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The way in which a social network is generated, in terms of how individuals attach to each other, determines the properties of the resulting network. Here we study an intuitively appealing `friend of a friend' model, where a network is formed by each newly added individual attaching first to a randomly chosen target and then to $n_q\geq 1$ randomly chosen friends of the target, each with probability $0

Building mathematical models of brains is difficult because of the sheer complexity of the problem. One potential starting point is through basal cognition, which give abstract representation of a range of organisms without central nervous systems, including fungi, slime moulds and bacteria. We propose one such model, demonstrating how a combination of oscillatory and current-based reinforcement processes can be used to couple resources in an efficient manner, mimicking the way these organisms function. A key ingredient in our model, not found in previous basal cognition models, is that we explicitly model oscillations in the number of particles (i.e. the nutrients, chemical signals or similar, which make up the biological system) and the flow of these particles within the modelled organisms. Using this approach, we find that our model builds efficient solutions, provided the environmental oscillations are sufficiently out of phase. We further demonstrate that amplitude differences can promote efficient solutions and that the system is robust to frequency differences. In the context of these findings, we discuss connections between our model and basal cognition in biological systems and slime moulds, in particular, how oscillations might contribute to self-organised problem-solving by these organisms.

There exist a wide range of single number metrics for assessing performance of classification algorithms, including AUC and the F1-score (Wikipedia lists 17 such metrics, with 27 different names). In this article, I propose a new metric to answer the following question: when an algorithm is tuned so that it can no longer distinguish labelled cats from real cats, how often does a randomly chosen image that has been labelled as containing a cat actually contain a cat? The steps to construct this metric are as follows. First, we set a threshold score such that when the algorithm is shown two randomly-chosen images -- one that has a score greater than the threshold (i.e. a picture labelled as containing a cat) and another from those pictures that really does contain a cat -- the probability that the image with the highest score is the one chosen from the set of real cat images is 50\%. At this decision threshold, the set of positively labelled images are indistinguishable from the set of images which are positive. Then, as a second step, we measure performance by asking how often a randomly chosen picture from those labelled as containing a cat actually contains a cat. This metric can be thought of as {\it precision at the indistinguishability threshold}. While this new metric doesn't address the tradeoff between precision and recall inherent to all such metrics, I do show why this method avoids pitfalls that can occur when using, for example AUC, and it is better motivated than, for example, the F1-score.

While mathematical models, in particular self-propelled particle (SPP) models, capture many of the observed properties of large fish schools, they do not always capture the interactions of smaller shoals. Nor do these models tend to account for the observation that, when swimming alone or in smaller groups, many species of fish use intermittent locomotion, often referred to as burst and coast or burst and glide. Recent empirical studies have suggested that burst and glide movement is indeed pivotal to the social interactions of individual fish. In this paper, we propose a model of social burst and glide motion by combining a well-studied model of neuronal dynamics, the FitzHugh-Nagumo model, with a model of fish motion. We begin by showing that the model can capture the motion of a single fish swimming down a channel. By then extending to a two fish model, where visual stimuli of the position of the other fish affect the internal burst or glide state of the fish, we find that our model captures a rich set of swimming dynamics found in many species of fish. These include: leader-follower behaviour; periodic changes in leadership; apparently random (i.e. chaotic) leadership change; and pendulum-like tit-for-tat turn taking. Unlike SPP models, which assume that fish move at a constant speed, the model produces realistic motion of individual fish. Moreover, unlike previous studies where a random component is used for leadership switching to occur, we show that leadership switching, both periodic and chaotic, can be the result of a deterministic interaction. We give several empirically testable predictions on how fish interact and discuss our results in light of recently established correlations between fish locomotion and brain activity.

We provide a critique of mathematical biology in light of rapid developments in modern machine learning. We argue that out of the three modelling activities -- (1) formulating models; (2) analysing models; and (3) fitting or comparing models to data -- inherent to mathematical biology, researchers currently focus too much on activity (2) at the cost of (1). This trend, we propose, can be reversed by realising that any given biological phenomena can be modelled in an infinite number of different ways, through the adoption of an open/pluralistic approach. We explain the open approach using fish locomotion as a case study and illustrate some of the pitfalls -- universalism, creating models of models, etc. -- that hinder mathematical biology. We then ask how we might rediscover a lost art: that of creative mathematical modelling. This article is dedicated to the memory of Edmund Crampin.

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how 'bottom up', individual-based models can be approximated by 'top down' models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterise these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

When mathematical modelling is applied to capture a complex system, multiple models are often created that characterize different aspects of that system. Often, a model at one level will produce a prediction which is contradictory at another level but both models are accepted because they are both useful. Rather than aiming to build a single unified model of a complex system, the modeller acknowledges the infinity of ways of capturing the system of interest, while offering their own specific insight. We refer to this pragmatic applied approach to complex systems -- one which acknowledges that they are incompressible, dynamic, nonlinear, historical, contextual, and value-laden -- as Open Machine Learning (Open ML). In this paper we define Open ML and contrast it with some of the grand narratives of ML of two forms: 1) Closed ML, ML which emphasizes learning with minimal human input (e.g. Google's AlphaZero) and 2) Partially Open ML, ML which is used to parameterize existing models. To achieve this, we use theories of critical complexity to both evaluate these grand narratives and contrast them with the Open ML approach. Specifically, we deconstruct grand ML `theories' by identifying thirteen 'games' played in the ML community. These games lend false legitimacy to models, contribute to over-promise and hype about the capabilities of artificial intelligence, reduce wider participation in the subject, lead to models that exacerbate inequality and cause discrimination and ultimately stifle creativity in research. We argue that best practice in ML should be more consistent with critical complexity perspectives than with rationalist, grand narratives., Comment: To appear in the HHAI 2022 conference proceedings

One of the best-known models in network science is preferential attachment. In this model, the probability of attaching to a node depends on the degree of all nodes in the population, and thus depends on global information. In many biological, physical, and social systems, however, interactions between individuals depend only on local information. Here, we investigate a truly local model of network formation, based on the idea of a friend of a friend, with the following rule: individuals choose one node at random and link to it with probability p, then they choose a neighbour of that node and link with probability q. Our model produces power laws with empirical exponents ranging from 1.5 upwards and clustering co-efficients ranging from 0 up to 0.5 (consistent with many real networks). For small p and q=1, the model produces super-hub networks, and we prove that for p=0 and q=1, the proportion of non-hubs tends to 1 as the network grows. We show that power-law degree distributions, small world clustering and super-hub networks are all outcomes of this, more general, yet conceptually simple model.

In this article we provide a systematic experimental method for sorting animals according to socially relevant traits, without assaying them or even tagging them individually. Instead, they are repeatedly subjected to behavioural assays in groups, between which the group memberships are rearranged, in order to test the effect of many different combinations of individuals on a group-level property or feature. We analyse this method using a general model for the group feature, and simulate a variety of specific cases to track how individuals are sorted in each case. We find that in the case where the members of a group contribute equally to the group feature, the sorting procedure increases the between-group behavioural variation well above what is expected for groups randomly sampled from a population. For a wide class of group feature models, the individual phenotypes are efficiently sorted across the groups and thus become available for further analysis on how individual properties affect group behaviour. We also show that the experimental data can be used to estimate the individual-level repeatability of the underlying traits., Comment: 16 pages, 3 figures + supplementary information (3 pages)

Collective movement can be achieved when individuals respond to the local movements and positions of their neighbours. Some individuals may disproportionately influence group movement if they occupy particular spatial positions in the group, for example, positions at the front of the group. We asked, therefore, what led individuals in moving pairs of fish (Gambusia holbrooki) to occupy a position in front of their partner. Individuals adjusted their speed and direction differently in response to their partner's position, resulting in individuals occupying different positions in the group. Individuals that were found most often at the front of the pair had greater mean changes in speed than their partner, and were less likely to turn towards their partner, compared to those individuals most often found at the back of the pair. The pair moved faster when led by the individual that was usually at the front. Our results highlight how differences in the social responsiveness between individuals can give rise to leadership in free moving groups. They also demonstrate how the movement characteristics of groups depend on the spatial configuration of individuals within them.

While a rich variety of self-propelled particle models propose to explain the collective motion of fish and other animals, rigorous statistical comparison between models and data remains a challenge. Plausible models should be flexible enough to capture changes in the collective behaviour of animal groups at their different developmental stages and group sizes. Here we analyse the statistical properties of schooling fish (Pseudomugil signifer) through a combination of experiments and simulations. We make novel use of a Boltzmann inversion method, usually applied in molecular dynamics, to identify the effective potential of the mean force of fish interactions. Specifically, we show that larger fish have a larger repulsion zone, but stronger attraction, resulting in greater alignment in their collective motion. We model the collective dynamics of schools using a self-propelled particle model, modified to include varying particle speed and a local repulsion rule. We demonstrate that the statistical properties of the fish schools are reproduced by our model, thereby capturing a number of features of the behaviour and development of schooling fish.

The escape trajectories animals take following a predatory attack appear to show high degrees of apparent 'randomness' - a property that has been described as 'protean behaviour'. Here we present a method of quantifying the escape trajectories of individual animals using a path complexity approach. When fish (Pseudomugil signifer) were attacked either on their own or in groups, we find that an individual's path rapidly increases in entropy (our measure of complexity) following the attack. For individuals on their own, this entropy remains elevated (indicating a more random path) for a sustained period (10 seconds) after the attack, whilst it falls more quickly for individuals in groups. The entropy of the path is context dependent. When attacks towards single fish come from greater distances, a fish's path shows less complexity compared to attacks that come from short range. This context dependency effect did not exist, however, when individuals were in groups. Nor did the path complexity of individuals in groups depend on a fish's local density of neighbours. We separate out the components of speed and direction changes to determine which of these components contributes to the overall increase in path complexity following an attack. We found that both speed and direction measures contribute similarly to an individual's path's complexity in absolute terms. Our work highlights the adaptive behavioural tactics that animals use to avoid predators and also provides a novel method for quantifying the escape trajectories of animals., Comment: 9 pages