Goodness of fit tests for random multigraph models

Goodness of fit tests for two probabilistic multigraph models are presented. The first model is random stub matching given fixed degrees (RSM) so that edge assignments to vertex pair sites are dependent, and the second is independent edge assignments (IEA) according to a common probability distribution. Tests are performed using goodness of fit measures between the edge multiplicity sequence of an observed multigraph, and the expected one according to a simple or composite hypothesis. Test statistics of Pearson type and of likelihood ratio type are used, and the expected values of the Pearson statistic under the different models are derived. Test performances based on simulations indicate that even for small number of edges, the null distributions of both statistics are well approximated by their asymptotic χ2-distribution. The non-null distributions of the test statistics can be well approximated by proposed adjusted χ2-distributions used for power approximations. The influence of RSM on both test statistics is substantial for small number of edges and implies a shift of their distributions towards smaller values compared to what holds true for the null distributions under IEA. Two applications on social networks are included to illustrate how the tests can guide in the analysis of social structure

Random multigraphs and aggregated triads with fixed degrees

AbstractRandom multigraphs with fixed degrees are obtained by the configuration model or by so called random stub matching. New combinatorial results are given for the global probability distribution of edge multiplicities and its marginal local distributions of loops and edges. The number of multigraphs on triads is determined for arbitrary degrees, and aggregated triads are shown to be useful for analyzing regular and almost regular multigraphs. Relationships between entropy and complexity are given and numerically illustrated for multigraphs with different number of vertices and specified average and variance for the degrees.</jats:p

Multiplexity analysis of networks using multigraph representations

From Springer Nature via Jisc Publications RouterHistory: accepted 2021-08-30, registration 2021-09-02, pub-electronic 2021-09-30, online 2021-09-30, pub-print 2021-12Publication status: PublishedAbstract: Multivariate networks comprising several compositional and structural variables can be represented as multigraphs by various forms of aggregations based on vertex attributes. We propose a framework to perform exploratory and confirmatory multiplexity analysis of aggregated multigraphs in order to find relevant associations between vertex and edge attributes. The exploration is performed by comparing frequencies of the different edges within and between aggregated vertex categories, while the confirmatory analysis is performed using derived complexity or multiplexity statistics under different random multigraph models. These statistics are defined by the distribution of edge multiplicities and provide information on the covariation and dependencies of different edges given vertex attributes. The presented approach highlights the need to further analyse and model structural dependencies with respect to edge entrainment. We illustrate the approach by applying it on a well known multivariate network dataset which has previously been analysed in the context of multiplexity

Trace-elemental and multi-isotopic (Sr-Nd-Pb) discrimination of jade in the circum-Caribbean: Implications for pre-colonial inter-island exchange networks

This research received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No 319209 (ERC-Synergy NEXUS 1492) and the European Union's Horizon 2020 research and innovation programme under grant agreement No 654208 (Europlanet 2020 RI) . We are grateful to the Museo del Hombre Dominicano for providing the Playa Grande samples. Thanks to Richard Smeets, Bas van der Wagt, Kirsten van Zuilen, Bouke Lacet, Eva Kelderman and Quinty Boosten for analytical assistance.Dense and strong, hydrothermal-metasomatic jadeitite and jadeite-omphacite rocks were used as tools and adornments throughout the wider Caribbean since initial inhabitation. Regionally, rich sources of jadeitite and jadeite-omphacite jade are known only in Guatemala (north and south of the Motagua Fault Zone), eastern Cuba and the northern Dominican Republic, establishing that humans transported jadeitic material over vast distances. This study validates that geochemical fingerprinting is a viable provenance method for Caribbean pre-colonial jadeitic lithologies. An assemblage of 101 source rocks has been characterised for trace element and combined Sr-Nd-Pb isotope compositions. Four statistical approaches (Principal Component Analysis, t-Distributed Stochastic Neighbour Embedding, Decision Tree, and Multiclass Regression) were assessed, employing sourcedistinct trace element ratios. A multiclass regression technique based on trace element ratios of immobile high field strength, light to medium rare earth and fluid-mobile, large-ion-lithophile elements is shown to be most effective in discriminating the four source regions. Ninety-one % of the Guatemalan samples can be discriminated from the Dominican and Cuban sources using La/Th, Zr/Hf and Y/Th ratios. Jadeitic rocks cropping out in the Dominican Republic can be distinguished from Cuban jades employing Er/Yb, Nb/Ta and Ba/Rb ratios with 71% certainty. Furthermore, the two Guatemala sources, north and south of the Motagua Fault Zone, can be discriminated by using (among others) Zr/Hf, Ta/Th, La/Sm and Dy/Y ratios with an 89% success rate. This raises the possibility of determining, in detail, former trading and mobility networks between different islands and the Meso- and Central American mainland within the Greater Caribbean. The provenance technique was applied to 19 pre-colonial jade celts excavated from the Late Ceramic Age Playa Grande archaeological site in the northern Dominican Republic. Three artefacts are discriminated as derived from the Guatemalan source, indicating that, despite a source of jade within 25 km, material was traded from Guatemala. The presence of Guatemalan jade in the Playa Grande lithic assemblage provides further evidence of large scale (>3000 km), regional trading and indigenous knowledge transfer networks.European Research Council (ERC) 319209European Union's Horizon 2020 research and innovation programme 65420

A Multigraph Approach to Social Network Analysis.

Multigraphs are graphs where multiple edges and edge loops are permitted. The main purpose of this article is to show the versatility of a multigraph approach when analysing social networks. Multigraph data structures are described and it is exemplified how they naturally occur in many contexts but also how they can be constructed by different kinds of aggregation in graphs. Special attention is given to a random multigraph model based on independent edge assignments to sites of vertex pairs and some useful measures of the local and global structure under this model are presented. Further, it is shown how some general measures of simplicity and complexity of multigraphs are easily handled under the present model.publishe

Random Multigraphs : Complexity Measures, Probability Models and Statistical Inference

This thesis is concerned with multigraphs and their complexity which is defined and quantified by the distribution of edge multiplicities. Two random multigraph models are considered.  The first model is random stub matching (RSM) where the edges are formed by randomly coupling pairs of stubs according to a fixed stub multiplicity sequence. The second model is obtained by independent edge assignments (IEA) according to a common probability distribution over the edge sites. Two different methods for obtaining an approximate IEA model from an RSM model are also presented. In Paper I, multigraphs are analyzed with respect to structure and complexity by using entropy and joint information. The main results include formulae for numbers of graphs of different kinds and their complexity. The local and global structure of multigraphs under RSM are analyzed in Paper II. The distribution of multigraphs under RSM is shown to depend on a single complexity statistic. The distributions under RSM and IEA are used for calculations of moments and entropies, and for comparisons by information divergence. The main results include new formulae for local edge probabilities and probability approximation for simplicity of an RSM multigraph. In Paper III, statistical tests of a simple or composite IEA hypothesis are performed using goodness-of-fit measures. The results indicate that even for very small number of edges, the null distributions of the test statistics under IEA have distributions that are  well approximated by their asymptotic χ2-distributions. Paper IV contains the multigraph algorithms that are used for numerical calculations in Papers I-III

Random Multigraphs : Complexity Measures, Probability Models and Statistical Inference

This thesis is concerned with multigraphs and their complexity which is defined and quantified by the distribution of edge multiplicities. Two random multigraph models are considered.  The first model is random stub matching (RSM) where the edges are formed by randomly coupling pairs of stubs according to a fixed stub multiplicity sequence. The second model is obtained by independent edge assignments (IEA) according to a common probability distribution over the edge sites. Two different methods for obtaining an approximate IEA model from an RSM model are also presented. In Paper I, multigraphs are analyzed with respect to structure and complexity by using entropy and joint information. The main results include formulae for numbers of graphs of different kinds and their complexity. The local and global structure of multigraphs under RSM are analyzed in Paper II. The distribution of multigraphs under RSM is shown to depend on a single complexity statistic. The distributions under RSM and IEA are used for calculations of moments and entropies, and for comparisons by information divergence. The main results include new formulae for local edge probabilities and probability approximation for simplicity of an RSM multigraph. In Paper III, statistical tests of a simple or composite IEA hypothesis are performed using goodness-of-fit measures. The results indicate that even for very small number of edges, the null distributions of the test statistics under IEA have distributions that are  well approximated by their asymptotic χ2-distributions. Paper IV contains the multigraph algorithms that are used for numerical calculations in Papers I-III