Network reconstruction from dynamics and behavior

In some cases, direct measurements of the edges of a network are either impossible, or can be done only at significant experimental cost. In such situations, we need to infer the network of interactions from the observed functional behavior [peixoto-network-2024]. In graph-tool this can be done for a variety of static and dynamical models, where the data \(\boldsymbol X\) is a \(N\times M\) matrix containing \(M\) samples (or “features”) for each node.

For static models we have that the likelihood is given by

\[P(\boldsymbol X | \boldsymbol W,\boldsymbol\theta) = \prod_m P(\boldsymbol X_{m}|\boldsymbol W, \boldsymbol\theta),\]

where each sample is assumed to be i.i.d., whereas for dynamic models \(\boldsymbol X\) is assumed to contain the states of a Markov chain with likelihood

\[P(\boldsymbol X | \boldsymbol W,\boldsymbol\theta) = \prod_m P(\boldsymbol X_{m}|X_{m-1},\boldsymbol W, \boldsymbol\theta).\]

In both cases above, \(\boldsymbol W\) represents the weighted adjacency matrix and \(\boldsymbol \theta\) are additional node parameters.

The inference procedure consists in the maximum a posteriori (MAP) estimation:

\[\begin{split}\begin{aligned} \widehat{\boldsymbol W}, \widehat{\boldsymbol\theta} &= \underset{\boldsymbol W, \boldsymbol\theta}{\operatorname{arg max}}\; P(\boldsymbol W,\boldsymbol\theta | \boldsymbol X) \\ &= \underset{\boldsymbol W, \boldsymbol\theta}{\operatorname{arg max}}\; \log P(\boldsymbol X | \boldsymbol W,\boldsymbol\theta) + \log P(\boldsymbol W,\boldsymbol\theta), \end{aligned}\end{split}\]

where the prior \(P(\boldsymbol W,\boldsymbol\theta)\) implements regularization.

In graph-tool, a variety of models are implemented, as given by the following table:

Name	State	Data domain	Kind
Multivariate normal	PseudoNormalBlockState	\(\mathbb{R}\)	Static
Dynamic multivariate normal	NormalGlauberBlockState	\(\mathbb{R}\)	Dynamic
Generalized Lotka-Volterra	LVBlockState	\(\mathbb{R}^+\)	Dynamic
Compartmental epidemics	EpidemicsBlockState	\(\{0, 1\}\)	Dynamic
Ising model	PseudoIsingBlockState	\(\{-1, +1\}\)	Static
Kinetic Ising model	IsingGlauberBlockState	\(\{-1, +1\}\)	Dynamic
Continuous Ising model	PseudoCIsingBlockState	\([-1,1]\)	Static
Continuous kinetic Ising model	CIsingGlauberBlockState	\([-1,1]\)	Dynamic

Reconstruction with synthetic data

Here we demonstrate how reconstruction is performed with a simple example, based on the simulation of the Kinetic Ising model on the dolphins network (we can use the IsingGlauberState class for this purpose):

Note

This is the dolphins network we will attempt to reconstruct from a the kinetic Ising model.

g = gt.collection.data["dolphins"]
E = g.num_edges()
N = g.num_vertices()
w = g.new_ep("double", vals=np.random.normal(N/(2*E), .05, E))  # True edge weights
istate = gt.IsingGlauberState(g, w=w)

M = 1000
X = []
for m in range(M):
    istate.iterate_sync()
    X.append(istate.get_state().a.copy())
X = np.array(X).T

At this point the array X contains our simulated data. We will now reconstruct the original network from it using a nonparametric approach, which consists of initializing an IsingGlauberBlockState object, and iterating over mcmc_sweep() until sufficient convergence is reached:

Note

When doing reconstruction, the default for mcmc_sweep() is the inverse “temperature” beta=1, meaning that this function performs sweeps of a MCMC. Here we are interested in a greedy optimization instead, so we need to pass beta=numpy.inf.

state = gt.IsingGlauberBlockState(X)

for i in range(10):
    delta, *_ = state.mcmc_sweep(beta=np.inf, niter=10)
    print(delta)

This outputs the following deltas, indicating that convergence was achieved.

-4374.171741316452
-3430.442095268032
-904.7176098519118
-190.28064153057187
-26.28956787113603
-1.9477590574467705
-0.4964012530509585
-0.0012287004910120913
-1.295718874416707e-07
-8.93862761586206e-12

We can compare the reconstructed network using the similarity() function, as follows

u = state.get_graph()      # reconstructed network
w_r = state.get_x()        # reconstructed weights
t_r = state.get_theta()    # reconstructed thetas

print(gt.similarity(g, u, w, w_r))

which gives:

0.907831...

We can visualize the reconstruction with:

cnorm = matplotlib.colors.Normalize(vmin=-abs(w.fa).max(), vmax=abs(w.fa).max())
# Original graph
gt.graph_draw(g, g.vp.pos, edge_pen_width=gt.prop_to_size(w.t(abs), 2, 8, power=1), edge_color=w,
              ecmap=matplotlib.cm.coolwarm_r, ecnorm=cnorm, output="dolphins-orig.svg")

# Inferred graph
gt.graph_draw(u, g.vp.pos, edge_pen_width=gt.prop_to_size(w_r.t(abs), 2, 8, power=1), edge_color=w_r,
              ecmap=matplotlib.cm.coolwarm_r, ecnorm=cnorm, output="dolphins-inf.svg")

# Inferred graph with SBM partition
state.bstate.draw(pos=g.vp.pos, edge_gradient=[],
                  edge_pen_width=gt.prop_to_size(w_r.t(abs), 2, 8, power=1),
                  edge_color=w_r, ecmap=matplotlib.cm.coolwarm_r, ecnorm=cnorm,
                  output="dolphins-inf-sbm.svg")

which outputs the networks below:

True network	Reconstructed	Reconstructed with SBM

Note

The algorithm implemented is the one described in [peixoto-scalable-2024] combined with the MDL regularization based on weight quantization from [peixoto-network-2024], which includes also a SBM as part of the prior for the latent edges.

The algorithm runs in time \(O(N\log^2 N)\) in the typical case, where \(N\) is the number of nodes, and if enabled during compilation will run in parallel using OpenMP, and therefore can be employed to reconstruct large networks.

\(L_1\) regularization

The default reconstruction methods employs a nonparametric regularization scheme [peixoto-network-2024], but optionally \(L_1\) can be used instead. This is done by initializing the reconstruction state as

lstate = gt.IsingGlauberBlockState(X, disable_xdist=True, disable_tdist=True)

This method requires a sparsity parameter \(\lambda\) to be set, e.g. the following sets \(\lambda=200\),

lstate.update_entropy_args(xl1=200)

and then we can proceed in the same way,

for i in range(10):
   delta, *_ = lstate.mcmc_sweep(beta=np.inf, niter=10)

which produces the network:

u = lstate.get_graph()
w_r = lstate.get_x()
gt.graph_draw(u, g.vp.pos, edge_pen_width=gt.prop_to_size(w_r.t(abs), 2, 8, power=1), edge_color=w_r,
             ecmap=matplotlib.cm.coolwarm_r, ecnorm=cnorm, output="dolphins-inf-l1.svg")

Network reconstructed using \(L_1\) regularization with \(\lambda=200\).

Caution

The \(L_1\) regularization method is faster than MDL, but the most appropriate value of \(\lambda\) is tricky to find, usually requiring cross-validation, which in the end makes it slower than MDL. It also introduces a shrinkage bias and tends to overfit when compared to MDL, which in general should be preferred. See [peixoto-network-2024] for details.

Reconstruction with empirical data

As an example of how to perform reconstructions on empirical data, we will consider the log-returns of the stock prices of the S&P 500 companies containing the 500 largest US companies.

We will begin by fetching the data and computing the log-returns:

import requests
from collections import defaultdict
import csv

r = requests.get('https://raw.githubusercontent.com/plotly/datasets/master/all_stocks_5yr.csv')
lines = r.iter_lines(decode_unicode=True)
next(lines)                                     # remove header
prices = defaultdict(list)
for vals in csv.reader(lines, delimiter=','):
    prices[vals[-1]].append(float(vals[4]))     # use closing price
stocks, s = zip(*[(stock, vals) for stock, vals in prices.items() if len(vals) == 1259])

# compute log-returns
s = [[log(p[i+1]) - log(p[i]) for i in range(len(p)-1)] for p in s]

s = array(s)

We will use a multivariate normal model via PseudoNormalBlockState:

state = gt.PseudoNormalBlockState(s)

delta = np.inf
while delta > 20:
    ret = state.mcmc_sweep(beta=np.inf, niter=1)
    delta = abs(ret[0])

We can now visualize the reconstructed network:

bstate = state.get_block_state().levels[0]
pos = gt.sfdp_layout(state.get_graph(), groups=bstate.get_blocks(), gamma=.02)
bstate.draw(pos=pos, vertex_color="white", edge_gradient=[],
            edge_pen_width=gt.prop_to_size(state.get_x().t(abs), .1, 3, power=1),
            output="SP500.svg")

Reconstructed network of interactions between the S&P500 companies according to their daily log-returns and a multivariate normal model. The edge thickness corresponds to the inferred negative inverse covariance (a.k.a. precision matrix entries).

References

[peixoto-network-2019]

Tiago P. Peixoto, “Network reconstruction and community detection from dynamics”, Phys. Rev. Lett. 123 128301 (2019), DOI: 10.1103/PhysRevLett.123.128301 [sci-hub, @tor], arXiv: 1903.10833

[peixoto-scalable-2024]

Tiago P. Peixoto, “Scalable network reconstruction in subquadratic time”, arXiv: 2401.01404

[peixoto-network-2024] (1,2,3,4)

Tiago P. Peixoto, “Network reconstruction via the minimum description length principle”, arXiv: 2405.01015