Predicting collapse of adaptive networked systems without knowing the network

L. Horstmeyer, T. Minh Pham, J. Korbel and S. Thurner

linear interaction model

$$\frac{\mathrm{d} X_i(t)}{\mathrm{d} t} = \sum_j M_{ij} X_j(t) - \Phi X_i(t) \qquad \qquad (1)$$

• $$X(t) = \{X_1(t),\dots,X_N(t)\}$$ - state vector
• $$M_{ij}$$ - interaction matrix

• $$\Phi$$ - decay rate

• Network $$G$$: $$N$$ nodes with adjacency matrix $$M$$

• Normalized state vector: $$x_i(t) = \frac{X_i(t)}{\sum_j X_j(t)}$$
• Fixed point: $$x_i = \lim_{t \rightarrow \infty} x_i(t)$$

eigenvector quantization

1. Convergence: For any initial condition $$x(0)$$ except a  set of points of Lebesgue-measure zero $$x(t)$$ converges to a  stable fixed point $$x$$ that is a non-negative eigenvector of $$M$$.

2. Eigenvector Quantization: Suppose $$G$$ contains a cycle,  and there is no node that is part of more than one cycle. Then any component $$x_i > 0$$ can be expressed as  $$x_i = n_i x_{min}$$ where $$x_{min}$$ is the minimal non-zero component and $$n_i \in M$$  is a natural number.
• $$x_{min}$$ is the value of cycle nodes
• $$n_i$$ equals the number of directed paths that lead from cycle-nodes to $$i$$.

sketch of the proof

Fixed point $$x$$ satisfies eigenvector equation

$$\sum_j M_{ij} x_j = \lambda x_i \qquad \qquad (2)$$

• $$\lambda = 0$$ - no cycle, $$\lambda = 1$$ - one cycle, $$\lambda > 1$$ - more cycles
• Cycle nodes - receive only one inlink $$\Rightarrow$$ all have the same value $$x_i = x_c$$
• Nodes without path from the cycle $$x_i = 0$$
• Nodes with path from the cycle $$x_i = n_i x_c$$
• $$n_i$$ number of paths from cycle (can be shown by induction)

Extension - weighted network

$$M_{ij} \in \{0,[1-\epsilon,1]\}$$

Extension - katz centrality

$$x^{(K)}_i = \alpha \sum_{ij} M_{ij} x_j^{(K)} + \beta \qquad \beta=1$$

Applications: Jain-Krishna model

• Slow-scale evolution of network ($$x(t) \approx x$$):
• Selection: choose one of the least fit species $$i_{\min} \in \arg\min_i x_i$$

• Mutation: delete all existing links from/to $$x_{i_{\min}}$$ and assign new directed links with each existing node with probability $$\frac{m}{N-1}$$

• Slow-scale dynamics:

growth phase $$\rightarrow$$ ordered phase $$\rightarrow$$ collapse to random phase (no cycles)

Applications: Jain-Krishna model

Observations:

1. Every collapse is preceeded by the single-cycle phase
2. After entering the single-cycle phase, the average time to collapse can be expressed as

$$\langle T \rangle = \frac{e}{m} \quad (\mathrm{for} \ N \gg 1)$$

Applications: SIS model

• Simple model for epidemic spreading
• Node states: S (susceptible), I (infected)
• Dynamics: $$S+I \stackrel{\beta}{\rightarrow} I+I$$, $$I \stackrel{r}{\rightarrow} S$$
• In the individual-based mean-field approximation the probability $$p_i(t)$$ that a node $$i$$ is infected is given by

$$\frac{\mathrm{d} p_i(t)}{\mathrm{d} t} = \beta \sum_j M_{ij} p_j (1-p_i) - r p_i$$

• For $$p_i \ll 1$$ is the equation approximately linear
• We compare the actual values of $$p_i$$ with eigenvector of $$M_{ij}$$

perspectives

• Selection pressure

Reference: L.H., T.M.P., J.K., S.T. Predicting collapse of adaptive networked systems without knowing the network, accepted to Sci. Rep.

By Jan Korbel

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