Quantifying Transient Dynamics in Heterogeneous Networks under Various Inputs

TL;DR

This work develops a general framework for quantifying transient dynamics in heterogeneous networks under diverse inputs by combining NSDD structure with Neumann-series expansions. It yields explicit, interpretable metrics for response strength and timing across constant, pulse, square, and white-noise inputs, linking node-level transients to directed walks on the network. The authors derive walk-based decompositions and motif-level contributions, revealing how mean degree and degree-variance shape propagation speed and amplification, and how chain, homogeneous-in-degree, and heterogeneous-in-degree topologies produce distinct scaling laws. The approach unifies deterministic and stochastic responses, extends conventional spectral analyses to finite, structured networks, and offers design principles for optimizing transient propagation in real-world networks such as neural and infrastructure systems.

Abstract

Understanding how transient dynamics unfold in response to localized inputs is central to predicting and controlling signal propagation in network systems, including neural processing, epidemic intervention, and power-grid resilience. Existing theoretical frameworks typically assume homogeneous network structures and constant or pulse-like inputs, overlooking how heterogeneity in structure and variety of input shape transient responses, often leading to discrepancies between theory and observation. Here, we develop a general theoretical framework that establishes quantitative relationships between the strength and timing of transient dynamics to various inputs in heterogeneous networks. Using a Neumann series expansion, we disentangle the distinct roles of self-dynamics and network structures beyond the scope of standard spectral theory, yielding intuitive and interpretable formulations. We show that node-to-node propagation can be represented as the cumulative effect of all directed walks, each weighted recursively by the self-dynamics of participating nodes. This framework further quantifies how heterogeneity, such as broad degree distributions or additional motifs, amplifies both response strength and time. Our results advance the understanding of transient dynamics across network structures and input types, extend the existing theory to more general settings, and provide practical guidance for optimizing transient responses.

Figures (16)

• Figure 1: Quantification of response strength and time in the general linear network model. (a) Schematic of node-to-node propagation under constant input. A single node receives a constant input, and the resulting transient responses are quantified for any single node in a heterogeneous network. (b) Summary of metrics quantifying node $i$’s response across four input types. Response strength is quantified by amplification and peak response, while response time is characterized by time constants and response times. Metrics marked with asterisks ($*$'s) indicate approximations derived from transcendental equations, as detailed in Appendix \ref{['sec:A']}. Constant input (first column): Amplification ($Z_i$, blue area above curve), peak response ($R_i$, maximum amplitude), time constant ($\tau_i$, rise to $(1 - 1/e)R_i$), and relative propagation time ($t_i$, time to $\eta R_i$). Pulse input (second column): Amplification ($R_i$, blue area under curve), peak response ($\widetilde{P}_i$, maximum amplitude), decay time constant ($D_i$, drop to $\widetilde{P}_i/e$), and time to peak ($\tilde{\tau}_i$). Square input (third column): Amplification ($R_i t_s$, blue area under curve), peak response ($R_i C(t_s, \tau_i)$), decay time constant ($\tau_i$, drop to $1/e$ of peak), and response time ($t_i$). Noise input (last two columns): Autocovariance ($i = j$): Amplification ($Z_{ii}$), peak response ($P_{ii}$, zero-lag), time constant ($\tau_{ii}$, decay to $P_{ii}/e$). Crosscovariance ($i \ne j$): Amplification ($Z_{ij}$), peak response ($P_{ij}$), and peak response time ($t_{ij}$, time to peak).
• Figure 2: Accuracy of estimated response metrics across classical network topologies. Network types from left to right: (i) Chain (directed for deterministic inputs; undirected for noise inputs), (ii) Regular lattice (average degree $\sim 4$), (iii) Erdős--Rényi (ER) random network (edge probability $\sim 0.02$), (iv) Small-world network (average degree $\sim 2$, rewiring probability $\sim 0.5$), (v) Scale-free network (preferential attachment parameter $\sim 1$), and (vi) Geometric network (connection radius $\sim 0.2$). Input types: (a) Constant; (b) Pulse; (c) Square; (d) Noise (for autocovariance); (e) Noise (for crosscovariance). The abscissa (x-axis) is shared across all panels. All networks contain $100$ nodes with uniform parameters: self-decay rate $\beta = 1$ and interaction weights set to $1$. Inputs are applied as follows: to the first node in the chain, randomly assigned in the lattice, and randomly assigned across $100$ independent instances for randomly generated networks (ER, small-world, scale-free, geometric). Time-related metrics (response time, time constant) use relative error $|t_{\text{sim}} - t_{\text{thr}}| / t_{\text{sim}}$ (left bars; $t_{\text{sim}}$: simulated, $t_{\text{thr}}$: theoretical). Strength-related metrics (peak response) use ratio $P_{\text{thr}} / P_{\text{sim}}$ (right bars). Error bars show mean $\pm$ variance across instances. Most time-related errors remain below $10^0$ ($100\%$), while strength ratios cluster near $\sim 1$ (within one order of magnitude), indicating consistent quantitative agreement. Gray labels indicate Spearman’s rank correlation for node-wise ordering preservation, with values close to $1$ reflecting strong rank consistency. In chain-like or sparse networks, nodes whose shortest path from the input node is $\geq 15$ are excluded to avoid numerical artifacts caused by rapid response decay.
• Figure 3: From directed chains to sparse random networks. (a) Schematic relationships: Directed chain topology with uniform self-decay rate $\beta$, uniform interaction weight $\alpha$, and distinct source-target shortest path length $d$. For the leading terms in metrics, strength metrics ($Z$, $R$, $P$) exhibit geometric decay with $d$, while temporal metrics ($\tau$, $t$) show linear path dependence. (b) Combined validation: Theoretical predictions (solid black: analytic leading terms for the chain; dotted: numerical chain simulations) and sparse ER random networks (dashed: ensemble mean of $100$ realizations; shading: $\pm 1$ SD; connection probability of random networks: $\sim 0.02$). Alignment enables direct structural comparison. Parameters: $\beta = \alpha = 1.0$, stimulus duration $t_s = 10$ for square input, input strength $I_0$ normalized to unity in simulation.
• Figure 4: Path length vs. truncation order in homogeneous in-degree networks. (a) Relations for dynamical metric dependence on path length $d$: Peak response $R_{im}$ (constant input) as weighted sums of $(D+\beta)^{-(d+1)} \times A^d$ terms, where $D$ (homogeneous in-degree) and $\beta$ (uniform self-decay) combine multiplicatively. The $A^d$ factor accounts for path multiplicity (all length-$d$ paths between nodes), while temporal metrics are derived from ratio relationships. (b) Truncation order effects: Strength metric ratios (circles, left axis; $\text{Ratio} = {P_{\text{trunc}}}/{P_{\text{sim}}}$) and temporal metric relative errors (circles, left axis; $\text{Error} = {|t_{\text{trunc}} - t_{\text{sim}}|} /{t_{\text{sim}}}$) vs. rank correlations (squares, right axis). Colors denote shortest path lengths (blue: $d=1$, green: $d=2$, red: $d=3$) and values are averaged from $100$ network realizations. Strength metrics require $\geq d$-order truncation (ratio $>0.9$, error $<10\%$, rank correlation $>0.9$); temporal metrics need $\geq(d+1)$-order (error $<10\%$, rank correlation $>0.8$). Crosscovariance $C_{im}^m(\tau)$ truncated separately in $\mathbf{H}$ and $\mathbf{P}_{\infty}$. Parameters: $\beta=10$ (uniform self-decay rate, satisfying $\beta > 2D$); $\alpha=0.1$ (identical interaction weight); $N=100$ (network size); $p=0.08$ (connection probability).
• Figure 5: Constant input propagation along a single path under heterogeneous degree configurations and triangle motifs. Constant input at the first node $m$ of the single path produces propagation laws: ${R(d+1)}/{R(d)} = {A_{d{\to}d+1}} /({\beta + D_{d+1}}),$ and $\quad (\tau(d+1) - \tau(d)) = {1}/({\beta + D_{d+1}})$, where $d$ is the shortest path length. (a) Path-degree means reduce $R,\tau$ while variance enhances them. (b, c) Feedforward/feedback triangles amplify $R_{FF/FB},\tau_{FF/FB} \propto n(\Delta)$ with distinct slopes, where $n(\Delta)$ represents the number of triangular motifs. Dots represent simulations, and dotted lines represent theory. Large triangle-node degrees $D_\Delta$ suppress motifs effects, recovering single-path dynamics $R,\tau$ (rows $2,4$, and $n(\Delta)=1$). Note that the y-axis does not start from zero for better visualization of slope differences. Parameters: self-decay rate $\beta=10$, total path length $D=5$, input strength $I_0 = 10^6$, and unit weight along the chain $A_{d{\to}d+1} = 1$, for all $d$.

Theorems & Definitions (8)

• Lemma 1: All nodal dynamics are positive under positive pulse and constant inputs; under constant inputs, they increase monotonically
• proof
• Lemma 2: Signatures of $\mathbf{H}$ at negative integer powers
• proof
• Lemma 3: Initial decrease of input node under pulse input
• proof
• Lemma 4: Autocovariance and crosscovariance are positive
• proof