Fuzzy community detection on the basis of similarities in structural/attribute in large-scale social networks

Abstract

Community detection aims to partition a set of nodes with more similarities in the set than out of it based on different criteria like neighborhood similarity or vertex connectivity. Most present day community detection methods principally concentrate on the topological structure, largely ignoring the heterogeneous properties of the vertex. This paper proposes a new community detection model, based on the possibilistic c-means model, by using structural as well as attribute similarities in a large scale in social networks. In the majority of real social networks, different clusters share nodes, resulting in the formation of overlapping communities. The proposed model, on the basis of structural and attribute similarity (PCMSA), serves as a fuzzy community detection model addressing the overlapping community detection problem, and detecting communities in a way that each community has a densely connected sub-graph with homogeneous attribute values. The function of the proposed model is assessed by a trade-off between intra-cluster and inter-cluster density and homogeneity. Therefore, to validate the proposed community detection algorithm (PCMSA) and its results, an index, compatible with the proposed model, is defined; and to assess the efficiency of the proposed fuzzy community detection, several experimental results in variety sizes from very small to very large sizes of real social networks are given, and the results are contrasted with other community detection models like FCAN, CODICIL, SA-cluster, K-SNAP and PCM. The experimental findings reveal the superiority of this novel model and its promising scalability and computational complexity over others.

Appendices

Appendix 1

Proof of Theorem

Theorem 1

All $u_{ik}$ in $U,\forall i,k$ are independent. Hence, minimization of $J(U,V)$ with regard to U is similar to that of $J(u_{ik} ,v_{i} )$ regarding $u_{ik}$. The gradient of $J(u_{ik} ,v_{i} )$ with respect to $u_{ik}$ is set to zero in an attempt to find the first-order necessary conditions for optimality:

$$\frac{\partial J}{{\partial u_{ik} }} = m{\text{ u}}_{ik}^{m - 1} D_{ik} - m \, \Delta_{i} (1 - u_{ik} )^{m - 1} = 0 \Rightarrow u_{ik} = \left( {1 + \left( {\frac{{D_{ik} }}{{\Delta_{i} }}} \right)^{1/(m - 1)} } \right)^{ - 1}$$

(30)

To find the most favorable node as the center of cluster, a node with the closest in structure to other members of the cluster considering their membership value $u_{ik}$ should be selected. As a result, the center of cluster i is defined as follows:

$$v_{i}^{ * } = \mathop {\arg \min }\limits_{{v_{i} \in [1,n]}} (\sum\limits_{k = 1}^{n} {\sum\limits_{j = 1}^{n} {u_{ik}^{m} D_{ik} } } )$$

(31)

Theorem 2

From (18), the necessary conditions considering partial derivatives are as follows:

$$\begin{gathered} \frac{\partial J}{{\partial u_{ik} }} = m(u_{ik} )^{m - 1} d_{ik} (\Omega_{i} ) + m\delta_{i} (1 - u_{ik} )^{m - 1} = 0 \hfill \\ u_{ik}^{*} = \left( {1 + \left( {\frac{{d_{ik} (\Omega_{i} )}}{{ - \delta_{i} }}} \right)^{1/(m - 1)} } \right)^{ - 1} \hfill \\ \end{gathered}$$

(32)

$$\begin{gathered} \left. {\frac{\partial J}{{\partial v_{i} }}} \right|_{*} = - 2\sum\limits_{k = 1}^{n} {(u_{ik} )^{m} H_{i} (x_{k} - v_{i}^{*} )} = 0 \, ; \, i = 1,2,...,c \hfill \\ v_{i}^{*} = \frac{{\sum\limits_{k = 1}^{n} {u_{ik}^{m} } x_{k} }}{{\sum\limits_{k = 1}^{n} {u_{ik}^{m} } }} \hfill \\ \end{gathered}$$

(33)

Note that $\frac{\partial }{\partial x}(x^{T} Hx) = 2Hx$ in which H is symmetric and is not a function of x.

and finally,

$$\begin{gathered} \left. {\frac{\partial J}{{\partial H_{i} }}} \right|_{*} = \sum\limits_{k = 1}^{n} {u_{ik}^{m} (x_{k} - v_{i} )(x_{k} - v_{i} )^{T} + \lambda_{i} \left| {H_{i}^{*} } \right|} \, H_{i}^{{*^{ - 1} }} = 0 \hfill \\ H_{i}^{{*^{ - 1} }} = \frac{1}{{\lambda_{i} \left| {H_{i}^{*} } \right|}}\sum\limits_{k = 1}^{n} {u_{ik}^{m} } (x_{k} - v_{i}^{*} )(x_{k} - v_{i}^{*} )^{T} \hfill \\ \end{gathered}$$

(34)

The identities $\frac{\partial }{\partial H}(x^{T} Hx) = xx^{T} { , }\frac{\partial }{\partial H}\left| H \right| = \left| H \right|H^{ - 1}$ are used for a non-singular matrix H and any compatible vector x (Gustafson and Kessel 1978).

Appendix 2

The required condition for converging of the algorithm proposed in Fig. 2 is met when:

$$\lim_{t \to \infty } \left\| {U^{(t)} - U^{(t - 1)} } \right\| = 0$$

(35)

The iterative formula for $u_{ik}$ originates from the classical gradient descent method (Newton–Raphson method) (Kelley 1999), with $J_{m}$ as the error function which should be minimized, namely:

$$u_{ik}^{(t)} = u_{ik}^{(t - 1)} - \tau^{(t)} (J_{m} (u_{ik} ,\Omega_{i} ,\delta_{i} )^{(t - 1)} )\left( {\frac{{\partial J_{m} (u_{ik} ,\Omega_{i} ,\delta_{i} )^{(t - 1)} }}{{\partial u_{ik} }}} \right)^{ - 1}$$

(36)

where $\tau^{(t)}$ denotes a positive parameter of learning rate and $2 \le c \le 20$ stands for the gradient of $J_{m}$ with respect to $u_{ik}$ at (t-1) iteration. Re-writing (36) for U renders:

$$U^{(t)} - U^{(t - 1)} = - \tau^{(t)} (J_{m} (U,\Omega ,\delta )^{(t - 1)} )\left( {\frac{{\partial J_{m} (U,\Omega ,\delta )^{(t - 1)} }}{\partial U}} \right)^{ - 1}$$

(37)

Now, putting (35) in (37):

$$\lim_{t \to \infty } \left\| {U^{(t)} - U^{(t - 1)} } \right\| = \lim_{t \to \infty } \left( {\left\| {\tau^{(t)} } \right\|\left\| {J_{m} (U,\Omega ,\delta )^{(t - 1)} } \right\|\left\| {\frac{{\partial J_{m} (U,\Omega ,\delta )^{(t - 1)} }}{\partial U}} \right\|^{ - 1} } \right)$$

(38)

By considering $\tau^{(t)} = \frac{{\psi^{(t)} }}{{\left\| {J_{m} (U,\Omega ,\delta )^{(t - 1)} } \right\|\left\| {\frac{{\partial J_{m} (U,\Omega ,\delta )^{(t - 1)} }}{\partial U}} \right\|^{ - 1} }}$, (38) becomes:

$$\lim_{t \to \infty } \left\| {U^{(t)} - U^{(t - 1)} } \right\| = \lim_{t \to \infty } \left\| {\psi^{(t)} } \right\| = 0$$

(39)

where, $\psi^{(t)} = \psi_{0}^{(t)} /t$ in which $\psi_{0}^{(t)}$ represents a constant value, and $\psi_{0}^{(t)} \to 0$ when $t \to \infty$. Hence, (B.5) is proved and, as a result, the suggested algorithm is convergent.

Appendix 3

NMI is a measurement index that measures the degree of matching between the communities identified by different algorithms and that of the expected. Consider $F = \{ F_{k} \} (1 < k < c\}$ is the expected communities. NMI is defined as follows (Hu and Chan 2016):

$$NMI = \frac{{\sum\limits_{{k_{1} = 1}}^{c} {\sum\limits_{{k_{2} = 1}}^{c} {n_{{C_{{k_{1} }} ,F_{{k_{2} }} }} \log \left(\frac{{n \, *n_{{C_{{k_{1} }} ,F_{{k_{2} }} }} }}{{n_{{C_{{k_{1} }} }} *n_{{F_{{k_{2} }} }} }}\right)} } }}{{\sqrt {\left(\sum\limits_{k = 1}^{c} {n_{{C_{k} }} \log \frac{{n_{{C_{k} }} }}{n}} \right)\left(\sum\limits_{k = 1}^{c} {n_{{F_{k} }} \log \frac{{n_{{F_{k} }} }}{n}} \right)} }}$$

(40)

where $n_{{C_{k} }}$ is the number of nodes in $C_{k}$, $n_{{F_{k} }}$ is the number of nodes in $F_{k}$, and $n_{{C_{{k_{1} }} ,F_{{k_{2} }} }}$ is the number of nodes discovered in both $C_{{k_{1} }}$ and $F_{{k_{2} }}$.

For Accuracy measure (Hu and Chan 2016), the mapping function $Z:C_{{k_{1} }} \to F_{{k_{2} }}$ is needed. To find Z, the $n_{{C_{{k_{1} }} ,F_{{k_{2} }} }}$ is determined for all combinations of $C_{{k_{1} }}$ and $F_{{k_{2} }}$ in C and F, respectively. Given that, this is an iterative process, for each iteration, starting from the largest $n_{{C_{{k_{1} }} ,F_{{k_{2} }} }}$, the $C_{{k_{1} }}$ that matches against $F_{{k_{2} }}$ is determined; then the mapping $C_{{k_{1} }} \to F_{{k_{2} }}$ to Z is added and $C_{{k_{1} }}$ and $F_{{k_{2} }}$ are ignored in future iterations. The process ends when all $C_{{k_{1} }}$ in C has found a match in F. This measurement is defined as follows:

$$Accuracy = \frac{1}{n}\sum\limits_{k = 1}^{c} {n_{{C_{k} ,Z(C_{k} )}} }$$

(41)

Therefore, the values of NMI and Accuracy are larger when $C_{k}$ matches better with the expected result F.