Assortative mixing in networks

A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. We define a measure of assortative mixing for networks and use it to show that social networks are often assortatively mixed, but that technological and biological networks tend to be disassortative. We propose a model of an assortative network, which we study both analytically and numerically. Within the framework of this model we find that assortative networks tend to percolate more easily than their disassortative counterparts and that they are also more robust to vertex removal.Comment: 5 pages, 1 table, 1 figur

Percolation transition in networks with degree-degree correlation

We introduce an exponential random graph model for networks with a fixed degree distribution and with a tunable degree-degree correlation. We then investigate the nature of a percolation transition in the correlated network with the Poisson degree distribution. It is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class of the uncorrelated network. Positive correlation turns out to be relevant. The percolation transition in the assortative network is characterized by the non-diverging mean size of finite clusters and power-law scalings of the density of the largest cluster and the cluster size distribution in the non-percolating phase as well as at the critical point. Our results suggest that the unusual type percolation transition in the growing network models reported recently may be inherited from the assortative degree-degree correlation.Comment: 7 pages, 11 figur

Assortative mixing in close-packed spatial networks

Background In recent years, there is aroused interest in expressing complex systems as networks of interacting nodes. Using descriptors from graph theory, it has been possible to classify many diverse systems derived from social and physical sciences alike. In particular, folded proteins as examples of self-assembled complex molecules have also been investigated intensely using these tools. However, we need to develop additional measures to classify different systems, in order to dissect the underlying hierarchy. Methodology and Principal Findings In this study, a general analytical relation for the dependence of nearest neighbor degree correlations on degree is derived. Dependence of local clustering on degree is shown to be the sole determining factor of assortative versus disassortative mixing in networks. The characteristics of networks constructed from spatial atomic/molecular systems exemplified by self-organized residue networks built from folded protein structures and block copolymers, atomic clusters and well-compressed polymeric melts are studied. Distributions of statistical properties of the networks are presented. For these densely-packed systems, assortative mixing in the network construction is found to apply, and conditions are derived for a simple linear dependence. Conclusions Our analyses (i) reveal patterns that are common to close-packed clusters of atoms/molecules, (ii) identify the type of surface effects prominent in different close-packed systems, and (iii) associate fingerprints that may be used to classify networks with varying types of correlations

Why social networks are different from other types of networks

We argue that social networks differ from most other types of networks, including technological and biological networks, in two important ways. First, they have non-trivial clustering or network transitivity, and second, they show positive correlations, also called assortative mixing, between the degrees of adjacent vertices. Social networks are often divided into groups or communities, and it has recently been suggested that this division could account for the observed clustering. We demonstrate that group structure in networks can also account for degree correlations. We show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real-world networks.Comment: 9 pages, 2 figure

Effect of degree correlations above the first shell on the percolation transition

The use of degree-degree correlations to model realistic networks which are characterized by their Pearson's coefficient, has become widespread. However the effect on how different correlation algorithms produce different results on processes on top of them, has not yet been discussed. In this letter, using different correlation algorithms to generate assortative networks, we show that for very assortative networks the behavior of the main observables in percolation processes depends on the algorithm used to build the network. The different alghoritms used here introduce different inner structures that are missed in Pearson's coefficient. We explain the different behaviors through a generalization of Pearson's coefficient that allows to study the correlations at chemical distances l from a root node. We apply our findings to real networks.Comment: In press EP

Resilience of the Interbank Network to Shocks and Optimal Bail-Out Strategy: Advantages of "Tiered" Banking Systems

This paper studies systemic risk and the scale of systemic breakdown in the frequently observed tiered banking system. The banking network is constructed from a number of banks which are linked by interbank exposures with a certain predefined probability. In this framework, the tiered structure is represented either by a network with negative correlation in connectivity of neighboring banks, or alternatively, by a network with a scale-free distribution of connectivity. The main findings of the paper highlight the advantages of tiering in terms of both the resilience of the banking network to systemic shocks and the extent of necessary government intervention should a crisis evolve.