Feature-Based Network Construction: From Sampling to What-if Analysis

TL;DR

FBNC tackles the inverse problem of constructing weighted digraphs that exactly realize chosen structural features by framing hard-constrained, gradient-based network construction. It introduces implicit regularization via steepest feasible descent, enabling what-if analyses that minimally perturb an initial network while achieving new feature values, and handles both bounded and Markovian weight regimes. The framework is demonstrated through hard-constraint sampling (including reconstructing confidential networks) and what-if scenarios in social and financial networks, highlighting differences from traditional ERGM-based weak sampling. Its contributions include a rigorous descent-theory for feasible directions, efficient computation of steeps for hypercube and Markov graphs, and practical pathways for network reconstruction and policy-oriented what-if analysis.

Abstract

Networks are characterized by structural features, such as degree distribution, triangular closures, and assortativity. This paper addresses the problem of reconstructing instances of continuously (and non-negatively) weighted networks from given feature values. We introduce the gradient-based Feature-Based Network Construction (FBNC) framework. FBNC allows for sampling networks that satisfy prespecified features exactly (hard constraint sampling). Initializing the FBNC gradient descent with a random graph, FBNC can be used as an alternative to exponential random graphs in sampling graphs conditional on given feature values. We establish an implicit regularization approach to the original feature-fitting loss minimization problem so that FBNC achieves a parsimonious change in the underlying graph, where the term "implicit" stems from using appropriate norms in the very construction of the FBNC gradient descent. In constructing the implicit regularization, we distinguish between the case where weights of a link can be chosen from a bounded range, and, the more demanding case, where the weight matrix of the graph constitutes a Markov chain. We show that FBNC expands to "what-if analysis" of networks, that is, for a given initial network and a set of features satisfied by this network, FBNC finds the network closest to the initial network with some of the feature values adjusted or new features added. Numerical experiments in social network management and financial network regulation demonstrate the value of FBNC for graph (re)construction and what-if analysis.

Figures (17)

• Figure 1: Schematic illustration of two networks obtained through adjusting link-weights so that target feature values are met. Both networks ${w}_1$ and ${w}_2$ depend on the initial network given to a gradient-based algorithm.
• Figure 2: Schematic illustration of two networks obtained through adjusting link-weights so that target feature values are met while applying explicit and implicit regularization. Here, $\hat{w}_{\mathrm{explicit}}$ is the result of using an explicit regularization and does not reach the target set. Instead, an implicitly regularized $\hat{w}_{\mathrm{implicit}}$ does reach the target set.
• Figure 3: Paths of algorithm (\ref{['eq:descentalgo']}) for the explicitly regularized loss function $\hat{J}_2 (w)$ using the $L^2$-descent, i.e., paths of ${\cal F}_{ ([0,1]^2 ,2, \hat{J}_2 )} ( w)$. It follows that ${\cal F}_{ ([0,1]^2, 2 , \hat{J}_2 )} ( 0,0 ) = (0.187, 0.374)$, denoted by the orange dot.
• Figure 4: Paths of algorithm (\ref{['eq:descentalgo']}) for basic loss function $J(w)$ using implicit regularization via the $L^2$-descent, i.e., paths of ${\cal F}_{ ( [0,1]^2, 2 , J ) } ( w )$. It follows that ${\cal F}_{ ([0,1]^2, 2 , J_2 )} ( 0,0 ) = (\frac{1}{5}, \frac{2}{5})$, denoted by the orange dot.
• Figure 5: Paths of algorithm (\ref{['eq:descentalgo']}) for the explicitly regularized loss function $\hat{J}_1(w)$ using the $L^2$-descent, i.e., paths of ${\cal F}_{ ( [0,1]^2, 2 ,\hat{J}_1 ) } ( w )$. It follows that ${\cal F}_{( [0,1]^2, 2 , \hat{J}_1 )} ( 0,0 ) = (0, \frac{3}{8})$, denoted by the orange dot.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Example 1
• Example 2