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

Convergence: For any initial condition \(x(0)\) except a set of points of Lebesguemeasure zero \(x(t)\) converges to a stable fixed point \(x\) that is a nonnegative eigenvector of \(M\).
 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 nonzero 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 cyclenodes 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: JainKrishna model
 Slowscale 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 \(\frac{m}{N1}\)

Slowscale dynamics:

growth phase \(\rightarrow\) ordered phase \(\rightarrow\) collapse to random phase (no cycles)
Applications: JainKrishna model
Observations:
 Every collapse is preceeded by the singlecycle phase
 After entering the singlecycle 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 individualbased meanfield approximation the probability \(p_i(t)\) that a node \(i\) is infected is given by
$$ \frac{\mathrm{d} p_i(t)}{\mathrm{d} t} = \beta \sum_j M_{ij} p_j (1p_i)  r p_i $$
 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)

Nonlinear 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.
Prediction collapse
By Jan Korbel
Prediction collapse
 150