Q UALITATIVE P ROPERTIES OF THE R ANDOM -F IELD I SING M ODEL

With the Ising model on a regular lattice and under uniform external loading, all phase transitions take place at 0 [166]. However, when some disorder or irregularity is incorporated into the model by including random components in the external fields, drastic qualitative changes occur in model behaviour [42], [65]. This model, called the random-field Ising model [128], comprising both uniform and random loading components, can demonstrate hysteresis, a phenomenon that occurs in various real systems, e.g., in the magnetisation of materials [98], in plastic materials [72], and in economics [39].

Hysteresis, which has been extensively studied in the random-field Ising model (e.g., [37], [42], [111], [128], [129], [130]), means that the system’s time-reversal symmetry is broken, and that the state of the network then depends on its history. Hysteresis exists even in the Monte Carlo

dy-namics of the Ising model under an oscillating loading, because the time variation of the network average state lags behind that of the oscillating field. This occurs when the relaxation time of the Monte Carlo dynamics of the network average state is slower than the frequency of the oscillat-ing field (see, e.g., [4], [5], [6], [27]). The random-field Isoscillat-ing model has shown hysteresis also on an irregular network topology, and the topological properties of an irregular network affect the hysteresis properties [42], [62], [93], [147]. On regular lattices, also network connectivity affects hysteresis properties [124].

Here hysteresis means that as is adiabatically varied, two separate paths exist for s : one is fol-lowed when initialising the model to state s 1 and as increasing , and the other when ini-tialising the model to state s 1 and decreasing , respectively. A range of external field val-ues exists now, where simultaneously exists two possible coherent metastable network states ( 1 and 1) and an unstable state, from which the network always transfers to either of the two co-herent states (Figure 4.3). Consequently, the JPD of the random-field Ising model becomes bi-modal with the two coherent state expectations corresponding to the two modes of distribution and unstable solutions represented by the improbable states between the two modes.

In the random-field Ising model, hysteresis is stimulated by randomness in node loadings break-ing the exact similarities of nodes. The model was studied in [128] at zero temperature for a regu-lar network topology (cubic lattice) of nodes and with nearest-neighbour interactions. The study of [128] is followed here. Based on Eq. (4.5), the random-field Ising model is obtained simply by replacing the term with a term , where the random-field loading component for each node is now included.

Let us now assume that the random components of the external node loadings are independ-ent and idindepend-entically distributed according to the Gaussian probability distribution with mean 0 and variance   . The hysteresis properties of the random-field Ising model depend on the of the Gaussian distribution. This deviation can be interpreted as the size of the fluctuations, or disor-der, caused by the random field. A critical value exists for , which is directly proportional to

Figure 4.3. Qualitative properties of the random-field Ising model. The average network node state s is given as a function of with three levels of interaction parameter (disorder ). The average state s changes continuously when  ( ) (solid red). Phase transition is continuous with the diverging first derivative when ( ) (dotted black). Phase transition is discontinuous when ( ) (dashed blue).

s

cr

cr

cr

the respective critical value of , which is here denoted by ; ~ [123]. In view of the quali-tative properties of the random-field Ising model, having is equal to choosing at

, because then ; i.e., in this case, node interactions are smaller and transitions smoother. Respectively, having is equal to choosing at , which implies that ; i.e., node interactions are now larger and transitions more abrupt.

The average node state s under a varying loading changes through a series of node ava-lanches, which in magnetism account for the phenomenon called the Barkhausen noise [35], [111], [123], [128], [129]. Accordingly, though changes in the average node state s may appear continuous, they are, in fact, discontinuous with nodes changing states in various-size clusters or domains. In a node, a state change can be triggered by either the external loading or the effect of its neighbouring nodes flipping state. That is, if a node flips its state, e.g., if the external load af-fecting it becomes large enough, it may cause its neighbouring nodes also to flip their states. The latter nodes may further cause their neighbours to flip their states, resulting in an avalanche of state changes [129], [130] (also [158]).

At the limit of infinite internode interaction, the Ising model exhibits first-order, or discontinu-ous, phase transitions, i.e., abrupt jumps from one coherent state into another as the uniform external load changes between positive and negative values. Figure 4.3 shows hysteresis proper-ties for the random-field Ising model with 0, 1, and 1, and with s again given as a function of adiabatically changing uniform external field. Figure 4.3 shows s in each case first from the initial state s 1 on to the final state s 1 under an increasing , and then from the initial state s 1 to the final state s 1 under a decreasing .

The Case with ( )

With , the external loading dominates the interaction term in the random-field Ising model with the node loadings experiencing large fluctuations because of large random com-ponents. Because the node states are rather independent of each other, the nodes undergo state changes nearly independently. Therefore, in transitions all avalanches are quite small.

Hysteresis occurs with s changing smoothly as a function of , as demonstrated schemati-cally in Figure 4.3 by the two solid red curves; the two transition curves are symmetrical with respect to 0. [128]

The Case with ( )

At , hysteresis occurs with continuous phase transitions, and correlations appear at all length scales. Consequently, also avalanches occur at all length scales, their size following a power-law distribution [123]. At this critical point, the disorder is just large enough for the state change of each node, on average, to trigger a state change in one of its neighbours [129].

Because these critical phenomena are universal, similar behaviour is expected to be prevalent and independent of the details of individual node interactions [128]. The universal behaviour of continuous phase transitions can be analysed with mean-field theory by analytical approxi-mate calculations, as demonstrated for the Ising model in Subsection 4.4.3 . The above case is shown in Figure 4.3 with dotted black curves. [128]

The Case with ( )

With , the external loading is similar for all the nodes with only small fluctuations.

The uniform loading must be greatly increased before considerable changes take place in s . When is finally large enough, nearly all nodes change states at the same time, causing an in-finite-size avalanche and potentially a set of smaller avalanches. Thus an abrupt change with a discontinuity in s occurs, that is, a discontinuous phase transition. This case is shown in Figure 4.3 with dashed blue curves. The two paths corresponding to increasing and decreas-ing are again separate at 0, and the system exhibits hysteresis with discontinuous phase transitions. [128]

In conclusion, the main difference in qualitative behaviour between the Ising model and the ran-dom-field Ising model is hysteresis. The qualitative behaviour of the ranran-dom-field Ising model is more relevant than that of the regular Ising model to MTNs for two reasons. First, in MTNs ex-ternal loadings vary between the network nodes, and their topology is irregular. Both facts cause disorder in the network, like random loadings in the random-field Ising model. Hence hysteresis is likely to appear in MTNs. Second, MTNs also tend to behave coherently; consequently, changes may occur rapidly in s , or even discontinuous phase transitions may take place. In an MTN under heavy external network loading, a discontinuous phase transition may mean a sud-den collapse of a finely performing network. Because such an occurrence may be costly and det-rimental to the quality of service, it is extremely important to study it. Furthermore, hysteresis together with discontinuous phase transitions is a particularly tricky phenomenon. In real net-works, it means that after a network has collapsed, the desired coherent network state cannot be regained simply by returning external loads to their pre-phase-transition values.