Simplicial complexes: higher-order spectral dimension and dynamics

Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the spectral properties of the simplicial complex model called"Network Geometry with Flavor"we provide evidence that the up and down higher-order Laplacians can have a finite spectral dimension whose value increases as the order of the Laplacian increases. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes.

The recently proposed non-equilibrium growing simplicial complex model called "Network Geometry with Flavor" (NGF) [15] is able to display emergent hyperbolic network geometry [16] together with the major universal properties of complex networks including scale-free degree distribution, small-word distance property, high clustering coefficient and significant modular structure. Interestingly, the simplicial complexes generated by the NGF model display also a finite spectral dimension [17,18,24,25]. The spectral dimension [27][28][29][30] characterises the spectrum of the graph Laplacian of network geometries and is well known to affect the return-time probability of classical [27] critical phenomena [31,32] and quantum diffusion [33]. Additionally the spectral dimension strongly affects the synchronization properties of the Kuramoto model which display a thermodynamically stable synchronized phase only if the spectral dimension d S is greater than four [17,18]. Finally the spectral dimension is also used in quantum gravity to probe the geometry of different model of quantum space-time [34][35][36][37][38][39] Recent works [19,[40][41][42] have emphasised that simplicial complexes can sustain dynamical processes whose variables can be located not only on their nodes but also on their higher dimensional simplices such as links, triangles and so on. In particular, in Ref. [19] the Kuramoto model has been extended to treat synchronization of phases located in higher-dimensional simplices. Additionally, a higher-order diffusion dynamics has been defined over simplicial complexes [42]. The higher-order diffusion dynamics and the higher-order Kuramoto model depend on the higher-order boundary maps of the simplices and the higher-order Laplacian matrix. The higher-order Laplacian matrix [41][42][43][44] of order n > 0 describes a diffusion dynamics taking place between simplices of order n and can be decomposed in the sum between the up-Laplacian and the down-Laplacian. The higher-order discrete Laplacian has been studied by several mathematicians [43,44] , however as far as we know, there is no previous result showing that the high-order Laplacian can display a finite spectral dimension.
In this work we investigate the spectral properties of the higher-order Laplacian on NGF. We show that the higher-order up and down-Laplacians have a finite spectral dimension that increases with their order n. By investigating the properties of higherorder diffusion on NGF we find that the higher-order spectral dimension has a significant effect on the return-time probability of the process. Therefore, we provide evidence that the diffusion, occurring on the same simplicial complex but taking place on simplices of different order n, can induce significantly different dynamical behavior.

Simplicial complexes
Simplicial complexes are able to capture higher-order interactions between two or more nodes described by simplices. A n-dimensional simplex µ is formed by n + 1 nodes Therefore, a 0-dimensional simplex is a node, a 1-dimensional simplex is a link, and so on. A face of n-dimensional simplex is a n -dimensional simplex formed by a proper subset of n + 1 nodes of the original simplex. Consequently, we necessarily have n < n. A simplicial complex K is a set of simplices that is closed under the inclusion of the faces of the simplices. We will indicate with d the dimension of the simplicial complex determining the maximum dimension of its simplices. Moreover, we will indicate with N [n] the number of n-dimensional simplices of the simplicial complex K. Therefore in the following N [0] indicates the number of nodes, N [1] the number of links, N [2] the number of triangles and so on. Simplicial complexes can sustain a diffusion dynamics occurring on its n-dimensional faces. This higher-order diffusion dynamics is determined by the properties of the higher-order Laplacians. In order to introduce here the higher-order Laplacian we will devote the next paragraph to some fundamental quantities in network topology.

Oriented simplices and boundary map
In topology each n-dimensional simplex µ has an orientation given by the sign of the permutation of the label of the nodes. Therefore, we have where σ(π) indicates the parity of the permutation π.
The boundary map ∂ n is a linear operator acting on linear combinations of ndimensional simplices and defined by its action on each of the simplices µ = [i 0 , i 2 . . . , i n ] of the simplicial complex as Therefore the boundary map of a link [i, j] is given by Similarly the boundary map of a triangle [i, j, k] is given by From this definition it follows directly that relations that is often referred to by saying that the "boundary of the boundary is zero". For instance, the reader can easily check using the above definitions that The boundary map ∂ n can also be represented by the incidence matrix . Then, Eq. (7) can be expressed using the incidence matrices as = 3 triangles whose incidence matrices B [1] and B [2] are given in Eqs. (9) and (10).

Higher-order Laplacians
The zero-order Laplacian L [0] is the usual graph Laplacian defined on networks and is a where here and in the following δ ij indicates the Kronecker delta, and a ij indicates the element (i, j) of the adjacency matrix. The graph Laplacian L [0] can be also expressed in terms of the incidence matrix B [1] as This definition can be extended to higher-order Laplacians using higher-order incidence matrices B [n] . In particular the higher-order Laplacian L [n] (with n > 0) [41][42][43] is the with L down The higher-order Laplacians are independent on the orientation of the simplices as long as the orientation of the simplices is induced by the label of the nodes. The degeneracy of the zero eigenvalue of the n Laplacian L [n] is equal to the Betti number β n . The eigenvectors associated to the zero eigenvalue of the n-Laplacian are localized on the corresponding n-dimensional cavities of the simplicial complex. Therefore, the higherorder Laplacians with n > 0 are not guaranteed to have a zero eigenvalue as simplicial complexes with β n = 0 for some n > 0 exist. In particular, if the topology of the simplicial complex is trivial, i.e. β 0 = 1 and β n = 0 for all n > 0, the Laplacians of order n > 0 do not admit a zero eigenvalue. Another important property of the n-Laplacian is that each non-zero eigenvalue is either a non-zero eigenvalue of the n-order up-Laplacian or is a non-zero eigenvalue of the n-order down-Laplacian. Consider an eigenvector v of the up-Laplacian with eigenvalue λ = 0. Then, we have Let us apply the down-Laplacian to the eigenvector v. Thus we obtain where we have used Eq. (8). It follows that if v is an eigenvector associated to a nonzero eigenvalue λ of the n-order up-Laplacian then it is an eigenvector of the n-order down-Laplacian with zero eigenvalue. Therefore, in this case v is an eigenvector of the n-order Laplacian with the eigenvalue λ. Similarly, it can be easily shown that if v is an eigenvector associated to a non-zero eigenvalue λ of the n-order down-Laplacian then it is also an eigenvector of the n-order Laplacian with the same eigenvalue. Consequently the spectrum of the n-order Laplacian includes all the eigenvalues of the n-order up-Laplacian and the n-order down-Laplacian. Another important property of the high-order up and down Laplacians is that the spectrum of the n-order up-Laplacian coincides with the spectrum of the (n + 1)-order down-Laplacian as the two are related by .
The n-Laplacian is positive semi-definite and, therefore, it has N [n] non negative eigenvalues that we indicate as Moreover, in the following we will indicate by v (r) the eigenvector corresponding to eigenvalue λ r of the n-Laplacian.

Spectral dimension of the graph Laplacian
The spectral dimension is defined for networks (1-dimensional simplicial complexes) with distinct geometrical properties, and determines the dimension of the network as "experienced" by a diffusion process taking place on it [17,18,[27][28][29]33]. The spectral dimension is traditionally defined starting from the density of eigenvalues ρ(λ) of the 0-Laplacian. We say that a network has spectral dimension d S if the density of eigenvalues ρ(λ) of the 0-Laplacian follows the scaling relation for λ 1. In d-dimensional Euclidean lattices d S is related to the Hausdorff dimension d H of the network by the inequalities [38,39] Therefore, for small-world networks, which have infinite Hausdorff dimension d H = ∞, it is only possible to have finite spectral dimension d S ≥ 2. If the density of eigenvalues ρ(λ) follows Eq. (20) it results that the cumulative distribution ρ c (λ) evaluating the density of eigenvalues λ ≤ λ satisfies for λ 1. In presence of a finite spectral dimension the Fiedler eigevalue λ 2 satisifies Therefore, the Fidler eigenvalue λ 2 → 0 as N [0] → ∞ and we say that in the large network limit the spectral gap closes. This is another distinct property of networks with a geometrical character, i.e. significantly different from random graphs and expanders. The spectral dimension has been proven to be essential to determine the stability of the synchronized state of the Kuramoto model which can be thermodynamically stable only if d In the next section we will show that the notion of spectral dimension can be generalized to order n > 0 with important consequences for higher-order simplicial complex dynamics.

Results
In this section we will investigate the spectral properties of a recently proposed model of simplicial complexes called "Network Geometry with Flavor". We will show that the higher-order up-Laplacians of these simplicial complexes display a finite spectral dimension depending on the order n of the up-Laplacian considered, the dimension of the simplicial complex d and a parameter of the model called flavor s. Therefore given a single instance of a NGF we can define different spectral dimensions d S for 0 < n < d−1.
Here we will show that this implies that the dynamics defined on simplices of different dimension n of the same simplicial complex can be significantly different.

Network Geometry with Flavor
The model "Network Geometry with Flavor" (NGF) [15,16] generates d-dimensional simplicial complexes. Each simplex is obtained by performing a non-equilibrium process consisting in the continuous addition of new d-simplices attached to the rest of the simplicial complex along a single (d − 1)-face. To every (d − 1)-face µ of the simplicial complex, (i.e. a link for d = 2, or a triangular face for d = 3) we associate an incidence number n µ given by the number of d-dimensional simplices incident to it minus one. The evolution of NGF depends on a parameter s ∈ {−1, 0, 1} called flavor. Starting from a single d-dimensional simplex, with d ≥ 2 at each time we add a d-dimensional simplex to a (d − 1)-face µ. The face µ is chosen randomly with probability Π µ given by According to a classical result in combinatorics, under this dynamics we obtain a discrete manifold only if n µ can take exclusively the values n µ = 0, 1. This occurs only for s = −1. In fact for n µ = 0 we obtain Π µ = 1/( ν 1 + sn ν ) > 0 but for n µ = 1 we obtain Π µ = 0. Therefore, the resulting simplicial complex is a discrete manifold, with each (d − 1)-face incident at most to two d-dimensional simplices, i.e. n µ = 0, 1. For s = 0 the attachment probability Π µ is uniform while for s = 1 the attachment probability Π µ increases linearly with the number of simplices already incident to the face µ implementing a generalized preferential attachment. Therefore for s = 0 as for s = 1 the incidence number n µ can take arbitrary large values n µ = 0, 1, 2, 3 . . ..
This model generates emergent hyperbolic geometry, and the underlying network is small-word (has infinite Hausdorff dimension, i.e. d H = ∞), has high clustering coefficient and high modularity. Interestingly, this model reduces to well known models in specific cases: for d = 1 and s = 1 it reduces to the tree Barabasi-Albert model [45], for d = 2 and s = 0 it reduces to the model first studied in Ref. [46] and finally for d = 3 and s = −1 it reduces to the random Apollonian model [47,48].

Spectral properties of NGF
The graph Laplacian of NGFs has been recently show to display a finite spectral dimension d S and localized eigenvectors with important consequences on dynamics [17,18,24]. Interestingly, here we show that also the higher-order up-Laplacians L up [n] and the higher-order down-Laplacians L down ). For NGFs, that have a trivial topology, the Hodge decomposition [41] guarantees that the number N [n] of non-zero eigenvalues of the n-order up-Laplacian with n > 0 achieves this limit and consequently we have In Figure 2 we plot the cumulative density of eigenvalues ρ c (λ) of the n-order Laplacian and the cumulative density of non-zero eigenvalues ρ up c (λ) of the n-order up-Laplacians of NGF with d = 3 and flavor s = −1, 0, 1. The n-order up-Laplacians display a finite spectral dimension, i.e. their cumulative density of eigenvalues obeys the scaling for λ 1. The fitted values of these higher-order spectral dimensions are reported in Table 1 for different values of the order n and the flavor s of the 3-dimensional NGF. From this table it can be clearly shown that the values of the higher-order spectral dimension d  Since the n-order up-Laplacian is the transpose matrix of the (n + 1)-order down-Laplacian (as given in Eq. (18)) the two matrices have the same spectrum. It follows directly that the (n + 1)-order down-Laplacian has spectral dimension d S . From these results on the higher-order up-Laplacian we can easily determine the scaling of the density of eigenvalues for the higher-order Laplacian of NGFs. In particular for 0 < n < d we have for n = 0 we have instead and for n = d we have Therefore the density of eigenvalues ρ(λ) of the n-order Laplacian reads S /2−1 for 0 < n < d, whereC [n] are constants. Therefore, the cumulative density of the eigenvalues of the higher-order Laplacian will asymptotically scale as a power-law dictated by the minimum between d

Diffusion using higher-order Laplacian
Higher-order Laplacians L [n] can be used to define a diffusion process defined over ndimensional simplices. For instance, one can consider a classical quantity x µ defined on the n-dimensional simplices µ of the simplicial complex and use the n-Laplacian L [n] to study its diffusion using the dynamical equation where with S n we indicate the set of all simplices of dimension n (of cardinality |S n | = N [n] ). For n = 0 there is always a stationary state as β 1 indicates at the same time the number of connected components of the simplicial complex (therefore we have β 1 ≥ 1) and the degeneracy of the zero eigenvalue of the Laplacian matrix L [0] . Additionally, for a connected network the stationary state is uniform over all the nodes of the network. However, Eq. (32) for n > 0 will have a stationary state only if the Betti number β n > 0, i.e. if there is at least a n-dimensional cavity in the simplicial complex. Note, however, that also if this stationary state exists the stationary state will be non-uniform over the network but localized on the n-dimensional cavities. In order to describe a diffusion equation that has a non trivial stationary state also when β n = 0 we can modify the diffusion equation and consider instead the dynamics This equation reduces to Eq. (32) if the smallest eigenvalue of the n-Laplacian is zero (i.e. λ 1 = 0) and admits always a non-trivial stationary state localized along the eigenvector v (1) corresponding to the smallest eigenvalue. The NGFs have Betti numbers β 0 = 1 and β n = 0 for every n > 0. In this case, when n > 0 the dynamics defined by Eq. (32) gives a transient to a vanishing solution x ν = 0 for every n-dimensional face ν.
On the contrary, the dynamics defined by Eq. (33) gives a transient to a non-vanishing steady state solution. The solution for the two dynamical equations (32) and (33) can be written as where for the dynamics defined in Eq. (32) we put c = 0 while for the dynamics defined in Eq.(33) we put c = 1. For both dynamics, we investigate the return-time probability P (t) as a function of time. The return-time probability P (t) is defined as the probability that the diffusion process starting from a localized configuration on a given simplex µ returns back to the simplex µ at time t, averaged over all simplices µ ∈ S n of the simplicial complex. Therefore P (t) is given by r=1 e −λr(1−cδ r,1 )t (35) where in the last expression we have used the normalization of the eigenvectors v (r) , i.e.
Interestingly, for large NGF the return-time probability P (t) decays in time at different rates depending on the dimension n over which the diffusion dynamics is defined. In particular, for a large simplicial complex when N [n] → ∞ we can approximate the return-time probability P (t) as