Entropy of network ensembles

By analogy to statistical mechanics, microcanonical ensembles and canonical ensembles of networks are introduced for the implementation. A partition function Z of an ensemble can be defined as:

$${\displaystyle Z=\sum _{\mathbf {a} }\delta \left[{\vec {F}}(\mathbf {a} )-{\vec {C}}\right]\exp \left(\sum _{ij}h_{ij}\Theta (a_{ij})+r_{ij}a_{ij}\right)}$$

where ${\displaystyle {\vec {F}}(\mathbf {a} )={\vec {C}}}$ is the constraint, and ${\displaystyle a_{ij}}$ (${\displaystyle a_{ij}\geq {0}}$) are the elements in the adjacency matrix, ${\displaystyle a_{ij}>0}$ if and only if there is a link between node i and node j. ${\displaystyle \Theta (a_{ij})}$ is a step function with ${\displaystyle \Theta (a_{ij})=1}$ if ${\displaystyle x>0}$, and ${\displaystyle \Theta (a_{ij})=0}$ if ${\displaystyle x=0}$. The auxiliary fields ${\displaystyle h_{ij}}$ and ${\displaystyle r_{ij}}$ have been introduced as analogy to the bath in classical mechanics.

For simple undirected networks, the partition function can be simplified as[6]

$${\displaystyle Z=\sum _{\{a_{ij}\}}\prod _{k}\delta ({\textrm {constraint}}_{k}(\{a_{ij}\}))\exp \left(\sum _{i<j}\sum _{\alpha }h_{ij}(\alpha )\delta _{a_{ij},\alpha }\right)}$$

where ${\displaystyle a_{ij}\in \alpha }$, ${\displaystyle \alpha }$ is the index of the weight, and for a simple network ${\displaystyle \alpha =\{0,1\}}$.

Microcanonical ensembles and canonical ensembles are demonstrated with simple undirected networks.

For a microcanonical ensemble, the Gibbs entropy ${\displaystyle \Sigma }$ is defined by:

$${\displaystyle {\begin{aligned}\Sigma &={\frac {1}{N}}\log {\mathcal {N}}\\&={\frac {1}{N}}\log Z|_{h_{ij}(\alpha )=0\forall (i,j,\alpha )}\end{aligned}}}$$

where ${\displaystyle {\mathcal {N}}}$ indicates the cardinality of the ensemble, i.e., the total number of networks in the ensemble.

The probability of having a link between nodes i and j, with weight ${\displaystyle \alpha }$ is given by:

$${\displaystyle \pi _{ij}(\alpha )={\frac {\partial \log Z}{\partial {h_{ij}}(\alpha )}}}$$

For a canonical ensemble, the entropy is presented in the form of a Shannon entropy:

$${\displaystyle {S}=-\sum _{i<j}\sum _{\alpha }\pi _{ij}(\alpha )\log \pi _{ij}(\alpha )}$$

Network ensemble ${\displaystyle G(N,L)}$ with given number of nodes ${\displaystyle N}$ and links ${\displaystyle L}$, and its conjugate-canonical ensemble ${\displaystyle G(N,p)}$ are characterized as microcanonical and canonical ensembles and they have Gibbs entropy ${\displaystyle \Sigma }$ and the Shannon entropy S, respectively. The Gibbs entropy in the ${\displaystyle G(N,p)}$ ensemble is given by:[7]

$${\displaystyle {N}\Sigma =\log \left({\begin{matrix}{\cfrac {N(N-1)}{2}}\\L\end{matrix}}\right)}$$

For ${\displaystyle G(N,p)}$ ensemble,

$${\displaystyle {p}_{ij}=p={\cfrac {2L}{N(N-1)}}}$$

Inserting ${\displaystyle p_{ij}}$ into the Shannon entropy:[6]

$${\displaystyle \Sigma =S/N+{\cfrac {1}{2N}}\left[\log \left({\cfrac {N(N-1)}{2L}}\right)-\log \left({\cfrac {N(N-1)}{2}}-L\right)\right]}$$

The relation indicates that the Gibbs entropy ${\displaystyle \Sigma }$ and the Shannon entropy per node S/N of random graphs are equal in the thermodynamic limit ${\displaystyle N\to \infty }$.

Von Neumann entropy is the extension of the classical Gibbs entropy in a quantum context. This entropy is constructed from a density matrix ${\displaystyle \rho }$: historically, the first proposed candidate for such a density matrix has been an expression of the Laplacian matrix L associated with the network. The average von Neumann entropy of an ensemble is calculated as:[8]

$${\displaystyle {S}_{VN}=-\langle \mathrm {Tr} \rho \log(\rho )\rangle }$$

For random network ensemble ${\displaystyle G(N,p)}$, the relation between ${\displaystyle S_{VN}}$ and ${\displaystyle S}$ is nonmonotonic when the average connectivity ${\displaystyle p(N-1)}$ is varied.

For canonical power-law network ensembles, the two entropies are linearly related.[6]

$${\displaystyle {S}_{VN}=\eta {S/N}+\beta }$$

Networks with given expected degree sequences suggest that, heterogeneity in the expected degree distribution implies an equivalence between a quantum and a classical description of networks, which respectively corresponds to the von Neumann and the Shannon entropy.[9]

This definition of the Von Neumann entropy can also be extended to multilayer networks with tensorial approach[10] and has been used successfully to reduce their dimensionality from a structural point of perspective.[11]

However, it has been shown that this definition of entropy does not satisfy the property of sub-additivity (see Von Neumann entropy's subadditivity), expected to hold theoretically. A more grounded definition, satisfying this fundamental property, has been introduced by De Domenico and Biamonte[12] as a quantum-like Gibbs state

$${\displaystyle \rho (\beta )={\frac {e^{-\beta L}}{Z(\beta )}}}$$

where

$${\displaystyle Z(\beta )=Tr[e^{-\beta L}]}$$

is a normalizing factor which plays the role of the partition function, and ${\displaystyle \beta }$ is a tunable parameter which allows multi-resolution analysis. If ${\displaystyle \beta }$ is interpreted as a temporal parameter, this density matrix is formally proportional to the propagator of a diffusive process on the top of the network.

This feature has been used to build a statistical field theory of complex information dynamics, where the density matrix can be interpreted in terms of the super-position of streams operators whose action is to activate information flows among nodes.[13] The framework has been successfully applied to analyze the protein-protein interaction networks of virus-human interactomes, including the SARS-CoV-2, to unravel the systemic features of infection of the latter at microscopic, mesoscopic and macroscopic scales,[14] as well as to assess the importance of nodes for integrating information flows within the network and the role they play in network robustness.[15]

This approach has been generalized to deal with other types of dynamics, such as random walks, on the top of multilayer networks, providing an effective way to reduce the dimensionality of such systems without altering their structure.[16] Using both classical and maximum-entropy random walks, the corresponding density matrices have been used to encode the network states of the human brain and to assess, at multiple scales, connectome’s information capacity at different stages of dementia.[17]

• Canonical ensemble
• Microcanonical ensemble
• Maximum-entropy random graph model
• Graph entropy