Resilience of networks with community structure behaves as if under an external field

Abstract

Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic- paramagnetic spin system. We also study universality class by defining the analogous critical exponents and, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

Structural resilience of spatial networks with inter-links behaving as an external field

“…Similar to our earlier studies [18,27], we find that the parameter r, governing the fraction of interconnected nodes, has effects analogous to a magnetic field in a spin system, near criticality. This analogy can be seen through the facts that: (i) the non-zero fraction of interconnected nodes destroys the original phase transition point of the single module; (ii) critical exponents (defined below) of values derived from percolation theory can be used to characterize the effect of external field on S(p, r).…”

“…In our earlier study [18], we mainly focused on the critical exponents of phase transition in networks with communities, where these communities were not spatially embedded. However, many real systems are spatially embedded.…”

“…Our model is also different from the interconnected modules model [25], where interconnected nodes are attacked. In our model, the interconnections between different communities are additional connectivity links [26] and randomly chosen nodes are attacked [18]. For a given set of parameters [p, r; L], we carried out 10 000 Monte Carlo realizations and took the average of these results to obtain S(p, r).…”

“…Percolation theory has demonstrated its great potential as a versatile tool for understanding system structural resilience based on both dynamical and structural properties [13,14], and has been applied to many real systems [15][16][17]. Recently, a theoretical framework has been developed to study the structural resilience of communities formed of either Erdős-Rényi (ER) and scale-free networks that have inter-links between them using percolation theory [18]. It has been found that the inter-links affect the percolation phase transition in a manner similar to an external field in a ferromagnetic-paramagnetic spin system.…”

Critical field-exponents for secure message-passing in modular networks

“…This is because even if one of the two modules disintegrates when the dominant color is removed from it, there always exists a finite fraction of its nodes which can communicate securely through external links to the other module. Thus k E >0 removes the transition by making the disconnected phase unreachable [22], just as an external magnetic field of magnitude H does with respect to the disordered phase in the Ising model [23]. In what follows we further support, both analytically and by extensive simulations, this intriguing analogy between spin models and secure message-passing on modular networks.…”