We quantify the notion of pattern and formalize the process of pattern discovery under the framework of binary bipartite networks. Patterns of particular focus are interrelated global interactions between clusters on its row and column axes. A binary bipartite network is built into a thermodynamic system embracing all up-and-down spin configurations defined by product-permutations on rows and columns. This system is equipped with its ferromagnetic energy ground state under Ising model potential. Such a ground state, also called a macrostate, is postulated to congregate all patterns of interest embedded within the network data in a multiscale fashion. A new computing paradigm for indirect searching for such a macrostate, called Data Mechanics, is devised by iteratively building a surrogate geometric system with a pair of nearly optimal marginal ultrametrics on row and column spaces. The coupling measure minimizing the Gromov-Wasserstein distance of these two marginal geometries is also seen to be in the vicinity of the macrostate. This resultant coupling geometry reveals multiscale block pattern information that characterizes multiple layers of interacting relationships between clusters on row and on column axes. It is the nonparametric information content of a binary bipartite network. This coupling geometry is then demonstrated to shed new light and bring resolution to interaction issues in community ecology and in gene-content-based phylogenetics. Its implied global inferences are expected to have high potential in many scientific areas.

The notion of dominance is ubiquitous across the animal kingdom, wherein some species/groups such relationships are strictly hierarchical and others are not. Modern approaches for measuring dominance have emerged in recent years taking advantage of increased computational power. One such technique, named Percolation and Conductance (Perc), uses both direct and indirect information about the flow of dominance relationships to generate hierarchical rank order that makes no assumptions about the linearity of these relationships. It also provides a new metric, known as ‘dominance certainty’, which is a complimentary measure to dominance rank that assesses the degree of ambiguity of rank relationships at the individual, dyadic and group levels. In this focused review, we will (i) describe how Perc measures dominance rank while accounting for both nonlinear hierarchical structure as well as sparsity in data—here we also provide a metric of dominance certainty estimated by Perc, which can be used to compliment the information dominance rank supplies; (ii) summarize a series of studies by our research team reflecting the importance of ‘dominance certainty’ on individual and societal health in large captive rhesus macaque breeding groups; and (iii) provide some concluding remarks and suggestions for future directions for dominance hierarchy research. This article is part of the theme issue ‘The centennial of the pecking order: current state and future prospects for the study of dominance hierarchies’.

Upon a matrix representation of a binary bipartite network, via the permutation invariance, a coupling geometry is computed to approximate the minimum energy macrostate of a network's system. Such a macrostate is supposed to constitute the intrinsic structures of the system, so that the coupling geometry can be taken as the information content of the network or even the nonparametric minimum sufficient statistics of the network data. Based on this, pertinent null and alternative hypotheses, such as nestedness, are to be formulated according to the macrostate. That is, any efficient testing statistic needs to be a function of this coupling geometry. These conceptual architectures and mechanisms are, by and large, still missing in community ecology literature and have rendered misconceptions to be prevalent in this research area. Here, the algorithmically computed coupling geometry is shown to consist of deterministic multiscale block patterns, which are framed by two marginal ultrametric trees on row and column axes, and stochastic uniform randomness within each block found on the finest scale. Functionally, a series of increasingly larger ensembles of matrix mimicries is derived by conforming to the multiscale block configurations. Here, matrix mimicking is meant to be subject to constraints of row and column sums sequences. Based on such a series of ensembles, a profile of distributions becomes a natural device for checking the validity of testing statistics or structural indexes. An energy-based index is used for testing whether network data indeed contain structural geometry. A new version of block-based nestedness index is also proposed. Its validity is checked and compared with the existing ones. A computing paradigm, called Data Mechanics, and its application in one real data network are illustrated throughout the developments and discussions in this paper.

In group-living animals, heterogeneity in individuals’ social connections may mediate the sharing of microbial infectious agents. In this regard, the genetic relatedness of individuals’ commensal gut bacteriumEscherichia colimay be ideal to assess the potential for pathogen transmission through animal social networks. Here we use microbial phylogenetics and population genetics approaches, as well as host social network reconstruction, to assess evidence for the contact-mediated sharing ofE. coliamong three groups of captively housed rhesus macaques (Macaca mulatta), at multiple organizational scales. For each group, behavioral data on grooming, huddling, and aggressive interactions collected for a six-week period were used to reconstruct social network communities via the Data Cloud Geometry (DCG) clustering algorithm. Further, anE. coliisolate was biochemically confirmed and genotypically fingerprinted from fecal swabs collected from each macaque. Population genetics approaches revealed that Group Membership, in comparison to intrinsic attributes like age, sex, and/or matriline membership of individuals, accounted for the highest proportion of variance inE. coligenotypic similarity. Social network approaches revealed that such sharing was evident at the community-level rather than the dyadic level. Specifically, although we found no links between dyadicE. colisimilarity and social contact frequencies, similarity was significantly greater among macaques within the same social network communities compared to those across different communities. Moreover, tests for one of our study-groups confirmed thatE. coliisolated from macaque rectal swabs were more genotypically similar to each other than they were to isolates from environmentally deposited feces. In summary, our results suggest that among frequently interacting, spatially constrained macaques with complex social relationships, microbial sharing via fecal-oral, social contact-mediated routes may depend on both individuals’ direct connections and on secondary network pathways that define community structure. They lend support to the hypothesis that social network communities may act as bottlenecks to contain the spread of infectious agents, thereby encouraging disease control strategies to focus on multiple organizational scales. Future directions includeincreasing microbial sampling effort per individual to better-detect dyadic transmission events, and assessments of the co-evolutionary links between sociality, infectious agent risk, and host immune function.

We demonstrate that gaps and distributional patterns embedded within real-valued measurements are inseparable biological and mechanistic information contents of the system. Such patterns are discovered through data-driven possibly gapped histogram, which further leads to the geometry-based analysis of histogram (ANOHT). Constructing a possibly gapped histogram is a complex problem of statistical mechanics due to the ensemble of candidate histograms being captured by a two-layer Ising model. This construction is also a distinctive problem of Information Theory from the perspective of data compression via uniformity. By defining a Hamiltonian (or energy) as a sum of total coding lengths of boundaries and total decoding errors within bins, this issue of computing the minimum energy macroscopic states is surprisingly resolved by applying the hierarchical clustering algorithm. Thus, a possibly gapped histogram corresponds to a macro-state. And then the first phase of ANOHT is developed for simultaneous comparison of multiple treatments, while the second phase of ANOHT is developed based on classical empirical process theory for a tree-geometry that can check the authenticity of branches of the treatment tree. The well-knownIrisdata are used to illustrate our technical developments. Also, a large baseball pitching dataset and a heavily right-censored divorce data are analysed to showcase the existential gaps and utilities of ANOHT.