Kirill Kovalenko’s research while affiliated with Sant'Anna School of Advanced Studies and other places

What is this page?


This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.

Publications (6)


Classes in the Master Stability Function
The maximum Lyapunov exponent Λ as a function of the parameter ν (see text for definitions), for a chaotic flow (Λ(0) > 0). The curve Λ(ν) is called the Master Stability Function (MSF). Given any pair of f and g, only three classes of systems are possible: Class I systems (yellow line) for which the MSF does not intercept the horizontal axis; Class II systems (violet line), for which the MSF has a unique intercept with the horizontal axis at ν = ν*; Class III systems (brown line), for which the MSF intercepts the horizontal axis at two critical points ν=ν1*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{1}^{*}$$\end{document} and ν=ν2*\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu={\nu }_{2}^{*}$$\end{document}.
Predicting the transition to synchronization
a An all-to-all connected, symmetric, weighted graph of N = 10 nodes is considered. The graph is endowed with three symmetry orbits: the one composed by the red nodes {1, 2, 3}, the one made of the orange nodes {4, 5, 6}, and the one made of the four yellow nodes {7, 8, 9, 10}. In the sketch, the widths of the links are proportional to the corresponding weights, and the sizes of the nodes are proportional to the corresponding strengths. b The entries of S10, which corresponds to λ10 = 6. cS8 (associated to λ8 = 6), where the entries equal to 2 clearly define a cluster formed by nodes {7, 8, 9, 10}. The first predicted event in the transition then consists in the synchronization of such nodes at d1 = ν*/λ8 = ν*/6. dS5 (related to λ5 = 4), where additional entries become equal to 2, indicating a second foreseen event in which nodes {4, 5, 6} join the existent synchronization cluster at d2 = ν*/λ5 = ν*/4. eS2 (corresponding to λ2 = 1) where it is seen that nodes {1, 2, 3} also join the existing synchronized cluster at d3 = ν*/λ2 = ν* in a third predicted event where complete synchronization of the network takes place. f The expected events (and their exact sequence) occurring in the path to synchrony of the network’s architecture depicted in a. The bar at the bottom of the Figure gives the color code used in panels (b-e) for matrices' entries.
The numerical verification of the predicted transition
a, b The normalized synchronization errors Ecl (see text for definition) as a function of d, for the Lorenz (a) and the Rössler (b) case. Data refers to ensemble averages over 500 different numerical simulations of the network sketched in (a) of Fig. 2. Cluster 1 (yellow line) is formed by nodes {7, 8, 9, 10}, Cluster 2 (orange line) is formed by nodes {4, 5, 6}, and Cluster 3 (red line) is formed by nodes {1, 2, 3}. The black dotted line refers to the synchronization error of the entire network (EN). In both panels it is seen that the expected sequence of events taking place during the transition is verified. Furthermore, the values d1 = 7.322/6 = 1.220 (d1 = 0.179/6 = 0.0298), d2 = 7.322/4 = 1.8305 (d2 = 0.179/4 = 0.04475) and d3 = 7.322 (d3 = 0.179) are marked in the horizontal axis respectively with a yellow, orange and red filled dot in (a) (in b)), indicating how accurate are the predictions and approximations made on the corresponding critical values for the coupling strength. For each interval, the arrow points to the composition of the synchronized cluster that is being observed, once again in perfect harmony with the predictions made. Finally, we have verified that no extra synchronization features emerge during the transition, other than those explicitly foreseen in Fig. 2. b1–b4 Temporal snapshots illustrating the evolution of the y variable of each of the 10 network’s nodes (see color code at the bottom of the four panels) during the transition to synchronization reported in b. At d = 0.01 (b1) the nodes display a fully uncorrelated dynamics. At d = 0.035 b2 the yellow nodes (7,8,9,10) are clustered and display a synchronous motion, whereas all other nodes feature a uncorrelated dynamics. At d = 0.1 (b3) the violet nodes (4,5,6) have joined the clustered evolution, while nodes (1,2,3) remains unsynchronized. Finally, at d = 0.2 (b4), all network’s nodes are synchronized.
Applications to large size synthetic networks
Ecl (see text for definition) vs. d, for the Rössler system (see the differential equations in the text). Data in a [in b] refer to ensemble averages over 50 (150) different numerical simulations of the graph G1 (G2) described in the main text. In both panels, the legend sets the color code for the curves corresponding to each of the existing clusters Ci and to the Entire Network (EN). Once again, the two predicted transitions are verified. a Cluster 1 synchronizes at d1 = 0.179/4 = 0.04475 (marked with a yellow dot), Cluster 2 synchronizes at d2 = 0.179/2 = 0.0895 (orange dot), and the entire network synchronizes at d3 = 0.179/0.4758 = 0.376 (black dot), as predicted. b The cluster with 30 nodes synchronizes at d1 = 0.179/5 = 0.0358 (yellow dot), the cluster with 100 nodes at d2 = 0.179/4 = 0.04475 (orange dot), the cluster with 300 nodes at d3 = 0.179/3 = 0.0597 (red dot), the cluster with 1,000 nodes at d4 = 0.179/2 = 0.0895 (violet dot), and the entire network at d5 = 0.179/0.6025 = 0.297 (black dot).
Applications to the PowerGrid network
Ecl (see text for definition) vs. d, for the Rössler system (see the differential equations in the text). Data in a [in b] refer to ensemble averages over 850 (200) different numerical simulations of the PowerGrid network. As in Fig. 4, the legends of both panels set the color code for the curves corresponding to each of the reported clusters Ci and to the Entire Network (EN). a reports the case of identical Rössler systems, and the error of 6 specific clusters is plotted (see the Supplementary Information for the composition of each of the 6 clusters Ci). The observed sequence of events perfectly matches the predicted one, with an excellent fit with the values d1, . . . , d6. In panel (b) the effects of heterogeneity in the network are reported. Namely, for each node i of the PowerGrid network, the parameter bi in the Rössler equations is randomly sorted from a uniform distribution in the interval [0.1 − ϵ, 0.1 + ϵ]. The curves plotted refer to ϵ = 0.01.

+1

The transition to synchronization of networked systems
  • Article
  • Full-text available

June 2024

·

379 Reads

·

4 Citations

·

·

·

[...]

·

Stefano Boccaletti

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network’s nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

Download

Distances in Higher-Order Networks and the Metric Structure of Hypergraphs

June 2023

·

247 Reads

·

7 Citations

Entropy

We explore the metric structure of networks with higher-order interactions and introduce a novel definition of distance for hypergraphs that extends the classic methods reported in the literature. The new metric incorporates two critical factors: (1) the inter-node distance within each hyperedge, and (2) the distance between hyperedges in the network. As such, it involves the computation of distances in a weighted line graph of the hypergraph. The approach is illustrated with several ad hoc synthetic hypergraphs, where the structural information unveiled by the novel metric is highlighted. Moreover, the method’s performance and effectiveness are shown through computations on large real-world hypergraphs, which indeed reveal new insights into the structural features of networks beyond pairwise interactions. Namely, using the new distance measure, we generalize the definitions of efficiency, closeness and betweenness centrality for the case of hypergraphs. Comparing the values of these generalized measures with their analogs calculated for the hypergraph clique projections, we show that our measures provide significantly different assessments on the characteristics (and roles) of the nodes from the information-transferability point of view. The difference is brighter for hypergraphs in which hyperedges of large sizes are frequent, and nodes relating to these hyperedges are rarely connected by other hyperedges of smaller sizes.


Figure 2. Predicting the transition to synchronization. (a) An all-to-all connected, symmetric, weighted graph of N = 10 nodes is considered. The graph is endowed with three symmetry orbits: the one composed by the red nodes {1, 2, 3}, the one made of the orange nodes {4, 5, 6}, and the one made of the four yellow nodes {7, 8, 9, 10}. In the sketch, the widths of the links are proportional to the corresponding weights, and the sizes of the nodes are proportional to the corresponding strengths. (b) The entries of S10, which corresponds to λ10 = 6. (c) S8 (associated to λ8 = 6), where the entries equal to 2 clearly define a cluster formed by nodes {7, 8, 9, 10}. The first predicted event in the transition then consists in the synchronization of such nodes at d1 = ν * /λ8 = ν * /6. (d) S5 (related to λ5 = 4), where additional entries become equal to 2, indicating a second foreseen event in which nodes {4, 5, 6} join the existent synchronization cluster at d2 = ν * /λ5 = ν * /4. (e) S2 (corresponding to λ2 = 1) where it is seen that nodes {1, 2, 3} also join the existing synchronized cluster at d3 = ν * /λ2 = ν * in a third predicted event where complete synchronization of the network takes place. (f) The expected events (and their exact sequence) occurring in the path to synchrony of the network's architecture depicted in panel (a). The bar at the bottom of the Figure gives the color code used in panels (b-e) for matrices' entries.
Figure 3. The numerical verification of the predicted transition. Panels (a,b): The normalized synchronization errors E cl (see text for definition) as a function of d, for the Lorenz [panel (a)] and the Rössler [panel (b)] case. Data refers to ensemble averages over 500 different numerical simulations of the network sketched in panel (a) of Fig. 2. Cluster 1 (yellow line) is formed by nodes {7, 8, 9, 10}, Cluster 2 (orange line) is formed by nodes {4, 5, 6}, and Cluster 3 (red line) is formed by nodes {1, 2, 3}. The black dotted line refers to the synchronization error of the entire network (EN). In both panels it is seen that the expected sequence of events taking place during the transition is verified. Furthermore, the values d1 = 7.322/6 = 1.220 (d1 = 0.179/6 = 0.0298), d2 = 7.322/4 = 1.8305 (d2 = 0.179/4 = 0.04475) and d3 = 7.322 (d3 = 0.179) are marked in the horizontal axis respectively with a yellow, orange and red filled dot in panel (a) (in panel (b)), indicating how accurate are the predictions and approximations made on the corresponding critical values for the coupling strength. For each interval, the arrow points to the composition of the synchronized cluster that is being observed, once again in perfect harmony with the predictions made. Finally, we have verified that no extra synchronization features emerge during the transition, other than those explicitly foreseen in Fig. 2. Panels (b1-b4): Temporal snapshots illustrating the evolution of the y variable of each of the 10 network's nodes (see color code at the bottom of the four panels) during the transition to synchronization reported in panel (b). At d = 0.01 (panel b1) the nodes display a fully uncorrelated dynamics. At d = 0.035 (panel b2) the yellow nodes (7,8,9,10) are clustered and display a synchronous motion, whereas all other nodes feature a uncorrelated dynamics. At d = 0.1 (panel b3) the violet nodes (4,5,6) have joined the clustered evolution, while nodes (1,2,3) remains unsynchronized. Finally, at d = 0.2 (panel b4), all network's nodes are synchronized.
Figure 4. Applications to large size synthetic networks. E cl (see text for definition) vs. d, for the Rössler system (see the differential equations in the text). Data in panel (a) [in panel (b)] refer to ensemble averages over 50 (150) different numerical simulations of the graph G1 (G2) described in the main text. In both panels, the legend sets the color code for the curves corresponding to each of the existing clusters Ci and to the Entire Network (EN). Once again, the two predicted transitions are verified. (a) Cluster 1 synchronizes at d1 = 0.179/4 = 0.04475 (marked with a yellow dot), Cluster 2 synchronizes at d2 = 0.179/2 = 0.0895 (orange dot), and the entire network synchronizes at d3 = 0.179/0.4758 = 0.376 (black dot), as predicted. (b) The cluster with 30 nodes synchronizes at d1 = 0.179/5 = 0.0358 (yellow dot), the cluster with 100 nodes at d2 = 0.179/4 = 0.04475 (orange dot), the cluster with 300 nodes at d3 = 0.179/3 = 0.0597 (red dot), the cluster with 1,000 nodes at d4 = 0.179/2 = 0.0895 (violet dot), and the entire network at d5 = 0.179/0.6025 = 0.297 (black dot).
Figure 5. Applications to the PowerGrid network. E cl (see text for definition) vs. d, for the Rössler system (see the differential equations in the text). Data in panel (a) [in panel (b)] refer to ensemble averages over 850 (200) different numerical simulations of the PowerGrid network. As in Fig. 4, the legends of both panels set the color code for the curves corresponding to each of the reported clusters Ci and to the Entire Network (EN). Panel (a) reports the case of identical Rössler systems, and the error of 6 specific clusters is plotted (see Table 1 of the Supplementary Information for the composition of each of the 6 clusters Ci). The observed sequence of events perfectly matches the predicted one, with an excellent fit with the values d1, ..., d6. In panel (b) the effects of heterogeneity in the network are reported. Namely, for each node i of the PowerGrid network, the parameter bi in the Rössler equations is randomly sorted from a uniform distribution in the interval [0.1 − ϵ, 0.1 + ϵ]. The curves plotted refer to ϵ = 0.01.
Figure 6. Application to biological and social networks. Synchronization error for the considered clusters C1, C2, C3 and C4 (see the color-code at the top of the panel) for the Yeast protein-protein interaction network [5] (panel a), and the ego-Facebook network [6] (panel b). In both panels, we report also the synchronization error of the entire network (EN, black dotted line).
The transition to synchronization of networked systems

March 2023

·

94 Reads

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.


Why are there six degrees of separation in a social network?

November 2022

·

352 Reads

A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-small world organization, whereby the graph's diameter is independent of the network size over several orders of magnitude, is still unknown. Here we show that the 'six degrees of separation' are the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.


FIG. 1: An illustrative example of a linegraph L(G) (center) of a higher-order network G = (V, E) with five nodes (on the left) and its projection network π2(G) (on the right). See text for specifications.
FIG. 3: µ100(c i , c j ) (see Eq. (2) of the text for definition). Reported values are limited to the first ten components, out of the 66, of the vector centrality. It is clearly seen that, in general, the values of µ100 are rather small for i = j.
Vector Centrality in Networks with Higher-Order Interactions

August 2021

·

181 Reads

Identifying the most influential nodes in networked systems is vital to optimize their function and control. Several scalar metrics have been proposed to that effect, but the recent shift in focus towards higher-order networks has rendered them void of performance guarantees. We propose a new measure of node's centrality, which is no longer a scalar value, but a vector with dimension one lower than the highest order of interaction in the graph. Such a vectorial measure is linked to the eigenvector centrality for networks containing only pairwise interactions, whereas it has a significant added value in all other situations where interactions occur at higher-orders. In particular, it is able to unveil different roles which may be played by a same node at different orders of interactions, an information which is impossible to be retrieved by single scalar measures.


Citations (3)


... 2 ⟩ T , for the two clusters C 1 and C 2 , i.e., h = {1, 2} vs. the coupling strength d. The critical values predicted by the approach described in Ref. [21] are also reported in Fig. 3(b) as blue (orange) triangles for C 1 (C 2 ). ...

Reference:

Taming Cluster Synchronization
The transition to synchronization of networked systems

... Despite these advances, characterising shortest paths and connectivity in systems with higher-order interactions remains an open problem. Recently, efforts have been devoted to characterise the concepts of distance [46] and walks [47] and networks in networks with non-dyadic ties, as well as proposing efficient algorithms [48] to extract shortest paths in hypergraphs, limiting the analysis to static systems. ...

Distances in Higher-Order Networks and the Metric Structure of Hypergraphs

Entropy

... One has empirically observed that in most real world networks τ ∈ (2, 3) and thus most effort has been directed to understanding the model in this particular regime. From the point of view of the graph colouring problem, the PA model was studied by Kovalenko in [12], who showed that for every ε > 0 there is m(ε) ∈ N such that for every m ≥ m(ε) asymptotically almost surely m (4 + ε) log(m) ≤ χ(P A t (m, δ)). ...

On the independence number and the chromatic number of generalized preferential attachment models
  • Citing Article
  • October 2020

Discrete Applied Mathematics