Unweighted Graphs Research Articles

A Hierarchy of Lower Bounds for Sublinear Additive Spanners

Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say $\tilde{O}(n^{1+\delta})$ bits. There is an inherent tradeoff between the sparsity parameter $\delta$ and the stretch function $f$ of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when $\delta\ge 1/3$ there are schemes with constant additive stretch (distance $d$ is stretched to at most $f(d) = d + O(1)$), and recent results of Abboud and Bodwin show that when $\delta < 1/3$ there are no such schemes. Thus, to get practically efficient graph compression with $\delta \to 0$ we must pay superconstant additive stretch, but exactly how much do we have to pay? In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function $f$ for sparsity parameter $\delta \in (0,1/3)$. Specifically, for any integer $k\ge 2$, any compression scheme using $O(n^{1+\frac{1}{2^k-1} - \epsilon})$ bits has a sublinear additive stretch function $f$: $f(d) = d + \Omega(d^{1-\frac{1}{k}})$. This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's $(1+\epsilon,\beta)$-spanners have an essentially optimal tradeoff between $\delta$, $\epsilon$, and $\beta$, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of $(1+\epsilon, O(k/\epsilon)^{k-1})$-spanners with size $O((k/\epsilon)^{h_k} kn^{1+\frac{1}{2^{k+1}-1}})$, where $h_k < 3/4$. This size bound improves on the spanners of Elkin and Peleg (2004), Thorup and Zwick (2006), and Pettie (2009). According to our lower bounds neither the size nor stretch function can be substantially improved. Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of $(\beta,\epsilon)$-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least $2\gamma+1$. One consequence is that Baswana et al.'s (2010) additive $O(\gamma)$-spanners with size $O(n^{1+\frac{1}{2\gamma+1}})$ cannot be improved in the exponent.

Method for Quantitative Estimation of the Risk Propagation Threshold in Electric Power CPS Based on Seepage Probability

Because of the non-uniformity of the electric power CPS network and the dynamic nature of the risk propagation process, it is difficult to quantify the critical point of a cyber risk explosion. From the perspective of the dependency network, this paper proposes a method for quantitative evaluation of the risk propagation threshold of power CPS networks based on the percolation theory. First, the power CPS network is abstracted as a dual-layered network-directed unweighted graph according to topology correlation and coupling logic, and the asymmetrical balls-into-bins allocation method is used to establish a "one-to-many" and "partially coupled" non-uniform power CPS characterization model. Then, considering the directionality between the cyber layer and the physical layer link, the probability of percolation flow is introduced to establish the propagation dynamic equations for the internal coupling relationship of each layer. Finally, the risk propagation threshold is numerically quantified by defining the survival function of power CPS network nodes, and the validity of the proposed method is verified by the IEEE 30-bus system and 150-node Barabsi-Albert model.

The Densest Subgraph Problem with a Convex/Concave Size Function

In the densest subgraph problem, given an edge-weighted undirected graph $G=(V,E,w)$, we are asked to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of weights of the edges in the subgraph induced by $S$. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of $S\subseteq V$. Specifically, we introduce the $f$-density of $S\subseteq V$, which is defined as $w(S)/f(|S|)$, where $f:\mathbb{Z}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}$ is a monotonically non-decreasing function. In the $f$-densest subgraph problem ($f$-DS), we aim to find $S\subseteq V$ that maximizes the $f$-density $w(S)/f(|S|)$. Although $f$-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex/concave size function $f$ appropriately. For $f$-DS with convex function $f$, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for $f$-DS with concave function $f$, we propose an LP-based exact algorithm, a flow-based $O(|V|^3)$-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.

Research on a large-scale community detection algorithm based on non-weighted graph

Community detection and partitioning has become a critical issue in large-scale social networks. However, the applicability of most existing methods is limited by computational costs. In order to improve the quality of community division and calculation efficiency, a community detection algorithm based on un-weighted graphs is proposed. This algorithm uses two parameters to measure the community to achieve community discovery, which is clustering coefficient and common neighbor similarity, and its effectiveness is proved by the academic formula. Experimental analysis was carried out using a real social network dataset, and compared with other algorithms proposed two methods. The experimental results show that the proposed method is more efficient and the computation time is linear. It is suitable for community detection in large-scale social networks.

Degree sums and dense spanning trees.

Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems.

Gröbner Bases Techniques for an $S$-Packing $k$-Coloring of a Graph

In this paper, polynomial ideal theory is used to deal with the problem of the $S$-packing coloring of a finite undirected and unweighted graph by introducing a family of polynomials encoding the problem. A method to find the $S$-packing colorings of the graph is presented and illustrated by examples.

Resistance characterizations of equiarboreal graphs

A weighted (unweighted) graph G is called equiarboreal if the sum of weights (the number) of spanning trees containing a given edge in G is independent of the choice of edge. In this paper, we give some resistance characterizations of equiarboreal weighted and unweighted graphs, and obtain the necessary and sufficient conditions for k-subdivision graphs, iterated double graphs, line graphs of regular graphs and duals of planar graphs to be equiarboreal. Applying these results, we obtain new infinite families of equiarboreal graphs, including iterated double graphs of 1-walk-regular graphs, line graphs of triangle-free 2-walk-regular graphs, and duals of equiarboreal planar graphs.

Eccentricity approximating trees

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that eccT(v)−eccG(v)≤2 for every vertex v, where eccG(v) (eccT(v)) is the eccentricity of a vertex v in G (in T, respectively). Using only metric properties of graphs, we extend that result to a much larger family of graphs containing among others chordal graphs, the underlying graphs of 7-systolic complexes and plane triangulations with inner vertices of degree at least 7. Furthermore, based on our approach, we propose two heuristics for constructing eccentricity k-approximating trees with small values of k for general unweighted graphs. We validate those heuristics on a set of real-world networks and demonstrate that all those networks have very good eccentricity approximating trees.

On the minimum eccentricity shortest path problem

In this paper, we introduce and investigate the Minimum Eccentricity Shortest Path (MESP) problem in unweighted graphs. It asks for a given graph to find a shortest path with minimum eccentricity. Let n and m denote the number of vertices and the number of edges of a given graph. We demonstrate that:•a minimum eccentricity shortest path plays a crucial role in obtaining the best to date approximation algorithm for a minimum distortion embedding of a graph into the line;•the MESP problem is NP-hard for planar bipartite graphs with maximum degree 3 and W[2]-hard for general graphs;•a shortest path of minimum eccentricity k can be computed in O(n2k+2m) time;•a 2-approximation, a 3-approximation, and an 8-approximation for the MESP problem can be computed in O(n3) time, in O(nm) time, and in O(m) time, respectively;•in a graph with a shortest path of eccentricity k, a k-dominating set can be found in nO(k) time.

Inverses of weighted graphs

We consider only connected bipartite graphs G with a unique perfect matching M. Let Gw be the weighted graph obtained from G by giving weights to its edges using the positive weight function w:E(G)→(0,∞) such that w(e)=1 for each e∈M. An unweighted graph G may be viewed as a weighted graph with the weight function w≡1. A weighted graph Gw is nonsingular if its adjacency matrix A(Gw) is nonsingular. The inverse of a nonsingular weighted graph Gw is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(Gw) via a diagonal matrix of ±1s. By G/M we denote the graph obtained from G by contracting each edge of M to a single vertex. It is known that if G/M is bipartite, then Gw is invertible for each weight function w. In this article, we consider the converse of this result and show that if Gw is invertible for each w, then the graph G/M is bipartite. We also supply exact conditions under which Gw is invertible for one w will force that the graph G/M is bipartite.

Shrink: Distance preserving graph compression

The ever increasing size of graphs makes them difficult to query and store. In this paper, we present Shrink, a compression method that reduces the size of the graph while preserving the distances between the nodes. The compression is based on the iterative merging of the nodes. During each merging, a system of linear equations is solved to define new edge weights in a way that the new weights have the least effect on the distances. Merging nodes continues until the desired size for the compressed graph is reached. The compressed graph, also known as the coarse graph, can be queried without decompression. As the complexity of distance-based queries such as shortest path queries is highly dependent on the size of the graph, Shrink improves the performance in terms of time and storage. Shrink not only provides the length of the shortest path but also identifies the nodes on the path. The approach has been applied to both weighted and unweighted graphs including road network, friendship network, collaboration network, web graph and social network. In the experiment, a road network with more than 2.5 million nodes is reduced to fifth while the average relative error is less than 1%.

Perfect quantum state transfer using Hadamard diagonalizable graphs

Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph. Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix. We give a simple eigenvalue characterization for when such a graph has perfect state transfer at time π/2; this characterization allows one to choose the correct eigenvalues to build graphs having perfect state transfer. We characterize the graphs that are diagonalizable by the standard Hadamard matrix, showing a direct relationship to cubelike graphs. We then give a number of constructions producing a wide variety of new graphs that exhibit perfect state transfer, and we consider several corollaries in the settings of both weighted and unweighted graphs, as well as how our results relate to the notion of pretty good state transfer. Finally, we give an optimality result, showing that among regular graphs of degree at most 4, the hypercube is the sparsest Hadamard diagonalizable connected unweighted graph with perfect state transfer.

Large-scale network organization of EEG functional connectivity in newborn infants.

The organization of functional brain networks changes across human lifespan. The present study analyzed functional brain networks in healthy full-term infants (N = 139) within 1-6 days from birth by measuring neural synchrony in EEG recordings during quiet sleep. Large-scale phase synchronization was measured in six frequency bands with the Phase Lag Index. Macroscopic network organization characteristics were quantified by constructing unweighted minimum spanning tree graphs. The cortical networks in early infancy were found to be significantly more hierarchical and had a more cost-efficient organization compared with MST of random control networks, more so in the theta and alpha than in other frequency bands. Frontal and parietal sites acted as the main hubs of these networks, the topological characteristics of which were associated with gestation age (GA). This suggests that individual differences in network topology are related to cortical maturation during the prenatal period, when functional networks shift from strictly centralized toward segregated configurations. Hum Brain Mapp 38:4019-4033, 2017. © 2017 Wiley Periodicals, Inc.

The Computational Complexity of Weighted Greedy Matching

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDYMATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bushgraphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is ρ, then for every ε &gt; 0 the randomized MRG algorithm of (Aronson et al. 1995) gives a (ρ − ε)-approximation for the maximum cardinality matching. We conjecture that a tightbound for ρ is 2/3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.

On the inverse of a class of weighted graphs

In this article, only connected bipartite graphs $G$ with a unique perfect matching $\c{M}$ are considered. Let $G_\w$ denote the weighted graph obtained from $G$ by giving weights to its edges using the positive weight function $\w:E(G)\ar (0,\ity)$ such that $\w(e)=1$ for each $e\in\c{M}$. An unweighted graph $G$ may be viewed as a weighted graph with the weight function $\w\equiv\1$ (all ones vector). A weighted graph $G_\w$ is nonsingular if its adjacency matrix $A(G_\w)$ is nonsingular. The {\em inverse} of a nonsingular weighted graph $G_\w$ is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix $A(G_\w)$ via a diagonal matrix whose diagonal entries are either $1$ or $-1$. In [S.K.~Panda and S.~Pati. On some graphs which possess inverses. {\em Linear and Multilinear Algebra}, 64:1445--1459, 2016.], the authors characterized a class of bipartite graphs $G$ with a unique perfect matching such that $G$ is invertible. That class is denoted by $\c{H}_{nmc}$. It is natural to ask whether $G_\w$ is invertible for each invertible graph $G\in\c{H}_{nmc}$ and for each weight function $\w\not\equiv\1$. In this article, first an example is given to show that there is an invertible graph $G\in\c{H}_{nmc}$ and a weight function$\w\not\equiv\1$ such that $G_\w$ is not invertible. Then the weight functions $\w$ for each graph $G\in\c{H}_{nmc}$ such that $G_\w$ is invertible, are characterized.

$\mathsf {pSCAN}$ : Fast and Exact Structural Graph Clustering

We study the problem of structural graph clustering, a fundamental problem in managing and analyzing graph data. Given an undirected unweighted graph, structural graph clustering is to assign vertices to clusters, and to identify the sets of hub vertices and outlier vertices as well, such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected. In this paper, we develop a new two-step paradigm for scalable structural graph clustering based on our three observations. Then, we present a $\mathsf {pSCAN}$ approach, within the paradigm, aiming to reduce the number of structural similarity computations, and propose optimization techniques to speed up checking whether two vertices are structure-similar. $\mathsf {pSCAN}$ outputs exactly the same clusters as the existing approaches $\mathsf {SCAN}$ and $\mathsf {SCAN\text{++}}$ , and we prove that $\mathsf {pSCAN}$ is worst-case optimal. Moreover, we propose efficient techniques for updating the clusters when the input graph dynamically changes, and we also extend our techniques to other similarity measures, e.g., Jaccard similarity. Performance studies on large real and synthetic graphs demonstrate the efficiency of our new approach and our dynamic cluster maintenance techniques. Noticeably, for the twitter graph with 1 billion edges, our approach takes 25 minutes while the state-of-the-art approach cannot finish even after 24 hours.

Altered brain structural networks in attention deficit/hyperactivity disorder children revealed by cortical thickness.

This study investigated the cortical thickness and topological features of human brain anatomical networks related to attention deficit/hyperactivity disorder. Data were collected from 40 attention deficit/hyperactivity disorder children and 40 normal control children. Interregional correlation matrices were established by calculating the correlations of cortical thickness between all pairs of cortical regions (68 regions) of the whole brain. Further thresholds were applied to create binary matrices to construct a series of undirected and unweighted graphs, and global, local, and nodal efficiencies were computed as a function of the network cost. These experimental results revealed abnormal cortical thickness and correlations in attention deficit/hyperactivity disorder, and showed that the brain structural networks of attention deficit/hyperactivity disorder subjects had inefficient small-world topological features. Furthermore, their topological properties were altered abnormally. In particular, decreased global efficiency combined with increased local efficiency in attention deficit/hyperactivity disorder children led to a disorder-related shift of the network topological structure toward regular networks. In addition, nodal efficiency, cortical thickness, and correlation analyses revealed that several brain regions were altered in attention deficit/hyperactivity disorder patients. These findings are in accordance with a hypothesis of dysfunctional integration and segregation of the brain in patients with attention deficit/hyperactivity disorder and provide further evidence of brain dysfunction in attention deficit/hyperactivity disorder patients by observing cortical thickness on magnetic resonance imaging.

Reconfiguration of Steiner Trees in an Unweighted Graph

We study a reconfiguration problem for Steiner trees in an unweighted graph, which determines whether there exists a sequence of Steiner trees that transforms a given Steiner tree into another one by exchanging a single edge at a time. In this paper, we show that the problem is PSPACE-complete even for split graphs (and hence for chordal graphs), while solvable in linear time for interval graphs.

Finitely dependent insertion processes

A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance greater than $k$ receive independent colorings. Holroyd and Liggett constructed the first stationary $k$-dependent $q$-colorings by introducing an insertion algorithm on the complete graph $K_q$. We extend their construction from complete graphs to weighted directed graphs. We show that complete multipartite analogues of $K_3$ and $K_4$ are the only graphs whose insertion process is finitely dependent and whose insertion algorithm is consistent. In particular, there are no other such graphs among all unweighted graphs and among all loopless complete weighted directed graphs. Similar results hold if the consistency condition is weakened to eventual consistency. Finally we show that the directed de Bruijn graphs of shifts of finite type do not yield $k$-dependent insertion processes, assuming eventual consistency.

Clustering Categorical Data Using Community Detection Techniques.

With the advent of the k-modes algorithm, the toolbox for clustering categorical data has an efficient tool that scales linearly in the number of data items. However, random initialization of cluster centers in k-modes makes it hard to reach a good clustering without resorting to many trials. Recently proposed methods for better initialization are deterministic and reduce the clustering cost considerably. A variety of initialization methods differ in how the heuristics chooses the set of initial centers. In this paper, we address the clustering problem for categorical data from the perspective of community detection. Instead of initializing k modes and running several iterations, our scheme, CD-Clustering, builds an unweighted graph and detects highly cohesive groups of nodes using a fast community detection technique. The top-k detected communities by size will define the k modes. Evaluation on ten real categorical datasets shows that our method outperforms the existing initialization methods for k-modes in terms of accuracy, precision, and recall in most of the cases.