arXiv subjects

Blair D. Sullivan

Publications and source records attributed to Blair D. Sullivan.

At least 37 records · Page 2Linked to original sources

arXiv2023

Information access representations and social capital in networks

Social network position confers power and social capital. In the setting of online social networks that have massive reach, creating mathematical representations of social capital is an important step towards understanding how network position can differentially confer advantage to different groups and how network position can itself be a source of advantage. In this paper, we use well established models for information flow on networks as a base to propose a formal descriptor of the network position of a node as represented by its information access. Combining these descriptors allows a full representation of social capital across the network. Using real-world networks, we demonstrate that this representation allows the identification of differences between groups based on network specific measures of inequality of access.

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arXiv2023

Hardness of the Generalized Coloring Numbers

The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as well, proving para-NP-hardness for all $r \geq 2$. Para-NP-hardness implies that no XP algorithm (runtime $O(n^{f(k)})$) exists for testing if a generalized coloring number is at most $k$. Moreover, there exists a constant $c$ such that it is NP-hard to approximate the generalized coloring numbers within a factor of $c$. To complement these results, we give an approximation algorithm for the generalized coloring numbers, improving both the runtime and approximation factor of the existing approach of Dvořák (EuJC 2013). We prove that greedily ordering vertices with small estimated backconnectivity achieves a $(k-1)^{r-1}$-approximation for the $r$-coloring number and an $O(k^{r-1})$-approximation for the weak $r$-coloring number.

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arXiv2022

Reducing Access Disparities in Networks using Edge Augmentation

In social networks, a node's position is a form of \it{social capital}. Better-positioned members not only benefit from (faster) access to diverse information, but innately have more potential influence on information spread. Structural biases often arise from network formation, and can lead to significant disparities in information access based on position. Further, processes such as link recommendation can exacerbate this inequality by relying on network structure to augment connectivity. We argue that one can understand and quantify this social capital through the lens of information flow in the network. We consider the setting where all nodes may be sources of distinct information, and a node's (dis)advantage deems its ability to access all information available on the network. We introduce three new measures of advantage (broadcast, influence, and control), which are quantified in terms of position in the network using \it{access signatures} -- vectors that represent a node's ability to share information. We then consider the problem of improving equity by making interventions to increase the access of the least-advantaged nodes. We argue that edge augmentation is most appropriate for mitigating bias in the network structure, and frame a budgeted intervention problem for maximizing minimum pairwise access. Finally, we propose heuristic strategies for selecting edge augmentations and empirically evaluate their performance on a corpus of real-world social networks. We demonstrate that a small number of interventions significantly increase the broadcast measure of access for the least-advantaged nodes (over 5 times more than random), and also improve the minimum influence. Additional analysis shows that these interventions can also dramatically shrink the gap in advantage between nodes (over \%82) and reduce disparities between their access signatures.

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arXiv2022

Improved Parameterized Complexity of Happy Set Problems

We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Maximum Edge Happy Set (MaxEHS)--also known as Densest k-Subgraph. Given a graph $G$ and an integer $k$, MaxHS asks for a set $S$ of $k$ vertices such that the number of $\textit{happy vertices}$ with respect to $S$ is maximized, where a vertex $v$ is happy if $v$ and all its neighbors are in $S$. We show that MaxHS can be solved in time $\mathcal{O}\left(2^\textsf{mw} \cdot \textsf{mw} \cdot k^2 \cdot |V(G)|\right)$ and $\mathcal{O}\left(8^\textsf{cw} \cdot k^2 \cdot |V(G)|\right)$, where $\textsf{mw}$ and $\textsf{cw}$ denote the $\textit{modular-width}$ and the $\textit{clique-width}$ of $G$, respectively. This resolves the open questions posed in literature. The MaxEHS problem is an edge-variant of MaxHS, where we maximize the number of $\textit{happy edges}$, the edges whose endpoints are in $S$. In this paper we show that MaxEHS can be solved in time $f(\textsf{nd})\cdot|V(G)|^{\mathcal{O}(1)}$ and $\mathcal{O}\left(2^{\textsf{cd}}\cdot k^2 \cdot |V(G)|\right)$, where $\textsf{nd}$ and $\textsf{cd}$ denote the $\textit{neighborhood diversity}$ and the $\textit{cluster deletion number}$ of $G$, respectively, and $f$ is some computable function. This result implies that MaxEHS is also fixed-parameter tractable by $\textit{twin cover number}$.

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arXiv2022

Minimizing Congestion for Balanced Dominators

A primary challenge in metagenomics is reconstructing individual microbial genomes from the mixture of short fragments created by sequencing. Recent work leverages the sparsity of the assembly graph to find $r$-dominating sets which enable rapid approximate queries through a dominator-centric graph partition. In this paper, we consider two problems related to reducing uncertainty and improving scalability in this setting. First, we observe that nodes with multiple closest dominators necessitate arbitrary tie-breaking in the existing pipeline. As such, we propose finding $\textit{sparse}$ dominating sets which minimize this effect via a new $\textit{congestion}$ parameter. We prove minimizing congestion is NP-hard, and give an $\mathcal{O}(\sqrt{Δ^r})$ approximation algorithm, where $Δ$ is the max degree. To improve scalability, the graph should be partitioned into uniformly sized pieces, subject to placing vertices with a closest dominator. This leads to $\textit{balanced neighborhood partitioning}$: given an $r$-dominating set, find a partition into connected subgraphs with optimal uniformity so that each vertex is co-assigned with some closest dominator. Using variance of piece sizes to measure uniformity, we show this problem is NP-hard iff $r$ is greater than $1$. We design and analyze several algorithms, including a polynomial-time approach which is exact when $r=1$ (and heuristic otherwise). We complement our theoretical results with computational experiments on a corpus of real-world networks showing sparse dominating sets lead to more balanced neighborhood partitionings. Further, on the metagenome $\textsf{HuSB1}$, our approach maintains high query containment and similarity while reducing piece size variance.

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arXiv2021

Parameterized algorithms for identifying gene co-expression modules via weighted clique decomposition

We present a new combinatorial model for identifying regulatory modules in gene co-expression data using a decomposition into weighted cliques. To capture complex interaction effects, we generalize the previously-studied weighted edge clique partition problem. As a first step, we restrict ourselves to the noise-free setting, and show that the problem is fixed parameter tractable when parameterized by the number of modules (cliques). We present two new algorithms for finding these decompositions, using linear programming and integer partitioning to determine the clique weights. Further, we implement these algorithms in Python and test them on a biologically-inspired synthetic corpus generated using real-world data from transcription factors and a latent variable analysis of co-expression in varying cell types.

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arXiv2021

An Updated Experimental Evaluation of Graph Bipartization Methods

We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal), motivated by recent advances in near-term quantum computing hardware and the related embedding problems. We assemble a preprocessing suite of fast input reduction routines from the Odd Cycle Transversal (OCT) and Vertex Cover (VC) literature, and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.

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arXiv2020

A color-avoiding approach to subgraph counting in bounded expansion classes

We present an algorithm to count the number of occurrences of a pattern graph $H$ as an induced subgraph in a host graph $G$. If $G$ belongs to a bounded expansion class, the algorithm runs in linear time. Our design choices are motivated by the need for an approach that can be engineered into a practical implementation for sparse host graphs. Specifically, we introduce a decomposition of the pattern $H$ called a counting dag $\vec C(H)$ which encodes an order-aware, inclusion-exclusion counting method for $H$. Given such a counting dag and a suitable linear ordering $\mathbb G$ of $G$ as input, our algorithm can count the number of times $H$ appears as an induced subgraph in $G$ in time $O(\|\vec C\| \cdot h \text{wcol}_{h}(\mathbb G)^{h-1} |G|)$, where $\text{wcol}_h(\mathbb G)$ denotes the maximum size of the weakly $h$-reachable sets in $\mathbb G$. This implies, combined with previous results, an algorithm with running time $O(4^{h^2}h (\text{wcol}_h(G)+1)^{h^3} |G|)$ which only takes $H$ and $G$ as input. We note that with a small modification, our algorithm can instead use strongly $h$-reachable sets with running time $O(\|\vec C\| \cdot h \text{col}_{h}(\mathbb G)^{h-1} |G|)$, resulting in an overall complexity of $O(4^{h^2}h \text{col}_h(G)^{h^2} |G|)$ when only given $H$ and $G$. Because orderings with small weakly/strongly reachable sets can be computed relatively efficiently in practice [11], our algorithm provides a promising alternative to algorithms using the traditional $p$-treedepth colouring framework [13]. We describe preliminary experimental results from an initial open source implementation which highlight its potential.

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arXiv2019

Approximating Vertex Cover using Structural Rounding

In this work, we provide the first practical evaluation of the structural rounding framework for approximation algorithms. Structural rounding works by first editing to a well-structured class, efficiently solving the edited instance, and "lifting" the partial solution to recover an approximation on the input. We focus on the well-studied Vertex Cover problem, and edit to the class of bipartite graphs (where Vertex Cover has an exact polynomial time algorithm). In addition to the naive lifting strategy for Vertex Cover described by Demaine et al., we introduce a suite of new lifting strategies and measure their effectiveness on a large corpus of synthetic graphs. We find that in this setting, structural rounding significantly outperforms standard 2-approximations. Further, simpler lifting strategies are extremely competitive with the more sophisticated approaches. The implementations are available as an open-source Python package, and all experiments are replicable.

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arXiv2019

Faster Biclique Mining in Near-Bipartite Graphs

Identifying dense bipartite subgraphs is a common graph data mining task. Many applications focus on the enumeration of all maximal bicliques (MBs), though sometimes the stricter variant of maximal induced bicliques (MIBs) is of interest. Recent work of Kloster et al. introduced a MIB-enumeration approach designed for "near-bipartite" graphs, where the runtime is parameterized by the size k of an odd cycle transversal (OCT), a vertex set whose deletion results in a bipartite graph. Their algorithm was shown to outperform the previously best known algorithm even when k was logarithmic in |V|. In this paper, we introduce two new algorithms optimized for near-bipartite graphs - one which enumerates MIBs in time O(M_I |V||E| k), and another based on the approach of Alexe et al. which enumerates MBs in time O(M_B |V||E| k), where M_I and M_B denote the number of MIBs and MBs in the graph, respectively. We implement all of our algorithms in open-source C++ code and experimentally verify that the OCT-based approaches are faster in practice than the previously existing algorithms on graphs with a wide variety of sizes, densities, and OCT decompositions.

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arXiv2019

Subgraph centrality and walk-regularity

Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between subgraph centrality, which is defined using the matrix exponential $f(x) = \exp(x)$, and the walk structure of a network. In a walk-regular graph, the number of closed walks of each length must be the same for all nodes, implying uniform $f$-subgraph centralities for any $f$ (or maximum $f$-$\textit{walk entropy}$). We consider when non--walk-regular graphs can achieve maximum entropy, calling such graphs $\textit{entropic}$. For parameterized measures, we are also interested in which values of the parameter witness this uniformity. To date, only one entropic graph has been identified, with only two witnessing parameter values, raising the question of how many such graphs and parameters exist. We resolve these questions by constructing infinite families of entropic graphs, as well as a family of witnessing parameters with a limit point at zero.

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arXiv2019

Mining Maximal Induced Bicliques using Odd Cycle Transversals

Many common graph data mining tasks take the form of identifying dense subgraphs (e.g. clustering, clique-finding, etc). In biological applications, the natural model for these dense substructures is often a complete bipartite graph (biclique), and the problem requires enumerating all maximal bicliques (instead of just identifying the largest or densest). The best known algorithm in general graphs is due to Dias et al., and runs in time O(M |V|^4 ), where M is the number of maximal induced bicliques (MIBs) in the graph. When the graph being searched is itself bipartite, Zhang et al. give a faster algorithm where the time per MIB depends on the number of edges in the graph. In this work, we present a new algorithm for enumerating MIBs in general graphs, whose run time depends on how "close to bipartite" the input is. Specifically, the runtime is parameterized by the size k of an odd cycle transversal (OCT), a vertex set whose deletion results in a bipartite graph. Our algorithm runs in time O(M |V||E|k^2 3^(k/3) ), which is an improvement on Dias et al. whenever k <= 3log_3(|V|). We implement our algorithm alongside a variant of Dias et al.'s in open-source C++ code, and experimentally verify that the OCT-based approach is faster in practice on graphs with a wide variety of sizes, densities, and OCT decompositions.

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arXiv2018

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to $\textit{edit}$ a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then $\textit{lift}$ the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, ($\ell$-)Dominating Set, Edge ($\ell$-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of several important graph classes (in some cases, also approximating the target parameter of the family). For bounded degeneracy, we obtain a bicriteria $(4,4)$-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria $(O(\log^{1.5} n), O(\sqrt{\log w}))$-approximation, and for bounded pathwidth, we obtain a bicriteria $(O(\log^{1.5} n), O(\sqrt{\log w} \cdot \log n))$-approximation. For treedepth $2$ (also related to bounded expansion), we obtain a $4$-approximation. We also prove complementary hardness-of-approximation results assuming $\mathrm{P} \neq \mathrm{NP}$: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor ($2$ assuming UGC).

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arXiv2018

Structural Sparsity of Complex Networks: Bounded Expansion in Random Models and Real-World Graphs

This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. We analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed s parse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabasi-Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks.

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arXiv2018

Polynomial Treedepth Bounds in Linear Colorings

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice. We introduce $p$-linear colorings as an alternative to the commonly used $p$-centered colorings. They can be efficiently computed in bounded expansion classes and use at most as many colors as $p$-centered colorings. Although a set of $k<p$ colors from a $p$-centered coloring induces a subgraph of treedepth at most $k$, the same number of colors from a $p$-linear coloring may induce subgraphs of larger treedepth. We establish a polynomial upper bound on the treedepth in general graphs, and give tighter bounds in trees and interval graphs via constructive coloring algorithms. We also give a co-NP-completeness reduction for recognizing $p$-linear colorings and discuss ways to overcome this limitation in practice. This preprint extends results that appeared in [9]; for full proofs omitted from [9], see previous versions of this preprint.

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arXiv2018

Benchmarking treewidth as a practical component of tensor-network--based quantum simulation

Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on the tensor ranks and the order in which they are contracted. Unfortunately, computing optimal contraction sequences (orderings) in general is known to be a computationally difficult (NP-complete) task. In 2005, Markov and Shi showed that optimal contraction sequences correspond to optimal (minimum width) tree decompositions of a tensor network's line graph, relating the contraction sequence problem to a rich literature in structural graph theory. While treewidth-based methods have largely been ignored in favor of dataset-specific algorithms in the prior tensor networks literature, we demonstrate their practical relevance for problems arising from two distinct methods used in quantum simulation: multi-scale entanglement renormalization ansatz (MERA) datasets and quantum circuits generated by the quantum approximate optimization algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms outperform domain-specific algorithms, while demonstrating that the optimal choice of algorithm has a complex dependence on the network density, expected contraction complexity, and user run time requirements. We further provide an open source software framework designed with an emphasis on accessibility and extendability, enabling replicable experimental evaluations and future exploration of competing methods by practitioners.

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arXiv2018

An Experimental Evaluation of a Bounded Expansion Algorithmic Pipeline

Previous work has suggested that the structural restrictions of graphs from classes of bounded expansion--locally dense pockets in a globally sparse graph--naturally coincide with common properties of real-world networks such as clustering and heavy-tailed degree distributions. As such, fixed-parameter tractable algorithms for bounded expansion classes may offer a promising framework for network analysis where other approaches have struggled to scale. However, there has been little work done in implementing and evaluating the performance of these structure-based algorithms. To this end we introduce CONCUSS, a proof-of-concept implementation of a generic algorithmic pipeline for classes of bounded expansion. In particular, we focus on using CONCUSS for subgraph isomorphism counting (also called motif or graphlet counting), which has been used extensively as a tool for analyzing biological and social networks. Through a broad set of experiments we first evaluate the interactions between implementation/engineering choices at multiple stages of the pipeline and their effects on overall run time. From there, we establish viability of the bounded expansion framework by demonstrating that in some scenarios CONCUSS achieves run times competitive with a popular algorithm for subgraph isomorphism counting that does not exploit graph structure. Finally, we empirically identify two particular ways in which future theoretical advances could alleviate bottlenecks in the algorithmic pipeline.

cs.DS↗

arXiv2018

Walk entropy and walk-regularity

A graph is said to be walk-regular if, for each $\ell \geq 1$, every vertex is contained in the same number of closed walks of length $\ell$. We construct a $24$-vertex graph $H_4$ that is not walk-regular yet has maximized walk entropy, $S^V(H_4,β) = \log 24$, for some $β>0$. This graph is a counterexample to a conjecture of Benzi [Linear Algebra Appl.~443 (2014), 395--399, Conjecture 3.1]. We also show that there exist infinitely many temperatures $β_0>0$ so that $S^V(G,β_0)=\log n_G$ if and only if a graph $G$ is walk-regular.

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