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Hidden geometry and dynamics of complex networks: Spin reversal in nanoassemblies with pairwise and triangle-based interactions

Recent studies of networks representing complex systems from the brain to social graphs have revealed their higher-order architecture, which can be described by aggregates of simplexes (triangles, tetrahedrons, and higher cliques). Current research aims at quantifying these hidden geometries by the algebraic topology methods and deep graph theory and understanding the dynamic processes on simplicial complexes. Here, we use the recently introduced model for geometrical self-assembly of cliques to grow nano-networks of triangles and study the filed-driven spin reversal processes on them. With the antiferromagnetic interactions between the Ising spins attached to the nodes, this assembly ideally supports the geometric frustration, which is recognized as the origin of some new phenomena in condensed matter physics. In the dynamical model, a gradual switching from the pairwise to triangle-based interactions is controlled by a parameter. Thus, the spin frustration effects on each triangle give way to the mesoscopic ordering conditioned by a complex arrangement of triangles. We show how the balance between these interactions changes the shape of the hysteresis loop. Meanwhile, the fluctuations in the accompanying Barkhausen noise exhibit robust indicators of self-organized criticality, which is induced by the network geometry alone without any magnetic disorder.

Current research of complex networks with hidden structures highlights three major directions. Specifically, • exploring empirical data to discover the higher-order structures and the related dynamical patterns in them (Q-analysis, persistent homology, topological information data analysis) [11,20,32]; • theoretical investigations of different stochastic processes taking part on such structures to reveal the impact of higher-order structures and new phenomena that they may induce [31,33,34]; • modelling the assembly of networks with simplicial complexes to control the appearance of specific struc-tural characteristics given altered function or dynamic processes that they can support. For example, preformatted groups, described as cliques (full graphs), are attached to a growing structure during the geometrically constrained process of self-assembly [1,35,36]. See the illustration in Fig. 1.
In materials science, the impact of the complex architecture of nano-materials to their functionality has been a subject of experimental investigations, which are often driven by the specific requirements for applications. A prominent example is the case of antiferromagnetic spintronics [37]. In this context, recent focus is on materials with complex architecture, like tetraborides [38] and artificial frustrated systems [39]. In such systems, the geometric constraints combined with the sign of the interactions prevent simultaneous minimisation of the global energy and each spin pair, leading to the spin frustration [40]. The geometric frustration effects have been shown to enable an unusual magnetic order and new dynamical phenomena in condensed matter physics [41][42][43][44] as well as in the Ising model applications [45,46]. For example, one of the prominent features of spin frustration is the appearance of the fractional magnetisation plateaus in the hysteresis loop [38,47,48]. In [34], the nanonetworks of self-assembled mono-disperse cliques were studied. It was shown that, when endowed with antiferromagnetic bonds between pairs of spins on triangle faces, these assemblies provide ideal conditions for geometric frustration. The size of building simplexes directly affects the shape of the hysteresis loop, and the fractional magnetisation plateaus.
In this work, we study the spin-reversal dynamics on selfassembled nanonetworks with simplicial complexes in the presence of simplex-based interactions. More precisely, the network is grown by geometrical self-assembly of triangles as pre-formatted groups of nanoparticles, based on the self-II THE SELF-ASSEMBLY OF SIMPLEXES assembly model introduced in [1], and the Ising spin variable is attached to each node (nanoparticle). With antiferromagnetic interactions among pairs of neighbouring spins, the spin frustration effects are fully supported by the geometry of these simplexes [34]. Here, in addition to the pairwise interactions, we introduce a simplex-based interaction involving three spins on a triangle. Our motivation is to assess the impact of the higher-order interactions, which are naturally enabled by the structure of the assembly, by balancing them (via a parameter α) against the strong spin frustration effects promoted by the antiferromagnetic pairwise interactions on the same structure. We analyse their combined influence on the shape of the hysteresis loop and the multi-scale structure of the magnetisation fluctuations during the reversal process.

II. THE SELF-ASSEMBLY OF SIMPLEXES
The above mentioned Q-analysis reveals the geometric constituents of a simplicial complex in a given network. The generative models, on the other hand, offer a bottom-up approach to grow a structure with simplicial complexes controlled by pertinent parameters. In the prototype model introduced in [1], which we use here, the self-assembly of cliques is constrained by the geometric compatibility and chemical affinity factors (see also [36] for an extended model with defect cliques). More precisely, a simplex (clique) of the size n ∈ [2, 10] is added by sharing one of its faces, i.e., sub-cliques of the size n q < n, with an already existing clique in the growing structure. The faces of the order q = 0, 1, 2, 3 · · · q max − 1 indicate a single node, link, triangle, tetrahedron, and so on up to the largest sub-clique, where q max = n − 1 is the order of the added clique. The probability of a clique of the order q max to attache along its q-face is given by where ν is the chemical affinity parameter, see Ref. [1] for a detailed description. The c q (t) is the number of geometrically compatible locations for docking a simplex of the order q, counted on the entire growing structure at the time t.
The geometrical factor is weighted by the chemical affinity of the system towards the addition of n a = q max − q new nodes. Specifically, for a large ν < 0, the system favours the addition of nodes. Thus the cliques preferably share a single node (see an example in Fig. 1 top panel). In the opposite limit, with increasing ν > 0, the cliques progressively share their larger face; eventually, a single node can be added, leading to highly compact structures, as shown in [1,36]. In the case when ν = 0, the process is governed by the geometric compatibility factors alone. Consequently, each face can be shared with a finite probability, depending on its size and the actual network structure. See online demo on this link [49]. For this work, we grow an assembly of triangles under the strictly geometrical compatibility conditions (q max = n − 1 = 2, and ν = 0). A close-up of the grown structure is depicted in the bottom panel of Fig. 1; it possesses 1000 nodes, 1737 FIG. 1: Top: Zoom-in nanoparticle network self-assembled by preformatted groups of particles as cliques of different sizes n ∈ [2, 10] for a negative chemical affinity ν (significant repulsion between the cliques). Bottom: Self-assembled network of triangles fo ν = 0 (geometrically constraint self-assembly). edges, and 738 triangles. As shown in [1,36], these networks exhibit a broad distribution of the node's degree, assortative mixing, and a wide range of shortest-path distances between nodes (see Fig. 2). They also have the hyperbolic geometry (a negative curvature in the shortest-path metric space), precisely, they are 1-hyperbolic. The origin of the small hyperbolicity parameter in these networks is in the attachment of new clique via shared face with an existing clique. Thus, the distance between individual cliques, which are ideal hyperbolic structures (δ clique = 0), is minimal leading to δ clique + 1hyperbolic clique-complexes [50,51]. We note that this attachment rule also prevents the appearance of holes in networks and that the order of a simplicial complex is equal to the order of the largest building simplex.
To study the magnetisation-reversal dynamics, here we adopt the network as shown in Fig. 1 bottom panel, and attach an Ising spin to each node (nanoparticle). The interactions among pairs of spins are enabled by the network links (adjacency matrix). Whereas, the triangle-based interactions are defined as tri-spin interactions among spins situated on a given triangle. In general, the triangle-adjacency matrix can be inferred by identifying all triangles, i.e., by performing the Q-analysis of the network [1,36,52]. In the present case, by growing the network in time steps t, we keep track of the nodes i 1 , i 2 , i 3 belonging to the added triangle. For example, in the first 11 steps, we can observe the following sequence of events (the variables in curly brackets are {t,N t ,Σ t ,n, n a , As mentioned in the Introduction, the antiferromagnetic spin-spin interactions in conjunction with a complex geometry of the assemblies provide ideal conditions for the spin frustration effects [34]. Here, we study the spin reversal dynamics driven by the external field h ext on an assembly of triangles, cf. Fig. 1. Each pair of adjacent spins obeys antiferromagnetic interaction, whereas the higher-order interaction involves the spins based on the same triangle. The Hamiltonian is given by where the Ising spins S i = ±1 are associated with the network vertices; the summation in the first term is over all pairs of spins that have a link in the network's adjacency matrix, whereas the sum in the second term runs over the identified triangles, as described above. The spin-spin interaction J i j = −J is antiferromagnetic, whereas K i jk = K 3 for the simplex-based interaction, where we consider both K 3 > 0 as well as K 3 < 0 cases. The parameter α ∈ [0, 1] is varied to provide a relative balance between the pairwise and higher-order interactions. We use the dimensionless units and set J = 1 and The process is driven by slow ramping of the external field along an ascending/descending branch of the hysteresis loop. A zero-temperature dynamics is applied, as it is a widely accepted approach in the study of Barkhausen noise in disordered magnetic systems, see [53] and references therein. A small increase of the field can trigger an avalanche of spin reversal, during which each spin aligns along its local fields to minimise the energy. In the disordered ferromagnets, an avalanche of spin flips stops when the magnetic defects pin the domain wall of the expanding domain of reversed spins. In contrast, in the spin assembles that we are studying here, there are no magnetic defects. Instead, the geometry of the network alone governs the progression of the spin reversal process (see also [34,54]). Specifically, for the ascending branch, we start from a state with all spins down, and a large negative filed h ext = −h max is applied. h max ≤ k max correlates with the number of connections of the leading hub in the network. The field changes are adiabatic, which means that the field is kept fixed during the avalanche propagation. The field is increased in small steps ∆ until it reaches the other limit h ext = +h max . Then the field is ramped by slow decreasing until it reaches −h max again, to complete the loop. Apart from the current value of the external field, the local field at the vertex i is given by the contribution of the actual states of the neighbouring spins according to the interactions in (2). In the present case, due to the fixed strength of the interactions, the local filed changes in the integer values, which depend on the node's pairwise and triangle-based connectivity, modified by the parameter α. In analogy to modelling the charged domain-wall motion during the ferroelectric switching [55], the frustration effects within the zero-temperature dynamics are emulated by the probabilistic alignment between the spin and its local field with a probability c < 1. The results for the hysteresis loop are shown in Fig. 3 for several values of the parameter α, interpolating between the strictly pairwise antiferromagnetic interactions, for α = 0, and strictly triangle-based interactions, for α = 1. As Fig. 3 shows, in the absence of the higher-order interactions, the hysteresis loop si symmetrical and, typically for the antiferromagnetic samples, splits into the positive and negative part. Moreover, the narrow tails at both high positive and high negative fields indicate the effects of a robust topological disorder, which permits only small avalanches of the reversed spins. The appearance of the higher-order interactions, however, induces a broadening and breaks the symmetry of the loop. In approaching the limiting case for α = 1, the triangle-based interactions cause an entirely different shape of the hysteresis loop with a large rectangular part (resembling a ferromagnetic case). However, it also shows a disorderconditioned tail, but on one side of the loop only. The implicated side depends on the sign of the K 3 , see also the inset in Fig. 4. Consequently, the increase of the magnetisation in time differs for the ascending and descending branch of the hysteresis whenever α is finite, as shown in the main panel of Fig. 4. For α = 1, a similarity occurs between the magnetisation curves in different branches, but for the reversed sign of K 3 . More precisely, the ascending line for K 3 > 0 is a mirror reflection of a descending line for K 3 < 0, and also the ascending line for the case K 3 < 0 mirrors the descending line for K 3 > 0, cf. full red lines and the corresponding dashed pale lines in Fig. 4. The magnetisation fluctuations comprising the Barkhausen noise signals during the reversal processes for different α are shown in Fig. 5. The bursting events (avalanches) triggered by the changed of the external field have a characteristic triangular shape caused by the underlying geometry in the absence of disorder. With the increased α, we notice that the signal becomes shorter as well as its most massive avalanches occur much earlier, following the shape of the hysteresis loop in Fig. 3. However, they all exhibit self-similarity with the longrange temporal correlations, as captured by the corresponding power-law behaviour of the power spectrum, S(i) ∼ i −φ , cf. lower panel in Fig. 5. Furthermore, we find that these noise signals possess the multifractal features, similar to the Barkhausen noise in disordered ferromagnets [56]. More precisely, the fluctuation function F q (n) dependence on the segment length n of the signal exhibits a power-law behaviour with a spectrum of the exponents. In particular, dividing the signal in N s segments and having defined the standard deviation F(µ, n) = 1 n ∑ n i=1 [Y ((µ − 1)n + i) − y µ (i)] 2 1/2 around a local trend y µ (i) in each segment µ = 1, 2 · · · N s , we have Here, a generalised Hurst exponent H q depends on the amplification factor q, as shown in Fig. 6. Intuitively, this means that the small fluctuations (amplified when q < 0) along the signal have different scaling properties than the large fluctuations (q > 0). In contrast, for the monofractal signals, H q = H 2 for all q, where H 2 is the standard Hurst exponent. We note that a small variation in H 2 agree with small variations in the exponent of the power spectrum, hence the expected scaling relation φ = 2H 2 − 1 is roughly satisfied (within the error bars).  The critical behaviour at the hysteresis loop in disordered ferromagnets is often related to the occurrence of a critical disorder point or a critical line [53] in the parameters space. Even though signatures of the self-organised criticality are present. For a recent review of systems exhibiting self-organised criticality in [57] and references therein. As stated above, the avalanching process that we study here is governed by the geometry of the assembly without any magnetic disorder. Nevertheless, Fig. 5 and Fig. 6 demonstrate robust features of criticality in the magnetisation fluctuations. A detailed analysis of scale invariance of the avalanches remains for separate work.

IV. CONCLUSIONS
We have briefly presented the subjects of current compelling research directions aiming to understand the role of the higher-order connectivity and interactions in the architecture and functional properties of complex systems. As a representative example, we have studied the nano-network grown by self-assembled simplexes (triangles), and the field-driven magnetisation reversal on them. With the antiferromagnetic interactions among the adjacent spin pairs and the trianglebased three-spin interactions, the underlying architecture enables the geometric frustration effects that have a pronounced impact on the hysteresis loop in these assemblies. By balancing between the pairwise and triangle-based interactions, the geometric frustration effects are gradually overridden by structural inhomogeneities due to the architecture of simplicial complexes. Moreover, the observed fluctuations of the magnetisation show a robust signature of self-organised criticality. The avalanching processes are conditioned solely by the geometry of the assembly without any magnetic disorder. These findings are in the line of research of complex nanoassembled materials with emerging functional properties.