Out-degree Sequence Research Articles

The maximum outdegree power of complete [formula omitted]-partite oriented graphs

Suppose that Gσ is an oriented graph with order n and orientation σ. For d1+≥d2+≥⋯≥dn+, the outdegree sequence of Gσ is denoted by (d1+,d2+,…,dn+). Similar to the definition of the degree power for a simple graph, define the outdegree power for an oriented graph Gσ by ∂q+(Gσ)=∑i=1n(di+)q where q is a positive integer. Obviously, ∂1+(Gσ)=|E(G)|. Denote by G(G) the set of oriented graphs whose underlying graph is isomorphic to G, where G is a simple graph. In the paper, we concentrate on the problem: How large the value of ∂q+(Gσ) could be among all oriented graphs in G(G)? Using majorization, we prove that Kn1,…,nk→ is the unique extremal graph which attains the maximum outdegree power ∂q+(Gσ) over all oriented graphs in G(Kn1,…,nk) for q>ln(n−n1+1)ln(1+1n−n1−1). Further, we present several results about the extrema of ∂q+(Gσ) over all complete k-partite oriented graphs.

A permutation method for network assembly.

We present a method for assembling directed networks given a prescribed bi-degree (in- and out-degree) sequence. This method utilises permutations of initial adjacency matrix assemblies that conform to the prescribed in-degree sequence, yet violate the given out-degree sequence. It combines directed edge-swapping and constrained Monte-Carlo edge-mixing for improving approximations to the given out-degree sequence until it is exactly matched. Our method permits inclusion or exclusion of ‘multi-edges’, allowing assembly of weighted or binary networks. It further allows prescribing the overall percentage of such multiple connections—permitting exploration of a weighted synthetic network space unlike any other method currently available for comparison of real-world networks with controlled multi-edge proportion null spaces. The graph space is sampled by the method non-uniformly, yet the algorithm provides weightings for the sample space across all possible realisations allowing computation of statistical averages of network metrics as if they were sampled uniformly. Given a sequence of in- and out- degrees, the method can also produce simple graphs for sequences that satisfy conditions of graphicality. Our method successfully builds networks with order O(107) edges on the scale of minutes with a laptop running Matlab. We provide our implementation of the method on the GitHub repository for immediate use by the research community, and demonstrate its application to three real-world networks for null-space comparisons as well as the study of dynamics of neuronal networks.

Asymptotic Enumeration of Orientations of a Graph as a Function of the Out-Degree Sequence

We prove an asymptotic formula for the number of orientations with given out-degree (score) sequence for a graph $G$. The graph $G$ is assumed to have average degrees at least $n^{1/3 + \epsilon}$ for some $\epsilon > 0$, and to have strong mixing properties, while the maximum imbalance (out-degree minus in-degree) of the orientation should be not too large. Our enumeration results have applications to the study of subdigraph occurrences in random orientations with given imbalance sequence. As one step of our calculation, we obtain new bounds for the maximum likelihood estimators for the Bradley-Terry model of paired comparisons.

Large-Scale Dynamic Social Network Directed Graph K-In&Out-Degree Anonymity Algorithm for Protecting Community Structure

Social network data publishing is dynamic, and attackers can perform association attacks based on social network directed graph data at different times. The existing social network privacy protection technology has low performance in dealing with large-scale dynamic social network directed graph data, and anonymous data publishing does not meet the needs of community structure analysis. A Dynamic Social Network Directed Graph K-In&Out-Degree Anonymity (DSNDG-KIODA) method to protect community structure is proposed. The method is based on the dynamic grouping anonymity rule to anonymize the dynamic K-in&out-degree sequence, and the virtual node distribution is added in parallel to construct an anonymous graph. The node information is transmitted based on the GraphX, and the virtual node pairs are selected and deleted according to the change of the directed graph modularity to reduce information loss. The experimental results show that the DSNDG-KIODA method improves the efficiency of processing large-scale dynamic social network directed graph data, and ensures the availability of community structure analysis when data is released.

Network and community structure in a scientific team with high creative performance

Current studies have indicated that the type and intensity of the relations in a social network can impact individuals’ creative performance. However, it is more practical to explore how to improve team’s creative performance from the perspective of network and community structure. We conduct a survey of a real scientific team network, in which each team member chooses five members (s)he regards as his (her) friends, five members (s)he always works with, and five members (s)he wants to work with. The members were asked to rank the five members they selected according to their closeness. We derived three networks: a friend network, a real cooperation network and a desired cooperation network. Additionally, we have constructed a directed-weighted network with the team members as nodes, the relations (friend, real cooperation, and cooperation desire) between members as edges, and the closeness of these relations as weights. The main results show the following. (1) This scientific team is a leader network with nodes with weighted degrees significantly larger than those of other nodes. The weighted in-degree sequence and the weighted out-degree sequence have different properties. (2) Creative performance is higher for the leader communities than for the self-organized communities. After removing the leader nodes of the leader communities, creative performance in the communities and in the whole network decreased. (3) Quadratic assignment procedure correlation (QAP) analysis shows that the similarity between the real cooperation network and the desired cooperation network is lowest in self-organized communities, and the creative performance is low if the similarity between the two networks is low; (4) creative performance is lower in a network that only takes into account the desired cooperation of members. This study sheds light on the potential application of complex networks in the issues of group division and team construction in the workplace and in the field of education.

Visibility graphs and symbolic dynamics

Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology nontrivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the time series. Hence, Pesin’s identity suggests that these block entropies are converging to the Kolmogorov–Sinai entropy of the physical measure, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k(x),x∈[0,1] that, for a given datum with value x, assigns in graph space a node with degree k. In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.

Bounds on the domination number of a digraph

A vertex subset S of a digraph D is called a dominating set of D if every vertex not in S is adjacent from at least one vertex in S. The domination number of D, denoted by $$\gamma (D)$$ , is the minimum cardinality of a dominating set of D. The Slater number $$s\ell (D)$$ is the smallest integer t such that t added to the sum of the first t terms of the non-increasing out-degree sequence of D is at least as large as the order of D. For any digraph D of order n with maximum out-degree $$\Delta ^+$$ , it is known that $$\gamma (D)\ge \lceil n/(\Delta ^++1)\rceil $$ . We show that $$\gamma (D)\ge s\ell (D)\ge \lceil n/(\Delta ^++1)\rceil $$ and the difference between $$s\ell (D)$$ and $$\lceil n/(\Delta ^++1)\rceil $$ can be arbitrarily large. In particular, for an oriented tree T of order n with $$n_0$$ vertices of out-degree 0, we show that $$(n-n_0+1)/2\le s\ell (T)\le \gamma (T)\le 2s\ell (T)-1$$ and moreover, each value between the lower bound $$s\ell (T)$$ and the upper bound $$2s\ell (T)-1$$ is attainable by $$\gamma (T)$$ for some oriented trees. Further, we characterize the oriented trees T for which $$s\ell (T)=(n-n_0+1)/2$$ hold and show that the difference between $$s\ell (T)$$ and $$(n-n_0+1)/2$$ can be arbitrarily large. Some other elementary properties involving the Slater number are also presented.

A note on graphically summable sequences

The notion of Jaco-type graphs has been introduced recently as the graphical embodiments of well-defined integer sequences. For a Jaco-type graph, the vertex out-degree d+(v) is well-defined. It is noted that for a given out-degree integer sequence, a specific in-degree sequence results. Graphically summable sequences are a pair of integer sequences, such that for a given out-degree sequence, say s1+, a specific in-degree sequence, say s2-, inevitably results. In this paper, we also report on the second and third known combinatorial interpretation of the Hofstadter G-sequence.

Using k-Mix-Neighborhood Subdigraphs to Compute Canonical Labelings of Digraphs

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 ) .

Locally Oriented Noncrossing Trees

We define an orientation on the edges of a noncrossing tree induced by the labels: for a noncrossing tree (i.e., the edges do not cross) with vertices $1,2,\ldots,n$ arranged on a circle in this order, all edges are oriented towards the vertex whose label is higher. The main purpose of this paper is to study the distribution of noncrossing trees with respect to the indegree and outdegree sequence determined by this orientation. In particular, an explicit formula for the number of noncrossing trees with given indegree and outdegree sequence is proved and several corollaries are deduced from it. Sources (vertices of indegree $0$) and sinks (vertices of outdegree $0$) play a special role in this context. In particular, it turns out that noncrossing trees with a given number of sources and sinks correspond bijectively to ternary trees with a given number of middle- and right-edges, and an explicit bijection is provided for this fact. Finally, the in- and outdegree distribution of a single vertex is considered and explicit counting formulas are provided again.

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D)=diag(D)+A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give a sharp bound on q(D) where D has a given outdegree sequence and compare the bound with known bounds. We establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the extremal graph. In addition, we also determine the unique digraph which achieves the minimum (or maximum), the second minimum (or maximum), the third minimum, the fourth minimum spectral radius and signless Laplacian spectral radius among all strongly connected digraphs, and answer the open problem proposed by Lin and Shu [14].

The Number of Euler Tours of Random Directed Graphs

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random $d$-in/$d$-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of an Eulerian directed graph with out-degree sequence $\mathbf{d}$ is the product of the number of arborescences and the term $\frac{1}{|V|}[\prod_{v\in V}(d_v-1)!]$. Therefore most of our effort is towards estimating the moments of the number of arborescences of a random graph with fixed out-degree sequence.

Degree sequences of k-multi-hypertournaments

Let n and k (n ≥ k > 1) be two non-negative integers. A k-multi-hypertournament on n vertices is a pair (V, A), where V is a set of vertices with |V| = n, and A is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V, A contains at least one (at most k!) of the k! k-tuples whose entries belong to S. The necessary and sufficient conditions for a non-decreasing sequence of non-negative integers to be the out-degree sequence (in-degree sequence) of some k-multi-hypertournament are given.