Classes of critical avalanche dynamics in complex networks

Dynamical processes exhibiting absorbing states are essential in the modeling of a large variety of situations from material science to epidemiology and social sciences. Such processes exhibit the possibility of avalanching behavior upon slow driving. Here, we study the distribution of sizes and durations of avalanches for well-known dynamical processes on complex networks. We find that all analyzed models display similar critical behavior, characterized by the presence of two distinct regimes. At small scales, sizes, and durations of avalanches exhibit distributions that are dependent on the network topology and the model dynamics. At asymptotically large scales instead—irrespective of the type of dynamics and of the topology of the underlying network—sizes and durations of avalanches are characterized by power-law distributions with the exponents of the standard mean-field critical branching process.

Figure 1

Models for avalanche dynamics on networks. The figure serves as a schematic illustration to emphasize differences and similarities between the various dynamical models considered in this paper. The upper-left panel depicts an initial configuration, where a single node is in the “active” state, whereas all other nodes are inactive. The rest of the panels display configurations reachable after one elementary reaction. Depending on the model, elementary reactions are triggered by randomly selected nodes, edges, or both. The elements triggering the elementary reactions are highlighted in the various panels. For clarity of the illustration, we report for each model only two of the possible configurations that can be reached after one elementary reaction. Specifically, configurations appearing in the left panels are reached after a recovery reaction, whereas configurations appearing in the right panels are obtained after a spreading reaction (in both cases the nodes or links triggering the reaction are highlighted with thick lines). Detailed definitions of all dynamical models can be found in Ref. [26].

Figure 2

Avalanche size in synthetic undirected scale-free networks. The degree exponent is $\gamma =2.1$. For clarity, we report results only for three dynamical models: (a) CIC, (b) IP, and (c) SIS. Results for all other models are in Ref. [26]. For each model and network, we measure, by means of numerical simulations, the probability distribution $P\left(S\right)$ of the total number of spreading events $S$ per avalanche. The dashed red line corresponds to MF exponents, i.e., Eq. (2); the full black line indicates anomalous BP scaling i.e., Eq. (4).

Figure 3

Avalanche size in synthetic directed networks for three of the considered models: (a) CIC, (b) IP, and (c) SIS. The description of the figure panels is as in Fig. 2. The networks analyzed here are instances of the directed configuration model with out-degree exponent $\gamma =2.1$ and maximum out-degree $k_{\max }^{\mathrm{out}}=N-1$.

Figure 4

Avalanche size in real-world networks. We consider the following networks: (a) undirected graph representing a snapshot of the Internet at the Autonomous system level [53]; (b) directed Twitter network of the Spanish 15M movement [54]; (c) directed graph representing a portion of the YouTube social network [55]. Different symbols and colors refer to different avalanche dynamical models. The red dashed line represent standard BP critical exponents, while the full black line indicates the power-law decay expected for anomalous BP. Note that the out-degree distributions of these networks are all well modeled by power laws with decay exponent $\gamma =2.1$ (see Ref. [26]).