Thermodynamic characterization of networks using graph polynomials

In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.

Figure 1

The scatter plot of Boltzmann partition function associated with normalized Laplacian operator in Eq. (6) and the normalized Laplacian quasicharacteristic polynomial given by Eq. (11) for different $\beta$ for Erdős-Rényi and Barabási-Albert random graphs. Black triangles: $\beta =0.01$; blue circles: $\beta =0.1$; red stars: $\beta =0.5.$

Figure 2

Mean and standard deviation of the temperature versus $\mathrm{\Delta }p$ and $\mathrm{\Delta }m$ for random graphs with different graph sizes. Red squares: 80 nodes; magenta triangles: 150 nodes; blue stars: 300 nodes.

Figure 3

The three-dimensional (3D) scatter plot of the dynamic stock correlation network in the thermodynamic space spanned by temperature, average energy, and entropy. Cyan dots: background; black downward-pointing triangles: Black Monday; green circles: Persian Gulf War; blue diamonds: Iraq War; red upward-pointing triangles: subprime mortgage crisis; magenta squares: bankruptcy of Lehman Brothers.

Figure 4

The temperature, average energy, and thermodynamic entropy versus time for the dynamic stock correlation network. The known financial crisis periods are identified by ellipses.

Figure 5

The individual time series of stock correlation network temperature, energy, and entropy for nine different global events that have been identified in Fig. 4.

Figure 6

The PCA plots of the dynamic stock correlation network characterization delivered by different signature methods. Top panel: heat kernel signature; bottom panel: wave kernel signature.

Figure 7

Path of the time-evolving stock correlation network in the entropy-energy-time space during different financial crises. Top panel: Black Monday; bottom panel: bankruptcy of Lehman Brothers. The color bar beside each plot represents the date in the time series.

Figure 8

The normalized histogram of $\beta$, defined as $\beta =1/kT$, for the dynamic stock correlation network.

Figure 9

The 3D scatter plot of the dynamic Drosophila gene regulatory network in the thermodynamic space spanned by temperature, average energy, and entropy. Cyan dots: steady development; black downward-pointing triangle: hatching from egg; magenta circles: first molt; blue diamonds: pupation; red upward-pointing triangles: adult emerging.

Figure 10

The temperature, average energy, and the thermodynamic entropy versus time for the dynamic Drosophila gene regulatory network. The important morphological changes are identified by arrows. Red line: embryonic; cyan line: larval; blue line: pupal: green line: adulthood.

Figure 11

The PCA plots of the dynamic Drosophila gene regulatory network characterization delivered by different signature methods. Top panel: heat kernel signature; bottom panel: wave kernel signature.

Figure 12

The normalized histogram of $\beta$, defined as $\beta =1/kT$, for the dynamic Drosophila gene regulatory network.