An Approximation Method for Improving Dynamic Network Model Fitting

There has been a great deal of interest recently in the modeling and simulation of dynamic networks, that is, networks that change over time. One promising model is the separable temporal exponential-family random graph model (ERGM) of Krivitsky and Handcock, which treats the formation and dissolution of ties in parallel at each time step as independent ERGMs. However, the computational cost of fitting these models can be substantial, particularly for large, sparse networks. Fitting cross-sectional models for observations of a network at a single point in time, while still a nonnegligible computational burden, is much easier. This article examines model fitting when the available data consist of independent measures of cross-sectional network structure and the duration of relationships under the assumption of stationarity. We introduce a simple approximation to the dynamic parameters for sparse networks with relationships of moderate or long duration and show that the approximation method works best in precisely those cases where parameter estimation is most likely to fail—networks with very little change at each time step. We consider a variety of cases: Bernoulli formation and dissolution of ties, independent-tie formation and Bernoulli dissolution, independent-tie formation and dissolution, and dependent-tie formation models.     

A Scoring Criterion For Learning Chain Graphs

A chain graph allows both directed and undirected edges, and contains the underlying mathematical properties of the two. An important method of learning graphical models is to use scoring criteria to measure how well the graph structures fit the data. In this paper, we present a scoring criterion for learning chain graphs based on the Kullback Leibler distance. It is score equivalent, that is, equivalent chain graphs obtain the same score, so it can be used to perform model selection and model averaging.     

A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs

Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this article, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest dataset of neural connectome graphs, we are able to distinguish between scans from different subjects; and in the C. elegans connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small-sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests. Supplementary materials for this article are available online.     

Algorithms for core stability,core largeness,exactness, and extendability of flow games

We study core stability and some related properties of flow games defined on simple networks (all edge capacities are equal) from an algorithmic point of view. We first present a sufficient and necessary condition that can be tested efficiently for a simple flow game to have a stable core. We also prove the equivalence of the properties of core largeness, extendability, and exactness of simple flow games and provide an equivalent graph theoretic characterization which allows us to decide these properties in polynomial time.     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

A Construction of Sequentially Cohen-Macaulay Graphs

For every simple graph G,a class of multiple clique cluster-whiskered graphs Geπm is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex IndGeπm is sequentially Cohen-Macaulay.The properties of the graphs Geπm and Gπ constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind Gπ and the calculation of Betti numbers of the cover ideal Ic(Geπ").We also prove that the complex △ =IndH is strongly shellable and pure for either a Boolean graph H =Bn or the full clique-whiskered graph H =Gw of G,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△) and the cover ideal Ic(H) have linear quotients.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

The clique-separator graph for chordal graphs

We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the clique-separator graph. We present an algorithm that constructs the clique-separator graph of a chordal graph in O(n³) time and of an interval graph in O(n²) time, where n is the number of vertices in the graph.     

Complexity of networks I: The set‐complexity of binary graphs

The balance between symmetry and randomness as a property of networks can be viewed as a kind of “complexity.” We use here our previously defined “set complexity” measure (Galas et al., IEEE Trans Inf Theory 2010, 56), which was used to approach the problem of defining biological information, in the mathematical analysis of networks. This information theoretic measure is used to explore the complexity of binary, undirected graphs. The complexities, Ψ, of some specific classes of graphs can be calculated in closed form. Some simple graphs have a complexity value of zero, but graphs with significant values of Ψ are rare. We find that the most complex of the simple graphs are the complete bipartite graphs (CBGs). In this simple case, the complexity, Ψ, is a strong function of the size of the two node sets in these graphs. We find the maximum Ψ binary graphs as well. These graphs are distinct from, but similar to CBGs. Finally, we explore directed and stochastic processes for growing graphs (hill‐climbing and random duplication, respectively) and find that node duplication and partial node duplication conserve interesting graph properties. Partial duplication can grow extremely complex graphs, while full node duplication cannot do so. By examining the eigenvalue spectrum of the graph Laplacian we characterize the symmetry of the graphs and demonstrate that, in general, breaking specific symmetries of the binary graphs increases the set‐based complexity, Ψ. The implications of these results for more complex, multiparameter graphs, and for physical and biological networks and the processes of network evolution are discussed. © 2011 Wiley Periodicals, Inc. Complexity, 17,51–64, 2011     

New constructions of hypohamiltonian and hypotraceable graphs

A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex‐deleted subgraphs of G are hamiltonian/traceable. All known hypotraceable graphs are constructed using hypohamiltonian graphs; here we present a construction that uses so‐called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex‐deleted subgraphs are hamiltonian with exactly one exception, see [15]). This construction is an extension of a method of Thomassen [11]. As an application, we construct a planar hypotraceable graph of order 138, improving the best‐known bound of 154 [8]. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method. We also prove that if G is a Grinbergian graph without a triangular region, then G is not maximal nonhamiltonian and using the proof method we construct a hypohamiltonian graph of order 36 with crossing number 1, improving the best‐known bound of 46 [14].     

Three theorems regarding testing graph properties

Property testing is a relaxation of decision problems in which it is required to distinguish YES ‐instances (i.e., objects having a predetermined property) from instances that are far from any YES ‐instance. We presents three theorems regarding testing graph properties in the adjacency matrix representation. More specifically, these theorems relate to the project of characterizing graph properties according to the complexity of testing them (in the adjacency matrix representation). The first theorem is that there exist monotone graph properties in ???? for which testing is very hard (i.e., requires to examine a constant fraction of the entries in the matrix). The second theorem is that every graph property that can be tested making a number of queries that is independent of the size of the graph can be so tested by uniformly selecting a set of vertices and accepting iff the induced subgraph has some fixed graph property (which is not necessarily the same as the one being tested). The third theorem refers to the framework of graph partition problems, and is a characterization of the subclass of properties that can be tested using a one‐sided error tester making a number of queries that is independent of the size of the graph. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 23–57, 2003     

连通的顶点可迁图的色唯一性

本文给出从一个已知的顶点可迁的非色唯一图出发,构造无穷多个顶点可迁的非色唯一图的一种方法,据此给出若干类无穷多个连通的顶点可迁,但不是色唯一的图簇,从而进一步否定地回答了Chia在[1]中提出的问题.     

Dynamic Scheduling on a Single Batch Processing Machine with Split Compatibility Graphs

We consider the problem of minimizing the makespan on a batch processing machine, in which jobs are not all compatible. Only compatible jobs can be included into the same batch. This relation of compatibility is represented by a split graph. Jobs have release dates. The capacity of the batch processing machine is finite or infinite. The processing time of a batch is given by the processing time of the longest job in the batch. We establish the NP-hardness of the general problem and present polynomial algorithms for several special cases. Relating scheduling theory and graph theory appears to be an interesting and important concept.     

Structural models with qualitative values

In fields of study from cognition to organizations and social networks , empirical structures have been formally represented in terms of graph theoretical models. When the empirical relationships can be seen as valued, a valued graph or digraph is called for. Values have been conventionally identified with real numbers, but other sorts of entities (most often signs) have been used. In this paper, we demonstrate a general system under which graphs and digraphs with values that are not numbers may be used to represent various important properties and features of empirical structures. Special cases include multiplexity of relationships, formal and informal linkages in organizational structures, systems and their environments, and structural consistency principles. The general system incorporates a matrix methodology which permits the convenient analysis of empirical structures. These cases are also intended to exemplify the ways in which valued relational models may be developed to extend this kind of formalization and its methodology to other areas of substantive interest.     

General results on preferential attachment and clustering coefficient

This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.     

A graph‐theoretic method to stabilize the delayed coupled systems on networks based on periodically intermittent control

This paper mainly focus on the exponential stabilization problem of coupled systems on networks with mixed time‐varying delays. Periodically intermittent control is used to control the coupled systems on networks with mixed time‐varying delays. Moreover, based on the graph theory and Lyapunov method, two different kinds of stabilization criteria are derived, which are in the form of Lyapunov‐type theorem and coefficients‐type criterion, respectively. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, a decision theorem is also presented. It is straightforward to show the stability of original system can be determined by that of modified system with added absolute value into the coupling weighted‐value matrix. Finally, the feasibility and validity of the obtained results are demonstrated by several numerical simulation figures.