Session G28: Statistical Physics of Networks: Theory and Applications to Complex Systems II

Romualdo Pastor-SatorrasOpinion depolarization transtions in interdependent topics on heterogeneous social networks

The presence of opinion polarization (i.e. two groups holding opposite and possibly extreme opinions in a population) has been extensively observed with  respect to several controversial topics, ranging from religion to political ideology. Modeling the process of reducing opinion polarization among the population, or depolarization, has been the object of much recent work. In most cases, such efforts address the simplest case of one-dimensional opinions with respect to a single topic. However, the process of opinion formation may invest multiple topics at the same time, requiring a proper multidimensional modeling framework for opinion dynamics. Here we present an analytically tractable model of opinion dynamics in a space of two interdependent topics, the so-called "Social Compass Model" (SCM). In the SCM, opinions are represented in polar coordinates, where the angle represents the orientation and the radius the conviction of individuals. We postulate a dynamics inspired by the classic Friedkin-Johnsen, in which the orientation of individuals, subject to an initial, preferred orientation and to social influence by peers, experience a depolarization phase transition for a sufficiently large social influence level. By means of a mean field analysis, we observe that the transition of the SCM is continuous for correlated initial opinions, while it has an discontinuous, explosive nature for uncorrelated initial opinions. These results are checked against numerical simulations using as initial opinions real data extracted form the American Nation Election Studies (ANES) surveys. Finally we discuss the effects of an complex networks pattern of contacts on the depolarization transition of the SCM model, and show how an heterogeneous pattern of interactions can decrease the over level of polarization.   

Phase transitions in the Potts model with hidden states: potential application to a shy voter model

A dark (or hidden) state in which a spin does not interact with any other spins contributes to the entropy of an interacting spin system. Here, we analytically demonstrate that the Potts model with hidden states exhibits a rich phase diagram comprising various types of phase transitions and critical points with different natures. Adopting the Ginzburg–Landau formalism, we reveal that when the number of visible states q is less than 2, and the number of hidden states is sufficiently large, the transition from the paramagnetic (disordered) to ferromagnetic (ordered) phase occurs as temperature decreases. The transition can be continuous, discontinuous, hybrid, or two consecutive transitions. There exist several characteristic points, explosive critical point, critical endpoint, and tricritical point, at which two different types of hybrid transitions, and supercritical behaviors occur, respectively. Thus, various of types of phase transitions that appear in nonequilibrium complex systems can be understood within an integrated scheme. Finally, we discuss the potential applications of the hidden Potts model to social opinion formation with shy voters.   

Direct measurements of the effects of complex time delay in a simple scattering network

The Wigner-Smith delay time is a measure of how long an excitation lingers in the vicinity of a scattering potential before leaving through the asymptotic scattering channels.  This delay time was originally defined only for unitary scattering systems, but over the past 10 years there has been increasing interest in defining a complex generalization of time delay (CTD) for non-Hermitian scattering systems [1].  Our earlier work has related the real and imaginary parts of CTD to the zeros and poles in the complex energy plane of the scattering matrix.  We now examine the influence of CTD on the propagation of Gaussian time-domain pulses through a simple scattering network, namely a ring graph [2].  We find that the time delay and carrier frequency shift of the scattered pulses are well described by the real and imaginary parts of CTD, in agreement with theory.

References

[1] Lei Chen, Steven M. Anlage, and Yan V. Fyodorov, “Generalization of Wigner Time Delay to Sub-Unitary Scattering Systems,” Phys. Rev. E 103, L050203 (2021).

[2] Lei Chen and Steven M. Anlage, “Use of Transmission and Reflection Complex Time Delays to Reveal Scattering Matrix Poles and Zeros: Example of the Ring Graph,” Phys. Rev. E 105, 054210 (2022).   

Network science Ising states of matter

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context.

Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D  Ising model at different temperatures, going across the phase transition.

Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the  IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. Importantly, we identified several indicators of the phase transition.

This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.    

Title: Persistence of chimera states and the challenge for synchronization in real-world networksOral

The emergence of order in nature manifests in different phenomena, with synchronization being one of the most representative examples. Understanding the role played by the interactions between the constituting parts of a complex system in synchronization has become a pivotal research question bridging network science and dynamical systems. Particular attention has been paid to the emergence of chimera states, where subsets of synchronized oscillations coexist with asynchronous ones. Such coexistence of coherence and incoherence is a perfect example where order and disorder can persist in a long-lasting regime. Although considerable progress has been made in recent years to understand such coherent and (coexisting) incoherent states, how they manifest in real-world networks remains to be addressed. Based on a pattern formation mechanism, in this paper, we shed light on the role that non-normality, a ubiquitous structural property of real networks, has in the emergence of several diverse dynamical phenomena, e.g., amplitude chimeras or oscillon patterns. Specifically, we demonstrate that the prevalence of source or leader nodes in networks leads to the manifestation of phase chimera states. Throughout the paper, we emphasize that non-normality poses ongoing challenges to global synchronization and is instrumental in the emergence of chimera states.   

Swift Consensus Formation on Scale-Free Networks

The spreading and evolution of opinions, ideas, and beliefs are crucial in grasping the complexities of social influence, information diffusion, and collective decision-making dynamics. We examine the processes that underlie opinion formation, transition, and consensus-building by investigating the short-time dynamics of the majority-vote evolution on scale-free networks. We employ short-time Monte Carlo simulations to examine the rapid consensus shift, measured using an order parameter from an almost entirely disordered state. Our analysis reveals the presence of a measurable initial critical slip regime during which consensus increases quickly to the stationary state. We show that the network density and size influence consensus evolution. This critical slip phenomenon opens up fascinating possibilities for application in decentralized consensus systems, such as consensus acceleration, suppression, and freezing.   

Epistasis in allosteric mechanical networks

Elastic networks of springs can be pruned to create a response between distant pairs of nodes that mimics allosteric behavior in proteins. We create pairs of such responses: one in which the source and target respond in phase with each other (+) and one in which they respond out-of-phase (-). These two networks differ by sets of “mutations” each of which removes or adds a single bond between network nodes. The effect of multiple mutations is epistatic, that is, it is not simply the sum of the effects due to each of the mutations considered separately. We generate ensembles of (+)/(-) network pairs that differ by a fixed number, N, of discrete mutations. Sampling thousands of mutational sequences from this ensemble, we study the dependence on N of the epistatic interactions. For this realistic mechanical system that is abstracted from biological and chemical complexity, we measure the statistics of local and global epistasis up to N=17 and study how epistasis is affected by the network structure (i.e., network coordination). We find that there is a large variation of response between different members of the ensemble.   

D-Mercator: multidimensional hyperbolic embedding of real networks

One of the pillars of the geometric approach to networks has been the development of model-based mapping tools that embed real networks in their latent geometry. In particular, the tool Mercator embeds networks into the hyperbolic plane. However, some real networks are better described by the multidimensional formulation of the underlying geometric model. Here, we introduce D-Mercator, a model-based embedding method that produces multidimensional maps of real networks into the (D+1)-hyperbolic space, where the similarity subspace is represented as a D-sphere. We used D-Mercator to produce multidimensional hyperbolic maps of real networks and estimated their intrinsic dimensionality in terms of navigability and community structure. Multidimensional representations of real networks are instrumental in the identification of factors that determine connectivity and in elucidating fundamental issues that hinge on dimensionality, such as the presence of universality in critical behavior.   

Characterizing universal intrinsic mesoscale heterogeneity in the structure and fluctuation correlations of equilibrium amorphous solids

The amorphous solid state (formed, e.g., through the sufficient random crosslinking of macromolecules) is an intrinsically heterogeneous state of matter. It is, therefore, appropriately characterized via various statistical distributions. These distributions are encoded in the order parameter for the amorphous solid state and its fluctuations. For example, it has long been known that the order parameter itself encodes both the fraction of localized particles and the distribution of equilibrium localization lengths [1], and that its fluctuations encode aspects of elastic heterogeneity [2]. Both cases have been addressed using a replica Landau-Wilson description of amorphous solidification [3] and, in the latter case, the Goldstone branch of low-energy fluctuations dictated by the pattern of spontaneous symmetry breaking [2]. We extend this Landau-Wilson-Goldstone approach to obtain a universal statistical characterization of the intrinsic mesoscale structure and fluctuations in the amorphous solid state. This characterization takes the form of a joint distribution governing the statistics of the localization lengths of pairs of particles at fixed mean spatial separation, together with the correlation characteristics of their equilibrium fluctuations. We focus attention on longer-scale particle separations, where the effects of the Goldstone fluctuations are expected to dominate.

[1] H. Castillo, P. M. Goldbart and A. Zippelius, Europhys. Lett. 28, 519-524 (1994).

[2] X. Mao, P.M. Goldbart, X. Xing and A. Zippelius, Europhys. Lett. 80, 26004 (2007).

[3] W. Peng, H.E. Castillo, P.M. Goldbart and A. Zippelius, Phys. Rev. B 57, 839 (1998).   

Spatial symmetry breaking in optimal networks

Despite its importance for practical applications, not much is known about the optimal shape of a network that connects in an efficient way a set of points. This problem can be formulated in terms of multiplex networks with a fast layer embedded in a slow one. To connect a pair of points, we can then use either the fast or slow layer, or both, with a switching cost when going from one layer to another. We consider the simpler problem of a distribution of points in the plane and ask for the fast layer network of a given length that minimizes the average time to reach a central node. We discuss the 1d case analytically and the 2d case numerically and show the existence of transitions when we vary the network length, the switching cost, and the relative speed of the two layers. Surprisingly, we see a transition characterized by a spatial symmetry breaking indicating that it is sometimes better to avoid serving a whole area in order to save on switching costs, at the expense of using more the slow layer. Moreover, we find that similar transitions are also observed in heterogeneous real-world population densities in cities. Our results underscore the importance of considering switching costs while studying the optimal subway structures, as small variations of the cost can lead to strikingly dissimilar optimal structures. Finally, we discuss real-world subways and their efficiency for Toronto, Boston, and Atlanta. We found that real subways are farther away from the optimal shapes as the car traffic congestion increases.   

Social interaction radius effects on three-state consensus resilience

The underlying mechanisms of human group behavior drive several aspects of social relations, organization, and dynamics, often exhibiting emergent properties that result from joint interactions. This work investigates collective decision-making processes via a three-state opinion formation model within the continuous network framework. We randomly place individuals as nodes on a unit square area with links representing social interactions within an influence radius, which we relate to the average connectivity 〈k〉 of the network. We implement a majority-vote dynamics in which individuals may assume one of three distinct opinions regarding a social subject under the influence of a social anxiety level q. We find a consensus-dissensus continuous transition responsive to the individual's social interaction radius. Using Monte Carlo simulations and finite-size scaling analysis, we obtain the order-disorder phase diagram q versus〈k〉and the critical exponents. Our findings suggest that larger communities exhibit greater resilience against the disruptive impact of disorder caused by social anxiety.   

Analyzing Local Information in Trainable Materials with Internal Prestress

We model amorphous materials via disordered elastic network models to understand the complex mechanics of non-crystalline systems and their use as platforms for training in function. A useful feature of disordered materials is that local stresses of each bond can be stored as information which the material can use to tune itself for desired mechanical properties. This behavior has only been studied in the case when some realistic features of materials, such as internal stresses, are neglected. We find that the generalization to include internal stresses obscures the relevant local stress information needed for self-tuning. As currently practiced, tuning requires knowing the forces along each bond or strut – that is how much each bond is stretched or compressed. When there is internal stress, however, orientational motion of the bonds also becomes important. Therefore, information about only bond-lengths becomes less useful and information about perpendicular motion becomes important as pre-stress is added to the system. We show how both modes of deformation contribute to the mechanics and training algorithms.