Graph Indexing for Efficient Evaluation of Label-constrained Reachability Queries

Given a directed edge labeled graph G , to check whether vertex v is reachable from vertex u under a label set S is to know if there is a path from u to v whose edge labels across the path are a subset of S . Such a query is referred to as a label-constrained reachability ( LCR ) query. In this article, we present a new approach to store a compressed transitive closure of G in the form of intervals over spanning trees (forests). The basic idea is to associate each vertex v with two sequences of some other vertices: one is used to check reachability from v to any other vertex, by using intervals, while the other is used to check reachability to v from any other vertex. We will show that such sequences are in general much shorter than the number of vertices in G. Extensive experiments have been conducted, which demonstrates that our method is much better than all the previous methods for this problem in all the important aspects, including index construction times, index sizes, and query times.

Direction Matters: On Influence-Preserving Graph Summarization and Max-Cut Principle for Directed Graphs

Summarizing large-scale directed graphs into small-scale representations is a useful but less-studied problem setting. Conventional clustering approaches, based on Min-Cut-style criteria, compress both the vertices and edges of the graph into the communities, which lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed-edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features, useful for efficient interventions of population behavior. In this letter, we present a model, based on minimizing reconstruction error with nonnegative constraints, which relates to a Max-Cut criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.

Detecting problematic transactions in a consumer-to-consumer e-commerce network

Providers of online marketplaces are constantly combatting against problematic transactions, such as selling illegal items and posting fictive items, exercised by some of their users. A typical approach to detect fraud activity has been to analyze registered user profiles, user’s behavior, and texts attached to individual transactions and the user. However, this traditional approach may be limited because malicious users can easily conceal their information. Given this background, network indices have been exploited for detecting frauds in various online transaction platforms. In the present study, we analyzed networks of users of an online consumer-to-consumer marketplace in which a seller and the corresponding buyer of a transaction are connected by a directed edge. We constructed egocentric networks of each of several hundreds of fraudulent users and those of a similar number of normal users. We calculated eight local network indices based on up to connectivity between the neighbors of the focal node. Based on the present descriptive analysis of these network indices, we fed twelve features that we constructed from the eight network indices to random forest classifiers with the aim of distinguishing between normal users and fraudulent users engaged in each one of the four types of problematic transactions. We found that the classifier accurately distinguished the fraudulent users from normal users and that the classification performance did not depend on the type of problematic transaction.

Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Hamiltonicity in random directed graphs is born resilient

Let $\{D_M\}_{M\geq 0}$ be the n-vertex random directed graph process, where $D_0$ is the empty directed graph on n vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $$\varepsilon > 0$$ , we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $1/2-\varepsilon$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$ -resiliently Hamiltonian. Furthermore, for each $\varepsilon > 0$ , we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$ -resiliently Hamiltonian.This improves a result of Ferber, Nenadov, Noever, Peter and Škorić who showed, for each $\varepsilon > 0$ , that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$ -resiliently Hamiltonian if $p=\omega(\log^8n/n)$ .

Stochastic Online Learning with Probabilistic Graph Feedback

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i – for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ {0,1}, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.

Safety evaluation of truss structures using nested discrete Bayesian networks

Truss structures have been widely adopted for civil structures such as long-span buildings and bridges. An actual truss system is usually statically indeterminate having numerous members and high redundancy. It is practically difficult to evaluate the truss safety through traditional reliability-based approaches in view of complex failure modes and uncertainties. Moreover, monitoring data are generally insufficient in reality due to limited sensors under cost consideration. Therefore, a nested discrete Bayesian network has been developed for safety evaluation of truss structures. A concept of member risk coefficient is first proposed based on the mechanical relationship between load effects and member resistance. According to the coefficients of all members, member risk sequences are found as the basis for establishing the topology of a member-level Bayesian network. Each network node represents a truss member and a nodal variable having three states: elasticity, plasticity, and failure. Two relevant member nodes are connected by a directed edge whose causality strength is expressed by a conditional probability table. Meanwhile, a system-level network topology is established to reflect the effects of member states on the truss system. The system is assigned with a node having two states: safety and failure. The directed edge of each member node directly points to the system node. Then, the two networks are combined to form a nested network topology. By this means, direct topology learning is avoided in order to find rational and concise topologies satisfying the mechanical characteristics of civil structures. After that, the conditional probability tables for the nested network are obtained through parameter learning on complete numerical observation data. The data acquirement procedure takes into account uncertainties by defining the randomness of cross-sectional areas and external loads. With the conditional probability tables, the nested network is ready for use. When new evidence from limited monitored members is input into the nested network, the state probabilities of the other members, as well as the system, are simultaneously updated using exact inference algorithms. The inference ability using insufficient information well accords with the demand of engineering practice. Finally, the proposed method has been successfully verified against both numerical and experimental truss structures. It was found that the network estimations could be further confirmed with more evidence.