Random 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.

Network data containing attributes of both vertices and vertex pairs can be analysed together using multivariate entropies. This general multivariate technique is here illustrated with data previously analysed by Lazega and others. We focus here on the role of triad count statistics for exploring the dependency structure of the network attributes.

We recently introduced a new way of using statistical entropies to capture interdependencies among vertex and edge variables in multivariate networks. These entropies are used to systematically check for tendencies in the multidimensional variable set concerning redundancies, functional relationships, independencies and conditional independencies among different variable combinations. An important use of this technique is to apply it for selection of good summary measures of network structure. For instance, there are many alternative network statistics available for measuring centrality, and it is not always easy to decide which one is appropriate for a current application. In this presentation, we consider different centrality statistics among the variables in the analysis. By using univariate and multivariate entropies, we aim to find the centrality measure that is most relevant for the network property of interest. Throughout this presentation, we use John Padgett’s extended Florentine network data consisting of 87 families where the vertices as well as the edges have numerical or qualitative attributes defined on them. We create edge and vertex variables that capture network information via the most common centrality measures. The dependence structure of the variables is then explored by entropy analysis and it is determined which centrality measure is most appropriate for representing political, social or economic influence among the Florentine families. Further, we demonstrate how divergence measures can be used to indicate and test structural tendencies with respect to centrality in this network.

Multigraphs with numerical or qualitative attributes defined on vertices and edges can benefit from systematic methods based on multivariate entropies for describing and analysing the interdependencies that are present between vertex and edge attributes. This is here illustrated by application of these tools to a subset of data on the social relations among Renaissance Florentine families collected by John Padgett. Using multivariate entropies we show how it is possible to systematically check for tendencies in data that can be described as independencies or conditional independencies, or as dependencies allowing certain combinations of variables to predict other variables. We also show how different structural models can be tested by divergence measures obtained from the multivariate entropies.

Estimation of global network properties by using local aggregated data.   We consider networks that can be modeled as random multigraphs and indicate various applications and appearances of multigraphs. Global properties are measured by entropy and complexity based on the edge multiplicities. Sample data give partial information about the network, and we consider a special kind of local information given by aggregation of vertices and counting of edges within and between blocks of vertices. Shafie (2012) gives a comprehensive analysis and comparison of random multigraphs of different kinds, and Frank and Shafie (2013) present special results about complexity and data aggregation in multigraphs.  

This article describes families of finite multigraphs with labeled or unlabeled edges and vertices. It shows how size and complexity vary for different types of equivalence classes of graphs defined by ignoring only edge labels or ignoring both edge and vertex labels. Complexity is quantified by the distribution of edge multiplicities, and different complexity measures are discussed. Basic occupancy models for multigraphs are used to illustrate different graph distributions on isomorphism and complexity. The loss of information caused by ignoring edge and vertex labels is quantified by entropy and joint information that provide tools for studying properties of and relations between different graph families.