A Generalized Configuration Model with Degree Correlations and Its Percolation Analysis

TL;DR

This paper introduces a generalized configuration model that incorporates degree correlations and allows for the creation of assortative or disassortative networks, with a detailed percolation analysis and validation through simulations.

Contribution

It extends the classical configuration model by partitioning stubs into blocks and using a permutation function to control degree correlations, enabling the construction of diverse network types.

Findings

Derived a closed-form joint degree distribution for the model

Showed linear relationship between Pearson correlation and parameter q

Validated results with extensive computer simulations

Abstract

In this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into b blocks of equal sizes and choose a permutation function h for these blocks. In each block, we randomly designate a number proportional to q of stubs as type 1 stubs, where q is a parameter in the range [0; 1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block i. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block h(i). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in q for any fixed b. By properly choosing h, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. We verify our results by extensive computer simulations.