Copy-Spread-Annihilate Dynamics in Degree-Assortative Networks

Abstract

In many systems, communication proceeds by broadcasting rather than single source-target routing, but network structures that maximize signal lifetime are not well understood. Degree correlations are known to influence robustness and spreading, yet their effect on signal persistence has remained unclear. Here we introduce Copy-Spread-Annihilate dynamics, a minimal synchronous broadcasting model with annihilation. We show that signal lifetimes vary non-monotonically with assortativity and are maximized near neutral assortativity, where hub-driven amplification is strong but annihilation via short cycles is still limited. Applying this framework to the mouse connectome suggests assortativity as a structural control parameter for broadcast signal persistence in brain-like and other complex networks.

Figures (10)

• Figure 1: Visualization of Copy-Spread-Annihilate dynamics implemented on a small network, shown in the single injection case for clarity. (a) demonstrates, frame-by-frame, how one message (injected to vertex 1) spreads, and how copies resulting from this single injection interfere with each other. (b) plots the node space versus time (space-time diagram). It illustrates the propagation of the same message and its copies through its lifetime until full self-annihilation. In this particular case, the initial message created 9 different walks that achieved the maximum distance $\tau_M = 6$.
• Figure 2: (a) Average message lifetime as a function of assortativity shown in dashed curves. Shaded areas indicate standard deviations. Results are collected from five different networks of 100 vertices, generated using Barabási-Albert (BA) algorithm, and $m$ = 2, 3, 4 as parameters of attachment (in green, blue, and red respectively). The spectrum of assortativities are achieved using a degree-based edge-switching algorithm trusina2004hierarchy in order to preserve degree sequence. (b.) Using the same graphs as in (a), the average neighbors' degrees of hub vertices, $k_{hn}$, and the number of 4-cycles, $C_4(G)$, are presented in dotted and solid curves, respectively. See supplementary material Sec. S1 for simulation details.
• Figure 3: Vertex participation frequency in walks that are above the 90th percentile in length, produced using five different BA graphs with 100 vertices and $m$ = 3. Vertices are sorted by degree. (a) high-degree vertices are preferred in disassortative graphs ($-0.85<r< -0.65$), (b) shows the vertex usage when assortativities are neutral ($-0.1<r< 0.1$), (c) low-degree vertices are preferred in assorted graphs ($0.6<r<0.8$). See supplementary material Sec. S1 for simulation details.
• Figure 4: Average message lifetime as a function of assortativity on empirical mouse connectome (asterisk) oh2014mesoscale and surrogate graphs (curve). Five sets of surrogate graphs are generated using the same edge-switching algorithm as in Fig. \ref{['fig:LifeTime1']} to achieve a spectrum of assortativities. The shaded area indicates standard deviations across five repetitions of all graphs.
• Figure S1: Numerical implementation of the Copy–Spread–Annihilate dynamics summarized in a flow chart.