Propagation dynamics on networks featuring complex topologies

Laurent Hébert-Dufresne, Pierre-André Noël, Vincent Marceau, Antoine Allard, and Louis J. Dubé

Phys. Rev. E 82, 036115 – Published 27 September 2010

Abstract

Analytical description of propagation phenomena on random networks has flourished in recent years, yet more complex systems have mainly been studied through numerical means. In this paper, a mean-field description is used to coherently couple the dynamics of the network elements (such as nodes, vertices, individuals, etc.) on the one hand and their recurrent topological patterns (such as subgraphs, groups, etc.) on the other hand. In a susceptible-infectious-susceptible (SIS) model of epidemic spread on social networks with community structure, this approach yields a set of ordinary differential equations for the time evolution of the system, as well as analytical solutions for the epidemic threshold and equilibria. The results obtained are in good agreement with numerical simulations and reproduce the behavior of random networks in the appropriate limits which highlights the influence of topology on the processes. Finally, it is demonstrated that our model predicts higher epidemic thresholds for clustered structures than for equivalent random topologies in the case of networks with zero degree correlation.

Received 21 May 2010

DOI:https://doi.org/10.1103/PhysRevE.82.036115

Authors & Affiliations

Laurent Hébert-Dufresne, Pierre-André Noël, Vincent Marceau, Antoine Allard, and Louis J. Dubé

Département de Physique, de Génie Physique, et d’Optique, Université Laval, Québec, Québec, Canada G1V 0A6

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 82, Iss. 3 — September 2010

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) Schematization of the particular topology studied in this paper. An open mark represents a susceptible individual; a shaded one, an infectious; and a black one, a group (or clique). The topology is constructed by allowing individuals to belong to a given number of cliques where they can be linked to other participants (solid lines) and then randomly assigning random exterior neighbors (dotted lines). Note that in the formalism, the cliques are differentiable by their exact population and state, while the precise connections between them remain unspecified and they are simply linked to a mean field.Reuse & Permissions
Figure 2
(Color online) Function $F (ξ^{*})$ is shown in shade on the topology defined in Eq. (32) for two different normalized propagations rates: $λ = 0.02$ in dotted line (under the threshold; no solution for $ξ^{*} > 0$ ) and $λ = 0.1$ in solid line (epidemic). The black solid line is the curve of slope 1, $F (ξ^{*}) = ξ^{*}$ .Reuse & Permissions
Figure 3
(Color online) Degree distribution in the infinite network limit of the chosen topologies: Eq. (33) is shown by a solid shaded line while Eq. (35) is shown by a dotted black line. Note the periodic local maxima corresponding to each $m$ value.Reuse & Permissions
Figure 4
(Color online) Comparisons of analytical and numerical results on a network defined by Eq. (32) using normalized dynamics ( $t \to r t$ and $λ = τ / r$ ). (a) Analytical stable states (curves) and epidemic thresholds (vertical lines at $λ_{c}^{CS} = 3.54 \times 10^{- 2}$ and $λ_{c}^{ERN} = 3.44 \times 10^{- 2}$ ). (b) Time evolution (curves) and analytical equilibrium (horizontal line) for $λ = 0.5$ . On both figures, the results are shown in solid shade for the CS and in dotted black line for the ERN. Numerical results are presented by markers and are averaged over 20 000 networks of 25 000 nodes. The standard deviation is smaller than the marker size.Reuse & Permissions
Figure 5
(Color online) Comparison between analytical and numerical results on a network with general community structure defined in Eq. (34) for a SIS model of propagation dynamics of parameter $λ = τ / r = 0.5$ under normalized time $t \to r t$ . (a) Time evolution of the global state (community structure in solid shade and equivalent random network in dotted black) and (b) time evolution for cliques of sizes 10, 15, 20, 25, 30, and 35 (lowest to highest curves). All numerical results are obtained via MC simulations on over 20 000 networks of 25 000 nodes and are presented by their mean value. Analytical predictions for the stable states are shown in horizontal dotted lines in both figures. Note that the deviation from the predictions is bigger for the smallest cliques than for the larger ones. This is a consequence of the mean-field description which is more accurate for large systems (or, in this case, subsystems) for which standard deviations are of lesser relative importance.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics