Characterization of Complex Networks for Epidemics Modeling

Abstract

Complex networks have certain properties that characterize them. These inherent properties can be measured and applied in other fields of research. To this end, we characterize several complex networks from different domains using concepts from graph theory. In particular, the node degrees, graph spectral radius, degree assortativity, and the entire topological structure of selected complex networks are studied on the Susceptible, Infectious, Recovered (SIR) epidemic model. The results show that various complex networks properties affect the epidemic model differently. For instance nodes with high average degrees with corresponding high clustering coefficients are seen to be effective in spreading epidemics. Also the degree distribution patterns have an effect on the spreading rate of epidemic, that is dis-assortative networks are good conduits for the epidemic spreading if low degree nodes are in turn connected to high degree nodes. These results have given us a new dimension on how epidemic spreadings can be studied using complex networks as these networks possessed in them certain properties that resemble that of human society. To the best of our knowledge, this is the first work that simulated the SIR model using heterogenous complex network properties in order to study their cumulative effects on epidemic spreading.

Appendix

The appendix shows some important definitions used in the analysis of the complex networks discussed in this paper.

Degree Sequence The degree sequence of a graph $G$ is obtained by listing its respective nodes degrees. The degree sequence has relevant data as information on the structure of a network can be derived from the network’s degree sequence. For example, the skewness of the degree sequence tells the presence or absence of hubs in the network. High skewed degree sequence shows the presence of few nodes with high degree, normally termed as hubs.

Degree Correlation The degree correlation of a graph $G$ can be defined based on the degree sequence as follows:

Let $G$ be a simple graph with degree sequence $d=[d_1,d_2,\ldots ,d_n]$ and let $A$ be an adjacency matrix of $G$. Let $V(G)=\lbrace v_1,v_2,\ldots ,v_n\rbrace$ be such that $\delta (v_i)=d_i$. The degree correlation of G is defined as [29]:

$$\begin{aligned} r_{deg(G)}\,\overset{def}{=} \,\frac{\sum _{i=1}^n \sum _{j=i+1} ^n((d_i - \bar{d})(d_j- \bar{d}).\;A[i, j])}{\sum _{i=1}^n(d_i- \bar{d})^2} \end{aligned}$$

(22)

where $\bar{d}$ denotes the average vertex degree.

Geodesic Distance Let $G$ be a directed or undirected graph and $u,v \in V(G)$. The geodesic distance between $u$ and $v$, denoted as $s(u,v)$, is the length of the shortest path between $u$ and $v$. The geodesic distance therefore have relevance to the spreading of epidemics in complex networks. Thus the extent to which short geodesics dominate a network is probably link to the role in epidemic spreading.

Vertex Eccentricity Consider a graph $G$ and assume that $G$ is connected. Let $s(u,v)$ be the geodesic distance between vertices $u$ and $v$. The eccentricity $\epsilon (u)$ of a vertex $u \in G$ is defined as [29]:

$$\begin{aligned} \epsilon (u)\,\overset{def}{=} \,\mathrm{max} \lbrace s(u,v)|v \in V(G)\rbrace \end{aligned}$$

(23)

The radius $Rad(G)$ of $G$ as [29]:

$$\begin{aligned} Rad(G)\,\overset{def}{=} \, \mathrm{min} \lbrace \epsilon (u)|u \in V(G)\rbrace \end{aligned}$$

(24)

The diameter $Diam(G)$ of $G$ is the maximal shortest path between the two vertices. This is defined as [29]:

$$\begin{aligned} Diam(G)\,\overset{def}{=} \, \mathrm{max} \lbrace s(u,v)|u,v \in V(G)\rbrace \end{aligned}$$

(25)

The $\epsilon (u)$ tells how far the farthest vertex from $u$ is positioned in the network. The $Rad(G)$ of a network is an indication of how dis-separate the vertices in a network actually are. The $Diam(G)$ tells what the maximal distance in network is. These distance measures dictate how fast information or epidemic can spread within a given network.

Shortest Path Length Let $G$ be a connected graph with vertex set $V$. Let $\bar{s}(u)$ denotes the average length of the shortest paths from vertex $u$ to any other vertex $v$ in $G$. Thus

$$\begin{aligned} \bar{s}(u)=\frac{1}{|V|-1}\sum _{v\in V,u\ne v}s(u,v) \end{aligned}$$

(26)

Then the Average Shortest Path Length of $G$ denotes as $\bar{S} (G)$ is defined as [29]:

$$\begin{aligned} \bar{S}(G) \,\overset{def}{=} \, \frac{1}{|V|}\sum _{u\in V}\bar{s}(u)=\frac{1}{|V|^2-|V|}\sum _{u,v \in V, u\ne v}s\left( u,v\right) \end{aligned}$$

(27)

The characteristic path length of $G$ is then the mean over all $\bar{s}(u)$.

Graph Transitivity Let $G$ be a simple, connected and undirected graph with vertex $v\in V(G)$. A triangle at vertex $v$ is a complete subgraph of $G$ with exactly three vertices, including $v$. A triple at vertex $v$ is a subgraph of exactly three vertices and two edges, where $v$ is incident with two edges. Let: $n_{\bigwedge }(v)$ be the number of triples at vertex $v$, $n_{\nabla }(v)$ be the number of triangles at vertex $v$, $n_{\nabla }(G)$ be the total number of distinct triangles of $G$ with $n_{\bigwedge }(G)$ of distinct triples. The Transitivity graph $G$ is then defined as [29]:

$$\begin{aligned} \tau (G)\,\overset{def}{=} \,\frac{n_\nabla (G)}{n_{\bigwedge }(G)} \end{aligned}$$

(28)

Transitivity is a global metric and some few instances $\tau (G)$ and $CC(G)$ are equal.

Graph Density Let $G$ be a simple, connected graph with $n$ vertices and $m$ edges. The graph density $\tau (G)$ is defined as

$$\begin{aligned} \rho (G)=\frac{m}{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \end{aligned}$$

(29)

The density of a graph tells to what extent a graph is complete or not complete. It value is important in deducing the extent information can get across a given graph.

Graph Modularity Graph Modularity $Q$ is the extent to which entities of the same type or from the same class link to each other. The value of $Q$ is always <1 and it approaches $1$ when there is more intragroup interaction than intergroup interaction within the network. Such a network has a strong community structure. Also, a positive value of $Q$ indicates more interaction among group members as compare with that of random and negative otherwise. The mathematical expression of modularity as defined by Newman [30] is as follows:

$$\begin{aligned} Q\,\overset{def}{=} \,\frac{1}{2L}\sum _{i j}\left( A_{i j}-\frac{d_id_j}{2.m}\right) \delta (c_i,c_j) \end{aligned}$$

(30)

where $\delta \left( c_i,c_j\right)$ defines the intragroup interaction within the network.

Assortativity Coefficient The assortativity Coefficient $AC$ is the normalized value of $Q$. This is defined as [30]:

$$\begin{aligned} AC\,\overset{def}{=} \,\frac{Q}{Q_{\mathrm{max}}}\,\overset{def}{=} \,\frac{\sum _{i j}\left( A_{i j}-\frac{d_id_j}{2.m}\right) \delta (c_i,c_j)}{2L-\sum _{i j}\left( \frac{d_id_j}{2.m}\right) \delta (c_i,c_j)} \end{aligned}$$

(31)

$AC$ attains a value of $1$ in a perfectly mixed network.

Graph Closeness Centrality Graph closeness centrality $c_C$ measures the mean geodesic distance between pair of nodes in a network. This is defined as follow [31]:

$$\begin{aligned} \ell _i\,\overset{def}{=} \,\frac{1}{n}\sum _{j}d_{i j} \end{aligned}$$

(32)

where $d_{i j}$ is the length of the geodesic distance between nodes $i$ and $j$ in the network. Nodes with small values of $\ell _i$ means that such nodes have more influence in the network as there have better access to information than others. Also epidemic and opinions can easily cascade to others through nodes with low values of $\ell _i$.