Ising model with variable spin/agent strengths

We introduce varying spin strengths to the Ising model, a central pillar of statistical physics. With inhomogeneous physical systems in mind, but also anticipating interdisciplinary applications, we present the model on network structures of varying degrees of complexity. We solve it for the generic case of power-law spin strength and find that, with a self-averaging free energy, the model has a rich phase diagram with new universality classes. Indeed, the degree of complexity added by variable spins is on a par to that added by endowing simple networks with increasingly realistic geometries. It is suitable for modeling emergent phenomena in many-body systems in contexts where non-identicality of spins or agents plays an essential role and for exporting statistical physics concepts beyond physics.

I t is difficult to overestimate the importance of the Ising model (IM) in physics and beyond. It was invented 100 years ago by W. Lenz (1) and solved in one dimension with nearest-neighbor interactions by E. Ising (2,3). It draws on works of R. Kirwan and W. Weber (4,5) who proposed that vanishing magnetization of macroscopic para-and ferromagnetic bodies originates from random disordering of identical elementary magnets (spins) at microscopic levels. While their ideas also explained magnetic saturation in strong external fields, they failed to explain gradual response to weak ones. E.A. Ewing (6) introduced interactions to mend this shortcoming and P. Weiss (7) used the idea in the mean-field approximation. The IM itself sits between the Kirwan-Weber non-interacting picture and the Curie-Weiss maximally interacting one and manifests spontaneous spin alignment as well. Yet it is its simplicity, coupled with the phenomenon of universality, that lies behind its applicability to many real-world many-body phenomena in physics and beyond (8).
More elaborate models describe specific physical phenomena with greater precision, using greater levels of sophistication either in spin variables themselves or interactions among them. For the former case, the simple polarity, wherein spins have two possible states, is maintained in the continuous spin Ising model (9,10). This is relaxed in the Potts model while maintaining spin discreetness (11,12) and the m-vector model goes further by introducing vector variables σi (13,14). Besides loosening Ising constraints on spins, they can be relaxed for interactions as well and simple extensions include nextnearest neighbor interactions, equivalent neighbours (15) or probabilistic long-range interactions (16).
These variants deliver new universality classes, and, while associated models have been successful in their respective contexts, they rest on the same concept of interacting entities being identical. But not all particles are identical -they may have different inhomogeneities in internal degrees of freedom manifestating, e.g., different magnetic moments. None of the above models deal with this feature.
The IM brings physics beyond its traditional realms too. E.g., it models phenomena as diverse as cancer cells' response to chemotherapy (17); yield patterns in trees (18); and advertising in duopoly markets (19). More recently, it has been used to model spread of the Covid-19 virus (20) which was accentuated by the slowness of authorities to act on prompt scientific advice (21).
There is a myriad of examples in which interacting entities are not identical spins and the term "agent" was introduced to reflect this. Ref. (22) recounts applications to social and political behavior where the agents are human and Ref. (23) discusses the rise of interdisciplinarity in physics. Still, a protest often encountered when exporting statistical concepts is "people are not atoms" (24). Assurances such as "the law of big numbers allows the application of statistical physics methods" (24) are often not understood, welcomed or accepted. Addressing these issues becomes important if we are to communicate physics to interdisciplinary colleagues or authorities; not all cells, trees or bankers are the same and there are different degrees of contagiousness among infected agents.
Inhomogeneities in physics coupled with the rise of interdisciplinarity and the need to communicate statistical physics concepts beyond physics suggest a new variant to the IM which, besides accounting for particle peculiarities, addresses non-identical agents. We introduce the model on network topologies for reasons applicable to both scenarios. In physics, there are nanosystems with topologies more akin to networks than lattices (25). Varying spin strength in such systems may serve to model polydispersity in elementary magnetic moments. Furthermore, perfect lattice structures are not common beyond physics and never encountered in sociophysics.

Significance Statement
The Ising model is a pillar on which statistical physics, sociophysics and econophysics stand. It relies on interactions between identical spins for critical phenomena to occur. Diverse agents replace identical spins in disciplines beyond physics and lack of diversity in spin models impairs export of fundamental Ising-model concepts to complex and emergent phenomena in other disciplines. We introduce variable agent strengths to the model on networks to compare individuality with interactions. We show that complexity introduced by spinstrength variability is on a par with complexity introduced by network-architecture variability; individual strength matters as much as connectivity. This individuality-interaction interplay delivers rich new physics which can impact on circumstances in physics and society where diversity plays essential roles.
There are no competing interests.

D R A F T
Here, we suggest and exactly solve an IM with variable spins/agents on networks of different degrees of complexity. Competition between individual agent strengths and collective connectivity generates a rich phase diagram suitable for the analysis of new universal emergent phenomena in manyagent systems (26).

Methods and Results
In the presence of a homogeneous field H, the Hamiltonian of the IM reads: For the standard version spins σi take values ±1, and coupling Jij is 1 for the nearest neighbor sites and 0 otherwise. In sociophysics individuals are represented as nodes with binary spins representing different social states. This limitation is precisely because of the duality of spins in the generic IM. It may well be convincing for "for" or "against" options in referendums (22) but not all societal activities are binary and our model introduces gradation for individual node features. We endow the spins σi with "strengths" which vary from site to site through a random variable |σi| ≡ Si drawn from probability distribution function q(Si). We chose a power-law decay: where cµ is a normalization constant, with µ > 2 ensuring finite mean strength S when Smax → ∞.
There are several obvious motivations for choosing a powerlaw form in the first instance. We don't expect to encounter physical phenomena with spin strengths distributed precisely in this manner, but it is an established tradition in physics to take idealised models to investigate the fundamentals of a concept. Indeed Lenz and Ising applied this strategy when introducing the model in the first place. To take examples beyond physics, variable agent strength can be used to represent degrees of contagiousness in pandemics (20) or opinion in social systems.
Because of the interactive nature of the IM, these strengths impact on nodes with which a given node interacts -the greater the value of Si the more contagious or persuasive node i is. This new element is closer to real social networks and more likely to be accepted beyond physics. (No more than 10% of world leaders have mathematical or scientific backgrounds (27).) Thus the introduction of variable spins to an established model opens new avenues to physics, interdisciplinarity and communication of same. As we shall see, these deliver very rich critical behavior and an onset of new accompanying phenomena which are of fundamental interest in their own right.
Since we are exploring new avenues, we consider three graph architectures which permit exact solutions: the complete graph, the Erdös-Rényi graph and scale-free networks. As in the Weiss model, every nodal pair {i, j}, is linked in the first case. The probability of connectedness is also the same for every nodal pair p = pi,j = c < 1 in the second case but it differs in that not every pair is linked. For the third case, the node degree distribution is governed by a power-law decay (28,29): for constant c λ and λ > 2. The adjacency matrix for an annealed random graph is (30)(31)(32): [4] where pij is the edge probability any pair i and j. For N nodes, one assigns a random degree ki to each, taken from the distribution with k = 1 N l k l . The expected value of the node degree is EKi = j pij = ki and its distribution is given by p(K) (31,32). The limiting cases pij = 1 and pij = c recover the complete and Erdös-Rényi random graph respectively.
Boltzmann averaging for the partition function is bond and spin-strength {S} configuration dependent. For annealed networks, averaging over links delivers equilibrium and is applied to the partition function vis.: The Since the spin product in (1) can attain only two values, we use the equality to get for the configuration-dependent partition function: Here, β is the inverse temperature, the trace is taken over all spins and we keep σi = ±1 representing each spin value as σiSi. The coefficients (9) implicitly depend on pij and Si via aij = 1−pij +pij cosh(βJSiSj), bij = pij sinh(βJSiSj). [10] The complete graph has pij = 1, from which cij = 1, dij = −2βSiSj, and averaging over spins gives: [11] having used (2) with Smax → ∞. We obtain for small magnetic fields Region V, Lines 5,6, Point B 0 0 1 2/3 1/2 3 Table 1. Critical indices governing temperature and field dependencies of the specific heat, susceptibility, and order parameter in different parts of the phase diagram of Fig. 1. with and ε = x √ N . In (11) and in all counterpart integral representations below, we omit irrelevant prefactors and numerical coefficients of associated response functions are presented elsewhere (34).
The partition function (11) is independent of {k}, {S}; for the random configuration {k} this is obvious since pij = 1 and for the random spin strength configuration {S} it is due to self-averaging. This is a generic feature of the model as we shall see below. With the asymptotic behavior of Iµ(ε) (13) to hand (34)(35)(36) one evaluates (11) in the thermodynamic limit N → ∞ and obtains for the free energy per spin: Here, m is the order parameter and τ the reduced temperature. A similar free energy describes the critical behavior of the standard IM (σi = ±1) on an annealed scale-free network, with decay exponent λ in that case (32) playing the role of µ in the current one. The system is ordered at any finite temperature when µ ≤ 3 and has a second order phase transition when µ > 3. All universal characteristics of the transition are µ-dependent when 3 < µ < 5 and the µ > 5 region is mean-field like. At µ = 5 logarithmic corrections feature (29)(30)(31)(32)(37)(38)(39), governed by exponents which adhere to the usual scaling relations (40). These and leading exponents are listed in the third row of Table 1 and discussed in further detail in that context. For Erdös-Rényi graphs one substitutes pij = c into (9). This delivers a similar partition function (8) to that of the complete graph up to renormalized interaction so that critical behaviors of both models are essentially equivalent.
For annealed scale-free networks, with pij given by (5), the thermodynamic limit N → ∞ (i.e. with small pij) applied to (9) gives dij ∼ pijβJSiSj. The Stratonovich-Hubbard transformation delivers the trace over spins in (8) and a partition function that has unary dependency on the random variables f (kiSi). It is convenient to pass from the summation over nodes i to summation over random variables ki, Si Considering the variables k and S as continuous and taking the thermodynamic limit N → ∞, we put kmax = Smax → ∞ and wlg choose the lower bounds kmin = Smin = 2 (34). It is straightforward to see that the partition function Z({S}, {k}) is independent of the random variables k and S and is selfaveraging. Indeed, when both distribution functions p(k), q(S) attain power-law forms (2), (3) we obtain [15] and [16] where with ϕ(k, S) = 1 k λ S µ ln cosh kS , [18] and the lower integration bound ε ∼ √ x √ T N tends to zero, when N → ∞.
As in the case of the model on the complete graph (13), the partition function (15) and hence the of the free energy is determined by the asymptotics of the integral (17) as ε → 0. This is governed by the interplay of the decay exponents λ and µ. In particular, for the diagonal case λ = µ we get: with [20] The constants in (19) can be readily evaluated and are presented elsewhere (34). For the non-diagonal case µ = λ, due to the symmetry I λ,µ = I µ,λ it is enough to evaluate the integral for λ > µ. With estimates available from (34), we apply the steepest descent method to get the exact solution for the partition function (15). The results that follow from the analysis of the free energy are summarized in Fig. 1 and Tables 1, 2. Fig. 1 presents the phase diagram of the model in the λ − µ plane. Behavior is controlled by the parameter with the smaller value. In Regions I and II, for which distributions are fat-tailed with λ < 3 or µ < 3, the system remains ordered at any finite temperature T . There, the order parameter decays with T as a power law, m ∼ T λ−2 λ−3 or m ∼ T µ−2 µ−3 for 2 < λ < 3 and 2 < µ < 3, correspondingly. Both asymptotics coincide along Line 1 in the figure.
The decay is exponential m ∼ T e −T along Lines 2 and 3 where λ = 3 or µ = 3 as well as at point A where they coincide. Second order phase transitions occur when both λ, µ > 3. In Region III where 3 < λ < 5 (µ > λ) the critical exponents are λ-dependent and in Region IV where 3 < µ < 5 (µ < λ) they are µ-dependent. ααcγγcβδ When λ = 5 or µ = 5 logarithmic corrections to scaling appear. For example, the order parameter in Lines 4-6 behaves as m ∼ τ β | ln τ | −β . The values of the logarithmic correction exponents in Lines 5 and 6 coincide with those for the IM on a scale-free network (32,(37)(38)(39). The additional richness of the phase diagram is characterised, for example, by new type of logarithmic corrections which emerge in Line 4 where 3 < λ = µ < 5 as well as Point B where λ = µ = 5. All of them obey the scaling relations for logarithmic corrections (40). Critical exponents are summarized in Tables 1 and  2.

Discussion
Thus introduction of variable spin or agent strengths to the IM on networks delivers rich new phase diagrams and universality classes relevant to circumstances wherein interacting spins and agents carry degrees of complexity over and above the binary features mostly considered in physics and often rejected in other disciplines.
The model with power-law decaying random strength distribution (2) on the complete graph has similar critical behavior to the standard IM on a scale-free network with random node degree distribution (3) with decay exponents µ and λ playing equivalent roles. This suggests that the level of complexity introduced by allowing spin strengths to vary is on a par to the level of complexity introduced by allowing network architecture to vary; i.e., individual strength matter as much as connectivity.
When introduced to already rich annealed scale-free network the complexity level is magnified yet more. Besides selfaveraging, it is governed by the concurrence of two parame-ters describing different phenomena arising from inherent interplay of two types of randomness. The full phase diagram of the model (Fig. 1) is symmetric under µ ↔ λ interchange, critical behavior governed by the smaller of the two parameters. Ordering, critical behaviour and logarithmic corrections all feature.
The IM itself taught us that interactions in physics play as important a role as spin properties, for without them we have no cooperative behavior or spontaneous magnetization. We have seen that spin strength and system architecture play similar counterbalancing roles and are tuned by the exponents µ and λ. Extending to sociophysics they capture the duality between strength of individual opinions and societal connectivity. To quote historian Yuval Noah Harari, 'We have a choice; we can go the way of the proletarian surveillance or we can go the way of empowering citizens, giving them good scientific education, giving them reliable information and trusting them to do the right thing" (41). The former is region V of the phase diagram with high values of µ (populism) or λ (weak society). The latter is Regions I and II -node empowerment and high social connectivity. The choice Harari gives us is to which side of the critical divide (Regions III and IV) humanity will move.