arXiv subjects

Daniel Allendorf

Publications and source records attributed to Daniel Allendorf.

6 recordsLinked to original sources

arXiv2024

Efficient Uniform Negative Edge Weights

We consider a maximum entropy edge weight model that allows for negative weights. Given a graph $G$ and possible weights $\mathcal{W}$ typically consisting of positive and negative values, the model selects edge weights $w \in \mathcal{W}^m$ uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that it converges to the required distribution. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search as well as an innovative resampling method. We empirically study the performance characteristics of these novel sampling algorithms as well as the output produced by the model.

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arXiv2023

Uniform Generation of Temporal Graphs with Given Degrees

Uniform sampling from the set $\mathcal{G}(\mathbf{d})$ of graphs with a given degree-sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ is a classical problem in the study of random graphs. We consider an analogue for temporal graphs in which the edges are labeled with integer timestamps. The input to this generation problem is a tuple $\mathbf{D} = (\mathbf{d}, T) \in \mathbb N^n \times \mathbb N_{>0}$ and the task is to output a uniform random sample from the set $\mathcal{G}(\mathbf{D})$ of temporal graphs with degree-sequence $\mathbf{d}$ and timestamps in the interval $[1, T]$. By allowing repeated edges with distinct timestamps, $\mathcal{G}(\mathbf{D})$ can be non-empty even if $\mathcal{G}(\mathbf{d})$ is, and as a consequence, existing algorithms are difficult to apply. We describe an algorithm for this generation problem which runs in expected time $O(M)$ if $\Delta^{2+\epsilon} = O(M)$ for some constant $\epsilon > 0$ and $T - \Delta = \Omega(T)$ where $M = \sum_i d_i$ and $\Delta = \max_i d_i$. Our algorithm applies the switching method of McKay and Wormald $[1]$ to temporal graphs: we first generate a random temporal multigraph and then remove self-loops and duplicated edges with switching operations which rewire the edges in a degree-preserving manner.

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arXiv2023

Maintaining Discrete Probability Distributions in Practice

A classical problem in random number generation is the sampling of elements from a given discrete distribution. Formally, given a set of indices $S = \{1, \dots, n\}$ and sequence of weights $w_1, \dots, w_n \in \mathbb{R}^+$, the task is to provide samples from $S$ with distribution $p(i) = w_i / W$ where $W = \sum_j w_j$. A commonly accepted solution is Walker's Alias Table, which allows for each sample to be drawn in constant time. However, some applications correspond to a dynamic setting, where elements are inserted or removed, or weights change over time. Here, the Alias Table is not efficient, as it needs to be re-built whenever the underlying distribution changes. In this paper, we engineer a simple data structure for maintaining discrete probability distributions in the dynamic setting. Construction of the data structure is possible in time $O(n)$, sampling is possible in expected time $O(1)$, and an update of size $\Delta$ can be processed in time $O(\Delta n / W)$. As a special case, we maintain an urn containing $W$ marbles of $n$ colors where with each update $O(W / n)$ marbles can be added or removed in $O(1)$ time per update. To evaluate the efficiency of the data structure in practice we conduct an empirical study. The results suggest that the dynamic sampling performance is competitive with the static Alias Table. Compared to existing more complex dynamic solutions we obtain a sampling speed-up of up to half an order of magnitude.

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arXiv2022

Parallel and I/O-Efficient Algorithms for Non-Linear Preferential Attachment

Preferential attachment lies at the heart of many network models aiming to replicate features of real world networks. To simulate the attachment process, conduct statistical tests, or obtain input data for benchmarks, efficient algorithms are required that are capable of generating large graphs according to these models. Existing graph generators are optimized for the most simple model, where new nodes that arrive in the network are connected to earlier nodes with a probability $P(h) \propto d$ that depends linearly on the degree $d$ of the earlier node $h$. Yet, some networks are better explained by a more general attachment probability $P(h) \propto f(d)$ for some function $f \colon \mathbb N~\to~\mathbb R$. Here, the polynomial case $f(d) = d^\alpha$ where $\alpha \in \mathbb R_{>0}$ is of particular interest. In this paper, we present efficient algorithms that generate graphs according to the more general models. We first design a simple yet optimal sequential algorithm for the polynomial model. We then parallelize the algorithm by identifying batches of independent samples and obtain a near-optimal speedup when adding many nodes. In addition, we present an I/O-efficient algorithm that can even be used for the fully general model. To showcase the efficiency and scalability of our algorithms, we conduct an experimental study and compare their performance to existing solutions.

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arXiv2021

Parallel Global Edge Switching for the Uniform Sampling of Simple Graphs with Prescribed Degrees

The uniform sampling of simple graphs matching a prescribed degree sequence is an important tool in network science, e.g. to construct graph generators or null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice. Given an arbitrary simple graph with the required degree sequence, ES-MC carries out a large number of small changes, called edge switches, to eventually obtain a uniform sample. In practice, reasonably short runs efficiently yield approximate uniform samples. In this work, we study the problem of executing edge switches in parallel. We discuss parallelizations of ES-MC, but find that this approach suffers from complex dependencies between edge switches. For this reason, we propose the Global Edge Switching Markov Chain (G-ES-MC), an ES-MC variant with simpler dependencies. We show that G-ES-MC converges to the uniform distribution and design shared-memory parallel algorithms for ES-MC and G-ES-MC. In an empirical evaluation, we provide evidence that G-ES-MC requires not more switches than ES-MC (and often fewer), and demonstrate the efficiency and scalability of our parallel G-ES-MC implementation.

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arXiv2021

Engineering Uniform Sampling of Graphs with a Prescribed Power-law Degree Sequence

We consider the following common network analysis problem: given a degree sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ return a uniform sample from the ensemble of all simple graphs with matching degrees. In practice, the problem is typically solved using Markov Chain Monte Carlo approaches, such as Edge-Switching or Curveball, even if no practical useful rigorous bounds are known on their mixing times. In contrast, Arman et al. sketch Inc-Powerlaw, a novel and much more involved algorithm capable of generating graphs for power-law bounded degree sequences with $\gamma \gtrapprox 2.88$ in expected linear time. For the first time, we give a complete description of the algorithm and add novel switchings. To the best of our knowledge, our open-source implementation of Inc-Powerlaw is the first practical generator with rigorous uniformity guarantees for the aforementioned degree sequences. In an empirical investigation, we find that for small average-degrees Inc-Powerlaw is very efficient and generates graphs with one million nodes in less than a second. For larger average-degrees, parallelism can partially mitigate the increased running-time.

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