arXiv subjects

Mark C. Wilson

Publications and source records attributed to Mark C. Wilson.

At least 19 recordsLinked to original sources

arXiv2023

A model for reference list length of scholarly articles

We introduce and analyse a simple probabilistic model of article production and citation behavior that explicitly assumes that there is no decline in citability of a given article over time. It makes predictions about the number and age of items appearing in the reference list of an article. The latter topics have been studied before, but only in the context of data, and to our knowledge no models have been presented. We then perform large-scale analyses of reference list length for a variety of academic disciplines. The results show that our simple model cannot be rejected, and indeed fits the aggregated data on reference lists rather well. Over the last few decades, the relationship between total publications and mean reference list length is linear to a high level of accuracy. Although our model is clearly an oversimplification, it will likely prove useful for further modeling of the scholarly literature. Finally, we connect our work to the large literature on "aging" or "obsolescence" of scholarly publications, and argue that the importance of that area of research is no longer clear, while much of the existing literature is confused and confusing.

cs.DL↗

arXiv2022

Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)

We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$ variables, and then apply ACSV theory to analyse the latter. This method allows us to give systematic results in the multivariate case, seems more promising than trying to derive analogs of the rational ACSV theory for algebraic GFs, and gives the prospect of further improvements as embedding methods are studied in more detail.

math.CO↗

arXiv2022

Asymptotic welfare performance of Boston assignment algorithms

We make a detailed analysis of three key algorithms (Serial Dictatorship and the naive and adaptive variants of the Boston algorithm) for the housing allocation problem, under the assumption that agent preferences are chosen iid uniformly from linear orders on the items. We compute limiting distributions (with respect to some common utility functions) as $n\to \infty$ of both the utilitarian welfare and the order bias. To do this, we compute limiting distributions of the outcomes for an arbitrary agent whose initial relative position in the tiebreak order is $θ\in[0,1]$, as a function of $θ$. The results for the Boston algorithms are all new, and we expect that these fundamental results on the stochastic processes underlying these algorithms will have wider applicability in future. Overall our results show that the differences in utilitarian welfare performance of the three algorithms are fairly small but still important. However, the differences in order bias are much greater. Also, Naive Boston beats Adaptive Boston, which beats Serial Dictatorship, on both utilitarian welfare and order bias.

econ.TH↗

arXiv2021

Non-cumulative measures of researcher citation impact

The most commonly used publication metrics for individual researchers are the the total number of publications, the total number of citations, and Hirsch's $h$-index. Each of these is cumulative, and hence increases throughout a researcher's career, making it less suitable for evaluation of junior researchers or assessing recent impact. Most other author-level measures in the literature share this cumulative property. By contrast, we aim to study non-cumulative measures that answer the question "in terms of citation impact, what have you done lately?" We single out six measures from the rather sparse literature, including Hirsch's $m$-index, a time-scaled version of the $h$-index. We introduce new measures based on the idea of "citation acceleration". After presenting several axioms for non-cumulative measures, we conclude that one of our new measures has much better theoretical justification. We present a small-scale study of its performance on real data and conclude that it shows substantial promise for future use.

cs.DL↗

arXiv2019

A modelling and computational study of the frustration index in signed networks

Computing the frustration index of a signed graph is a key step toward solving problems in many fields including social networks, political science, physics, chemistry, and biology. The frustration index determines the distance of a network from a state of total structural balance. Although the definition of the frustration index goes back to the 1950's, its exact algorithmic computation, which is closely related to classic NP-hard graph problems, has only become a focus in recent years. We develop three new binary linear programming models to compute the frustration index exactly and efficiently as the solution to a global optimisation problem. Solving the models with prioritised branching and valid inequalities in Gurobi, we can compute the frustration index of real signed networks with over 15000 edges in less than a minute on inexpensive hardware. We provide extensive performance analysis for both random and real signed networks and show that our models outperform all existing approaches by large factors. Based on solve time, algorithm output, and effective branching factor we highlight the superiority of our models to both exact and heuristic methods in the literature.

cs.SI↗

arXiv2019

Balance and Frustration in Signed Networks

The frustration index is a key measure for analysing signed networks, which has been underused due to its computational complexity. We use an exact optimisation-based method to analyse frustration as a global structural property of signed networks coming from diverse application areas. In the classic friend-enemy interpretation of balance theory, a by-product of computing the frustration index is the partitioning of nodes into two internally solidary but mutually hostile groups. The main purpose of this paper is to present general methodology for answering questions related to partial balance in signed networks, and apply it to a range of representative examples that are now analysable because of advances in computational methods. We provide exact numerical results on social and biological signed networks, networks of formal alliances and antagonisms between countries, and financial portfolio networks. Molecular graphs of carbon and Ising models are also considered. The purpose served by exploring several problems in this paper is to propose a single general methodology for studying signed networks and to demonstrate its relevance to applications. We point out several mistakes in the signed networks literature caused by inaccurate computation, implementation errors or inappropriate measures.

cs.SI↗

arXiv2019

Quiver Asymptotics: $\mathcal{N}=1$ Free Chiral Ring

The large N generating functions for the counting of chiral operators in $\mathcal{N}=1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with this asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine $\mathbb{C}^3/\hat{A}_n$ orbifold quivers.

hep-th↗

arXiv2019

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps $\mathcal{S}$ is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions with non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets which are symmetric over all but one axis. As a special case, we apply our results when $d=2$ to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.

math.CO↗

arXiv2018

Measuring Partial Balance in Signed Networks

Is the enemy of an enemy necessarily a friend? If not, to what extent does this tend to hold? Such questions were formulated in terms of signed (social) networks and necessary and sufficient conditions for a network to be "balanced" were obtained around 1960. Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas. However, investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice. We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on synthetic datasets, and investigate their axiomatic properties. The synthetic data involves Erdős-Rényi and specially structured random graphs. We show that some measures behave better than others in terms of axioms and ability to differentiate between graphs. We also use well-known datasets from the sociology and biology literature, such as Read's New Guinean tribes, gene regulatory networks related to two organisms, and a network involving senate bill co-sponsorship. Our results show that substantially different levels of partial balance is observed under cycle-based, eigenvalue-based, and frustration-based measures. We make some recommendations for measures to be used in future work.

cs.SI↗

arXiv2018

Distance rationalization of anonymous and homogeneous voting rules

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the $\ell^1$-norm is not strictly convex.

cs.GT↗

arXiv2018

Computing the Line Index of Balance Using Integer Programming Optimisation

An important measure of signed graphs is the line index of balance which has several applications in many fields. However, this graph-theoretic measure was underused for decades because of the inherent complexity in its computation which is closely related to solving NP-hard graph optimisation problems like MAXCUT. We develop new quadratic and linear programming models to compute the line index of balance exactly. Using the Gurobi integer programming optimisation solver, we evaluate the line index of balance on real-world and synthetic datasets. The synthetic data involves Erdős-Rényi graphs, Barabási-Albert graphs, and specially structured random graphs. We also use well known datasets from the sociology literature, such as signed graphs inferred from students' choice and rejection as well as datasets from the biology literature including gene regulatory networks. The results show that exact values of the line index of balance in relatively large signed graphs can be efficiently computed using our suggested optimisation models. We find that most real-world social networks and some biological networks have small line index of balance which indicates that they are close to balanced.

cs.SI↗

arXiv2017

Multi-district preference modelling

Generating realistic artificial preference distributions is an important part of any simulation analysis of electoral systems. While this has been discussed in some detail in the context of a single electoral district, many electoral systems of interest are based on multiple districts. Neither treating preferences between districts as independent nor ignoring the district structure yields satisfactory results. We present a model based on an extension of the classic Eggenberger-Pólya urn, in which each district is represented by an urn and there is correlation between urns. We show in detail that this procedure has a small number of tunable parameters, is computationally efficient, and produces "realistic-looking" distributions. We intend to use it in further studies of electoral systems.

cs.GT↗

arXiv2017

New algorithms for matching problems

The standard two-sided and one-sided matching problems, and the closely related school choice problem, have been widely studied from an axiomatic viewpoint. A small number of algorithms dominate the literature. For two-sided matching, the Gale-Shapley algorithm; for one-sided matching, (random) Serial Dictatorship and Probabilistic Serial rule; for school choice, Gale-Shapley and the Boston mechanisms. The main reason for the dominance of these algorithms is their good (worst-case) axiomatic behaviour with respect to notions of efficiency and strategyproofness. However if we shift the focus to fairness, social welfare, tradeoffs between incompatible axioms, and average-case analysis, it is far from clear that these algorithms are optimal. We investigate new algorithms several of which have not appeared (to our knowledge) in the literature before. We give a unified presentation in which algorithms for 2-sided matching yield 1-sided matching algorithms in a systematic way. In addition to axiomatic properties, we investigate agent welfare using both theoretical and computational approaches. We find that some of the new algorithms are worthy of consideration for certain applications. In particular, when considering welfare under truthful preferences, some of the new algorithms outperform the classic ones.

cs.GT↗

arXiv2016

Manipulability of consular election rules

The Gibbard-Satterthwaite theorem is a cornerstone of social choice theory, stating that an onto social choice function cannot be both strategy-proof and non-dictatorial if the number of alternatives is at least three. The Duggan-Schwartz theorem proves an analogue in the case of set-valued elections: if the function is onto with respect to singletons, and can be manipulated by neither an optimist nor a pessimist, it must have a weak dictator. However, the assumption that the function is onto with respect to singletons makes the Duggan-Schwartz theorem inapplicable to elections which necessarily select a committee with multiple members. In this paper we make a start on this problem by considering elections which elect a committee of size two (such as the consulship of ancient Rome). We establish that if such a consular election rule cannot be expressed as the union of two disjoint social choice functions, then strategy-proofness implies the existence of a dictator. Although we suspect that a similar result holds for larger sized committees, there appear to be many obstacles to proving it, which we discuss in detail.

cs.GT↗

arXiv2016

Distance rationalization of social rules

The concept of distance rationalizability of social choice rules has been explored in recent years by several authors. We deal here with several foundational questions, and unify, correct, and generalize previous work. For example, we study a new question involving uniqueness of representation in the distance rationalizability framework, and present a counterexample. For rules satisfying various axiomatic properties such as anonymity, neutrality and homogeneity, the standard profile representation of input can be compressed substantially. We explain in detail using quotient constructions and symmetry groups how distance rationalizability is interpreted in this situation. This enables us to connect the theory of distance rationalizability with geometric concepts such as Earth Mover distance and optimal transportation. We expect this connection to prove fruitful in future work. We improve on the best-known sufficient conditions for rules rationalized via votewise distances to satisfy anonymity, neutrality, homogeneity, consistency and continuity. This leads to a class of well-behaved rules which deserve closer scrutiny in future.

cs.GT↗

arXiv2016

Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Padé-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

math.CO↗

arXiv2013

Congruence successions in compositions

A \emph{composition} is a sequence of positive integers, called \emph{parts}, having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair of adjacent parts $x$ and $y$ within a composition such that $x\equiv y(\text{mod} m)$. Here, we consider the problem of counting the compositions of size $n$ according to the number of $m$-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case $m=2$, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size $n$ having no $m$-congruence successions.

math.CO↗

arXiv2012

Power measures derived from the sequential query process

We study a basic sequential model for the discovery of winning coalitions in a simple game, well known from its use in defining the Shapley-Shubik power index. We derive in a uniform way a family of measures of collective and individual power in simple games, and show that, as for the Shapley-Shubik index, they extend naturally to measures for TU-games. In particular, the individual measures include all weighted semivalues. We single out the simplest measure in our family for more investigation, as it is new to the literature as far as we know. Although it is very different from the Shapley value, it is closely related in several ways, and is the natural analogue of the Shapley value under a nonstandard, but natural, definition of simple game. We illustrate this new measure by calculating its values on some standard examples.

math.CO↗