A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs

In this study, we present a class of directed graphs with bounded degree sequences, which embodies the physical phenomenon of numerosity found in the collective behavior of large animal groups. Behavioral experiments show that an animal’s perception of number is capped by a critical limit, above which an individual perceives a nonspecific “many”. This species-dependent limit plays a pivotal role in the decision making process of large groups, such as fish schools and bird flocks. Here, we consider directed graphs whose edges model information-sharing between individual vertices. We incorporate the numerosity phenomenon as a critical limit on the intake of information by bounding the degree sequence and include the variability of cognitive processes by using a random variable in the network construction. We analytically compute measures of the expected structure of this class of graphs based on cycles, clustering, and sorting among vertices. Theoretical results are verified with numerical simulation.

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New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

Combinatorics Probability Computing ◽

10.1017/s0963548317000499 ◽

2017 ◽

Vol 27 (2) ◽

pp. 186-207

Author(s):

PÉTER L. ERDŐS ◽

ISTVÁN MIKLÓS ◽

ZOLTÁN TOROCZKAI

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Directed Graphs ◽

Degree Sequence ◽

Degree Sequences ◽

Network Modelling ◽

Split Graph ◽

Simple Graphs ◽

Large Classes ◽

Special Degree

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

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Win-Win Kernelization for Degree Sequence Completion Problems

Algorithm Theory – SWAT 2014 - Lecture Notes in Computer Science ◽

10.1007/978-3-319-08404-6_17 ◽

2014 ◽

pp. 194-205 ◽

Cited By ~ 2

Author(s):

Vincent Froese ◽

André Nichterlein ◽

Rolf Niedermeier

Keyword(s):

Degree Sequence ◽

Completion Problems

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The Number of Euler Tours of Random Directed Graphs

The Electronic Journal of Combinatorics ◽

10.37236/2377 ◽

2013 ◽

Vol 20 (3) ◽

Author(s):

Páidí Creed ◽

Mary Cryan

Keyword(s):

Asymptotic Distribution ◽

Random Graph ◽

Directed Graph ◽

Directed Graphs ◽

Degree Sequence ◽

Simple Approach ◽

Uniform Sampling ◽

Concentration Result ◽

De Bruijn ◽

Euler Tours

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.

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A Simple Havel–Hakimi Type Algorithm to Realize Graphical Degree Sequences of Directed Graphs

The Electronic Journal of Combinatorics ◽

10.37236/338 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 18

Author(s):

Péter L. Erdős ◽

István Miklós ◽

Zoltán Toroczkai

Keyword(s):

Greedy Algorithm ◽

Directed Graphs ◽

Degree Sequence ◽

Finite Sequence ◽

Simple Graph ◽

Degree Sequences ◽

Mcmc Algorithm ◽

Type Algorithm ◽

Positive Integers ◽

The Greedy Algorithm

One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.

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Win-win kernelization for degree sequence completion problems

Journal of Computer and System Sciences ◽

10.1016/j.jcss.2016.03.009 ◽

2016 ◽

Vol 82 (6) ◽

pp. 1100-1111 ◽

Cited By ~ 3

Author(s):

Vincent Froese ◽

André Nichterlein ◽

Rolf Niedermeier

Keyword(s):

Degree Sequence ◽

Completion Problems

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Half-Graphs, Other Non-stable Degree Sequences, and the Switch Markov Chain

The Electronic Journal of Combinatorics ◽

10.37236/9652 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Péter L. Erdős ◽

Ervin Győri ◽

Tamás Róbert Mezei ◽

István Miklós ◽

Dániel Soltész

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Rate Of Convergence ◽

Uniform Distribution ◽

Directed Graphs ◽

Degree Sequence ◽

Strong Stability ◽

Degree Sequences ◽

Markov Chain Method ◽

Stable Degree

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called P-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not P-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.

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A Polynomial Bound on the Mixing Time of a Markov Chain for Sampling Regular Directed Graphs

The Electronic Journal of Combinatorics ◽

10.37236/721 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 7

Author(s):

Catherine Greenhill

Keyword(s):

Markov Chain ◽

Directed Graphs ◽

Mixing Time ◽

Degree Sequence ◽

Negative Eigenvalue ◽

Multicommodity Flow ◽

Degree Sequences ◽

Regular Graphs ◽

Significant Extension ◽

Second Largest Eigenvalue

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree $d$, where $d$ is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock [Annals of Applied Probability 1991], and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill [Combinatorics, Probability and Computing 2007], but with significant extension required.

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