In this paper, we present a multidimensional fuzzy analysis of the levels and the patterns of poverty and social fragility of migrants' families, in the Italian region of Lombardy, in year 2014. Migrants' poverty emerges as a complex trait, better described as a stratification of nuanced patterns than in black and white terms; Lombard migrants are in fact affected, to different extents, by "a diffused sharing of deprivation facets" and cannot be trivially split into deprived and non-deprived. The paper employs innovative data analysis tools from the Theory of Partially Ordered Sets; compared to mainstream monetary approaches, this leads to more realistic estimates of poverty diffusion and eliminates some well-known biases of standard evaluation procedures, providing strong support to the use of partial order concepts and tools in social evaluation studies.

Assortativity was first introduced by Newman and has been extensively studied and applied to many real world networked systems since then. Assortativity is a graph metrics and describes the tendency of high degree nodes to be directly connected to high degree nodes and low degree nodes to low degree nodes. It can be interpreted as a first order measure of the connection between nodes, i.e. the first autocorrelation of the degree-degree vector. Even though assortativity has been used so extensively, to the author's knowledge, no attempt has been made to extend it theoretically. This is the scope of our paper. We will introduce higher order assortativity by extending the Newman index based on a suitable choice of the matrix driving the connections. Higher order assortativity will be defined for paths, shortest paths, random walks of a given time length, connecting any couple of nodes. The Newman assortativity is achieved for each of these measures when the matrix is the adjacency matrix, or, in other words, the correlation is of order 1. Our higher order assortativity indexes can be used for describing a variety of real networks, help discriminating networks having the same Newman index and may reveal new topological network features.Comment: 24 pages, 16 figure

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A network effect is introduced taking into account competition, cooperation, and mixed-type interaction among agents along a generalized Verhulst-Lotka-Volterra model. It is also argued that the presence of a market capacity undoubtedly enforces a definite limit on the agent's size growth. The state stability of triadic agents, i.e., the most basic network plaquette, is investigated analytically for possible scenarios, through a fixed-point analysis. It is discovered that: (i) market demand is only satisfied for full competition when one agent monopolizes the market; (ii) growth of agent size is encouraged in full cooperation; (iii) collaboration among agents to compete against one single agent may result in the disappearance of this single agent out of the market; and (iv) cooperating with two rivals may become a growth strategy for an intelligent agent.

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