. Our simulations show that Ising models on directed triangular and simple cubic lattices undergo a phase transition that most likely belongs to the Ising universality class. On the directed square lattice the model remains paramagnetic at any positive temperature as already reported in some previous studies. We also examine random directed graphs and show that contrary to undirected ones, percolation of directed bonds does not guarantee ferromagnetic ordering. Only above a certain threshold can a random directed graph support finite-temperature ferromagnetic ordering. Such behavior is found also for out-homogeneous random graphs, but in this case the analysis of magnetic and percolative properties can be done exactly. Directed random graphs also differ from undirected ones with respect to zero-temperature freezing. Only at low connectivity do they remain trapped in a disordered configuration. Above a certain threshold, however, the zero-temperature dynamics quickly drives the model toward a broken symmetry (magnetized) state. Only above this threshold, which is almost twice as large as the percolation threshold, do we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behavior on directed random graphs is reproduced within a certain approximate scheme.

Unsupervised Object Segmentation with a Hybrid Graph Model (HGM)

In this work, we address the problem of performing class-specific unsupervised object segmentation, i.e., automatic segmentation without annotated training images. Object segmentation can be regarded as a special data clustering problem where both class-specific information and local texture/color similarities have to be considered. To this end, we propose a hybrid graph model (HGM) that can make effective use of both symmetric and asymmetric relationship among samples. The vertices of a hybrid graph represent the samples and are connected by directed edges and/or undirected ones, which represent the asymmetric and/or symmetric relationship between them, respectively. When applied to object segmentation, vertices are superpixels, the asymmetric relationship is the conditional dependence of occurrence, and the symmetric relationship is the color/texture similarity. By combining the Markov chain formed by the directed subgraph and the minimal cut of the undirected subgraph, the object boundaries can be determined for each image. Using the HGM, we can conveniently achieve simultaneous segmentation and recognition by integrating both top-down and bottom-up information into a unified process. Experiments on 42 object classes (9,415 images in total) show promising results.

LibDAI: A Free and Open Source C++ Library for Discrete Approximate Inference in Graphical Models

This paper describes the software package libDAI, a free & open source C++ library that provides implementations of various exact and approximate inference methods for graphical models with discrete-valued variables. libDAI supports directed graphical models (Bayesian networks) as well as undirected ones (Markov random fields and factor graphs). It offers various approximations of the partition sum, marginal probability distributions and maximum probability states. Parameter learning is also supported. A feature comparison with other open source software packages for approximate inference is given. libDAI is licensed under the GPL v2+ license and is available at http://www.libdai.org.

Motion Planning for Carlike Robots Using a Probabilistic Learning Approach

In this article, a recently developed learning approach for robot motion planning is extended and applied to two types of carlike robots: normal ones, and robots that can only move for ward. In this learning approach the motion planning process is split into two phases: the learning phase and the query phase. In the learning phase, a probabilistic roadmap is incremen tally constructed in configuration space. This roadmap is an undirected graph where nodes correspond to randomly chosen configurations in free space and edges correspond to simple collision-free paths between the nodes. These simple motions are computed using a fast local method. In the query phase, this roadmap can be used to find paths between different pairs of configurations. The approach can be applied to normal carlike robots (with nonholonomic constraints) by using suitable local methods that compute paths feasible for the robots. Application to carlike robots that can move only forward demands a more fundamen tal adaptation of the learning method; that is, the roadmaps must be stored in directed graphs instead of undirected ones. We have implemented the planners, and we present experi mental results that demonstrate their efficiency for both robot types, even in cluttered workspaces. Also, we prove probabilis tic completeness of the planners. 1. A robot is fully controllable iff the existence of a path in the open free C space implies the existence of a feasible path. 2. By "metric," we simply mean a function of type C x C → R+ without any restrictions. 3. Formally, this requires a minor adaptation of the general outline of the learning algorithm, as presented in Section 3.1. Line 7. will be: if -connected(c, n) Λ L( c, n) ⊂ free C space then E = E U {( c, n)}. 4. This does not necessarily hold if P1 consists of just one or two circular arcs of maximal curvature. In this case, however, P1 can be found directly with the local planner. 5. We say a path Q lies within distance ∈ of a path R, iff ∀ q ∈ Q : ∃r ∈ R : |q - r| ≤ ∈ (in C space).