On the Random Satisfiable Process

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

In this paper, we give structural and degree sequence characterizations for a new class of digraphs called directed 3-cycle anchored. A digraph in this class has the property that, for every realization of its degree sequence, there is a directed 3-cycle through each vertex of a labeled vertex set. We end by illustrating their use in the uniform sampling of simple directed graph realizations from a fixed degree sequence.

We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and mean-field theories of spreading processes. Its structure is well understood for large, sparse, simple graphs: the expected number of edges between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many network data sets are neither large, sparse, nor simple, and in these cases the standard approximation no longer applies. We derive a novel estimator using a dynamical approach: the estimator emerges from the stationarity conditions of a class of Markov Chain Monte Carlo algorithms for graph sampling. We derive error bounds for this estimator and provide an efficient scheme with which to compute it. We test the estimator on synthetic and empirical degree sequences, finding that it enjoys relative error against ground truth a full order of magnitude smaller than the standard approximation. We then compare modularity maximization techniques using both the standard and novel estimators, finding that the qualitative structure of the optimization landscape depends significantly on the estimator choice. Our results emphasize the importance of using carefully specified random graph models in data scientific applications.

Let $\operatorname{Ord}(\tau)$ be the order of an element $\tau$ in the group $S_n$ of permutations of an $n$-element set $X$. The present paper is concerned with the so-called general parametric model of a random permutation; according to this model an arbitrary fixed permutation $\tau$ from $S_n$ is observed with the probability $\theta_1^{u_1}\dotsb\theta_n^{u_n}/H(n)$, where $u_i$ is the number of cycles of length $i$ of the permutation $\tau$, $\{\theta_i,\ i\in \mathbf{N}\}$ are some nonnegative parameters (the weights of cycles of length $i$ of the permutation $\tau$), and $H(n)$ is the corresponding normalizing factor. We assume that an arbitrary permutation $\tau_n$ has such a distribution. The function $p(n)=H(n)/n!$ is assumed to be $\mathrm{RO}$-varying at infinity with the lower index exceeding $-1$ (in particular, it can vary regularly), and the sequence $\{\theta_i,\ i\in \mathbf N\}$ is bounded. Under these assumptions it is shown that the random variable $\ln\operatorname{Ord}(\tau_n)$ is asymptotically normal with mean $\sum_{k=1}^n\theta_k\ln (k)/k$ and variance $\sum_{k=1}^n\theta_k\ln^2(k)/k$. In particular, this scheme subsumes the class of random $A$-permutations (i.e., when $\theta_i=\chi\{i\in A\}$), where $A$ is an arbitrary fixed subset of the positive integers. This scheme also includes the Ewens model of random permutation, where $\theta_i\equiv\theta>0$ for any $i\in\mathbf N$. The limit theorem we prove here extends some previous results for these schemes. In particular, with $\theta_i\equiv1$ for any $i\in\mathbf N$, the result just mentioned implies the well-known Erdös--Turán limit theorem.

We study large deviations principles for N random processes on the lattice Z d with finite time horizon (0 ,β ) under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permu- tation σ of N elements and a vector (x1 ,...,x N )o fN initial points we let the random processes terminate in the points (xσ(1) ,...,x σ(N )) and then sum over all possible per- mutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviations principles for the mean of empirical path measures, for the mean of paths and for the mean of occupation local times under this symmetrised measure. The symmetrised measure cannot be written as any product of single random process distributions. We show a couple of important applications of these results in quantum statistical mechanics using the Feynman-Kac formulae representing traces of certain trace class operators. In particular we prove a non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein statistics and mean field interactions. A special case of our large deviations principle for the mean of occupation local times of N simple random walks has the Donsker-Varadhan rate function as the rate function for the limit N →∞ but for finite time β. We give an interpretation in quantum statistical mechanics for this surprising result.

Random graph models can help us assess the significance of the structural properties of real complex systems. Given the value of a graph property and its value in a randomized ensemble, we can determine whether the property is explained by chance by comparing its real value to its value in the ensemble. The conclusions drawn with this approach obviously depend on the choice of randomization. We argue that keeping graphs in one connected piece, or component, is key for many applications where complex graphs are assumed to be connected either by definition (e.g. the Internet) or by construction (e.g. a crawled subset of the World-Wide Web obtained only by following hyperlinks). Using an heuristic to quickly sample the ensemble of small connected simple graphs with a fixed degree sequence, we investigate the significance of the structural patterns found in real connected graphs. We find that, in sparse networks, the connectedness constraint changes degree correlations, the outcome of community detection with modularity, and the predictions of percolation on the ensemble.

This paper investigates the relationship between spectral graph theory and graph properties, specifically focusing on the spectral radius, which is the largest eigenvalue of the adjacency matrix of a graph. Our problem is finding sharp upper bounds for the spectral radius of bipartite graphs with given bounde vertex sets. We first review existing inequalities, noting and discuss their limitations, noting particularly that some, like Hong's inequality, are not always sharp for all graph types. Our primary contribution is an elementary and direct approach to solving an optimization problem: finding a bipartite graph with a bounded number of vertices that maximizes its spectral radius. We prove that for any bipartite graph with bounded vertex sets of \(n_1\) and \(n_2\), the spectral radius \(\rho(G)\) is bounded by \(\sqrt{n_1n_2}\). We demonstrate that this inequality is sharp, with equality holding exclusively for the complete bipartite graph \(K_{n_1,n_2}\).

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and n−m vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization–to–delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.

Let ω(G) be the clique number of a graph G. We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values ω(G) completely cover a line segment [a,b] of positive integers. For an arbitrary graphic degree sequence d, we define min(ω,d) and max(ω,d) as follows: $$$$ where \(\) is the graph of realizations of d.

We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values χ(G) of the function chromatic number completely cover a line segment [a,b] of positive integers. Thus for an arbitrary graphical sequence d, two invariants minχ(d):=a and maxχ(d):=b naturally arise. For a regular graphical sequence d=rn:=(r,r,…,r) where r is the degree and n is the number of vertices, the exact values of a and b are found in all situations, except the case where n and r are both even and n<2r.

This paper presents a general recursive formula to estimate the number of labeled graphs as well as details to evaluate the formula for the following graph properties: number of edges (graph density), degree sequence, degree distribution, classification mixing, and degree mixing, i.e., the formula estimates the number of labeled graphs that have given values for graph properties. The proposed approach can be extended to additional graph properties (e.g., number of triangles) as well as properties of bipartite graphs. For special settings in which formulas exist from previous research, simulation studies demonstrate the validity of the proposed approach. In addition, we demonstrate how our approach can be used to quantify the level of variability in values of a graph property in the subset of graphs that hold a specified value of a different graph property (or properties) constant.

Given a set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> distinct values, the operation of shuffling those elements and creating a random order is used in a wide range range of applications, including (but not limited to) statistical analysis, machine learning, games, and bootstrapping. The operation of shuffling elements is equivalent to generating a random permutation and applying the permutation. For example, the random permutation of an input allows splitting into two or more subsets without bias. This operation is repeated for machine learning applications when both a train and test data set are needed. In this paper we describe a new method for creating random permutations that is scalable, efficient, and simple. We show that the operation of generating a random permutation shares traits with building a hashtable. Our method uses a fairly new hash table, called HashGraph, to generate the permutation. HashGraph's unique data-structure ensures easy generation and retrieval of the permutation. HashGraph is one of the fastest known hash-tables for the GPU and also outperforms many leading CPU hash-tables. We show the performance of our new permutation generation scheme using both Python and CUDA versions of HashGraph. Our CUDA implementation is roughly 10% faster than our Python implementation. In contrast to the shuffle operation in NVIDIA's Thrust and cuPy frameworks, our new permutation generation algorithm is 2.6 x and 1.73 x faster, respectively and up to 150 x faster than numPy.

Several attributes of a bipartite graph are exploited in designing switching systems. First, in a network based on bipartite graphs, no two paths are allowed to intersect at a vertex. This attribute is used to design a directional coupler-based photonic switching network with very low crosstalk. Since crosstalk is the most limiting factor in constructing a large directional-coupler-based photonic switching network, crosstalk reduction on the device and architecture levels is an important design issue. Secondly, broadcast, which is an intrinsic property of a bipartite graph (the same is always true with a crossbar representation), is used in designing multiconnection switching networks. Thirdly, it is shown that the nonplanar nature of bipartite graphs makes them well suited for designing three-dimensional free-space-transmission photonic switching systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Bipartite bithreshold graphs

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.