Finding and tracking local communities by approximating derivatives in networks

Abstract

Since various complex systems are represented by networks, detecting and tracking local communities has become a crucial task nowadays. Local community detection methods are getting much attention because they can address large networks. One famous class of local community detection is to find communities around a seed node. In this research, a novel local community detection method, inspired by geometric active contours, is proposed for finding a community surrounding an initial seed. While most of real world networks are dynamic and the majority of local community detection cannot tackle dynamic networks, the proposed model has the ability to track a local community in a dynamic network. The proposed model introduces and uses the derivative-based concepts curvature and gradient of the boundary of a connected sub-graph in networks. Then, a velocity function based on curvature and gradient is proposed to determine if the boundary of a community should evolve to include a neighbouring candidate. Approximating derivatives in discrete Euclidean space has a long history. However, compared to Euclidean space, graphs follow a non-uniform space in which the dimensionality, given by the the fluctuation in degrees of nodes, fluctuates from one node to another. This complexity complicates the approximation of derivatives which are needed for defining the curvature and gradient of a node in the boundary of a community. A new framework to approximate derivatives in graphs is proposed for such a purpose. For finding local communities, benchmarking our method against two recent methods indicates that it is capable of finding communities with equal or better conductance; and, for tracking dynamic local communities, benchmarking of the proposed method against ground-truth dataset shows a noticeable level of accuracy.

Appendices

Appendix A: Error analysis

Suppose F and G are two functions defined with $F^{\prime }(v_{c})$ and $G^{\prime }(v_{c})$ as the first order derivative in node C. Then:

$$ \begin{array}{@{}rcl@{}} \text{Addition rule: }(F \pm G)^{\prime}(v_{c})=F^{\prime}(v_{c}) \pm G^{\prime}(v_{c}) \end{array} $$

(50)

$$ \begin{array}{@{}rcl@{}} \text{Multiplication rule: }[F(v_{c}).G(v_{c})]^{\prime}=F^{\prime}(v_{c}).G(v_{c})+F(v_{c}).G^{\prime}(v_{c}) \end{array} $$

(51)

$$ \begin{array}{@{}rcl@{}} \text{Division rule: }[\frac{F}{G}(v_{c})]^{\prime}=\frac{F^{\prime}(v_{c}).G(v_{c})-F(v_{c}).G^{\prime}(v_{c})}{[G(v).G(v_{c})]^{2}} \end{array} $$

(52)

Proof: Suppose D is the derivative matrix in (30). We define $D_{1}^{(-1)}$ as the first row of matrix D^(− 1), and H = F + G:

$$DH^{(n)} = H(v)-H(v_{c}); \text{where } H^{(n)} \text{is the vector of derivatives}$$

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[H(v)-H(v_{c})] \end{array} $$

(53)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[F(v)+G(v)-F(v_{c})-G(v_{c})] \end{array} $$

(54)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[F(v)-F(v_{c})]+D_{1}^{(-1)}[G(v)-G(v_{c})]=F^{\prime}(v_{c})+G^{\prime}(v_{c}) \end{array} $$

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Differentation rule’s proof is similar to addition.

Multiplication rule: We define H = F.G where . is the inner product operator.

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[H(v)-H(v_{c})] \end{array} $$

(56)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[F(v).G(v)-F(v_{c}).G(v_{c})] \end{array} $$

(57)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[F(v).G(v)-F(v_{c}).G(v)+F(v_{c}).G(v)-F(v_{c}).G(v_{c})] \end{array} $$

(58)

$$ \begin{array}{@{}rcl@{}} D_{1}^{(-1)}[G(v).(F(v)-F(v_{c}))]+D_{1}^{(-1)}[F(v_{c}).(G(v)-G(v_{c}))] \end{array} $$

(59)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=F^{\prime}(v_{c}).G(v)+F(v_{c}).G^{\prime}(v_{c}) \end{array} $$

(60)

Division rule:

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[\frac{F(v)}{G(v)}-\frac{F(v_{c})}{G(v_{c})}] \end{array} $$

(61)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=D_{1}^{(-1)}[\frac{F(v).G(v_{c})-F(v_{c}).G(v_{c})+F(v_{c}).G(v_{c})-F(v_{c}).G(v)}{G(v).G(v_{c})}] \end{array} $$

(62)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=\frac{F^{\prime}(v_{c}).G(v_{c})}{G(v).G(v_{c})}-\frac{F(v_{c}).G^{\prime}(v_{c})}{G(v).G(v_{c})} \end{array} $$

(63)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=\frac{F^{\prime}(v_{c}).G(v_{c})-F(v_{c}).G^{\prime}(v_{c})}{G(v).G(v_{c})} \end{array} $$

(64)

Depending on the neighbors of node C, the term G(v), appeared in the right hand side of multiplication and division rules, can be approximated differently. For a simple case where node C is connected to just one node V, multiplication is rewritten:

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=F^{\prime}(v_{c}).[G(v_{c})+LG^{\prime}(v_{c})]+F(v_{c}).G^{\prime}(v_{c}) \end{array} $$

(65)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=F^{\prime}(v_{c}).G(v_{c})+F(v_{c}).G^{\prime}(v_{c})+LF^{\prime}(v_{c}).G^{\prime}(v_{c}) \end{array} $$

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and division rule is simplified:

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=\frac{F^{\prime}(v_{c}).G(v_{c})-F(v_{c}).G^{\prime}(v_{c})}{[G(v_{c})+LG^{\prime}(v_{c})].G(v_{c})} \end{array} $$

(67)

$$ \begin{array}{@{}rcl@{}} H^{\prime}(v_{c})=\frac{F^{\prime}(v_{c}).G(v_{c})-F(v_{c}).G^{\prime}(v_{c})}{G^{2}(v_{c})+L.G(v_{c}).G^{\prime}(v_{c})} \end{array} $$

(68)

Appendix B: List of symbols

C:

community

t:

time step

∇:

differential operator

κ:

curvature

∂:

partial derivative

n :

normal direction

V:

set of graph nodes

G:

graph

χ:

graph shape

E:

set of graph nodes

ν:

set of shape nodes

ξ:

set of shape edges

O:

truncation error

F:

similarity function

d:

node difference

e:

approximation error

N:

set of a node’s neighbours

s:

velocity