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Dynamic information routing in complex networks

Christoph Kirst, Marc Timme, Demian Battaglia

arXiv:1510.05033v1nlin.AOmath.DSphysics.bio-phq-bio.MNq-bio.NC

TL;DR

The paper develops a theoretical basis for information routing through collective dynamical states in complex networks. It shows that local unit features, global topology, and collective state jointly organize information sharing and transfer, including non-local communication in oscillatory modular networks.

  • Problem

    The paper addresses how collective states in complex networks give rise to specific information-sharing and routing patterns.

  • Method

    The approach analyzes delayed mutual information and transfer entropy across oscillatory modular networks at multiple scales, relating routing to local features, topology, and collective dynamics.

  • Results

    Information routing patterns self-organize through the joint action of local unit features, global interaction topology, and collective dynamical state, while local interventions can regulate non-local whole-network routing.

  • Takeaways & Limitations

    Collective dynamical states provide a theoretical basis for regulating information-sharing and routing patterns across scales in complex networks.

  • Takeaways & Limitations

    The discussion includes a derivation simplified under the assumption that, without noise, every community X has a specified property.

Abstract

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Flexible information routing fundamentally underlies the function of many biological and artificial networks. Yet, how such systems may specifically communicate and dynamically route information is not well understood. Here we identify a generic mechanism to route information on top of collective dynamical reference states in complex networks. Switching between collective dynamics induces flexible reorganization of information sharing and routing patterns, as quantified by delayed mutual information and transfer entropy measures between activities of a network's units. We demonstrate the power of this generic mechanism specifically for oscillatory dynamics and analyze how individual unit properties, the network topology and external inputs coact to systematically organize information routing. For multi-scale, modular architectures, we resolve routing patterns at all levels. Interestingly, local interventions within one sub-network may remotely determine non-local network-wide communication. These results help understanding and designing information routing patterns across systems where collective dynamics co-occurs with a communication function.

I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS

The paper routes information on collective dynamical reference states by measuring delayed information sharing and transfer between network units. Because these measures depend on the reference state, manipulating collective dynamics can flexibly change communication patterns.

  • I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS: The network model treats intrinsic dynamics and stochastic external inputs as jointly determining fluctuating signals that carry information.A deterministic reference state is defined in the absence of external signals, while noise drives instantaneous state fluctuations.
  • I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS: Information routing is quantified between network units using time-delayed mutual information and delayed transfer entropy.These measures assess shared and transferred information between node signals and reveal dominant communication direction.
  • I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS: A small-noise expansion conditions the information measures on a specific collective reference state, making its role in routing analytically accessible.Although limited to the vicinity of one reference state, this restriction enables extraction of how that state shapes emergent routing.
  • I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS: The resulting delayed mutual information and transfer entropy depend on the collective reference state, providing a mechanism for changing network communication by manipulating collective dynamics.The paper focuses this general mechanism on oscillatory dynamics.
  • I. INFORMATION ROUTING VIA COLLECTIVE DYNAMICS: The framework develops mechanisms for flexibly changing information routing in networks with oscillatory collective dynamics.The analysis targets oscillatory phenomena commonly observed in networks with communication functions.

II. INFORMATION EXCHANGE IN PHASE SIGNALS

Information routing patterns summarize the strength and direction of information sharing in oscillatory networks. Distinct collective states, intrinsic properties, connectivity changes, and external interventions can produce different network-wide patterns.

  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: Information routing patterns comprise pairwise measures that capture both the strength and directionality of information sharing or transfer.They are constructed from asymmetries in delayed mutual information or delayed transfer entropy and are analogous to functional connectivity patterns.
  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: Multiple oscillatory networks with different physical interactions, topologies, and external inputs support distinct information routing patterns.The framework therefore links routing patterns to both network structure and dynamical inputs.
  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: In a two-oscillator example, asymmetry in delayed mutual-information curves indicates directed information sharing, and the theoretical prediction agrees with numerical data.The directed pattern is summarized graphically by arrow thickness.
  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: Small changes in connection weights can generate different network-wide information routing patterns in larger coupled-oscillator networks.The example uses oscillators close to a Hopf bifurcation.
  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: Switching between collective dynamical states changes information routing patterns in modular neuronal networks.Different initial conditions lead to different collective states and specific changes in the resulting patterns.
  • II. INFORMATION EXCHANGE IN PHASE SIGNALS: Network-wide routing can be modified by changing individual unit properties, system connectivity, or the selected dynamical state of an unchanged structure.These options are illustrated across oscillator and modular-network examples.

III. THEORY OF PHASE INFORMATION ROUTING

The paper derives analytical information-routing measures for weakly coupled oscillatory networks by separating deterministic phase-locked dynamics from stochastic fluctuations. The predictions match numerical estimates and typically yield similar patterns for delayed mutual information and transfer entropy under independent inputs.

  • III. THEORY OF PHASE INFORMATION ROUTING: The theory represents oscillator phases using intrinsic frequencies, phase-difference coupling functions, and stochastic external inputs.For weak coupling, the effective phase dynamics depend on phase differences and noise covariance.
  • III. THEORY OF PHASE INFORMATION ROUTING: A small-noise expansion around deterministic phase-locked reference dynamics gives a first-order approximation for joint phase probabilities.The dynamics are decomposed into deterministic reference and stochastic parts.
  • III. THEORY OF PHASE INFORMATION ROUTING: The resulting analytical expressions for delayed mutual information and transfer entropy depend on local dynamics, inputs, coupling functions, and interaction topology.These dependencies are contained in the inverse-variance quantity used in the phase-distribution ansatz and in the derived transfer-entropy expressions.
  • III. THEORY OF PHASE INFORMATION ROUTING: Theoretical predictions match numerical estimates across the illustrated oscillatory-network examples.The agreement is reported for the delayed mutual-information and transfer-entropy calculations.
  • III. THEORY OF PHASE INFORMATION ROUTING: For independent input signals, delayed mutual information and transfer entropy typically produce similar information routing patterns.This correspondence is reported for independent noise inputs.

IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING

Anisotropic information routing arises from symmetry-broken collective states and can switch without changing physical network structure. Local state-dependent interactions and noise characteristics determine routing direction, allowing a state change to reorganize communication across the network.

  • IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING: Two stable phase-locked reference states in a symmetric neuronal network produce different information routing patterns under weak noise.Each state has its own phase-difference fluctuations and associated routing pattern.
  • IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING: Switching between collective states reverses the dominant communication direction without changing structural network properties.The reversal is visible in delayed mutual information and is more pronounced in delayed transfer entropy.
  • IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING: In the symmetric two-unit example, different coupling slopes at the phase-locking offsets generate directionality even though the units and coupling architecture are identical.The phase-coupling function and its antisymmetric part identify the two stable offsets associated with the alternative routing directions.
  • IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING: The direction of information transfer need not follow the order in which oscillators phase-lock.Either the phase-advanced oscillator can pull the lagging one, or the lagging oscillator can push the leading one toward equilibrium.
  • IV. MECHANISM OF ANISOTROPIC INFORMATION ROUTING: Effective interactions local in state space, together with the reference state and noise characteristics, determine the network’s information routing pattern.Symmetry-broken dynamical states therefore induce anisotropic and switchable routing without physical rewiring.

V. INFORMATION ROUTING IN NETWORKS OF NETWORKS

The framework lifts node-level information-routing measures to community and network-of-networks scales. In modular networks, fine-scale routing patterns determine larger-scale routing, while local changes can invert remote information flow.

  • A two-community network can exhibit oppositely directed information-sharing patterns after changing one oscillator’s intrinsic frequency.The change reorganizes equilibrium phase differences and reverses routing between node pairs in different clusters.
  • Hierarchical reduction recovers inversion of fine-scale information-routing patterns across configurations and parameter changes.The reduced system reflects the inversion observed between node pairs and as the oscillator-frequency change varies.
  • A local link-strength change in one module can invert information-routing direction between two remote modules.In the three-module example, changing a connection within sub-network A changes routing between B and C.
  • Inter-node dMI and dTE measures can be lifted to community-level quantities by replacing node properties with community-related quantities.The procedure can be iterated to networks of networks.
  • At larger scales, routing direction reflects the majority and relative strengths of information-routing patterns at finer scales.This provides a multiscale interpretation of routing patterns resolved across hierarchical network levels.

VI. NON-LOCAL INFORMATION REROUTING VIA LOCAL INTERVENTIONS

Local properties of modular networks can control network-wide information routing. Small interventions alter collective dynamics and effective couplings, producing remote and switch-like routing changes.

  • Network-level information-routing patterns depend on local community properties, including local dynamical states and phase responses.The effective coupling between communities also depends on local interactions and cluster-specific properties.
  • A frequency change in one Hopf oscillator induces a non-local inversion of routing between its own cluster and another cluster.The intervention is local to sub-network A, while the routing inversion occurs between clusters A and B.
  • Increasing a local link strength in module A remotely changes routing direction between sub-networks B and C.The affected communication lies outside the module containing the intervention.
  • These interventions alter cluster frequency, local dynamical state, phase response, effective noise, couplings, and inter-cluster phase locking.Together, these changes produce the observed routing inversions.
  • Routing transitions show a switch-like dependence on the changed parameter, enabling digital-like changes in information-routing direction.The reported transitions are not described as continuous.

VII. COMBINATORIAL INFORMATION ROUTING PATTERNS

Switching local dynamical states in a modular network can generate combinatorially many global information-routing patterns. Local multistability can also yield multiple global collective states and time-dependent routing.

  • Varying local dynamical states in a hierarchical network produces a combinatorial number of information-routing patterns in the same physical network.The routing patterns arise from alternative local states rather than physical network rewiring.
  • For three modules with two local phase-locked states each, each local-state combination gives rise to at least one network-wide collective state.Under sufficiently weak coupling, local multistability is preserved across the modular network.
  • Some local-state combinations support multiple globally phase-locked states, whereas others support only one.The same local dynamical configuration can correspond to more than one globally locked collective state.
  • The configuration [βAαBβC] produces a periodic global state whose hierarchically reduced routing pattern becomes time-dependent.The routing pattern changes while the global dynamics remains periodic.

VIII. TIME-DEPENDENT INFORMATION ROUTING

Information routing can vary over time when collective reference dynamics are non-stationary. Periodic states, transients, and heteroclinic switching generate temporally structured routing sequences.

  • For non-stationary reference dynamics, delayed mutual information and transfer entropy become dependent on time.This makes information-routing patterns explicitly time-resolved.
  • Periodic reference states produce information-routing patterns that undergo cyclic changes.Figure 4c illustrates cyclic routing changes driven by an underlying periodic state.
  • Different starting positions in a system with a global fixed point can produce stochastic transients with different, time-dependent routing patterns.The routing differences arise across transient trajectories from different initial positions.
  • Switching dynamics along heteroclinic orbits provides another way to generate specific progressions of reference dynamics.These non-stationary configurations yield temporally structured sequences of information-routing patterns.

IX. DISCUSSION

The paper establishes collective dynamics as a generic basis for flexible, state-dependent information routing without physical network changes. It develops theoretical measures and shows that local properties, topology, inputs, and collective states jointly organize routing across network scales.

  • Collective dynamical states provide a theoretical basis for emergent information routing across complex networks.
  • Delayed mutual information and transfer entropy quantify information sharing and transfer, allowing routing patterns to be derived across network scales.The analysis uses theoretical results based on delayed mutual-information and transfer-entropy curves.
  • Information-routing patterns self-organize according to general principles linking routing control to network dynamical bifurcations.
  • The routing mechanism is determined by the entire network’s collective dynamics rather than by a communication protocol or local propagation rule.
  • The framework is limited by its small-noise, phase-focused analysis, although higher-order expansions and amplitude variables are proposed extensions.Higher-order diagrammatic approaches address recurrent-network interactions, while amplitude degrees of freedom could broaden applications to activity- and phase-based signaling.
  • Local interventions can modify non-local information sharing and transfer, enabling context-dependent processing and routing across distant sub-networks.The paper also identifies routing paradigms capable of regulation at the non-local level across the whole network.
  • Switching collective states can reroute information without physically changing the network, including through externally controlled inputs.In the analyzed networks, injected signals are encoded in oscillator-frequency fluctuations and decoded along pathways predicted by the current state-dependent routing pattern.
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