# Multiscale Information Propagation in Emergent Functional Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Propagator of Information Dynamics

#### 2.2. Emergent Functional States

#### 2.3. Emergent Functional Modules

#### 2.4. Statistical Physics of Complex Information Dynamics

#### 2.5. Functional Diversity

#### 2.6. Rescaling the Temporal Scales across Networks

#### 2.7. Fungal Networks

#### 2.8. Randomized Networks

## 3. Results

#### Functional Networks

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Rosvall, M.; Bergstrom, C.T. Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. USA
**2008**, 105, 1118–1123. [Google Scholar] [CrossRef][Green Version] - De Domenico, M.; Lancichinetti, A.; Arenas, A.; Rosvall, M. Identifying Modular Flows on Multilayer Networks Reveals Highly Overlapping Organization in Interconnected Systems. Phys. Rev. X
**2015**, 5, 011027. [Google Scholar] [CrossRef][Green Version] - Salnikov, V.; Schaub, M.T.; Lambiotte, R. Using higher-order Markov models to reveal flow-based communities in networks. Sci. Rep.
**2016**, 6, 23194. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lambiotte, R.; Rosvall, M.; Scholtes, I. From networks to optimal higher-order models of complex systems. Nat. Phys.
**2019**, 15, 313. [Google Scholar] [CrossRef] - Carletti, T.; Fanelli, D.; Lambiotte, R. Random walks and community detection in hypergraphs. J. Phys. Complex.
**2021**, 2, 015011. [Google Scholar] [CrossRef] - Lambiotte, R.; Delvenne, J.C.; Barahona, M. Random walks, Markov processes and the multiscale modular organization of complex networks. IEEE Trans. Netw. Sci. Eng.
**2014**, 1, 76–90. [Google Scholar] [CrossRef][Green Version] - De Domenico, M. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Phys. Rev. Lett.
**2017**, 118, 168301. [Google Scholar] [CrossRef][Green Version] - Liu, Z.; Barahona, M. Geometric multiscale community detection: Markov stability and vector partitioning. J. Complex Netw.
**2018**, 6, 157–172. [Google Scholar] [CrossRef] - Bertagnolli, G.; De Domenico, M. Diffusion geometry of multiplex and interdependent systems. Phys. Rev. E
**2021**, 103, 042301. [Google Scholar] [CrossRef] [PubMed] - Estrada, E.; Hatano, N. Communicability in complex networks. Phys. Rev. E
**2008**, 77, 036111. [Google Scholar] [CrossRef][Green Version] - Grindrod, P.; Parsons, M.C.; Higham, D.J.; Estrada, E. Communicability across evolving networks. Phys. Rev. E
**2011**, 83, 046120. [Google Scholar] [CrossRef][Green Version] - Estrada, E.; Hatano, N.; Benzi, M. The physics of communicability in complex networks. Phys. Rep.
**2012**, 514, 89–119. [Google Scholar] [CrossRef][Green Version] - Estrada, E. Informational cost and networks navigability. Appl. Math. Comput.
**2021**, 397, 125914. [Google Scholar] [CrossRef] - Courtney, O.T.; Bianconi, G. Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E
**2016**, 93, 062311. [Google Scholar] [CrossRef][Green Version] - Battiston, F.; Cencetti, G.; Iacopini, I.; Latora, V.; Lucas, M.; Patania, A.; Young, J.G.; Petri, G. Networks beyond pairwise interactions: Structure and dynamics. Phys. Rep.
**2020**, 874, 1–92. [Google Scholar] [CrossRef] - Skardal, P.S.; Arenas, A. Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes. Phys. Rev. Lett.
**2019**, 122, 248301. [Google Scholar] [CrossRef][Green Version] - Millán, A.P.; Torres, J.J.; Bianconi, G. Explosive higher-order Kuramoto dynamics on simplicial complexes. Phys. Rev. Lett.
**2020**, 124, 218301. [Google Scholar] [CrossRef] - Skardal, P.S.; Arenas, A. Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching. Commun. Phys.
**2020**, 3, 218. [Google Scholar] [CrossRef] - Iacopini, I.; Petri, G.; Barrat, A.; Latora, V. Simplicial models of social contagion. Nat. Commun.
**2019**, 10, 2485. [Google Scholar] [CrossRef] [PubMed] - Matamalas, J.T.; Gómez, S.; Arenas, A. Abrupt phase transition of epidemic spreading in simplicial complexes. Phys. Rev. Res.
**2020**, 2, 012049. [Google Scholar] [CrossRef][Green Version] - Gambuzza, L.; Di Patti, F.; Gallo, L.; Lepri, S.; Romance, M.; Criado, R.; Frasca, M.; Latora, V.; Boccaletti, S. Stability of synchronization in simplicial complexes. Nat. Commun.
**2021**, 12, 1255. [Google Scholar] [CrossRef] - Lee, S.H.; Fricker, M.D.; Porter, M.A. Mesoscale analyses of fungal networks as an approach for quantifying phenotypic traits. J. Complex Netw.
**2017**, 5, 145–159. [Google Scholar] [CrossRef][Green Version] - Tero, A.; Takagi, S.; Saigusa, T.; Ito, K.; Bebber, D.P.; Fricker, M.D.; Yumiki, K.; Kobayashi, R.; Nakagaki, T. Rules for biologically inspired adaptive network design. Science
**2010**, 327, 439–442. [Google Scholar] [CrossRef][Green Version] - Reid, C.R.; Latty, T.; Dussutour, A.; Beekman, M. Slime mold uses an externalized spatial “memory” to navigate in complex environments. Proc. Natl. Acad. Sci. USA
**2012**, 109, 17490–17494. [Google Scholar] [CrossRef] [PubMed][Green Version] - Blondel, V.D.; Guillaume, J.L.; Lambiotte, R.; Lefebvre, E. Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp.
**2008**, 2008, P10008. [Google Scholar] [CrossRef][Green Version] - Ghavasieh, A.; Nicolini, C.; De Domenico, M. Statistical physics of complex information dynamics. Phys. Rev. E
**2020**, 102, 052304. [Google Scholar] [CrossRef] [PubMed] - De Domenico, M.; Biamonte, J. Spectral Entropies as Information-Theoretic Tools for Complex Network Comparison. Phys. Rev. X
**2016**, 6, 041062. [Google Scholar] [CrossRef][Green Version] - Ghavasieh, A.; De Domenico, M. Enhancing transport properties in interconnected systems without altering their structure. Phys. Rev. Res.
**2020**, 2, 013155. [Google Scholar] [CrossRef][Green Version] - Ghavasieh, A.; Bontorin, S.; Artime, O.; Verstraete, N.; Domenico, M.D. Multiscale statistical physics of the pan-viral interactome unravels the systemic nature of SARS-CoV-2 infections. Commun. Phys.
**2021**, 4, 83. [Google Scholar] [CrossRef] - De Domenico, M.; Solé-Ribalta, A.; Cozzo, E.; Kivelä, M.; Moreno, Y.; Porter, M.A.; Gómez, S.; Arenas, A. Mathematical Formulation of Multilayer Networks. Phys. Rev. X
**2013**, 3, 041022. [Google Scholar] [CrossRef][Green Version] - Newman, M.E.J. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA
**2006**, 103, 8577–8582. [Google Scholar] [CrossRef][Green Version] - Rossetti, G.; Cazabet, R. Community Discovery in Dynamic Networks: A Survey. ACM Comput. Surv.
**2018**, 51, 1–37. [Google Scholar] [CrossRef][Green Version] - Cimini, G.; Squartini, T.; Saracco, F.; Garlaschelli, D.; Gabrielli, A.; Caldarelli, G. The statistical physics of real-world networks. Nat. Rev. Phys.
**2019**, 1, 58–71. [Google Scholar] [CrossRef][Green Version] - Radicchi, F.; Krioukov, D.; Hartle, H.; Bianconi, G. Classical information theory of networks. J. Phys. Complex.
**2020**, 1, 025001. [Google Scholar] [CrossRef] - Passerini, F.; Severini, S. The von Neumann Entropy of Networks. SSRN Electron. J.
**2008**. [Google Scholar] [CrossRef][Green Version] - De Domenico, M.; Nicosia, V.; Arenas, A.; Latora, V. Structural reducibility of multilayer networks. Nat. Commun.
**2015**, 6, 6864. [Google Scholar] [CrossRef] [PubMed][Green Version] - Biamonte, J.; Faccin, M.; De Domenico, M. Complex networks from classical to quantum. Commun. Phys.
**2019**, 2, 53. [Google Scholar] [CrossRef][Green Version] - Nicolini, C.; Forcellini, G.; Minati, L.; Bifone, A. Scale-resolved analysis of brain functional connectivity networks with spectral entropy. NeuroImage
**2020**, 211, 116603. [Google Scholar] [CrossRef] - Benigni, B.; Ghavasieh, A.; Corso, A.; d’Andrea, V.; Domenico, M.D. Persistence of information flow: A multiscale characterization of human brain. Netw. Neurosci.
**2021**, 5, 831–850. [Google Scholar] [CrossRef] - Ghavasieh, A.; Stella, M.; Biamonte, J.; Domenico, M.D. Unraveling the effects of multiscale network entanglement on empirical systems. Commun. Phys.
**2021**, 4, 129. [Google Scholar] [CrossRef] - Kötter, R.; Stephan, K.E. Network participation indices: Characterizing component roles for information processing in neural networks. Neural Netw.
**2003**, 16, 1261–1275. [Google Scholar] [CrossRef] [PubMed] - Maslov, S.; Sneppen, K. Specificity and stability in topology of protein networks. Science
**2002**, 296, 910–913. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 1.**Emergent functional state. Emergent functional states associated with four different network types, at four different temporal scales, $\tau $. On the left, the Structure column represents the considered synthetic networks exhibiting random geometric, Barabasi Albert, Watts Strogatz, and Stochastic Blocks topology. In front of each network type, its emergent functional state is visualized at four temporal scales: $\tau =0.1,1,10,100$. Subplots show the emergent functional state of a (

**A**) random geometric network with radius $0.18$, (

**B**) Barabasi Albert network with $m=1$, (

**C**) Watts Strogatz network with mean degree of 4 and rewiring probability of $0.05$, and (

**D**) a Stochastic Blocks network having four communities with intra-community connectivity probability of ${10}^{-3}$ and inter-community connectivity probability of $0.5$. The intensity of each link in the emergent functional state, characterized by the flow exchange between the nodes at the specific temporal scale, is colored from low (dark blue) to high (yellow).

**Figure 2.**Connectivity distribution in emergent functional states of fungal networks. The emergent functional states corresponding to one realization of Resinicium bicolor networks, at 3 different scales (Markov time: $\tau =0.0001,0.01,1.5$) are shown. Below each functional state, the corresponding strength distribution is reported. Tuning the temporal parameter from small to large values, one reaches a state where the functional state is fully entangled.

**Figure 3.**Average connectivity of emergent functional network states. The average strength of nodes $\overline{k}$ in the emergent functional state (see Methods) is plotted against the rescaled temporal parameter $\tau /{\tau}_{d}$ (see Methods). The relevant temporal scales for analysis of information dynamics depend on a variety of parameters, such as the number of nodes and their topology, reflected in the spectrum of the Laplacian matrix. During the transition from the extremely small rescaled temporal scale to the large rescaled temporal scale, it is observed that the average strength of the emergent functional state (see Methods) goes from zero, where no exchange flow is possible between the nodes, to ${\varphi}_{0}$, where all the field originated from each node is completely distributed to the others. Here, we numerically observe that the relevant temporal scale for analysis of information dynamics in all three fungal species lays in $\tau /{\tau}_{d}\in [{10}^{-10},10]$.

**Figure 4.**Multiscale functional fungal networks. Mesoscale organization of emergent functional states and the Von Neumann entropy of fungal networks, with species such as Physarum polycephalum (Pp), Phanerochaete velutina (Pv), and Resinicium bicolor (Rb) is illustrated. (

**a**) Functional networks corresponding to a Physarum polycephalum network, at 5 different scales $\tau =0.001,0.009,0.079,0.708,6.31$. The functional modules captured by the Louvain algorithm are colored differently. (

**b**) Average entropy of all fungal networks considered in this study plotted as a function of rescaled temporal parameter $\tau /{\tau}_{d}$. (

**c**) Average number of functional modules at each rescaled temporal parameter over all the fungal networks considered in this study. (

**d**) Relation between the average number of functional modules with the Von Neumann entropy is shown.

**Figure 5.**Entropy analysis. The average Von Neumann entropy for Physarum polycephalum (Pp), Phanerochaete velutina (Pv), and Resinicium bicolor (Rb) are shown as red lines. Their corresponding null models RWCM are plotted as blue lines. The Von Neumann entropy of the original networks and RWCM are indistinguishable when extremely small and large temporal scales are considered, while at the middle scales the Von Neumann entropy of original networks is often higher than the corresponding RWCM. The black dashed lines show the difference, to highlight the advantage of the real complex topology in keeping the functional diversity high.

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Ghavasieh, A.; De Domenico, M. Multiscale Information Propagation in Emergent Functional Networks. *Entropy* **2021**, *23*, 1369.
https://doi.org/10.3390/e23101369

**AMA Style**

Ghavasieh A, De Domenico M. Multiscale Information Propagation in Emergent Functional Networks. *Entropy*. 2021; 23(10):1369.
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**Chicago/Turabian Style**

Ghavasieh, Arsham, and Manlio De Domenico. 2021. "Multiscale Information Propagation in Emergent Functional Networks" *Entropy* 23, no. 10: 1369.
https://doi.org/10.3390/e23101369