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Let A = (a , a , ..., a ) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n ), where n is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, that has a running time of O(ma ), where and a is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by and

The authors in the paper [15] presented an algorithm that generates uniformly all the bipartite realizations and the other algorithm that generates uniformly all the simple bipartite realizations whenever A is a bipartite degree sequence of a simple graph. The running time of both algorithms is (m),where m=12∑i=1nai. Let A =(A1 : A2 : ... : Ak) be a k-partite degree sequence of a simple graph, where Ai has ni entries such that ∑ni=n. In the present article, we give a generalized algorithm that generates uniformly all the k-partite realizations of A and another algorithm that generates uniformly all the simple k-partite realizations of A. The running time of both algorithms is (m), where m=12∑i=1nai.

This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to N a u t y ’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its o p e n k- n e i g h b o r h o o d s u b g r a p h . It defines d i f f u s i o n d e g r e e s e q u e n c e s and e n t i r e d i f f u s i o n d e g r e e s e q u e n c e . For each node of a graph G, it designs a characteristic m _ N e a r e s t N o d e to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of M a x Q ( G ) . Another theorem presents how to determine the second nodes of M a x Q ( G ) . When computing C m a x ( G ) , if M a x Q ( G ) already holds the first i nodes u 1 , u 2 , ⋯ , u i , Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of M a x Q ( G ) . Further, it also establishes two theorems to determine the C m a x ( G ) of disconnected graphs. Four algorithms implemented here demonstrate how to compute M a x Q ( G ) of a graph. From the results of the software experiment, the accuracy of our algorithms is preliminarily confirmed. Our method can be employed to mine the frequent subgraph. We also conjecture that if there is a node v ∈ S ( G ) meeting conditions C m a x ( G − v ) ⩽ C m a x ( G − w ) for each w ∈ S ( G ) ∧ w ≠ v , then u 1 = v for M a x Q ( G ) .

Given a set F of graphs, a graph G is F -free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F -free graphs, every realization of the degree sequence of G is also in C . A degree-sequence-forcing set is minimal if no proper subset is degree-sequence-forcing. We characterize the non-minimal degree-sequence-forcing sets F when F has size 3 .

This paper presents a novel theory and method to calculate the canonical labelings of digraphs whose definition is entirely different from the traditional definition of Nauty. It indicates the mutual relationships that exist between the canonical labeling of a digraph and the canonical labeling of its complement graph. It systematically examines the link between computing the canonical labeling of a digraph and the k-neighborhood and k-mix-neighborhood subdigraphs. To facilitate the presentation, it introduces several concepts including mix diffusion outdegree sequence and entire mix diffusion outdegree sequences. For each node in a digraph G, it assigns an attribute m_NearestNode to enhance the accuracy of calculating canonical labeling. The four theorems proved here demonstrate how to determine the first nodes added into M a x Q ( G ) . Further, the other two theorems stated below deal with identifying the second nodes added into M a x Q ( G ) . When computing C m a x ( G ) , if M a x Q ( G ) already contains the first i vertices u 1 , u 2 , ⋯ , u i , Diffusion Theorem provides a guideline on how to choose the subsequent node of M a x Q ( G ) . Besides, the Mix Diffusion Theorem shows that the selection of the ( i + 1 ) th vertex of M a x Q ( G ) for computing C m a x ( G ) is from the open mix-neighborhood subdigraph N + + ( Q ) of the nodes set Q = u 1 , u 2 , ⋯ , u i . It also offers two theorems to calculate the C m a x ( G ) of the disconnected digraphs. The four algorithms implemented in it illustrate how to calculate M a x Q ( G ) of a digraph. Through software testing, the correctness of our algorithms is preliminarily verified. Our method can be utilized to mine the frequent subdigraph. We also guess that if there exists a vertex v ∈ S + ( G ) satisfying conditions C m a x ( G − v ) ⩽ C m a x ( G − w ) for each w ∈ S + ( G ) ∧ w ≠ v , then u 1 = v for M a x Q ( G ) .

The β-model of random graphs is an exponential family model with the degree sequence as a sufficient statistic. In this paper, we contribute three key results. First, we characterize conditions that lead to a quadratic time algorithm to check for the existence of MLE of the β-model, and show that the MLE never exists for the degree partition β-model. Second, motivated by privacy problems with network data, we derive a differentially private estimator of the parameters of β-model, and show it is consistent and asymptotically normally distributed—it achieves the same rate of convergence as the nonprivate estimator. We present an efficient algorithm for the private estimator that can be used to release synthetic graphs. Our techniques can also be used to release degree distributions and degree partitions accurately and privately, and to perform inference from noisy degrees arising from contexts other than privacy. We evaluate the proposed estimator on real graphs and compare it with a current algorithm for releasing degree distributions and find that it does significantly better. Finally, our paper addresses shortcomings of current approaches to a fundamental problem of how to perform valid statistical inference from data released by privacy mechanisms, and lays a foundational groundwork on how to achieve optimal and private statistical inference in a principled manner by modeling the privacy mechanism; these principles should be applicable to a class of models beyond the β-model.

Degree-based graph construction is a fundamental problem in network science. A graph is simple if there are no self-loops and no multiple links between any pair of nodes in the graph. A degree sequence d=(d1,d2,⋯,dN) is graphical if d can be represented as the degree sequence of at least one simple graph, where the graph is called a realization of the sequence d. In this work, we introduce a novel method (LSFGR) for generating simple graphs from graphical degree sequences, focusing additionally on connectedness and on assortativity. LSFGR guarantees connected graphs for all potentially connected degree sequences. In the case where a degree sequence has no simple realization, LSFGR produces graphs with at most one node with self-loops. In addition, the graphs generated from LSFGR characterize real-world networks with medium assortativity.

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of d -regular graphs for all d , as nınfty . The key to our results is a new approach to estimating ratios between point probabilities in the space of degree sequences of the random graph, including analysis of fixed points of the associated operators.

Let d₁ ≥ d₂ ≥ ··· ≥ dn be the degree sequence of a graph G of order n and μ₁ ≥ μ₂ ≥ ··· ≥ μn = 0 be the Laplacian eigenvalues of G. In this paper, we propose a new conjecture that for any graph G except for C 4k+1(k ∈ Z⁺), ∑ μ i ≥ 2 ( μ i − 2 ) 2 ≤ ( 1 − 1 d 1 ) ∑ i = 1 n d i ( d i − 1 ) . We also prove this conjecture is true for the star, the path, the strongly regular graph, the threshold graph, the barbell graph and the complete bipartite graph, respectively.

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.