Threshold Graphs Research Articles

Intergroup networks as random threshold graphs.

Similar-minded people tend to form social groups. Due to pluralistic homophily as well as a sort of heterophily, people also participate in a wide variety of groups. Thus, these groups generally overlap with each other; an overlap between two groups can be characterized by the number of common members. These common members can play a crucial role in the transmission of information between the groups. As a step towards understanding the information dissemination, we perceive the system as a pruned intergroup network and show that it maps to a very basic graph theoretic concept known as a threshold graph. We analyze several structural properties of this network such as degree distribution, largest component size, edge density, and local clustering coefficient. We compare the theoretical predictions with the results obtained from several online social networks (LiveJournal, Flickr, YouTube) and find a good match.

On a class of graphs between threshold and total domishold graphs

A total dominating set in a graph is a subset of vertices such that every vertex in the graph has a neighbor in it. A graph is said to be total domishold if it admits a total domishold structure, that is, a hyperplane with non-negative coefficients that separates the characteristic vectors of its total dominating sets from the characteristic vectors of other vertex subsets. Hereditary total domishold graphs, that is, graphs every induced subgraph of which is total domishold, were recently characterized in terms of forbidden induced subgraphs; this characterization leads to an O(|V(G)|6) recognition algorithm of hereditary total domishold graphs.In this paper, we study a subclass G of the hereditary total domishold graphs obtained by replacing two of the forbidden induced subgraphs of order 6 with two of their proper induced subgraphs of order 5. We show that every connected graph in G is a leaf extension of a threshold graph, that is, it can be obtained from some threshold graph G by attaching some pendant vertices to each vertex of G. Using this property, we develop a structural characterization of graphs in G, which implies a linear time recognition algorithm for graphs in this class. We also give a linear time algorithm for computing a total domishold structure of a graph in G. In contrast to the algorithm for the general case of (hereditary) total domishold graphs, which relies on solving a linear program, our algorithm is purely combinatorial and works directly with the graph.

Cointegration-based financial networks study in Chinese stock market

We propose a method based on cointegration instead of correlation to construct financial complex network in Chinese stock market. The network is obtained starting from the matrix of p-value calculated by Engle–Granger cointegration test between all pairs of stocks. Then some tools for filtering information in complex network are implemented to prune the complete graph described by the above matrix, such as setting a level of statistical significance as a threshold and Planar Maximally Filtered Graph. We also calculate Partial Correlation Planar Graph of these stocks to compare the above networks. Last, we analyze these directed, weighted and non-symmetric networks by using standard methods of network analysis, including degree centrality, PageRank, HITS, local clustering coefficient, K-shell and strongly and weakly connected components. The results shed a new light on the underlying mechanisms and driving forces in a financial market and deepen our understanding of financial complex network.

The Star and Biclique Coloring and Choosability Problems

A biclique of a graph G is an induced complete bipartite graph. A star of G is a biclique contained in the closed neighborhood of a vertex. A star (biclique) k-coloring of G is a k-coloring of G that contains no monochromatic maximal stars (bicliques). Similarly, for a list assignment L of G, a star (biclique) L-coloring is an L-coloring of G in which no maximal star (biclique) is monochromatic. If G admits a star (biclique) L-coloring for every k-list assignment L, then G is said to be star (biclique) k-choosable. In this article we study the computational complexity of the star and biclique coloring and choosability problems. Specifically, we prove that the star (biclique) k-coloring and k-choosability problems are \Sigma_2^p-complete and \Pi_3^p-complete for k > 2, respectively, even when the input graph contains no induced C_4 or K_{k+2}. Then, we study all these problems in some related classes of graphs, including H-free graphs for every H on three vertices, graphs with restricted diamonds, split graphs, threshold graphs, and net-free block graphs.

Computing the Characteristic Polynomial of Threshold Graphs

We present an algorithm for constructing the characteristic polynomial of a threshold graph’s adjacency matrix. The algorithm is based on a diagonalization procedure that is easy to describe. It can be implemented using O(n) space and with running time O(n 2 ).

On pairwise compatibility graphs having Dilworth number two

A graph G=(V,E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge-weight function w on T, and two non-negative real numbers dmin and dmax, dmin⩽dmax, such that V coincides with the set of leaves of T, and there is an edge (u,v)∈E if and only if dmin⩽dT,w(u,v)⩽dmax where dT,w(u,v) is the sum of the weights of the edges on the unique path from u to v in T. When the constraints on the distance between the pairs of leaves concern only dmax or only dmin the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined.The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another.It is known that LPG∩mLPG is not empty and that threshold graphs, i.e. Dilworth one graphs, are contained in it. In this paper we prove that Dilworth two graphs belong to the set LPG∩mLPG, too. Our proof is constructive since we show how to compute all the parameters T, w, dmax and dmin exploiting the usual representation of Dilworth two graphs in terms of node weight function and thresholds. For graphs with Dilworth number two that are also split graphs, i.e. split permutation graphs, we provide another way to compute T, w, dmin and dmax when these graphs are given in terms of their intersection model.

Blockers for the Stability Number and the Chromatic Number

Given an undirected graph G = (V, E) and two positive integers k and d, we are interested in finding a set of edges (resp. non-edges) of size at most k to delete (resp. to add) in such a way that the chromatic number (resp. stability number) in the resulting graph will decrease by at least d compared to the original graph. We investigate these two problems in various classes of graphs (split graphs, threshold graphs, bipartite graphs and their complements) and determine their computational complexity. In some of the polynomial-time solvable cases, we also give a structural description of a solution.

Graphs with Dilworth Number Two are Pairwise Compatibility Graphs

A graph G=(V,E) is called a pairwise compatibility graph (PCG) if there exists a tree T, a positive edge-weight function w on T, and two non-negative real numbers dmin and dmax, dmin⩽dmax, such that V coincides with the set of leaves of T, and there is an edge (u,v)∈E if and only if dmin⩽dT,w(u,v)⩽dmax where dT,w(u,v) is the sum of the weights of the edges on the unique path from u to v in T. When the constraints on the distance between the pairs of leaves concern only dmax or only dmin the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined.It is known that LPG∩mLPG is not empty and that threshold graphs are one of the classes contained in it. It is also known that threshold graphs are all the graphs with Dilworth number one, where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another.In this paper we do one step ahead toward the comprehension of the structure of the set LPG∩mLPG, proving that graphs with Dilworth number two belong to this intersection.

SVM 기반의 걸음 검출 분석기의 구현

본 연구에서는 걸음 검출 알고리즘들의 걸음 검출률과 걸음 검출 결과를 상호 비교 분석할 수 있는 걸음검출 분석기를 구현하였다. 걸음 검출 분석기는 3축 가속도 센서의 데이터를 SVM 연산을 통하여 연속적인 에너지값으로 변환하고 걸음 측정 장치들에 사용되고 있는 걸음 검출 알고리즘들을 적용하여 걸음 검출결과를 동시에 비교할 수 있도록 함으로써 걸음 검출의 과정과 검출 형태, 그리고 정확성을 분석할 수 있게 하였다. 이를 위하여 걸음 검출 분석기는 에너지값을 에너지 파형 그래프로 나타내고, 걸음 검출 알고리즘을 에너지값에 적용하여, 걸음으로 검출된 Peak값들과 위치를 확인하고, 실제 걸음이 정상적으로 검출되지 않은 경우에는 원인을 시각적으로 분석할 수 있도록 하였다. 또한 걸음 검출 알고리즘마다 걸음 검출에 적용된 임계값을 그래프화하여 같이 표시함으로써, 걸음으로 인정되지 않는 경우에 사용된 임계값을 바로 확인하고, 이를 통하여 적절한 임계값을 구하는 수식의 조정에 필요한 요소들을 파악하는 데에 도움이 될 수 있도록 하였다. 본 연구에서 구현된 걸음 검출 분석기는 여러 걸음 검출 알고리즘을 수평적으로 비교, 분석함으로써 측정 장치에 가장 적합한 걸음 검출 알고리즘을 선택하거나, 기존의 걸음 검출 알고리즘의 성능 개선 방안을 파악하는 데에 효과적으로 활용할 수 있으므로 걸음 검출 알고리즘에 대한 연구, 또는 걸음 측정 장치의 설계에 폭넓게 활용될 수 있다. In this study, we designed and implemented a step detection analyzer that can compare and analyze the step detection rates and results among the step detection algorithms. The step detection analyzer converts 3-axes accelerometer data into continuous energy stream through SVM operation, shows the horizontal comparison among the step detection results for each step detection algorithms, and can make elemental detection analyses. For these processes, the step detection analyzer presents the continuous energy stream as energy waveform, checks the peak values and time location of the detected steps with step detection algorithms, and gives visual interface to get some possible causes in cases of step detection miss. It can also give the threshold graph for each algorithm to check the threshold value on missed cases directly and can help to get more appropriate threshold values or other adjustable parameters in step detection algorithm. This step detection analyzer can be applied efficiently on performance enhancement of step detection algorithm, on deciding an appropriate algorithm for a specific step counter system in the various step counter filed operations.

Betti diagrams from graphs

The emergence of Boij-S\"oderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a k-linear minimal resolution arises from that of the Stanley-Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of 2-linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with 2-linear resolutions, threshold graphs, and anti-lecture hall compositions. Moreover, we prove that any Betti diagram of a module with a 2-linear resolution is realized by a direct sum of Stanley-Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.

Perfectness of clustered graphs

Given a clustered graph (G,V), that is, a graph G=(V,E) together with a partition V of its vertex set, the selective coloring problem consists in choosing one vertex per cluster such that the chromatic number of the subgraph induced by the chosen vertices is minimum. This problem can be formulated as a covering problem with a 0–1 matrix M(G,V). Nevertheless, we observe that, given (G,V), it is NP-hard to check if M(G,V) is conformal (resp. perfect). We will give a sufficient condition, checkable in polynomial time, for M(G,V) to be conformal that becomes also necessary if conformality is required to be hereditary. Finally, we show that M(G,V) is perfect for every partition V if and only if G belongs to a superclass of threshold graphs defined with a complex function instead of a real one.

Chordal 2‐Connected Graphs and Spanning Trees

AbstractWe present a transformation on a chordal 2‐connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with , the threshold graph having n vertices and m edges that consists of an ‐clique and vertices of degree 2 is the only graph with the fewest spanning trees among all 2‐connected chordal graphs on n vertices and m edges.

Scaling Laws for Connectivity in Random Threshold Graph Models with Non-Negative Fitness Variables

We explore the scaling properties for graph connectivity in random threshold graphs. In the many node limit, we provide a complete characterization for the existence and type of the underlying zero-one laws, and identify the corresponding critical scalings. These results are consequences of well-known facts in Extreme Value Theory concerning the asymptotic behavior of running maxima on i.i.d. random variables. In the important special case of exponentially distributed fitness, we show that the (essentially unique) critical scaling which ensures a power-law degree distribution, does not result in graph connectivity in the asymptotically almost sure (a.a.s.) sense.

On the adjacency matrix of a threshold graph

A threshold graph on n vertices is coded by a binary string of length n−1. We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases.

Eigenvalue location in threshold graphs

Let G be a threshold graph of order n with adjacency matrix A. We present an O(n) algorithm for constructing a diagonal matrix congruent to Bx=A+xI for any x. An application, using Sylvesterʼs Law of Inertia, can determine how many eigenvalues lie in an interval, allowing efficient approximation. We prove that eigenvalues of threshold graphs, other than 0 or −1, are simple. We give the spectrum for threshold graphs G(k,j), represented by 0k1j. When n⩾3, G(n−⌊n3⌋,⌊n3⌋) has the minimum eigenvalue λmin,n among threshold graphs of order n, and a formula for λmin,n is given. There is one more graph if n≡2mod3.

Hereditary unigraphs and Erdős–Gallai equalities

We give characterizations of the structure and degree sequence of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs.The degree sequence characterization of hereditary unigraphs makes use of the list of values k for which the kth Erdős–Gallai inequality holds with equality for a graphic sequence. Using the canonical decomposition of Tyshkevich, we show how this list describes structure common among all realizations of an arbitrary graphic sequence.

An integrated model to assess critical rainfall thresholds for run-out distances of debris flows

A dramatic increase in debris flows occurred in the years after the 2008 Wenchuan earthquake in SW China due to the deposition of loose co-seismic landslide material. This paper proposes a preliminary integrated model, which describes the relationship between rain input and debris flow run-out in order to establish critical rain thresholds for mobilizing enough debris volume to reach the basin outlet. The model integrates in a simple way rainfall, surface runoff, and concentrated erosion of the loose material deposited in channels, propagation, and deposition of flow material. The model could be calibrated on total volumes of debris flow materials deposited at the outlet of the Shuida catchment during two successive rain events which occurred in August 2011. The calibrated model was used to construct critical rainfall intensity-duration graphs defining thresholds for a run-out distance until the outlet of the catchment. Model simulations show that threshold values increase after successive rain events due to a decrease in erodible material. The constructed rainfall intensity-duration threshold graphs for the Shuida catchment based on the current situation appeared to have basically the same exponential value as a threshold graph for debris flow occurrences, constructed for the Wenjia catchment on the basis of 5 observed triggering rain events. This may indicate that the triggering mechanism by intensive run-off erosion in channels in this catchment is the same. The model did not account for a supply of extra loose material by landslips transforming into debris flow or reaching the channels for transportation by run-off. In August 2012, two severe rain events were measured in the Shuida catchment, which did not produce debris flows. This could be confirmed by the threshold diagram constructed by the model.

Formula omitted]-product of graphs, [formula omitted]-threshold graphs and threshold-width of graphs

One method of graph decomposition is to define a binary operation on the set of graphs and to represent graphs as products of prime elements with respect to this operation. We consider a graph together with a partition of its vertex set into n subsets (n-partitioned graph). On the set of all isomorphism classes of n-partitioned graphs we consider a binary algebraic operation ∘H (H-product of graphs), determined by a digraph H. We prove that every operation ∘H defines the unique factorization as a product of prime factors. We define H-threshold graphs as graphs that can be represented as products of one-vertex factors under the operation ∘H. Threshold-width of a graph G is the minimum number of vertices in a digraph H such that G is an H-threshold graph. In particular, threshold graphs and difference graphs are H-threshold graphs for certain digraphs H. Threshold-width is naturally related to clique-width and NLC-width of graphs. Graphs with bounded threshold-width generalize threshold graphs and difference graphs and extend their properties. We characterize graphs with bounded threshold-width and study in detail graphs with threshold-width at most 2.

Rainfall thresholds for debris flow initiation in the Wenchuan earthquake-stricken area, southwestern China

The Wenchuan earthquake-stricken area is frequently hit by heavy rainfall, which often triggers sediment-related disasters, such as shallow landslides, debris flows, and related natural events, sometimes causing tremendous damage to lives, property, infrastructure, and environment. The assessment of the rainfall thresholds for debris flow occurrence is very important in order to improve forecasting and for risk management. In the context of the Wenchuan earthquake-stricken area, however, the rainfall thresholds for triggering debris flows are not well understood. With the aim of defining the critical rainfall thresholds for this area, a detailed analysis of the main rainstorm events was carried out. This paper presents 11 rainfall events that induced debris flows which occurred between 2008 and 2012 after the Wenchuan earthquake. The rainfall thresholds were defined in terms of mean rainfall intensity I, rainfall duration D, and normalized using the mean annual precipitation (MAP). An ID threshold and a normalized I MAP D threshold graph could be set up for the Wenchuan earthquake-stricken area which forms the lower boundary of the domain with debris flow-triggering rainfall events. The rainfall threshold curves obtained for the study area were compared with the local, regional, and global curves proposed by various authors. The results suggest that debris flow initiation in the study area almost requires a higher amount of rainfall and greater intensity than elsewhere. The comparison of rainfall intensity prior to and after the earthquake clearly indicates that the critical rainfall intensity necessary to trigger debris flows decreased after the earthquake. Rainfall thresholds presented in this paper are generalized, so that they can be used in debris flow warning systems in areas with the same geology as the Wenchuan earthquake-stricken area.

Extremal Graphs for Homomorphisms II

AbstractExtremal problems for graph homomorphisms have recently become a topic of much research. Let denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize . We prove that if H is loop‐threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges that maximizes . Similarly, we show that loop‐quasi‐threshold image graphs have quasi‐threshold extremal graphs. In the case , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for . Also in this article, using similar techniques, we determine a set of extremal graphs for “the fox,” a graph formed by deleting the loop on one of the end‐vertices of . The fox is the unique connected loop‐threshold image graph on at most three vertices for which the extremal problem was not previously solved.