On the geometry of exponential random graphs and applications (2404.09680v1)

Abstract: In a seminal paper in 2009, Borcea, Br\"and\'en, and Liggett described the connection between probability distributions and the geometry of their generating polynomials. Namely, they characterized that stable generating polynomials correspond to distributions with the strongest form of negative dependence. This motivates us to investigate other distributions that can have this property, and our focus is on random graph models. In this article, we will lay the groundwork to investigate Markov random graphs, and more generally exponential random graph models (ERGMs), from this geometric perspective. In particular, by determining when their corresponding generating polynomials are either stable and/or Lorentzian. The Lorentzian property was first described in 2020 by Br\"and\'en and Huh and independently by Anari, Oveis-Gharan, and Vinzant where the latter group called it the completely log-concave property. The theory of stable polynomials predates this, and is commonly thought of as the multivariate notion of real-rootedness. Br\"and\'en and Huh proved that stable polynomials are always Lorentzian. Although it is a strong condition, verifying stability is not always feasible. We will characterize when certain classes of Markov random graphs are stable and when they are only Lorentzian. We then shift our attention to applications of these properties to real-world networks.

• The paper explores the geometry of exponential random graph models (ERGMs) by analyzing the stability and Lorentzian property of their generating polynomials to understand negative dependence.
• It links polynomial stability to strong negative dependence (strongly Rayleigh measures) and identifies specific parameter conditions, showing cubic ERGMs are stable only if the triangle parameter is zero.
• The study introduces the Lorentzian property as a more practical test for negative dependence and applies it to real social networks, finding they often do not exhibit strongly Rayleigh characteristics.

Geometry of Exponential Random Graphs and Their Applications

The exploration of probability distributions via the geometric lens of their generating polynomials has, over time, paved the way for insightful revelations in various domains, notably in random graph theory. This paper presents an in-depth investigation into the geometry of exponential random graph models (ERGMs), underpinned by an analysis of their generating polynomials’ stability and the Lorentzian property.

Fundamental Concepts and Framework

The paper builds on foundational work identifying a link between stability in multivariate polynomials and negative dependence in probability distributions. Specifically, stable generating polynomials correspond to distributions characterized by the strongest negative dependence, a property captured under the strongly Rayleigh classification. In the context of random graphs, particularly Markov random graphs, these algebraic properties translate to analyzing complex dependencies within a graph’s structure.

Multiaffine Polynomials and Stability

The property of stability in multiaffine polynomials signifies that no roots exist within the complex upper half-plane. This condition, a multivariate analogue of real-rootedness, when translated into the probabilistic domain, designates the strongly Rayleigh measures. The paper identifies conditions under which Markov random graphs possess stable generating polynomials, thereby conforming to this strongest form of negative dependence.

Notably, the investigation reveals critical conditions for the stability of generating polynomials in cubic Markov random graphs. It is shown that the edge-triangle cubic Markov random graph model is stable if and only if the parameter governing triangles is zero (Prop. 5.4). Extending to general cubic Markov random graphs, parameters describing higher-order interactions further delineate this stability.

Lorentzian Polynomials: A Practical Approach

Given the complexity inherent in establishing polynomial stability, the paper pivots to the Lorentzian property, a less stringent condition still indicative of negative dependence. Lorentzian polynomials, introduced and formalized in recent mathematical advances, are evaluated via their homogenization and satisfying certain eigenvalue criteria of their derivatives. Through an algorithmic approach, this analysis extends to practical applications in modeling social networks.

The transition from stable to Lorentzian random graph models broadens the scope for identifying negative dependence, allowing many more configurations to be evaluated as practically and theoretically significant. It is notably established that the Lorentzian property, unlike strong stability, tolerates certain parameter configurations otherwise restrictive under stability constraints.

Empirical Applications to Social Networks

The research highlights practical applications of its theoretical outcomes by modeling real-world networks. Analysis on historic and contemporary social networks, such as the Medici business network and modern corporate structures, showcases that these networks typically fail to align with strongly Rayleigh measures, evidenced by the lack of strong negative dependencies. This observation is intuitively consistent with social dynamics, where transitive relations (e.g., a friend of a friend becoming a friend) contradict negative dependence paradigms.

Implications and Future Directions

The findings carry robust implications for both theoretical and applied network studies. As real-world social networks frequently elude strong Rayleigh structures, alternative formulations capturing complex associative relations are warranted. In the computational domain, the development and refinement of algorithms that efficiently test for the Lorentzian property further bridge theoretical constructs and empirical modeling, offering versatile tools for future network analyses.

This paper contributes substantially to understanding how geometric properties of polynomials can elucidate the underlying behavior of complex network models, setting the stage for expanded exploration across diverse applications in network theory and beyond.