Showing papers on "Connectivity published in 2022"

Distantly Supervised Relation Extraction using Global Hierarchy Embeddings and Local Probability Constraints

Abstract: To find relational facts of interest from plain texts, distantly supervised relation extraction (DSRE) has drawn significant attention. Recent works exploit relation hierarchies to mine more clues for long-tail relations and achieve good performance. However, they ignore or underutilize the correlation of relations in the hierarchical structure. Empirically, the correlation facilitates knowledge transfer between different relations to further handle long-tail relations and improves inter-relational discrimination. In this paper, we devise an end-to-end network to model the correlation of relations from two perspectives. Globally, we construct an undirected connected graph according to the relation hierarchies, and employ Graph Attention Networks (GATs) to aggregate node information and generate correlation-aware Global Hierarchy Embeddings (GHE). Locally, we assume that along the relation hierarchies, the classification results of adjacent levels should be highly interdependent, and introduce a constraint called Local Probability Constraints (LPC) to take it into account. LPC is then combined with a branch network for both sentence-level and bag-level classification. Experimental results on the popular New York Times (NYT) dataset show that, our model GHE-LPC outperforms other state-of-the-art baselines in terms of AUC, Top-N precision, accuracy of Hits@K, etc.

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

Abstract: The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

Sharp Thresholds in Random Simple Temporal Graphs

Abstract: A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erdös-Rényi random graph G ~ Gn,p by considering a random permutation π of the edges and interpreting the ranks in π as presence times. We give a thorough study of the temporal connectivity of such graphs and derive implications for the existence of several kinds of sparse spanners. It turns out that temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that, at p = log $n$ /n, any fixed pair of vertices can a.a.s. reach each other; at 2 log $n$ /n, at least one vertex (and in fact, any fixed vertex) can a.a.s. reach all others; and at 3 log $n$ /n, all the vertices can a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size 2n + o(n) as soon as it becomes temporally connected, which is nearly optimal as 2n - 4 is a lower bound. This result is quite significant because temporal graphs do not admit spanners of size O(n) in general (Kempe, Kleinberg, Kumar, STOC 2000). In fact, they do not even always admit spanners of size o( $n$ ² ) (Axiotis, Fotakis, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, any non-negligible obstruction is statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners a step further, we show that pivotal spanners-i.e., spanners of size 2n - 2 made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently)-exist a.a.s. at 4 log $n$ / n, this threshold being also sharp. Finally, we show that optimal spanners (of size 2n - 4) also exist a.a.s. at p = 4 log $n$ /n, Whether this value is a sharp threshold is open, we conjecture that it is. For completeness, we compare the above results to existing results in related areas, including edge-ordered graphs, gossip theory, and population protocols, showing that our results can be interpreted in these settings as well, and that in some cases, they improve known results therein. Finally, we discuss an intriguing connection between our results and Janson's celebrated results on percolation in weighted graphs.

On the Spectral Radius of Minimally 2-(Edge)-Connected Graphs with Given Size

Abstract: A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting any arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). Let $m= \binom{d}{2}+t$, $1\leq t\leq d$ and $G_m$ be the graph obtained from the complete graph $K_d$ by adding one new vertex of degree $t$. Let $H_m$ be the graph obtained from $K_d\backslash\{e\}$ by adding one new vertex adjacent to precisely two vertices of degree $d-1$ in $K_d\backslash\{e\}$. Rowlinson [Linear Algebra Appl., 110 (1988) 43--53.] showed that $G_m$ attains the maximum spectral radius among all graphs of size $m$. This classic result indicates that $G_m$ attains the maximum spectral radius among all $2$-(edge)-connected graphs of size $m=\binom{d}{2}+t$ except $t=1$. The next year, Rowlinson [Europ. J. Combin., 10 (1989) 489--497] proved that $H_m$ attains the maximum spectral radius among all $2$-connected graphs of size $m=\binom{d}{2}+1$ ($d\geq 5$), this also indicates $H_m$ is the unique extremal graph among all $2$-connected graphs of size $m=\binom{d}{2}+1$ ($d\geq 5$). Observe that neither $G_m$ nor $H_m$ are minimally $2$-(edge)-connected graphs. In this paper, we determine the maximum spectral radius for the minimally $2$-connected ($2$-edge-connected) graphs of given size; moreover, the corresponding extremal graphs are also characterized.

Synchronization of Continuous-Time Linear Systems With Time-Varying Output Couplings

Abstract: In this article, we investigate the synchronization problem of continuous-time linear systems with time-varying output couplings. Both bidirectional and unidirectional couplings are studied. In the former one, each pair of connected subsystems can both measure the outputs consisting of the relative information, whereas in the unidirectional case, it is not necessarily the case and a specific scenario when the output matrices have a common matrix factor is considered. The distributed synchronization protocols are designed and the convergence conditions of the resulting system are established making use of switched systems theory based on the detectability assumption and appropriate graph connectivity conditions. Finally, numerical examples are given to verify the derived results.

The Upper and Forcing Connected Outer Connected Geodetic Numbers of a Graph

Abstract: For a connected graph [Formula: see text] of order at least two, a connected outer connected geodetic set [Formula: see text] of [Formula: see text] is called a minimal connected outer connected geodetic set if no proper subset of [Formula: see text] is a connected outer connected geodetic set of [Formula: see text]. The upper connected outer connected geodetic number [Formula: see text] of [Formula: see text] is the maximum cardinality of a minimal connected outer connected geodetic set of [Formula: see text]. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for any two integers [Formula: see text], [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is the connected outer connected geodetic number of a graph. Also, another parameter forcing connected outer connected geodetic number [Formula: see text] of a graph [Formula: see text] is introduced and several interesting results on this parameter are studied.

A study of connectivity on dynamic graphs: computing persistent connected components

Abstract: This work focuses on connectivity in a dynamic graph. An undirected graph is defined on a finite and discrete time interval. Edges can appear and disappear over time. The first objective of this work is to extend the notion of connected component to dynamic graphs in a new way. Persistent connected components are defined by their size, corresponding to the number of vertices, and their length, corresponding to the number of consecutive time steps they are present on. The second objective of this work is to develop an algorithm computing the largest, in terms of size and length, persistent connected components in a dynamic graph. PICCNIC algorithm (PersIstent Connected CompoNent InCremental Algorithm) is a polynomial time algorithm of minimal complexity. Another advantage of this algorithm is that it works online: knowing the evolution of the dynamic graph is not necessary to execute it. PICCNIC algorithm is implemented using the GraphStream library and experimented in order to carefully study the outcome of the algorithm according to different input graph types, as well as real data networks, to verify the theoretical complexity, and to confirm its feasibility for graphs of large size.

Consensus of double integrators over a chain with reverse edges*

Abstract: Consensus in directed networks, comprising agents modelled as double integrators, is dependent on eigenvalues of the graph Laplacian matrix of the network. Second order consensus is always guaranteed if there is a directed spanning tree without any directed cycles. However, in the presence of directed cycles, consensus cannot be guaranteed. Although the necessary and sufficient condition for consensus then depends on both the real and imaginary part of the eigenvalues of the graph Laplacian, the eigenvalue with smallest real part paves the way towards a sufficient condition for consensus. This paper will first investigate how the presence of reverse edges (leading to directed cycles) affects the Laplacian spectra of a chain network, and subsequently present a sufficient condition for consensus among double integrators in the presence of two directed cycles in a chain network.

2-Vertex Self Switching of Trees

Abstract: For a finite undirected graph \(G(V,E)\) and a non empty subset \(\sigma \subseteq V\), the switching of \(G\) by \(\sigma\) is defined as the graph \(G^\sigma (V, E')\) which is obtained from \(G\) by removing all edges between \(\sigma\) and its complement \(V\)-\(\sigma\) and adding as edges all non-edges between \(\sigma\) and \(V\)-\(\sigma\). For \(\sigma=\{v\}\), we write \(G^v\) instead of \(G^{\{v\}}\) and the corresponding switching is called as vertex switching. We also call it as \(|\sigma|\)-vertex switching. When \(|\sigma|=2\), it is termed as 2-vertex switching. If \(G \cong G^\sigma\), then it is called self vertex switching. A subgraph \(B\) of \(G\) which contains \(G[\sigma]\) is called a joint at \(\sigma\) in \(G\) if \(B-\sigma\) is connected and maximal. If \(B\) is connected, then we call \(B\) as a \(c\)-joint and otherwise a \(d\)-joint. A graph with no cycles is called an acyclic graph. A connected acyclic graph is called a tree. In this paper, we give necessary and sufficient conditions for a graph \(G\), for which \(G^{\sigma}\) at \(\sigma =\{u, v\}\) to be connected and acyclic when \(uv\in E(G)\) and \(uv otin E(G)\). Using this, we characterize trees with a 2-vertex self switching.

Extreme graphs on the Sombor indices

Abstract: Gutman proposed the concept of Sombor index. It is defined via the term $ \sqrt{d_F(v_i)^2+d_F(v_j)^2} $, where $ d_F(v_i) $ is the degree of the vertex $ v_i $ in graph $ F $. Also, the reduced Sombor index and the Average Sombor index have been introduced recently, and these topological indices have good predictive potential in mathematical chemistry. In this paper, we determine the extreme molecular graphs with the maximum value of Sombor index and the extremal connected graphs with the maximum (reduced) Sombor index. Some inequalities relations among the chemistry indices are presented, these topology indices including the first Banhatti-Sombor index, the first Gourava index, the Second Gourava index, the Sum Connectivity Gourava index, Product Connectivity Gourava index, and Eccentric Connectivity index. In addition, we characterize the graph where equality occurs.

Bounds on the General Eccentric Connectivity Index

Abstract: The general eccentric connectivity index of a graph R is defined as ξec(R)=∑u∈V(G)d(u)ec(u)α, where α is any real number, ec(u) and d(u) represent the eccentricity and the degree of the vertex u in R, respectively. In this paper, some bounds on the general eccentric connectivity index are proposed in terms of graph-theoretic parameters, namely, order, radius, independence number, eccentricity, pendent vertices and cut edges. Moreover, extremal graphs are characterized by these bounds.

Prototypical Graph Neural Network for Few-Shot Learning

Abstract: The graph neural network (GNN) can significantly improve the performance of few-shot learning due to its ability to automatically aggregate sample node information. However, many previous GNN works are sensitive to noise. In this paper, a few-shot image classification algorithm (Proto-GNN) based on the prototypical graph neural network is presented. First, convolutional neural network (CNN) is used to obtain the feature vectors of the support set samples, which can be used to calculate the prototype of each category. Then, the feature vectors of the support set, prototype vectors and the feature vectors of the query set are used to form a completely connected graph. Finally, Proto-GNN is built and trained. Due to the addition of the prototype vectors, the query set samples obtain more category information, which is more conducive to classification. The experimental results show that the model in this paper has a better classification performance on the Omniglot data set.

Graph Realizations Constrained by Connected Local Dimensions and Connected Local Bases

Abstract: For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.

On the 2-independence subdivision number of graphs

Abstract: A subset S of vertices in a graph G = (V;E) is 2-independent if every vertexof S has at most one neighbor in S: The 2-independence number is the maximumcardinality of a 2-independent set of G: In this paper, we initiate the study of the2-independence subdivision number sdβ2(G) defined as the minimum numberof edges that must be subdivided (each edge in G can be subdivided at mostonce) in order to increase the 2-independence number. We first show that forevery connected graph G of order at least three, 1≤sdβ2(G)≤2; and we give anecessary and sufficient condition for graphs G attaining each bound. Moreover,restricted to the class of trees, we provide a constructive characterization of alltrees T with sdβ2(T)= 2; and we show that such a characterization suggestsan algorithm that determines whether a tree T has sdβ2(T)= 2 or sdβ2(T) = 1in polynomial time.

A Polynomial Delay Algorithm for Enumerating 2-Edge-Connected Induced Subgraphs

Abstract: The problem of enumerating connected induced subgraphs of a given graph is classical and studied well. It is known that connected induced subgraphs can be enumerated in constant time for each subgraph. In this paper, we focus on highly connected induced subgraphs. The most major concept of connectivity on graphs is vertex connectivity. For vertex connectivity, some enumeration problem settings and enumeration algorithms have been proposed, such as k-vertex connected spanning subgraphs. In this paper, we focus on another major concept of graph connectivity, edge-connectivity. This is motivated by the problem of finding evacuation routes in road networks. In evacuation routes, edge-connectivity is important, since highly edge-connected subgraphs ensure multiple routes between two vertices. In this paper, we consider the problem of enumerating 2-edge-connected induced subgraphs of a given graph. We present an algorithm that enumerates 2-edge-connected induced subgraphs of an input graph G with n vertices and m edges. Our algorithm enumerates all the 2-edge-connected induced subgraphs in O(n3m|SG|) time, where SG is the set of the 2-edge-connected induced subgraphs of G. Moreover, by slightly modifying the algorithm, we have a O(n3m)-delay enumeration algorithm for 2-edge-connected induced subgraphs.