Predicting collapse of adaptive networked systems without knowing the network
L. Horstmeyer, T. Minh Pham, J. Korbel and S. Thurner
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Introduction
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Complex systems are prone to collapse
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typical example = networked systems
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Prediction of collapses is
notorously difficult -
requires the full structural information - knowledge of all network links
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we present a novel precursor called eigenvector quantization
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Corollary of perron-
frobenius theorem -
it does not require any structural information
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for certain cases we can also predict the time to collapse
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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
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\(M_{ij}\) - interaction matrix
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\( \Phi\) - decay rate
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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
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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 \(M\).
- 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 \(i\).
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154709/onecycle.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154710/twocycles.png)
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)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154802/weighted-1.png)
Extension - weighted network
\(M_{ij} \in \{0,[1-\epsilon,1]\}\)
Extension - katz centrality
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154817/katz-1.png)
\(x^{(K)}_i = \alpha \sum_{ij} M_{ij} x_j^{(K)} + \beta \qquad \beta=1\)
Applications: Jain-Krishna model
- Slow-scale evolution of network (\(x(t) \approx x\)):
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Selection: choose one of the least fit species \( i_{\min} \in \arg\min_i x_i\)
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Mutation: delete all existing links from/to \(x_{i_{\min}}\) and assign new directed links with each existing node with probability \(\frac{m}{N-1}\)
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Slow-scale dynamics:
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growth phase \(\rightarrow\) ordered phase \(\rightarrow\) collapse to random phase (no cycles)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154927/samplerun6JunAsp05.png)
Applications: Jain-Krishna model
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154955/ExpCollTimesAsp34Size8-1.png)
Observations:
- Every collapse is preceeded by the single-cycle phase
- 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
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154971/histogram05R17_test-1.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/924506/images/6154985/SISquantpointplot-1.png)
perspectives
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Weighted networks (Leslie model, Leontief model)
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Non-linear dynamics (SIS model and generalizations)
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Other centrality measures (Katz,...)
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More complicated network dynamics
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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
- 151