Complexity

Volume 2018, Article ID 2845031, 11 pages

https://doi.org/10.1155/2018/2845031

## Dilution of Ferromagnets via a Random Graph-Based Strategy

^{1}School of Computing, University of Kent, Chatham Maritime, UK^{2}nChain Ltd., London W1W 8AP, UK^{3}School of Computer Science, University of Hertfordshire, Hatfield AL10 9AB, UK^{4}Department of Data Analysis, Faculty of Psychology and Educational Sciences, University of Ghent, Ghent, Belgium

Correspondence should be addressed to Marco Alberto Javarone; moc.liamg@enoravajocram

Received 21 September 2017; Revised 6 February 2018; Accepted 10 February 2018; Published 15 April 2018

Academic Editor: Lucia Valentina Gambuzza

Copyright © 2018 Marco Alberto Javarone and Daniele Marinazzo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamics and behavior of ferromagnets have a great relevance even beyond the domain of statistical physics. In this work, we propose a Monte Carlo method, based on random graphs, for modeling their dilution. In particular, we focus on ferromagnets with dimension , which can be approximated by the Curie-Weiss model. Since the latter has as graphic counterpart, a complete graph, a dilution can be in this case viewed as a pruning process. Hence, in order to exploit this mapping, the proposed strategy uses a modified version of the Erdős-Renyi graph model. In doing so, we are able both to simulate a continuous dilution and to realize diluted ferromagnets in one step. The proposed strategy is studied by means of numerical simulations, aiming to analyze main properties and equilibria of the resulting diluted ferromagnets. To conclude, we also provide a brief description of further applications of our strategy in the field of complex networks.

#### 1. Introduction

The study of diluted ferromagnets [1–6] dates back to several years ago, following two main paths sometimes overlapping, that is, the statistical mechanics approach to lattices and the graph theory approach to networks [7, 8]. A notable result, coming from their combination, is the modern network theory [9–11]. In particular, the latter extends the classical graph theory to the analysis of networks characterized by nontrivial topologies and containing a big amount of nodes. So, the role of statistical mechanics is to offer methods and strategies for investigating the properties and the dynamics of these “complex networks” [12, 13]. Usually, investigations on ferromagnets are performed using the Ising model [14], mainly because the latter constitutes a simple and powerful tool for studying phase transitions and further applications, also beyond the domain of statistical mechanics (e.g., Data Science [15] and Machine Learning [16, 17]). Despite its simplicity, the Ising model becomes, itself, a very hard problem (not yet solved) when studied in dimensions greater than 3. In those cases, the Curie-Weiss [18, 19] model allows approximating its behavior, with the advantage of being also analytically tractable (i.e., it can be exactly solved for any size of system). As a result, in some conditions, solving the Ising model might require performing numerical simulations using Monte Carlo methods [20]. For instance, the Metropolis algorithm [21] constitutes one of the early, and most adopted, strategies for simulating thermalization processes over a lattice. This latter algorithm is based on the optimization of the Hamiltonian function representing the energy of the system. Notably, the Hamiltonian of the Ising model readswhere the summation is extended to all the nearest neighbors in the lattice (realized with periodic boundary conditions, so actually becoming, in topological terms, a toroid). As a result, the value of Hamiltonian 1 depends on the set , that is, the configuration of spins in the lattice. Accordingly, the two ground states of the system correspond to the spin configurations and . Therefore, considering a lattice with sites and starting with a random configuration , defined as , the Metropolis algorithm leads the system towards a state of equilibrium which, for a temperature , corresponds to one of the two ground states. This algorithm is based on two simple steps: (1)Randomly select a site , and compute the local associated with its spin flip.(2)If , accept the flip; else, accept the flip with probability .

They are repeated until the equilibrium state is reached. We recall that and , appearing in the probability shown in step of the Metropolis algorithm, refer to the Boltzmann constant and to the system temperature, respectively. In addition, the term “local” , used in step , indicates that the difference in energy is computed considering only the site and its nearest neighbors. Thus, in principle, some flips may increase the global energy of the whole system. In general, the process simulated by the Metropolis algorithm takes into account the fact that the ferromagnetic interactions are* quenched*, that is, the thermalization is fast enough to allow considering the interactions as constant. In the opposite case, that is, with nonconstant interactions, we have different scenarios. For instance, a spin system can become glassy by introducing antiferromagnetic interactions (i.e., ) or can undergo a dilution process by removing interactions (i.e., setting ). In this work, we focus on dilution of ferromagnets introducing a strategy, based on the Erdős-Renyi model [22], for modeling this process. It is worth recalling that previous investigations (e.g., [23–26]) highlighted the critical behavior of diluted ferromagnets, including, for example, the ergodicity breaking and the vanishing of a giant component. So, beyond providing a novel method for dilution, we give also a description of some statistical properties of the resulting system, of the dynamical processes living on it, and on potential applications. To this end, the analyses are performed in two different conditions: for introducing the dilution strategy and studying some properties of the ferromagnets, the spin variables (i.e., ) are considered* quenched*, while for studying thermalization processes after a dilution, the* quenched* variables are the interactions . Finally, the proposed strategy and the related analyses are performed by means of numerical simulations. Beyond describing the behavior of our model, we emphasize that the achieved results allow also envisioning potential applications in the area of complex networks. The reminder of the paper is organized as follows: Section 2 introduces the proposed strategy. Section 3 shows results of numerical simulations. Eventually, Section 4 provides a description of the main findings.

#### 2. Modeling Dilution on Ferromagnets

Let us consider ferromagnets of dimension , modeled via the Curie-Weiss (CW hereinafter) model. The latter is composed of sites, with a position and a spin . Here, the interactions are not limited to the nearest neighbors (like in the Ising model) but are extended to all the system; that is, every site interacts with all the others. Accordingly, the Hamiltonian of the CW model readswith , that is, combination of spins . Then, like in the Ising model, the two ground states correspond to the spin configurations and , that is, those that minimize the value of (see (2)). It is worth highlighting that the CW model can be represented as a complete (i.e., fully connected) graph, where each site corresponds to a node and each interaction corresponds to an edge. In addition, the number of interactions is equal to . The mapping from the physical object (i.e., the ferromagnet) to the mathematical entity (i.e., the graph) allows mapping a dilution to a pruning process. However, before presenting the dilution strategy developed with the framework of graph theory, we discuss the application of a more classical method, that is, the previously mentioned Metropolis algorithm.

##### 2.1. Dilution by the Metropolis Algorithm

In principle, the Metropolis algorithm, and similar methods, may be used for modeling the dilution of ferromagnets. Notably, since this algorithm modifies spins from to , and vice versa, according to the energy difference resulting from the spin flipping, an opportune variant—say Metropolis-like—might be used for flipping the interaction variables . In the case of spin flipping the possible values that can take are whereas, in the case of interactions , the latter may take three different values: (i.e., ferromagnetic), (i.e., antiferromagnetic), (i.e., removal). Thus, a Metropolis-like algorithm devised for flipping interactions may, in principle, generate a spin glass [27–29] (flipping from to ) and perform a dilution (flipping from to ). In addition, both processes (i.e., from to and to ) can be combined, modeling the emergence of a diluted spin glass. Hence, focusing on dilution, from now on, we consider only the case . In doing so, starting with a random distribution of spins, a Metropolis-like algorithm (M-L hereinafter) can be defined as follows:(1)Randomly select an interaction between two sites, and compute the local associated with its flip to .(2)If , accept the flip; else, accept the flip with probability .

As in the thermalization processes, the M-L strategy depends on the Hamiltonian of the system. Furthermore, one might consider also flipping of from to , that is, modeling a kind of (edge) repopulation. However, since the addition of interactions between inverse spins would increase the Hamiltonian, the actual realization of flipping would be quite rare.

##### 2.2. Dilution via a Random Graphs-Based Strategy

As mentioned above, modern network theory and its methods are spreading in many other scientific fields. Then, it is interesting to see whether and how network theory can be useful for facing the problem of diluting ferromagnets. Notably, our work, beyond to introduce a further method for this task, allows also proving the effectiveness of network theory in a further application. As a result, the proposed model has a double valence; that is, the process of ferromagnet dilution can be analyzed by the tools developed in network theory and allows envisioning new applications. For instance, as shown later, the subfield of community analysis can benefit from the proposed strategy. Given this premise, we can now proceed with a brief description of ferromagnets with the formal language of graph theory. In general, a graph is an entity composed of two sets: (i.e., nodes) and (i.e., edges). As above reported, the maximum number of edges (i.e., ) depends on . In addition, the edges can be provided with some properties, as a direction, a weight, and so on, in order to represent specific characteristics of the object they refer to (a ferromagnet or a real network as a social network [30], a biological network [31], an immune network [32, 33], a financial network [34], and many others). In the proposed model, edges have no particular properties (i.e., they are indirect and unweighted), and the graph is implemented via the E-R model. The latter is realized by defining a number of nodes and a parameter , which represents the probability of each edge to exist. Thus, the expected number of edges in an E-R graph is equal to . Notably, decreasing (increasing) entails removing (add) edges in the graph. The algorithm for generating an E-R graph is very simple:(1)Define the number of of nodes and the probability .(2)Draw each edge with probability .

In particular, in step (2), all possible edges are considered. Therefore, an E-R graph generated with contains exactly edges (i.e., it is complete) and constitutes the graphical counterpart of the CW model previously described. The tuning of the parameter allows representing ferromagnets with different amounts of interactions. Thus graphs generated with (i.e., having ) represent diluted ferromagnets. This last observation constitutes the base of our model, that is, an E-R-like (ER-L hereinafter) model devised for dilution processes. For the sake of clarity, now we provide a pictorial representation for highlighting their main differences between two mentioned strategies, that is, M-L and ER-L—see Figure 1. So, a quick glance to the pictorial representation allows observing what follows: (i) the ER-L strategy starts with nonconnected nodes and then populates the graphs with new edges, while the M-L strategy starts with a complete graph and then removes the edges; (ii) the ER-L strategy allows obtaining more configurations than the M-L strategy, the latter being “Hamiltonian-dependent.” In particular, once the Hamiltonian has been optimized, further actions (i.e., edge removal) have very low probability. On the other hand, the ER-L strategy, being (partially) “Hamiltonian-independent,” allows the realization of ferromagnets with higher degree of dilution. For this reason, M-L is closer to a physical realization of a dilution than ER-L.