Spatiotemporal signal propagation in complex networks

Spatiotemporal signal propagation in complex networks

A major achievement in the study of complex networks is the realization that diverse systems, from sub-cellular biology to social networks, exhibit universal topological characteristics. Yet, such universality does not naturally translate to the dynamics of these systems, as dynamic behavior cannot be uniquely predicted from topology alone. Rather, it depends on the interplay of the network’s topology with the dynamic mechanisms of interaction between the nodes. Hence, systems with similar structure may exhibit profoundly different dynamic behavior. We therefore seek a general theoretical framework to help us systematically translate topological elements into their predicted dynamic outcome. Here, we offer such a translation in the context of signal propagation, linking the topology of a network to the observed spatiotemporal spread of perturbative signals across it, thus capturing the network’s role in propagating local information. For a range of nonlinear dynamic models, we predict that the propagation rules condense into three highly distinctive dynamic regimes, characterized by the interplay between network paths, degree distribution and the interaction dynamics. As a result, classifying a system’s intrinsic interaction mechanisms into the relevant dynamic regime allows us to systematically translate topology into dynamic patterns of information propagation.

Diversity in signal propagation  

Networks represent a powerful tool to visualize and predict information propagation in complex systems, from viral spread, to neuronal or biochemical signals. In some cases the network topology exposes the natural geometry of the propagation, with the more distant nodes impacted at later times. In others, however, the translation is less transparent, a consequence of the diverse forms of nonlinear interactions that may take place between the nodes. Therefore, advances are often system dependent, with each dynamics warranting its own dedicated analysis, allowing limited insight to transfer across domains. To overcome this diversity we seek general tools to translate a network’s topology into predictions on its observed propagation patterns—often characterized through the system’s response to local perturbations. A single component is perturbed, and as a result all other network components are impacted, leading, in the long term, to a cascade of responses, whose size and penetration depth help characterize the network’s dynamic behavior. Such predictions, however, tracking the system’s long-term response, provide little insight into its temporal propagation, leaving open time-related questions, such as, how much time it will take the cascade to build up, which nodes respond first and which later, or what percentage of nodes are impacted at any given point in time. We address these questions here by predicting an individual node’s degree-dependent response time, allowing us, by piecing together all local responses, to predict the complete network spatiotemporal propagation patterns. Analytically tractable for a family of commonly encountered steady-state dynamics, our formalism allows us to systematically translate topology into signal propagation, in effect, reestablishing networks as predictors of information spread. Diversity in signal propagation to illustrate the challenge we begin with an artificial N-node protein interaction network Aij, which we use to simulate the propagation of biochemical signals in a sub-cellular environment (Fig. 1a). Denoting the abundance of the ith protein by xi (t), we capture the system’s dynamics through x_i = -B_ix_i^a +  ∑ⁿ_j=1A_ij x H(x_j) , in which the first term describes a protein’s self-dynamics, for example, degradation (a=1), dimerization (a=2) or a more complicated chain reaction (fractional a, Supplementary Section 2.2), and the second term depicts i’s regulation by its interacting partners, often captured by a Hill function of the form H( x_j)  = x_j^h  ∕ (1+x_j^h ). Changes in the abundance of one protein propagate, through Aij, to affect the abundance levels of all other proteins, representing a spread of biochemical information in the system3 . Hence we initiate a biochemical signal by introducing a perturbation Δxj  to the steady-state abundance of the source j (black node), and then track its propagation, as it penetrates the network, for illustration, focusing on four selected target nodes i=1, 2, 3, 4 (coloured nodes). In Fig. 1d–i we track the resulting propagation under different dynamics, by controlling the values of a and h and monitoring the sequence of responses Δxi (t) of the four selected target nodes. We also measured the propagation times T(j→i) for the signal in j to travel to i through

Δ = x t( ( T j → = i x )) ηΔ →( ) t ∞ (1)

i namely T(j→i) represents the time when i has reached an η-fraction of its final response to the j-signal (typically setting η~1/2, the half-life of i’s response, Supplementary Section 3.2). For a=h=1 the propagation seems predictable—first impacting the nearest neighbors 1, 2, then reaching the farther nodes. This clean picture, however, is violated as the dynamics is changed. For instance, under a=1/2 we find that the nearest neighbor 1 is impacted significantly later than all other nodes, seemingly skipped over by the more distant 3 and 4 (Fig. 1f,g). The opposite occurs when we set a=3, h=1/2, as now 1 becomes the first to receive the signal, preceding 4 by two orders of magnitude Propagation of signals in a complex network environment. To model network dynamics we use a two-layer description. a, The first layer is the topology, captured by the weighted network Aij. b, The second layer is the system’s dynamics, designed to capture the inner mechanisms driving the system’s observed behaviour. Here, for illustration, we consider protein interactions in sub-cellular networks: proteins are depleted at a rate proportional to (setting Bi= 1 for all i) and activated by their network neighbors via the Hill function H( ) x  = ∕ (1 + x )  The dynamic behavior of the system (a,b) can be captured through its patterns of information, or signal, propagation. A local signal in the form of an activity perturbation Δxj  is applied to the source node j (black), spreading through the network to impact all other nodes 1, 2, …. c, The spatiotemporal propagation of this signal is captured by the response of all target nodes, i, via Δxi (t), which we normalize to satisfy Δxi (t→∞) = 1. The propagation time, T(j→i), captures the time for Δxi  to reach an η-fraction of its final response, here illustrated for η= 1/2 (half-life). We tested the propagation from j under different dynamics, varying the exponents a and h. d, The response Δxi (t) versus t, as obtained for a=h= 1 for nodes i= 1, 2, 3 and 4 (coloured in Aij). 1 and 2 respond almost simultaneously, hence their overlapping plots. e, The propagation time T(j→i) for these four nodes. As expected we find that the signal first reaches the neighbouring nodes 1,2, then propagates to impact the next neighbours 3, 4. f,g, Changing the value of a to 1/2 has a pronounced impact on the propagation pattern, as now 1 seems to be skipped by the more distant 3 and 4. Moreover, T(j→ 1) is now two orders of magnitude greater than T(j→ 2), despite being at equal distance from j, and in sharp contrast with their previously simultaneous responses. h,i, The propagation sequence is further scrambled under a= 3, h= 1/2, as are the propagation timescales, with 1 now preceding the others by two orders of magnitude. Hence, Aij alone is insufficient to predict information spread, as indeed, changes in the dynamic equation in b translate to profound and seemingly unpredictable consequences on the observed propagation.