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A measure of individual role in collective dynamics
Konstantin Klemm, M. Angeles Serrano, Victor M. Eguiluz, Maxi San Miguel
TL;DR
Characterizing node importance in collective dynamics requires accounting for the interplay between network topology and the dynamics. The paper defines dynamical influence as a centrality measure for diverse network processes, including epidemic, Ising, and diffusive dynamics. For critical epidemic spreading and the Ising model, dynamical influence predicts spreading capabilities; its rank order correlation with driving efficiency is 0.97 versus 0.66 with degree ratio.
Problem
Characterizing node importance in collective dynamics requires accounting for the interplay between network topology and the dynamics.
Source: Discussion · p. 8Method
The paper defines dynamical influence as a centrality measure for diverse network processes, including epidemic, Ising, and diffusive dynamics.
Source: Discussion · p. 8Results
For critical epidemic spreading and the Ising model, dynamical influence predicts spreading capabilities; its rank order correlation with driving efficiency is 0.97 versus 0.66 with degree ratio.
Source: Discussion · p. 8, Efficient driving of complex systems · p. 8Takeaways & Limitations
DI provides a proxy for identifying target nodes for controlling global behavior in the studied systems.
Source: Efficient driving of complex systems · p. 8, Discussion · p. 8The nodes with greatest dynamical influence are generally not the same as those targeted to raise the epidemic threshold through vaccination.
Source: Epidemic spreading · p. 6
Abstract
from arXiv · showhide
Identifying key players in collective dynamics remains a challenge in several research fields, from the efficient dissemination of ideas to drug target discovery in biomedical problems. The difficulty lies at several levels: how to single out the role of individual elements in such intermingled systems, or which is the best way to quantify their importance. Centrality measures describe a node's importance by its position in a network. The key issue obviated is that the contribution of a node to the collective behavior is not uniquely determined by the structure of the system but it is a result of the interplay between dynamics and network structure. We show that dynamical influence measures explicitly how strongly a node's dynamical state affects collective behavior. For critical spreading, dynamical influence targets nodes according to their spreading capabilities. For diffusive processes it quantifies how efficiently real systems may be controlled by manipulating a single node.
Results
The paper defines dynamical influence (DI) from the dynamics of a networked system and tests its use across spreading, Ising, voter, and oscillator models. Its predictive value depends on the process and regime, while in diffusion it also characterizes how initial states shape the final state.
Defining dynamical influence
- DI is the leading left-eigenvector weight that quantifies how strongly an element’s initial state affects the system’s long-time state.Power iteration connects DI to counts of increasingly long walks departing from each node.Source: Defining dynamical influence · p. 2
Epidemic spreading
- At β=βc, DI’s rank correlation with SIR spreading efficiency is 0.97, versus 0.86 for degree, 0.82 for shell index, and 0.79 for betweenness.These values are for Zachary’s karate-club network.Source: Figure 1 · p. 2, Epidemic spreading · p. 4
- Across larger real-world and Barabási-Albert networks, DI predicts SIR spreading efficiency well near β/βc ≈ 1 and outperforms other measures across a broad interval of infection probabilities.The reported interval includes average outbreaks reaching up to 10% of network nodes.Source: Epidemic spreading · p. 4
- Away from criticality, degree predicts spreading efficiency better; at criticality, length-2 or length-3 walk counts predict it about as well as DI.Results also remain qualitatively similar for stochastic-recovery SIR and SIS dynamics.Source: Epidemic spreading · pp. 4–5
- When degree and DI are homogeneous, betweenness can resolve small spreading differences, while shell-index degeneracy limits its ability to distinguish nodes.The authors caution that efficient spreaders are generally not the same nodes selected to raise the epidemic threshold through vaccination.Source: Epidemic spreading · p. 6
Ising model
- At the Ising order-disorder transition, Ising spreading efficiency correlates more strongly with DI than with the other tested centrality measures.The comparison is made at time lag τ = 10.Source: Figure 4 · p. 5, Ising model · p. 6
Diffusive processes: the voter model
- In voter dynamics, link update gives every node equal DI and conserves average magnetization, whereas node update makes DI proportional to degree and conserves weighted magnetization.DI also weights each node’s initial state in the probability of reaching the all-positive absorbing state.Source: Diffusive processes: the voter model · pp. 6–7
Efficient driving of complex systems
- For phase oscillators on a macaque-cortex network, driving-efficiency rank correlation is 0.97 with DI, versus 0.66 with degree ratio, −0.14 with shell index, and −0.09 with betweenness.The authors present DI as a proxy for identifying target nodes for controlling global behavior.Source: Efficient driving of complex systems · pp. 7–8
Discussion
Dynamical influence incorporates the interplay of topology and dynamics and applies across many network processes. Its demonstrated uses include identifying influential nodes in nonlinear systems and assessing targets for controlling collective behavior.
Discussion
- Dynamical influence is a centrality measure that accounts for the interplay between topology and dynamics across a wide range of network processes.Its motivation and rigorous analysis use linear systems, while its practical use extends to inherently nonlinear systems.Source: Discussion · p. 8
- The measure applies to stochastic equilibrium and nonequilibrium models, as well as deterministic continuous-state systems, including Kuramoto and chaotic Rössler dynamics.The demonstrated examples include the Ising, epidemic, and voter models.Source: Discussion · p. 8
- For diffusion, dynamical influence assesses single-node impacts on the asymptotic homogeneous state and serves as a proxy for driving efficiency in real networks.Identifying nodes to force toward specific states has potential applications from social influence to biomedical responses.Source: Discussion · p. 8
Methods
The methods describe epidemic dynamics and spreading efficiency, define rank-order correlation for comparing node rankings, and distinguish degree from in- and out-degree in directed networks.
Epidemic models
- In SIR and SIS models, spreading efficiency is the average outbreak size when a node is the sole infected seed; SIS-infected nodes can become susceptible again, unlike SIR-infected nodes.Outbreak size counts nodes infected at least once before the system reaches an absorbing configuration.Source: Epidemic models · p. 8
Rank order correlation
- Rank-order correlation is the Pearson correlation of two vectors’ rank vectors, ranging from −1 to +1 for strictly decreasing and increasing relations, respectively.A value of +1 or −1 occurs if and only if the vectors are in a strictly increasing or decreasing relation.Source: Rank order correlation · p. 8
Degree and degree ratio
- A node’s degree is the number of nodes it connects to, with in-degree and out-degree distinguished in directed networks.Source: Degree and degree ratio · p. 9
SUPPLEMENTARY MATERIAL
Supplementary analyses examine short-walk predictors and vaccination strategies for epidemic spreading, then test dynamical influence against driving efficiency in oscillator systems.
Epidemic spreading
- The number of walks of length l ≥2 departing from a node is a good predictor of spreading efficiency in the critical regime, even for moderate l.Source: Epidemic spreading · p. 11
- Supplementary tests compare SIR predictors across recovery probabilities and report analogous SIS results; vaccination results concern critical SIR spreading at β = 1/λmax.The supplied passages describe these comparisons but do not state a comparative outcome for the recovery-probability or SIS tests.Source: Epidemic spreading · p. 11, Figure S5 · p. 16
- Across four networks, spreading efficiency varies among top spreaders sharing the maximum shell index, including by more than fivefold among 26 top spreaders in the Internet graph.The cited cases also include top spreaders that do not attain the maximum shell index in the email network.Source: Epidemic spreading · p. 11
- Degree and betweenness centrality reduce outbreak size most when used to select vaccinated nodes, whereas shell index and dynamical influence provide less guidance.All four measures perform better than random assignment; the discussion attributes the vaccination distinction to removals raising the epidemic threshold and making the remaining network subcritical, where degree best predicts spreading.Source: Epidemic spreading · p. 11
Driving coupled oscillators
- Kuramoto tests compare driving-efficiency rankings with Laplacian influence, degree ratio, and betweenness on randomly grown networks with geometric or scale-free out-degree.The passages describe the network variants and comparison but do not report the correlation outcomes.Source: Driving coupled oscillators · p. 11, Figure S6 · p. 17
- Driving efficiency has rank-order correlations of 0.957 ± 0.010 with dynamical influence and 0.646 ± 0.012 with degree ratio, reported as mean values and standard deviations over 10 independent realizations.Source: Driving coupled oscillators · p. 11