Dynamics on Modular Networks with Heterogeneous Correlations

I. INTRODUCTION

• The ensemble assigns each module its own joint degree-degree distribution and specifies inter-module edges through a joint degree-degree distribution for each module pair.Source: §I · p. 1
• The model generalizes the configuration model, Pk,k′ networks, uncorrelated Ei,i′ networks, and LFR benchmark networks.Source: §I · p. 1
• The analytical method targets a broad class of binary-state dynamics, and simulations are used to test its accuracy on synthetic and real-world networks.Source: §I · p. 1

II. DEFINITION AND CONSTRUCTION OF P i,i′

• P i,i′k,k′ is represented as an M-by-M block matrix whose diagonal blocks encode within-module degree correlations and off-diagonal blocks encode inter-module connections.Source: §II · p. 2
• The model can be constructed by calculating edge-type counts and randomly grouping the corresponding edge ends by module and degree.Source: §II · p. 2
• Its defining P i,i′k,k′ distribution can also be obtained from an existing network by rewiring edges while preserving that distribution.Source: §II · p. 2

III. ANALYTICAL CALCULATION OF DYNAMICS ON P i,i′

• Neighbor activation probabilities are obtained from the degree- and module-specific edge connections encoded by the network matrix.Source: §III · pp. 2–3
• The method supports binary dynamics with module-specific, non-decreasing response functions Fi(m, k), including bond percolation, site percolation, Watts threshold dynamics, and k-core sizes.Source: §III · p. 2
• The recurrence tracks the probability that a degree-k node in module i is active at each discrete time step, conditional on at least one inactive neighbor.Source: §III · p. 3
• Equation (3) models an initially inactive node becoming active when it has sufficiently many active neighbors while one neighbor remains inactive.Source: §III · p. 3
• The resulting equations calculate active-node fractions separately for each module and for the aggregate network.Source: §III · p. 3

IV. APPLICATION TO Pk,k′ AND Ei,i′ NETWORKS

• The equations apply directly to Pk,k′ networks and to Ei,i′ networks after constructing an uncorrelated P i,i′k,k′ matrix from their mixing matrices and degree distributions.Source: §IV · p. 3
• The framework is expected to match numerical simulations better for P i,i′k,k′ networks because they provide more information about network structure.Source: §IV · p. 3
• P i,i′k,k′ theory can distinguish nodes with the same degree in different modules, unlike single-module Pk,k′ theory.Source: §IV · p. 3
• Unlike Ei,i′ theory, P i,i′k,k′ theory captures degree-degree correlations both within modules and between modules.Source: §IV · p. 3

V. EXAMPLES OF DYNAMICS ON MODULAR SYNTHETIC Ei,i′ NETWORKS

• The examples compare computational results with predictions from the analytical equations using Ei,i′ networks, with more complicated specifications reserved for later examples.Source: §V · p. 4
• The synthetic network has two modules of 10,000 nodes each, with module 1 containing only degree-4 nodes.Source: §V · p. 4
• Module 2 contains degree-4 and degree-12 nodes in a 1:1 proportion, and the mixing matrix specifies inter-module connections.Source: §V · p. 4
• The resulting graph is a (k1, k2)-regular graph in which nodes have one of two possible degrees.Source: §V · p. 4

A. Bond Percolation

• The analytical calculation uses bond occupation probability p as the response-function parameter for predicting GCC size from network degree correlations.Classical bond percolation assumes the same occupation probability in every module; module-specific probabilities instead define semidirected bond percolation.Source: A. Bond Percolation · p. 4
• The test network has 10,000-node modules, with one module containing only degree-4 nodes and the other mixing degree-4 and degree-12 nodes equally.This construction exposes differences between degree classes that are hidden when modules are averaged together.Source: A. Bond Percolation · p. 5, Figure 3 · p. 5
• The P^i,i′_k,k′ theory is indistinguishable from simulations and predicts the GCC correctly, while P_k,k′ fails on the two-module example.The network contains degree-4 nodes in both modules, but their neighboring degree distributions differ.Source: A. Bond Percolation · pp. 4–5
• The GCC curve is dominated by degree-12 nodes in module 2, shifts slightly from 1/11, and shows a module-1 step at p = 1/3.The shift reflects degree-4 nodes in module 2, while the step corresponds to percolation of module 1.Source: A. Bond Percolation · p. 5

B. Site Percolation

• Site percolation removes nodes and their incident edges, with occupied nodes activating when they have at least one active neighbor.The response function assigns occupation probability Q^i_k to degree-k nodes in module i.Source: B. Site Percolation · p. 5
• Figure 4 plots GCC size against the occupation probability of degree-12 nodes on the same E_i,i′ network used for bond percolation.The first module contains only degree-4 nodes, while the second mixes degree-4 and degree-12 nodes.Source: Figure 4 · p. 5, B. Site Percolation · p. 5
• The P^i,i′_k,k′ theory approximates direct site-percolation calculations very well, whereas P_k,k′ theory does not.The comparison varies the occupation probability of degree-12 nodes in module 2 while fixing all other occupation probabilities at one.Source: B. Site Percolation · p. 5

C. Watts Threshold Model

• The Watts model activates a degree-k node when at least a threshold fraction R of its neighbors are active, with response function C_i(m/k).C_i is the cumulative distribution function of thresholds in module i; for a common threshold, the response reduces to the corresponding cumulative threshold rule.Source: C. Watts Threshold Model · pp. 5–6
• The P^i,i′_k,k′ theory captures direct simulations accurately, whereas P_k,k′ theory does not, showing that modular structure matters for the Watts process.The experiment activates 0.1% of nodes initially and varies a common threshold R on the same network used for the percolation tests.Source: C. Watts Threshold Model · p. 6
• The module-resolved curve has transitions at R = 1/4 and R = 1/6, corresponding respectively to degree-4 activation in module 1 and subsequent degree-12 cascades.At R = 1/4, module 1 cascades while module 2 remains inactive; at R = 1/6, both modules eventually cascade.Source: C. Watts Threshold Model · p. 6
• The averaged P_k,k′ theory produces only the R = 1/4 transition because it mixes degree-4 and degree-12 nodes into one module.It consequently predicts an erroneous activation cascade among degree-12 nodes surrounded by more active degree-4 nodes than in the actual network.Source: C. Watts Threshold Model · p. 6

A. Synthetic P_i,i′

• The P_i,i′ network has two modules: module 1 contains degree-3 nodes, while module 2 contains degree-3 and degree-11 nodes with structured cross-module connections.Degree-3 nodes in module 2 connect exclusively to degree-11 nodes, while the other node types have distinct connection ratios.Source: A. Synthetic P i,i′ · pp. 6–7
• P_i,i′ theory is required for networks whose degree-degree structure cannot be accurately represented by either P_k,k′ or E_i,i′ models.The constructed matrix provides an example that simpler single-module and uncorrelated multi-module descriptions cannot capture.Source: A. Synthetic P i,i′ · p. 6
• P_i,i′ theory is indistinguishable from simulations across bond-occupation probabilities, while E_i,i′ predicts p_c too low and P_k,k′ predicts it too high.The theory agrees for all values of the bond-occupation probability, including the percolation threshold.Source: A. Synthetic P i,i′ · p. 7
• P_i,i′ theory better matches Watts-threshold dynamics than P_k,k′ and E_i,i′ theories on the same heterogeneous modular network.The comparison uses temporal activation from an initial 0.1% seed set in module 2 with threshold R = 0.18.Source: A. Synthetic P i,i′ · p. 7

B. LFR Benchmark Networks

• LFR networks assign power-law node degrees and community sizes, with mixing parameter µ controlling the fraction of edges connecting different communities.The model also uses γ, β, kmax, cmin, and cmax to specify degree and community-size distributions.Source: B. LFR Benchmark Networks · p. 7
• LFR networks are atypical E_i,i′ examples because each degree-k node has an externally connected-edge count concentrated near µk rather than binomially distributed.This additional constraint helps explain why E_i,i′ predictions do not exactly reproduce LFR behavior.Source: B. LFR Benchmark Networks · p. 8
• All theories accurately predict GCC size on the examined LFR network as bond-occupation probability varies.The agreement is attributed to statistically similar modules with the same degree distribution and inter-module connection process.Source: B. LFR Benchmark Networks · p. 8, Figure 9 · p. 8
• The Watts model exhibits two transitions: increasing µ enables propagation beyond the seed module, whereas sufficiently large µ prevents activation even within that module.The first transition reflects enough inter-module connectivity for spread; the second reflects seed-module nodes having too many edges to inactive external nodes.Source: B. LFR Benchmark Networks · p. 8, Figure 10 · p. 8
• P_i,i′ and E_i,i′ theories are reasonable but imperfect on LFR networks, with discrepancies attributed to finite size and ensemble differences.On corresponding P_i,i′ networks, the theory performs very well, while P_k,k′ misses modular propagation and predicts almost no activation beyond seeds.Source: B. LFR Benchmark Networks · p. 8

• 7000 conjoining edges make the network essentially random and render all three theories adequate, whereas too few edges produce finite-size failures.Source: C. Conjoined Real-World Networks · p. 9
• P_i,i′k,k′ theory captures bond-percolation properties better than Pk,k′, Ei,i′, and P_i,i′k,k′ alternatives on the conjoined Caltech-protein network.Source: C. Conjoined Real-World Networks · p. 9
• All theories accurately predict GCC size on the multi-university Facebook network, although Ei,i′ performs slightly worse than the others.Source: C. Conjoined Real-World Networks · p. 9
• The multi-university network contains substantial inter-university connectivity, with 290278 cross-module edges comprising about 11.2% of all edges.Source: C. Conjoined Real-World Networks · p. 9
• For Watts threshold dynamics, P_i,i′k,k′ theory accurately predicts both initial temporal evolution and final active fraction, while other theories can diverge substantially.Source: C. Conjoined Real-World Networks · pp. 9–10
• Identical-threshold Watts dynamics on real-world networks remain difficult to fit analytically, and all theories predict slower activation than observed in this example.Source: C. Conjoined Real-World Networks · p. 10

• The ensemble permits different within-module degree-degree correlations and pair-specific joint degree-degree distributions between modules.Source: §VII · p. 10
• The P_i,i′k,k′ model generalizes the configuration, Pk,k′, Ei,i′, and LFR network models.Source: §VII · p. 10
• Synthetic and real-world examples show that P_i,i′k,k′ theory can explain dynamics that Pk,k′ and Ei,i′ models cannot capture.Source: §VII · p. 10
• The analytical approach allows dynamics to differ across modules and may support benchmarking community-detection methods.Source: §VII · p. 10

• Updating a fraction τ of nodes interpolates between synchronous updating at τ = 1 and completely asynchronous updating at τ = 1/N.Source: Appendix: Continuous-Time Analytical… · p. 11
• For monotonic dynamics, synchronous and asynchronous updating produce the same final state, although their transient dynamics can differ.Source: Appendix: Continuous-Time Analytical… · p. 11
• Setting Δt = τ provides a common time scale across updating schemes and permits continuous approximations for sufficiently small τ.Source: Appendix: Continuous-Time Analytical… · p. 11
• The resulting ordinary differential equations generate the continuous-theory curves used for the Watts threshold model in Figs. 8 and 13.Source: Appendix: Continuous-Time Analytical… · p. 11