Weighted growing simplicial complexes

Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial complexes are weighted. Here we propose a nonequilibrium model for weighted growing simplicial complexes. The proposed dynamics is able to generate weighted simplicial complexes with a rich interplay between weights and topology emerging not just at the level of nodes and links, but also at the level of faces of higher dimension.

Figure 1

Graphical representation of process A [panel (a)] and process B [panel (b)] for a two-dimensional simplicial complex with $m=3$ and $m^{\prime }=4$ starting from a given initial condition.

Figure 2

Skeleton networks of simplicial complexes generated by the model for $d=3,N=100$, and $s=-1$. Node sizes indicate their degrees while link widths indicate their generalized strength. Node and edge colorings indicate community structure calculated according to the Louvain algorithm [45]. Panel (a) shows the skeleton of a simplicial complex with $m=1$ and $m^{\prime }=2$, while panel (b) shows the skeleton of a simplicial complex with $m=2$ and $m^{\prime }=1$.

Figure 3

Generalized degree distributions $P_{d,\delta }\left(k_{d,\delta }\right)$ are shown for simplicial complexes of dimension $d=3$ and flavor $s=-1$ [panels (a), (d), and (g)], $s=0$ [panels (b), (e), and (h)], and $s=1$ [panels (c), (f), and (i)], and for faces of dimension $\delta =2$ [panels (a), (b), and (c)], $\delta =1$ [panels (d), (e), and (f)], and $\delta =0$ [panels (g), (h), and (i)]. The results of simulations are shown for $m=1$, 2, and 3 (blue circles, red squares, and yellow triangles, respectively). The simulated simplicial complexes have $N={10}^5$ nodes, and the results are averaged over 10 simplicial complex realizations.

Figure 4

The average generalized strengths $s_{d,\delta }(t,t_{\alpha })$ of the $\delta$-faces arrived in the network at time $t_{\alpha }$ are shown for simplicial complexes of dimension $d=3$, and flavor $s=-1$ [panels (a), (d), and (g)], $s=0$ [panels (b), (e), and (h)], and $s=1$ [panels (c), (f), and (i)] and for faces of dimension $\delta =2$ [panels (a), (b), and (c)], $\delta =1$ [panels (d), (e), and (f)], and $\delta =0$ [panels (g), (h), and (i)]. The results of simulations are shown for $(m=1,m^{\prime }=5),(m=2,m^{\prime }=5),(m=2,m^{\prime }=3)$, and $(m=3,m^{\prime }=1)$ (blue circles, red squares, yellow crosses, and purple triangles, respectively). The simulated simplicial complexes have $N={10}^5$ nodes, and the results are averaged over 10 simplicial complex realizations.

Figure 5

The exponents ${\lambda }_{\text{fitted}}$ obtained by fitting Eq. (58) to the data in Fig. 4 are shown vs the predicted exponents ${\lambda }_{\text{predicted}}$ given by Eq. (59) for different $\delta$-faces. The panels (a), (b), and (c) refer, respectively, to triangles $\left(\delta =2\right)$, links $(\delta =1)$, and nodes $\left(\delta =0\right)$. The blue stars, red squares, and yellow circles indicate the data obtained, respectively, for the the flavors $s=-1$, 0, and 1.

Figure 6

The average generalized strengths of $\delta$-faces as a function of their corresponding generalized degree $s_{d,\delta }\left(k_{d,\delta }\right)$ are shown for simplicial complexes of dimension $d=3$ and flavor $s=-1$ [panels (a), (d), and (g)], $s=0$ [panels (b), (e), and (h)], and $s=1$ [panels (c), (f), and (i)] and for faces of dimension $\delta =2$ [panels (a), (b), and (c)], $\delta =1$ [panels (d), (e), and (f)], and $\delta =0$ [panels (g), (h), and (i)]. The results of simulations are shown for $(m=1,m^{\prime }=5),(m=2,m^{\prime }=5),(m=2,m^{\prime }=3)$, and $(m=3,m^{\prime }=1)$ (blue circles, red squares, yellow crosses, and purple triangles, respectively). The simulated simplicial complexes have $N={10}^5$ nodes, and the results are averaged over 10 simplicial complex realizations.