Predicting collapse of adaptive networked systems without knowing the network

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

Introduction

• Complex systems are prone to collapse

• typical example = networked systems

• Prediction of collapses is notorously difficult

• requires the full structural information                         - knowledge of all network links

• we present a novel precursor called            eigenvector quantization 

  • Corollary of perron-frobenius theorem

  • it does not require any structural information

  • for certain cases we can also predict the time to collapse

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 \(i\).

 eigenvector quantization

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 \(M_{ij}\)

Applications: SIS model

perspectives

• Weighted networks (Leslie model, Leontief model)

• Non-linear dynamics (SIS model and generalizations)

• Other centrality measures (Katz,...)

• More complicated network dynamics

• 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.