Structural stability of interaction networks against negative external fields

We explore structural stability of weighted and unweighted networks of positively interacting agents against a negative external field. We study how the agents support the activity of each other to confront the negative field, which suppresses the activity of agents and can lead to collapse of the whole network. The competition between the interactions and the field shape the structure of stable states of the system. In unweighted networks (uniform interactions) the stable states have the structure of $k$-cores of the interaction network. The interplay between the topology and the distribution of weights (heterogeneous interactions) impacts strongly the structural stability against a negative field, especially in the case of fat-tailed distributions of weights. We show that apart from critical slowing down there is also a critical change in the system structure that precedes the network collapse. The change can serve as an early warning of the critical transition. To characterize changes of network structure we develop a method based on statistical analysis of the $k$-core organization and so-called “corona” clusters belonging to the $k$-cores.

Figure 1

Example of an unweighted network of interacting agents in negative external field, Eq. (4). (a) In the case $k=1$ ($U=-\delta$, where $\delta <1/2$), all agents forming finite clusters (nodes 10–13 and 14–15) and a “giant” connected component (nodes 1–9) are active, while isolated agents 16 and 17 are inactive (open circles). (b) At $k=2$ ($U=-1-\delta$ where $0<\delta <1$), only agents forming 2-core are active. (c) At $k=3$ ($U=-2-\delta$), only agents forming 3-core are active.

Figure 2

Fraction of active agents $M$ versus the field magnitude $\left|U\right|$ in the ground state of the ER network of size $N={10}^4$ and the mean degree $\langle q\rangle =10$. Jumps occur at $\left|U\right|=2,3,\cdots ,7$. Results are averaged over 100 realizations.

Figure 3

(a) Fraction $M$ of active agents versus the occupation probability $p$ in the ground state of a randomly damaged ER network in the negative field $U_i=-(k-1+\delta )$ at $k=3$ (red triangles) and $k=4$ (green triangles). (b) The parameter ${\chi }_{\text{cr}}\left(k\right)$ versus $p$ at $k=3$ (red triangles) and $k=4$ (green triangles). In simulation we studied ER networks of size $N={10}^5$ and the mean degree $\langle q\rangle =10$. Results are averaged over 500 realizations.

Figure 4

(a) ER network of interacting agents with Gaussian weights in a negative field $U$. $M$ is a fraction of active agents (open circles), $M_7$ is the size of the highest 7-core (rhombi), $M_{\text{gc}}$ is the size of giant connected component (stars). Inset represents zoom of the region near the critical point. (b) The ${\chi }_{\text{cr}}\left(k\right)$ for $k=7$ (open triangles) and $M-M_{\text{gc}}$ (filled triangles) versus $\left|U\right|$. The vertical dot-dashed line shows the field at which the highest $k$-core ($k_h=7$) collapses and ${\chi }_{\text{cr}}\left(7\right)$ has a peak. The vertical dashed line is the point $M_{\text{gc}}=0$. Inset represent zoom of the region near the critical point. Parameters in simulations: the network size $N={10}^4$, the mean degree $\langle q\rangle =10$, the mean weight $\langle w\rangle =1$, the variance ${\sigma }^2=0.01$. The number of realizations is 500.

Figure 5

(a) The fraction $M$ of active agents, the fraction $M_7$ of the highest 7-core, and the fraction $M_{\text{gc}}$ of the giant connected component versus $\left|U\right|$ in the ER networks with a fat-tailed distribution of weights. (b) Field dependence of the fraction $M-M_{\text{gc}}$ of finite clusters of active agents. (c) The index $k_h$ of the highest $k$-core and ${\chi }_{\text{cr}}\left(k_h\right)$ versus the field strength $\left|U\right|$. The vertical dash-dotted line on the left corresponds to the critical field of the collapse of the highest $k$-core. The vertical dash-dotted line on the right corresponds to the critical field of disappearance of $M_{\text{gc}}$. Other parameters: the network size $N={10}^5$, the mean degree $\langle q\rangle =10$, $\alpha =2.5$ is the exponent of the distribution function of weights. The number of realizations is 500.

Figure 6

Schematic representation of the evolution of the $k$-core organization when increasing the field strength $\left|U\right|$: (a) unweighted networks, (b) weighted networks. The $k$-cores are represented as nested circles. The largest circle is the 2-core, which includes the higher cores with $k=3,4$. The layers $k=2$ and $k=3$ are 2- and 3-shells, respectively. There is also 1-shell, which is not shown. In unweighted networks, when increasing $\left|U\right|$ the highest $k$-core disappears last. In the weighted networks with a fat-tailed distribution of weights, the 2-core disappears last. At sufficiently large fields ($\left|U\right|>|U_4|$), only finite disjoint clusters of strongly interacting agents may exist.

Figure 7

(a) Fraction of active agents $M$ versus the field strength $\left|U\right|$ at the temperatures $T=0.1$ (diamonds) and 1.0 (stars). (b) The parameter ${\chi }_{\text{cr}}\left(k\right)$ versus $\left|U\right|$ at $T=0.1$ (diamonds) and $T=1$ (stars). The unweighted ER network in Fig. 2 is used. The number of realizations is 500.

Figure 8

(a) Fraction $M$ of active agents versus temperature $T$ in the unweighted ER network of interacting agents Eq. (5) in the uniform negative field $U=-5.001$. Black and red arrows show the directions of the temperature increase and decrease, respectively. (b) The parameter ${\chi }_{\text{cr}}(k=6)$ versus $T$. The unweighted ER network with the mean degree $\langle q\rangle =10$ and the network size of $N={10}^6$ and $N={10}^5$ were used for (a) and (b), respectively. The number of realizations is 500.

Figure 9

(a) The fraction $M$ of active agents versus the negative field strength $\left|U\right|$ in the unweighted plant-pollinator networks for 4 years [27]. The numbers of plants-pollinators are (79,126) at 2007, (88,128) at 2008, (100,165) at 2009, and (108,165) at 2010. (b) The fraction $M$ of active agents versus $\left|U\right|$ in the plant-fungus network (weighted, 33 plants and 387 fungus) with cutoff level 0.95 of DNA sequence similarity of fungus [52]. The inset represents the zoom of the low field region. Panels (c) and (d) represent the index $k_h$ of the highest core versus $\left|U\right|$ in the plant-pollinator and plant-fungus networks, respectively.