arXiv subjects

Jonathan Jordan

Publications and source records attributed to Jonathan Jordan.

17 recordsLinked to original sources

arXiv2026

Critical branching processes in random environment with immigration and an application to randomised reproducing graphs

We study branching processes in an i.i.d.\ random environment with immigration in the critical regime, where the underlying offspring mechanism satisfies the critical condition that the log of the average population growth, across environments, and before immigration, is zero. In this setting environmental fluctuations are balanced on average, and the long-term behaviour is determined by the interaction between these fluctuations and the immigration sequence. While recurrence and transience criteria for critical BPREI were established by Bauernschubert (2014), the possibility of null recurrence remained unresolved. We show that, under natural integrability assumptions on the offspring and immigration distributions, a critical BPREI is null recurrent. In particular, the process returns to zero infinitely often but admits no stationary distribution. Our results close a gap in the classification of the critical regime and provide a structural understanding of the balance between environmental variability and immigration. As an application, we resolve the open critical case of the Randomised Reproducing Graph (`RRG') model introduced by Jordan (2011), showing that in the critical regime the proportion of vertices of a fixed degree admits no limiting distribution.

math.PR↗

arXiv2025

A Third Paradigm for LLM Evaluation: Dialogue Game-Based Evaluation using clembench

There are currently two main paradigms for evaluating large language models (LLMs), reference-based evaluation and preference-based evaluation. The first, carried over from the evaluation of machine learning models in general, relies on pre-defined task instances, for which reference task executions are available. The second, best exemplified by the LM-arena, relies on (often self-selected) users bringing their own intents to a site that routes these to several models in parallel, among whose responses the user then selects their most preferred one. The former paradigm hence excels at control over what is tested, while the latter comes with higher ecological validity, testing actual use cases interactively. Recently, a third complementary paradigm has emerged that combines some of the strengths of these approaches, offering control over multi-turn, reference-free, repeatable interactions, while stressing goal-directedness: dialogue game based evaluation. While the utility of this approach has been shown by several projects, its adoption has been held back by the lack of a mature, easily re-usable implementation. In this paper, we present clembench, which has been in continuous development since 2023 and has in its latest release been optimized for ease of general use. We describe how it can be used to benchmark one's own models (using a provided set of benchmark game instances in English), as well as how easily the benchmark itself can be extended with new, tailor-made targeted tests.

cs.CL↗

arXiv2025

Plant in Cupboard, Orange on Rably, Inat Aphone. Benchmarking Incremental Learning of Situation and Language Model using a Text-Simulated Situated Environment

Large Language Models (LLMs) serve not only as chatbots but as key components in agent systems, where their common-sense knowledge significantly impacts performance as language-based planners for situated or embodied action. We assess LLMs' incremental learning (based on feedback from the environment), and controlled in-context learning abilities using a text-based environment. We introduce challenging yet interesting set of experiments to test i) how agents can incrementally solve tasks related to every day objects in typical rooms in a house where each of them are discovered by interacting within the environment, ii) controlled in-context learning abilities and efficiency of agents by providing short info about locations of objects and rooms to check how faster the task can be solved, and finally iii) using synthetic pseudo-English words to gauge how well LLMs are at inferring meaning of unknown words from environmental feedback. Results show that larger commercial models have a substantial gap in performance compared to open-weight but almost all models struggle with the synthetic words experiments.

cs.CL↗

arXiv2024

How Many Parameters Does it Take to Change a Light Bulb? Evaluating Performance in Self-Play of Conversational Games as a Function of Model Characteristics

What makes a good Large Language Model (LLM)? That it performs well on the relevant benchmarks -- which hopefully measure, with some validity, the presence of capabilities that are also challenged in real application. But what makes the model perform well? What gives a model its abilities? We take a recently introduced type of benchmark that is meant to challenge capabilities in a goal-directed, agentive context through self-play of conversational games, and analyse how performance develops as a function of model characteristics like number of parameters, or type of training. We find that while there is a clear relationship between number of parameters and performance, there is still a wide spread of performance points within a given size bracket, which is to be accounted for by training parameters such as fine-tuning data quality and method. From a more practical angle, we also find a certain degree of unpredictability about performance across access methods, possible due to unexposed sampling parameters, and a, very welcome, performance stability against at least moderate weight quantisation during inference.

cs.CL↗

arXiv2022

Competing types in preferential attachment graphs with community structure

We extend the two-type preferential attachment model of Antunovi\'c, Mossel and R\'acz, where each new vertex takes its type according to a defined rule based on the types of its neighbours, to incorporate community structure, and investigate whether the proportions of vertices of each type synchronise between communities. The behaviour depends both on the choice of community structure and on the type assignment rule. For essentially all cases where the single community model has more than one possible limit, communities may fail to synchronise for weakly interacting communities. Even when the single community model almost surely converges to a deterministic limit, synchronisation is not guaranteed. However, we give natural conditions on the assignment rule and, for two communities, on the structure, either of which will imply synchronisation to this limit, and each of which is essentially best possible. We also give an example where the proportions of types almost surely do not converge, which is impossible in the single community model.

math.PR↗

arXiv2021

Fragility of non-convergence in preferential attachment graphs with three types

Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random, and are more likely to connect to those that already have many connections. We investigate further a family of models introduced by Antunovi\'{c}, Mossel and R\'{a}cz where each vertex in a preferential attachment graph is assigned a type, based on the types of its neighbours. Instances of this type of process where the proportions of each type present do not converge over time seem to be rare. Previous work found that a "rock-paper-scissors" setup where each new node's type was determined by a rock-paper-scissors contest between its two neighbours does not converge. Here, two cases similar to that are considered, one which is like the above but with an arbitrarily small chance of picking a random type and one where there are four neighbours which perform a knockout tournament to determine the new type. These two new setups, despite seeming very similar to the rock-paper-scissors model, do in fact converge, perhaps surprisingly.

math.PR↗

arXiv2020

Phase transitions in non-linear urns with interacting types

We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented.

math.PR↗

arXiv2018

Extensive Condensation in a model of Preferential Attachment with Fitnesses

We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between the model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation phenomenon, in which unusually fit vertices may obtain abnormally high degree: it arises from a growth-extinction dichotomy within the branching part of the dual. We show further that the condensation is extensive. As the graph grows, unusually fit vertices become, each only for a limited time, neighbouring to a non-vanishing proportion of the current graph.

math.PR↗

arXiv2018

Condensation in preferential attachment models with location-based choice

We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting $r$ vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant $\alpha$. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in $\alpha$ below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

math.PR↗

arXiv2018

Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

We consider the preferential attachment model with multiple vertex types introduced by Antunovi\'c, Mossel and R\'acz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunovi\'c, Mossel and R\'acz. We also consider another family of examples where we show that the conjecture does hold.

math.PR↗

arXiv2018

Preferential attachment graphs with co-existing types of different fitnesses

We extend the work of Antunovi\'{c}, Mossel and R\'{a}cz on competing types in preferential attachment models to include cases where the types have different fitnesses, which may be either multiplicative or additive. We will show that, depending on the values of the parameters of the models, there are different possible limiting behaviours depending on the zeros of a certain function. In particular we will show the existence of choices of the parameters where one type is favoured both by having higher fitness and by the type attachment mechanism, but the other type has a positive probability of dominating the network in the limit.

math.PR↗

arXiv2017

The connected component of the partial duplication graph

We consider the connected component of the partial duplication model for a random graph, a model which was introduced by Bhan, Galas and Dewey as a model for gene expression networks. The most rigorous results are due to Hermann and Pfaffelhuber, who show a phase transition between a subcritical case where in the limit almost all vertices are isolated and a supercritical case where the proportion of the vertices which are connected is bounded away from zero. We study the connected component in the subcritical case, and show that, when the duplication parameter $p<e^{-1}$, the degree distribution of the connected component has a limit, which we can describe in terms of the stationary distribution of a certain Markov chain and which follows an approximately power law tail, with the power law index predicted by Ispolatov, Krapivsky and Yuryev. Our methods involve analysing the quasi-stationary distribution of a certain continuous time Markov chain associated with the evolution of the graph.

math.PR↗

arXiv2014

Preferential attachment with choice

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.

math.PR↗

arXiv2013

Phase transitions for random geometric preferential attachment graphs

We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab\'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab\'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors.

math.PR↗

arXiv2013

Random walk attachment graphs

We consider the random walk attachment graph introduced by Saram\"{a}ki and Kaski and proposed as a mechanism to explain how behaviour similar to preferential attachment may appear requiring only local knowledge. We show that if the length of the random walk is fixed then the resulting graphs can have properties significantly different from those of preferential attachment graphs, and in particular that in the case where the random walks are of length 1 and each new vertex attaches to a single existing vertex the proportion of vertices which have degree 1 tends to 1, in contrast to preferential attachment models. AMS 2010 Subject Classification: Primary 05C82. Key words and phrases:random graphs; preferential attachment; random walk.

math.PR↗

arXiv2012

Geometric preferential attachment in non-uniform metric spaces

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of $S$ of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.

math.PR↗

arXiv2011

Randomised reproducing graphs

We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, $\alpha$, $\beta$ and $\gamma$, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If $(1+\alpha)(1+\gamma)<1$ then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on $\alpha$ and $\gamma$. If $(1+\alpha)(1+\gamma)>1$ then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree $d$ tends to zero. We also give some results on the number of edges and on the spectral gap.

math.PR↗