In-degree Sequence Research Articles

Methods for the graph realization problem

Abstract The graph realization problem seeks an answer to how and under what conditions a graph can be constructed if we know the degrees of its vertices. The problem was widely studied by many authors and in many ways, but there are still new ideas and solutions. In this sense, the paper presents the known necessary and su cient conditions for realization with the description in pseudocode of the corresponding algorithms. Two cases to solve the realization problem are treated: finding one solution, and finding all solutions. In this latter case a parallel approach is presented too, and how to exclude isomorphic graphs from solutions. We are also discussing algorithms using binary integer programming and flow networks. In the case of a bigraphical list with equal out- and in-degree sequences a modified Edmonds–Karp algorithm is presented such that the resulting graph will be always symmetric without containing loops. This algorithm solves the problem of graph realization in the case of undirected graphs using flow networks.

Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz

The long-standing Erdős–Faber–Lovász conjecture states that every n-uniform linear hypergaph with n edges has a proper vertex-coloring using n colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős–Faber–Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

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.

Stationary distribution and cover time of sparse directed configuration models

We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent gamma ge 1 of the logarithm and show that the cover time grows as nlog ^{gamma }(n), where n is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution pi is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of pi have the form frac{1}{n}log ^{1-gamma }(n), while the maximal values of pi behave as frac{1}{n}log ^{1-kappa }(n) for some other exponent kappa in [0,1]. In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices.

A Limit Theorem for the Six-length of Random Functional Graphs with a Fixed Degree Sequence

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v, f(v), \ldots, f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.

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.

Partial graph orientations and the Tutte polynomial

Gessel and Sagan investigated the Tutte polynomial, TG(x,y) using depth-first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2m−n+1TG(3,1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2n−1TG(1/2,3). We then prove that the number of partial orientations modulo cycle reversals is 2gTG(3,1) and the number of partial orientations modulo cut reversals is 2n−1TG(1,3). To prove these results, we introduce cut and cycle-minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals, extending known representatives for full orientations introduced by Greene and Zaslavksy. We then introduce distinguished partial orientations representing a given indegree sequence. We utilize these partial orientations to derive the Ehrhart polynomial of the win vector polytope, and give a combinatorial interpretation of its volume, thus answering a question of Bartels, Mount, and Welsh. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities p with 0<p<1/2.

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.

Network of likes and dislikes: Conflict and membership

Abstract We all have friends and foes. In the study of complex networks, such a pairwise interaction is described by a directed link since the relation is not necessarily symmetric. We study a real network constructed from a survey in which each individual chooses five members (s)he wants to work with, and other five (s)he does not like to work together. Although everyone’s outdegrees for such like and dislike links are fixed to five, respectively, it is found that indegree sequence for each type of links exhibits very different behaviors. We also pursue to answer the question of proper divisions of the organization based on the concept of happiness defined for each directed relation. For example, two individuals connected by like (dislike) links in both directions are happy if they belong to the same (different) group(s). We then adopt the framework of the q -state Potts model with long-ranged ferromagnetic and antiferromagnetic interactions and discuss the group structure in the organization that minimizes a suitably defined unhappiness.

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.

Expressive efficiency of two kinds of specific CP-nets

CP-nets (conditional preference networks) are a graphical model for compactly expressing conditional ceteris paribus (all other things being equal) preference statements on multi-attribute domains. In this paper, we investigate the expressive efficiency of two kinds of binary-valued CP-nets, the first kind is set-structured CP-nets, and the second kind is equal difference CP-nets. For the first kind, we prove that it can express 3n-2n preference relations with n preference rules, and has an expressive efficiency of (3n-2n)/n. For the second kind, we show that it can express a total order of 2n-1∗(2n-1) preference relations with 2n-1 preference rules, and has an expressive efficiency of 2n-1. For the future research, we propose an open problem: given an acyclic CP-net N with the in-degree sequence of (d1,d2,…,dn), how many preference relations can be expressed by N?

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, T n D̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂ 1,D̂ 2,…,D̂ n ). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂ 1,D̂ 2,…,D̂ n . As an application of these results, we consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

Labeled Trees, Functions, and an Algebraic Identity

We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher label. This solves a problem posed by Shin and Zeng in a recent article. We also provide a generalization of this identity that translates to a formula for the number of rooted spanning forests with given indegree sequence.

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set { 1 , 2 , … , n } , the local direction of each edge ( i j ) is from i to j if i < j . For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ = 1 e 1 2 e 2 … of a tree on the vertex set { 1 , 2 , … , n } is a partition of n − 1 . We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.

A Semiexact Degree Condition for Hamilton Cycles in Digraphs

We show that for each $\beta > 0$, every digraph G of sufficiently large order n whose outdegree and indegree sequences $d_1^+ \leq \cdots \leq d_n^+$ and $d_1^- \leq \cdots \leq d_n^-$ satisfy $d_i^+, d_i^- \geq \min{\{i + \beta n, n/2\}}$ is Hamiltonian. In fact, we can weaken these assumptions to (i) $d_i^+ \geq \min{\{i + \beta n, n/2\}}$ or $d^-_{n - i - \beta n} \geq n-i$, (ii) $d_i^- \geq \min{\{i + \beta n, n/2\}}$ or $d^+_{n - i - \beta n} \geq n-i$, and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of Kühn, Osthus, and Treglown.

Hamiltonian degree sequences in digraphs

We show that for each η > 0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d 1 + ⩽ ⋯ ⩽ d n + and d 1 − ⩽ ⋯ ⩽ d n − satisfy (i) d i + ⩾ i + η n or d n − i − η n − ⩾ n − i and (ii) d i − ⩾ i + η n or d n − i − η n + ⩾ n − i for all i < n / 2 . This gives an approximate solution to a problem of Nash-Williams (1975) [22] concerning a digraph analogue of Chvátal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.

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.

Counting labelled trees with given indegree sequence

For a labelled tree on the vertex set [ n ] : = { 1 , 2 , … , n } , define the direction of each edge ij to be i → j if i < j . The indegree sequence of T can be considered as a partition λ ⊢ n − 1 . The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on [ n ] with indegree sequence corresponding to a partition λ. In this paper we give two proofs of Cotterill's conjecture: one is “semi-combinatorial” based on induction, the other is a bijective proof.

Minimum entropy orientations

We study graph orientations that minimize the entropy of the in-degree sequence. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit.