arXiv subjects

Steve Lawford

Publications and source records attributed to Steve Lawford.

5 recordsLinked to original sources

arXiv2026

Counting five-node subgraphs

For a simple undirected graph $G$, we derive exact formulae for the number of copies of each of the 21 non-isomorphic connected graphs on five vertices. For nine of these graphs (the stingray, spinning top, kite, ufo, crown, envelope, lamp, arrowhead, and cat's cradle) the resulting analytical formulae appear to provide substantially new formulations; we also identify and correct five erroneous published formulae for the 5-path, banner, and lollipop. The proofs use elementary combinatorial arguments, organized around recurring constructions based on incident structures, walks, common neighbourhoods, and neighbourhood subgraphs, several of which extend naturally to larger subgraphs. We illustrate the formulae by deriving analytical results for several regular graphs and by applying them to a real-world network, where induced five-node subgraph counts, obtained as linear combinations of general counts, are compared with Erdős-Rényi and degree-preserving null ensembles.

math.CO↗

arXiv2026

Gaussian Boundary Inference in a Hypergeometric Heavy-Tailed Family

This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family \[ G_c(z) = \frac12 + z\,\frac{Γ(c-1/2)}{2\sqrt2\,Γ(c)}\,{}_1F_1\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \] contains the standard normal distribution at the boundary $c=3/2$. Away from the boundary, the density has algebraic tail behaviour $g_c(z)\sim(c-3/2)|z|^{-3}$, so the parameter $c$ indexes a directed heavy-tailed deformation of the Gaussian law. The distribution also admits an equivalent beta-precision normal scale-mixture representation, in which the Gaussian boundary corresponds to degenerate unit precision. I establish the admissibility of the hypergeometric family, derive minimum-distance estimators, and obtain both regular interior asymptotics and nonstandard boundary asymptotics under the normal null. The standardised fit-improvement statistic converges to the mixture distribution $\tfrac12δ_0+\tfrac12χ_1^2$. I extend the theory to plug-in location-scale procedures, including a robust median/IQR version. Simulations document accurate null size and directed power against heavy-tailed alternatives. An application to daily S&P 500 returns, both unconditionally and after GARCH(1,1) filtering, illustrates the empirical implications for tail fitting and risk quantiles, and documents that GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test.

stat.ME↗

arXiv2025

Network effects and incumbent response to entry threats: empirical evidence from the airline industry

I investigate how incumbents in the U.S. airline industry respond to threatened and actual route entry by Southwest Airlines. I use a two-way fixed effects and event study approach, and the latest available data from 1999-2022, to identify a firm's price and quantity response. I find evidence that incumbents cut fares preemptively (post-entry) by 6-8% (16-18%) although the significance, pattern, and timing of the preemptive cuts are quite different to Goolsbee and Syverson's (2008) earlier results. Incumbents increase capacity preemptively by 10-40%, up to six quarters before the entry threat is established, and by 27-46% post-entry. My results suggest a clear shift in firms' strategic response from price to quantity. I also investigate the impact of an incumbent's network structure on its preemptive and post-entry behaviour. While the results on price are unclear, a firm's post-entry capacity reaction depends strongly on its global network structure as well as the local importance (centrality) of the route.

econ.GN↗

arXiv2020

Subgraphs and motifs in a dynamic airline network

How does the small-scale topological structure of an airline network behave as the network evolves? To address this question, we study the dynamic properties of small undirected subgraphs using 15 years of data on Southwest Airlines' domestic route service. We use exact enumeration formulae to identify statistically over- and under-represented subgraphs, known as motifs and anti-motifs. We discover substantial topology transitions in Southwest's network and provide evidence for time-varying power-law scaling between subgraph counts and the number of edges in the network. We also suggest a node-ranking measure that can identify important nodes relative to specific local topologies. Our results extend the toolkit of subgraph-based methods and provide new insight into transportation networks and the strategic behaviour of firms.

cs.SI↗

arXiv2020

Cliques and a new measure of clustering: with application to U.S. domestic airlines

We propose a higher-order generalization of the well-known overall clustering coefficient for triples $C(3)$ to any number of nodes. We give analytic formulae for the special cases of three, four, and five nodes and show that they have very fast runtime performance for small graphs. We discuss some theoretical properties and limitations of the new measure, and use it to provide insight into dynamic changes in the structure of U.S. airline networks.

cs.SI↗