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Degree is a fundamental property of nodes in networks. However, computing the degree distribution of nodes in probabilistic networks is an expensive task for large networks. To overcome this difficulty, expected degree is commonly utilized in the literature. However, in this article, we show that in some cases expected degree does not allow us to evaluate the probability of two nodes having the same degree or one node having higher degree than another. This suggests that expected degree in probabilistic networks does not completely play the same role as degree in deterministic networks. For each node, we define a reference node with the same expected degree but the least possible variance, corresponding to the least uncertain degree distribution. Then, we show how the probability of a node’s degree being higher or equal to the degree of its reference node can be approximated by using variance and skewness of the degree distribution in addition to expected degree. Experimental results on a real dataset show that our approximation functions produce accurate probability estimations in linear computational complexity, while computing exact probabilities is polynomial with order of 3.

Analyzing ego networks to investigate local properties and behaviors of individuals is a fundamental task in social network research. In this paper we show that there is not a unique way of defining ego networks when the existence of edges is uncertain, since there are two different ways of defining the neighborhood of a node in such network models. Therefore, we introduce two definitions of probabilistic ego networks, called V-Alters-Ego and F-Alters-Ego, both rooted in the literature. Following that, we investigate three fundamental measures (degree, betweenness and closeness) for each definition. We also propose a method to approximate betweenness of an ego node among the neighbors which are connected via shortest paths with length 2. We show that this approximation method is faster to compute and it has high correlation with ego betweenness under the V-Alters-Ego definition in many datasets. Therefore, it can be a reasonable alternative to represent the extent to which a node plays the role of an intermediate node among its neighbors.

Sparsification is the process of decreasing the number of edges in a network while one or more topological properties are preserved. For probabilistic networks, sparsification has only been studied to preserve the expected degree of the nodes. In this work we introduce a sparsification method to preserve ego betweenness. Moreover, we study the effect of backboning and density on the resulting sparsified networks. Our experimental results show that the sparsification of high density networks can be used to efficiently and accurately estimate measures from the original network, with the choice of backboning algorithm only partially affecting the result.